https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Shusterman&feedformat=atomUW-Math Wiki - User contributions [en]2024-03-28T17:59:36ZUser contributionsMediaWiki 1.39.5https://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19385NTS ABSTRACTSpring20202020-04-29T20:43:40Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Mathilde Gerbelli-Gauthier'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Cohomology of Arithmetic Groups and Endoscopy<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== March 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jessica Fintzen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | From representations of p-adic groups to congruences of automorphic forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).<br />
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.<br />
This is joint work with Sug Woo Shin. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrea Dotto'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Functoriality of Serre weights<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D^*$ of $GL(n)$ to a maximal compact subgroup of $GL(n)$. Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to $D^*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Litt'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The section conjecture at the boundary of moduli space<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Grothendieck's section conjecture predicts that over<br />
arithmetically interesting fields (e.g. number fields or <math>p</math>-adic fields),<br />
rational points on a smooth projective curve X of genus at least 2 can be<br />
detected via the arithmetic of the etale fundamental group of X. We<br />
construct infinitely many curves of each genus satisfying the section<br />
conjecture in interesting ways, building on work of Stix, Harari, and<br />
Szamuely. The main input to our result is an analysis of the degeneration<br />
of certain torsion cohomology classes on the moduli space of curves at<br />
various boundary components. This is (preliminary) joint work with<br />
Padmavathi Srinivasan, Wanlin Li, and Nick Salter.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rachel Pries'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Galois action on the étale fundamental group of the Fermat curve<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
If <math>X</math> is a curve over a number field <math>K</math>, then we are motivated to understand the action of the absolute Galois group <math>G_K</math> on the étale fundamental group <math>\pi_1(X)</math>. When <math>X</math> is the Fermat curve of degree <math>p</math> and <math>K</math> is the cyclotomic field containing a <math>p</math>th root of unity, Anderson proved theorems about this action on the homology of <math>X</math>, with coefficients mod <math>p</math>. In earlier work, we extended Anderson's results to give explicit formulas for this action on the homology. Recently, we use a cup product in cohomology to determine the action of <math>G_K</math> on the lower central series of <math>\pi_1(X)</math>, with coefficients mod <math>p</math>. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 9 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Dalal'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Statistics of Automorphic Representations through the Stable Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Since individual automorphic representations are very difficult to access, it is helpful to study families of them instead. In particular, statistics over families can be computed through various forms of Arthur's trace formula. One refinement---the stable trace formula---has recently developed to the point where explicit computations with it are feasible. I will present one result on statistics that this new technique can prove. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jayce Getz'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On triple product $L$-functions<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Establishing the conjectured analytic properties of triple product $L$-functions is a crucial case of Langlands functoriality. However, little is known. I will present work in progress on the case of triples of automorphic representations on $\mathrm{GL}_3$; in some sense this is the smallest case that appears out of reach via standard techniques. The approach involves a relative trace formula and Poisson summation on spherical varieties in the sense of Braverman-Kazhdan, Ngo, and Sakellaridis. <br />
<br />
https://zoom.us/j/96864496800 <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Noah Taylor'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The Sato Tate Conjecture on Abelian Surfaces<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Sato-Tate conjecture says that the normalized point counts of genus $g$ curves are equidistributed with respect to a certain measure. We will construct the Sato-Tate group, state the conjecture precisely, prove a case, and in the cases where not everything is known, we will discuss how much we can say about the point counts anyway.<br />
<br />
https://us02web.zoom.us/j/81526488221<br />
<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=19384NTS2020-04-29T20:38:47Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_6 On Shimura varieties for unitary groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_20 Ramanujan Conjectures and Density Theorems]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | Mathilde Gerbelli-Gauthier<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_27 Cohomology of Arithmetic Groups and Endoscopy]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_6 From representations of p-adic groups to congruences of automorphic forms]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto (CANCELED)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_12 Functoriality of Serre weights]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt (ONLINE)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_26 The section conjecture at the boundary of moduli space]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Rachel Pries (Note different time: 2 pm)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_2 Galois action on the etale fundamental group of the Fermat curve]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_9 Statistics of Automorphic Representations through the Stable Trace Formula]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| CANCELLED<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_23 On triple product L-functions]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | Noah Taylor <br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_30 The Sato Tate Conjecture on Abelian Surfaces]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | May 7<br />
| bgcolor="#F0B0B0" align="center" | Aaron Landesman <br />
| bgcolor="#BCE2FE"| ONLINE<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
<br />
= VaNTAGe =<br />
This is a virtual math seminar on open conjectures in<br />
number theory and arithmetic geometry. The seminar will be presented in English at (1 pm Eastern time)=(10 am Pacific time), every first and third Tuesday of the month. The Math Department of UW, Madison broadcasts the seminar in the math lounge room at Room 911, Van Vleck Building.<br />
For more information, please visit the official website: <br />
[https://sites.google.com/view/vantageseminar]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19379NTS ABSTRACTSpring20202020-04-23T15:19:17Z<p>Shusterman: /* April 23 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Mathilde Gerbelli-Gauthier'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Cohomology of Arithmetic Groups and Endoscopy<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== March 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jessica Fintzen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | From representations of p-adic groups to congruences of automorphic forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).<br />
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.<br />
This is joint work with Sug Woo Shin. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrea Dotto'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Functoriality of Serre weights<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D^*$ of $GL(n)$ to a maximal compact subgroup of $GL(n)$. Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to $D^*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Litt'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The section conjecture at the boundary of moduli space<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Grothendieck's section conjecture predicts that over<br />
arithmetically interesting fields (e.g. number fields or <math>p</math>-adic fields),<br />
rational points on a smooth projective curve X of genus at least 2 can be<br />
detected via the arithmetic of the etale fundamental group of X. We<br />
construct infinitely many curves of each genus satisfying the section<br />
conjecture in interesting ways, building on work of Stix, Harari, and<br />
Szamuely. The main input to our result is an analysis of the degeneration<br />
of certain torsion cohomology classes on the moduli space of curves at<br />
various boundary components. This is (preliminary) joint work with<br />
Padmavathi Srinivasan, Wanlin Li, and Nick Salter.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rachel Pries'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Galois action on the étale fundamental group of the Fermat curve<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
If <math>X</math> is a curve over a number field <math>K</math>, then we are motivated to understand the action of the absolute Galois group <math>G_K</math> on the étale fundamental group <math>\pi_1(X)</math>. When <math>X</math> is the Fermat curve of degree <math>p</math> and <math>K</math> is the cyclotomic field containing a <math>p</math>th root of unity, Anderson proved theorems about this action on the homology of <math>X</math>, with coefficients mod <math>p</math>. In earlier work, we extended Anderson's results to give explicit formulas for this action on the homology. Recently, we use a cup product in cohomology to determine the action of <math>G_K</math> on the lower central series of <math>\pi_1(X)</math>, with coefficients mod <math>p</math>. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 9 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Dalal'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Statistics of Automorphic Representations through the Stable Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Since individual automorphic representations are very difficult to access, it is helpful to study families of them instead. In particular, statistics over families can be computed through various forms of Arthur's trace formula. One refinement---the stable trace formula---has recently developed to the point where explicit computations with it are feasible. I will present one result on statistics that this new technique can prove. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jayce Getz'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On triple product $L$-functions<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Establishing the conjectured analytic properties of triple product $L$-functions is a crucial case of Langlands functoriality. However, little is known. I will present work in progress on the case of triples of automorphic representations on $\mathrm{GL}_3$; in some sense this is the smallest case that appears out of reach via standard techniques. The approach involves a relative trace formula and Poisson summation on spherical varieties in the sense of Braverman-Kazhdan, Ngo, and Sakellaridis. <br />
<br />
https://zoom.us/j/96864496800 <br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19367NTS ABSTRACTSpring20202020-04-20T23:10:14Z<p>Shusterman: /* April 23 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Mathilde Gerbelli-Gauthier'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Cohomology of Arithmetic Groups and Endoscopy<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== March 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jessica Fintzen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | From representations of p-adic groups to congruences of automorphic forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).<br />
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.<br />
This is joint work with Sug Woo Shin. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrea Dotto'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Functoriality of Serre weights<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D^*$ of $GL(n)$ to a maximal compact subgroup of $GL(n)$. Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to $D^*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Litt'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The section conjecture at the boundary of moduli space<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Grothendieck's section conjecture predicts that over<br />
arithmetically interesting fields (e.g. number fields or <math>p</math>-adic fields),<br />
rational points on a smooth projective curve X of genus at least 2 can be<br />
detected via the arithmetic of the etale fundamental group of X. We<br />
construct infinitely many curves of each genus satisfying the section<br />
conjecture in interesting ways, building on work of Stix, Harari, and<br />
Szamuely. The main input to our result is an analysis of the degeneration<br />
of certain torsion cohomology classes on the moduli space of curves at<br />
various boundary components. This is (preliminary) joint work with<br />
Padmavathi Srinivasan, Wanlin Li, and Nick Salter.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rachel Pries'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Galois action on the étale fundamental group of the Fermat curve<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
If <math>X</math> is a curve over a number field <math>K</math>, then we are motivated to understand the action of the absolute Galois group <math>G_K</math> on the étale fundamental group <math>\pi_1(X)</math>. When <math>X</math> is the Fermat curve of degree <math>p</math> and <math>K</math> is the cyclotomic field containing a <math>p</math>th root of unity, Anderson proved theorems about this action on the homology of <math>X</math>, with coefficients mod <math>p</math>. In earlier work, we extended Anderson's results to give explicit formulas for this action on the homology. Recently, we use a cup product in cohomology to determine the action of <math>G_K</math> on the lower central series of <math>\pi_1(X)</math>, with coefficients mod <math>p</math>. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 9 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Dalal'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Statistics of Automorphic Representations through the Stable Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Since individual automorphic representations are very difficult to access, it is helpful to study families of them instead. In particular, statistics over families can be computed through various forms of Arthur's trace formula. One refinement---the stable trace formula---has recently developed to the point where explicit computations with it are feasible. I will present one result on statistics that this new technique can prove. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jayce Getz'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On triple product $L$-functions<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Establishing the conjectured analytic properties of triple product $L$-functions is a crucial case of Langlands functoriality. However, little is known. I will present work in progress on the case of triples of automorphic representations on $\mathrm{GL}_3$; in some sense this is the smallest case that appears out of reach via standard techniques. The approach involves a relative trace formula and Poisson summation on spherical varieties in the sense of Braverman-Kazhdan, Ngo, and Sakellaridis. <br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19366NTS ABSTRACTSpring20202020-04-20T23:09:42Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Mathilde Gerbelli-Gauthier'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Cohomology of Arithmetic Groups and Endoscopy<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== March 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jessica Fintzen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | From representations of p-adic groups to congruences of automorphic forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).<br />
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.<br />
This is joint work with Sug Woo Shin. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrea Dotto'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Functoriality of Serre weights<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D^*$ of $GL(n)$ to a maximal compact subgroup of $GL(n)$. Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to $D^*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Litt'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The section conjecture at the boundary of moduli space<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Grothendieck's section conjecture predicts that over<br />
arithmetically interesting fields (e.g. number fields or <math>p</math>-adic fields),<br />
rational points on a smooth projective curve X of genus at least 2 can be<br />
detected via the arithmetic of the etale fundamental group of X. We<br />
construct infinitely many curves of each genus satisfying the section<br />
conjecture in interesting ways, building on work of Stix, Harari, and<br />
Szamuely. The main input to our result is an analysis of the degeneration<br />
of certain torsion cohomology classes on the moduli space of curves at<br />
various boundary components. This is (preliminary) joint work with<br />
Padmavathi Srinivasan, Wanlin Li, and Nick Salter.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rachel Pries'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Galois action on the étale fundamental group of the Fermat curve<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
If <math>X</math> is a curve over a number field <math>K</math>, then we are motivated to understand the action of the absolute Galois group <math>G_K</math> on the étale fundamental group <math>\pi_1(X)</math>. When <math>X</math> is the Fermat curve of degree <math>p</math> and <math>K</math> is the cyclotomic field containing a <math>p</math>th root of unity, Anderson proved theorems about this action on the homology of <math>X</math>, with coefficients mod <math>p</math>. In earlier work, we extended Anderson's results to give explicit formulas for this action on the homology. Recently, we use a cup product in cohomology to determine the action of <math>G_K</math> on the lower central series of <math>\pi_1(X)</math>, with coefficients mod <math>p</math>. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 9 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Dalal'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Statistics of Automorphic Representations through the Stable Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Since individual automorphic representations are very difficult to access, it is helpful to study families of them instead. In particular, statistics over families can be computed through various forms of Arthur's trace formula. One refinement---the stable trace formula---has recently developed to the point where explicit computations with it are feasible. I will present one result on statistics that this new technique can prove. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Dalal'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On triple product $L$-functions<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Establishing the conjectured analytic properties of triple product $L$-functions is a crucial case of Langlands functoriality. However, little is known. I will present work in progress on the case of triples of automorphic representations on $\mathrm{GL}_3$; in some sense this is the smallest case that appears out of reach via standard techniques. The approach involves a relative trace formula and Poisson summation on spherical varieties in the sense of Braverman-Kazhdan, Ngo, and Sakellaridis. <br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=19365NTS2020-04-20T23:06:34Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_6 On Shimura varieties for unitary groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_20 Ramanujan Conjectures and Density Theorems]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | Mathilde Gerbelli-Gauthier<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_27 Cohomology of Arithmetic Groups and Endoscopy]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_6 From representations of p-adic groups to congruences of automorphic forms]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto (CANCELED)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_12 Functoriality of Serre weights]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt (ONLINE)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_26 The section conjecture at the boundary of moduli space]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Rachel Pries (Note different time: 2 pm)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_2 Galois action on the etale fundamental group of the Fermat curve]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_9 Statistics of Automorphic Representations through the Stable Trace Formula]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| CANCELLED<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_23 On triple product L-functions]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | Noah Taylor <br />
| bgcolor="#BCE2FE"| ONLINE<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | May 7<br />
| bgcolor="#F0B0B0" align="center" | Aaron Landesman <br />
| bgcolor="#BCE2FE"| ONLINE<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
<br />
= VaNTAGe =<br />
This is a virtual math seminar on open conjectures in<br />
number theory and arithmetic geometry. The seminar will be presented in English at (1 pm Eastern time)=(10 am Pacific time), every first and third Tuesday of the month. The Math Department of UW, Madison broadcasts the seminar in the math lounge room at Room 911, Van Vleck Building.<br />
For more information, please visit the official website: <br />
[https://sites.google.com/view/vantageseminar]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19335NTS ABSTRACTSpring20202020-04-07T14:52:34Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Mathilde Gerbelli-Gauthier'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Cohomology of Arithmetic Groups and Endoscopy<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== March 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jessica Fintzen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | From representations of p-adic groups to congruences of automorphic forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).<br />
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.<br />
This is joint work with Sug Woo Shin. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrea Dotto'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Functoriality of Serre weights<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D^*$ of $GL(n)$ to a maximal compact subgroup of $GL(n)$. Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to $D^*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Litt'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The section conjecture at the boundary of moduli space<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Grothendieck's section conjecture predicts that over<br />
arithmetically interesting fields (e.g. number fields or <math>p</math>-adic fields),<br />
rational points on a smooth projective curve X of genus at least 2 can be<br />
detected via the arithmetic of the etale fundamental group of X. We<br />
construct infinitely many curves of each genus satisfying the section<br />
conjecture in interesting ways, building on work of Stix, Harari, and<br />
Szamuely. The main input to our result is an analysis of the degeneration<br />
of certain torsion cohomology classes on the moduli space of curves at<br />
various boundary components. This is (preliminary) joint work with<br />
Padmavathi Srinivasan, Wanlin Li, and Nick Salter.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== April 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rachel Pries'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Galois action on the étale fundamental group of the Fermat curve<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
If <math>X</math> is a curve over a number field <math>K</math>, then we are motivated to understand the action of the absolute Galois group <math>G_K</math> on the étale fundamental group <math>\pi_1(X)</math>. When <math>X</math> is the Fermat curve of degree <math>p</math> and <math>K</math> is the cyclotomic field containing a <math>p</math>th root of unity, Anderson proved theorems about this action on the homology of <math>X</math>, with coefficients mod <math>p</math>. In earlier work, we extended Anderson's results to give explicit formulas for this action on the homology. Recently, we use a cup product in cohomology to determine the action of <math>G_K</math> on the lower central series of <math>\pi_1(X)</math>, with coefficients mod <math>p</math>. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 9 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Dalal'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Statistics of Automorphic Representations through the Stable Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Since individual automorphic representations are very difficult to access, it is helpful to study families of them instead. In particular, statistics over families can be computed through various forms of Arthur's trace formula. One refinement---the stable trace formula---has recently developed to the point where explicit computations with it are feasible. I will present one result on statistics that this new technique can prove. <br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=19334NTS2020-04-07T14:50:27Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_6 On Shimura varieties for unitary groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_20 Ramanujan Conjectures and Density Theorems]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | Mathilde Gerbelli-Gauthier<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_27 Cohomology of Arithmetic Groups and Endoscopy]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_6 From representations of p-adic groups to congruences of automorphic forms]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto (CANCELED)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_12 Functoriality of Serre weights]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt (ONLINE)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_26 The section conjecture at the boundary of moduli space]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Rachel Pries (Note different time: 2 pm)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_2 Galois action on the etale fundamental group of the Fermat curve]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#April_9 Statistics of Automorphic Representations through the Stable Trace Formula]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| CANCELLED<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | Noah Taylor <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | May 7<br />
| bgcolor="#F0B0B0" align="center" | Aaron Landesman <br />
| bgcolor="#BCE2FE"| ONLINE<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
<br />
= VaNTAGe =<br />
This is a virtual math seminar on open conjectures in<br />
number theory and arithmetic geometry. The seminar will be presented in English at (1 pm Eastern time)=(10 am Pacific time), every first and third Tuesday of the month. The Math Department of UW, Madison broadcasts the seminar in the math lounge room at Room 911, Van Vleck Building.<br />
For more information, please visit the official website: <br />
[https://sites.google.com/view/vantageseminar]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=19241NTS2020-03-12T12:50:43Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_6 On Shimura varieties for unitary groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_20 Ramanujan Conjectures and Density Theorems]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | Mathilde Gerbelli-Gauthier<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_27 Cohomology of Arithmetic Groups and Endoscopy]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_6 From representations of p-adic groups to congruences of automorphic forms]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto (CANCELED)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_12 Functoriality of Serre weights]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt (CANCELED)<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| CANCELED<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| CANCELED<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | Noah Taylor <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
<br />
= VaNTAGe =<br />
This is a virtual math seminar on open conjectures in<br />
number theory and arithmetic geometry. The seminar will be presented in English at (1 pm Eastern time)=(10 am Pacific time), every first and third Tuesday of the month. The Math Department of UW, Madison broadcasts the seminar in the math lounge room at Room 911, Van Vleck Building.<br />
For more information, please visit the official website: <br />
[https://sites.google.com/view/vantageseminar]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=19231NTS2020-03-11T14:54:30Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_6 On Shimura varieties for unitary groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_20 Ramanujan Conjectures and Density Theorems]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | Mathilde Gerbelli-Gauthier<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_27 Cohomology of Arithmetic Groups and Endoscopy]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_6 From representations of p-adic groups to congruences of automorphic forms]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto (CANCELED)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_12 Functoriality of Serre weights]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt (CANCELED)<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | Noah Taylor <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
<br />
= VaNTAGe =<br />
This is a virtual math seminar on open conjectures in<br />
number theory and arithmetic geometry. The seminar will be presented in English at (1 pm Eastern time)=(10 am Pacific time), every first and third Tuesday of the month. The Math Department of UW, Madison broadcasts the seminar in the math lounge room at Room 911, Van Vleck Building.<br />
For more information, please visit the official website: <br />
[https://sites.google.com/view/vantageseminar]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19206NTS ABSTRACTSpring20202020-03-05T16:17:57Z<p>Shusterman: /* March 12 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Mathilde Gerbelli-Gauthier'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Cohomology of Arithmetic Groups and Endoscopy<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== March 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jessica Fintzen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | From representations of p-adic groups to congruences of automorphic forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).<br />
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.<br />
This is joint work with Sug Woo Shin. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrea Dotto'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Functoriality of Serre weights<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D^*$ of $GL(n)$ to a maximal compact subgroup of $GL(n)$. Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to $D^*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.<br />
<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19205NTS ABSTRACTSpring20202020-03-05T16:17:43Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Mathilde Gerbelli-Gauthier'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Cohomology of Arithmetic Groups and Endoscopy<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== March 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jessica Fintzen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | From representations of p-adic groups to congruences of automorphic forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).<br />
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.<br />
This is joint work with Sug Woo Shin. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrea Dotto'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Functoriality of Serre weights<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form $D*$ of $GL(n)$ to a maximal compact subgroup of $GL(n)$. Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to $D*$ (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.<br />
<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19204NTS ABSTRACTSpring20202020-03-05T16:15:38Z<p>Shusterman: /* March 5 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Mathilde Gerbelli-Gauthier'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Cohomology of Arithmetic Groups and Endoscopy<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
How fast do Betti numbers grow in a congruence tower of compact arithmetic manifolds? The dimension of the middle degree of cohomology is proportional to the volume of the manifold, but away from the middle the growth is known to be sub-linear in the volume. I will explain how automorphic representations and the phenomenon of endoscopy provide a framework to understand and quantify this slow growth. Specifically, I will discuss how to obtain explicit bounds in the case of unitary groups using the character identities appearing in Arthur’s stable trace formula. This is joint work in progress with Simon Marshall.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
== March 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jessica Fintzen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | From representations of p-adic groups to congruences of automorphic forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of automorphic forms and the global Langlands program have been very active research areas for the past 30 years. Significant progress has been achieved by developing intricate geometric methods, but most results to date are restricted to general linear groups (and general unitary groups).<br />
In this talk I will present new results about the representation theory of p-adic groups and demonstrate how these can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.<br />
This is joint work with Sug Woo Shin. <br />
<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=19203NTS2020-03-05T16:14:53Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_6 On Shimura varieties for unitary groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_20 Ramanujan Conjectures and Density Theorems]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | Mathilde Gerbelli-Gauthier<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_27 Cohomology of Arithmetic Groups and Endoscopy]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_6 From representations of p-adic groups to congruences of automorphic forms]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#March_12 Functoriality of Serre weights]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | Noah Taylor <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
<br />
= VaNTAGe =<br />
This is a virtual math seminar on open conjectures in<br />
number theory and arithmetic geometry. The seminar will be presented in English at (1 pm Eastern time)=(10 am Pacific time), every first and third Tuesday of the month. The Math Department of UW, Madison broadcasts the seminar in the math lounge room at Room 911, Van Vleck Building.<br />
For more information, please visit the official website: <br />
[https://sites.google.com/view/vantageseminar]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19069NTS ABSTRACTSpring20202020-02-19T23:13:44Z<p>Shusterman: /* Feb 20 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of $GL(n)$. One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=19048NTS ABSTRACTSpring20202020-02-17T22:32:50Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Smithling'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On Shimura varieties for unitary groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Shimura varieties attached to unitary similitude groups are a well-studied class of Shimura varieties of PEL type (i.e., admitting moduli interpretations in terms of abelian varieties with additional structure). There are also natural Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, but they have other advantages (e.g., they give rise to interesting cycles of the sort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W. Zhang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Feb 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shai Evra'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Ramanujan Conjectures and Density Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The Generalized Ramanujan Conjecture (GRC) for the group $GL(n)$ is a central open problem in modern number theory. The (GRC) is known to imply the solution of many Diophantine problems associated to arithmetic congruence subgroups of GL(n). One can also state analogous (Naive) Ramanujan Conjectures (NRC) for other reductive groups $G$, whose validity would imply various applications for the congruence subgroups of $G$. However, already in the 70s Howe and Piatetski-Shapiro proved that the (NRC) fails even for the class of classical split and quasi-split groups. In the 90s Sarnak and Xue put forth the conjecture that a Density Hypothesis (DH) version of the (NRC) should hold, and that these Density Hypotheses can serve as a replacement of the (NRC) in many applications. In this talk I will describe the (GRC), (NRC) and (DH), and explain how to prove the (DH) for certain classical groups, by invoking deep and recent results coming from the Langlands program. <br />
<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=19047NTS2020-02-17T22:25:31Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_6 On Shimura varieties for unitary groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_20 Ramanujan Conjectures and Density Theorems]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | Mathilde Gerbelli-Gauthier<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | Noah Taylor <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
<br />
= VaNTAGe =<br />
This is a virtual math seminar on open conjectures in<br />
number theory and arithmetic geometry. The seminar will be presented in English at (1 pm Eastern time)=(10 am Pacific time), every first and third Tuesday of the month. The Math Department of UW, Madison broadcasts the seminar in the math lounge room at Room 911, Van Vleck Building.<br />
For more information, please visit the official website: <br />
[https://sites.google.com/view/vantageseminar]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=19006NTS2020-02-12T14:11:03Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Feb_6 On Shimura varieties for unitary groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | Mathilde Gerbelli-Gauthier<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | Noah Taylor <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
<br />
= VaNTAGe =<br />
This is a virtual math seminar on open conjectures in<br />
number theory and arithmetic geometry. The seminar will be presented in English at (1 pm Eastern time)=(10 am Pacific time), every first and third Tuesday of the month. The Math Department of UW, Madison broadcasts the seminar in the math lounge room at Room 911, Van Vleck Building.<br />
For more information, please visit the official website: <br />
[https://sites.google.com/view/vantageseminar]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_Semester_2020&diff=18791NTS Spring Semester 20202020-01-25T23:49:49Z<p>Shusterman: /* Schedule */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our last semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS Fall 2019].<br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_2018_Semester Spring 2018].<br />
<br />
<br />
= Schedule =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna (Northwestern)<br />
| bgcolor="#BCE2FE"| A relative trace formula comparison for the global Gross-Prasad conjecture for Orthogonal groups<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley (U. of Chicago)<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | '''Friday Jan 31, 4-5 pm'''<br />
| bgcolor="#F0B0B0" align="center" | Lillian Pierce (Duke) <br /> (joint analysis / NT seminar)<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=18779NTS ABSTRACTSpring20202020-01-24T16:29:07Z<p>Shusterman: /* Jan 30 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the $p$-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N = 1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar $p$-th power conditions, and we give exact characterizations of the rank for small $p$. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology. This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=18778NTS ABSTRACTSpring20202020-01-24T16:27:07Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Jan 30 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Eric Stubley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Class Groups, Congruences, and Cup Products<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the p-part of the class group of the p-th cyclotomic field to congruences of Bernoulli numbers mod p. For p and N prime with N = 1 mod p, a similar result of Calegari and Emerton relates the rank of the p-part of the class group of Q(N^1/p) to whether or not a certain quantity (Merel's number) is a p-th power mod N. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar p-th power conditions, and we give exact characterizations of the rank for small p. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology This is joint work with Karl Schaefer.<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18777NTS2020-01-24T16:25:12Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_30 Class Groups, Congruences, and Cup Products]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | '''Friday Jan 31, 3-4 pm'''<br />
| bgcolor="#F0B0B0" align="center" | Lillian Pierce <br /> (joint analysis / NT seminar)<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTSpring2020&diff=18736NTS ABSTRACTSpring20202020-01-22T18:16:17Z<p>Shusterman: Created page with "Return to [https://www.math.wisc.edu/wiki/index.php/NTS ] == Jan 23 == <center> {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspa..."</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Jan 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rahul Krishna'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.<br />
<br />
|} <br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18735NTS2020-01-22T18:07:48Z<p>Shusterman: /* Spring 2020 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2020 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our Fall 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_Semester_2019 Fall 2019]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Spring 2020 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2020#Jan_23 A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | '''Friday Jan 31, 3-4 pm'''<br />
| bgcolor="#F0B0B0" align="center" | Lillian Pierce <br /> (joint analysis / NT seminar)<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | Jayce Getz<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18493NTS2019-11-25T22:11:07Z<p>Shusterman: /* Fall 2019 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Fall_2019 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our next semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2020 Spring 2020]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Fall 2019 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Sep 5<br />
| bgcolor="#F0B0B0" align="center" | Will Sawin (Columbia)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_5 The sup-norm problem for automorphic forms over function fields and geometry]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 12<br />
| bgcolor="#F0B0B0" align="center" | Yingkun Li (Darmstadt)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_12 CM values of modular functions and factorization]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 19<br />
| bgcolor="#F0B0B0" align="center" | Soumya Sankar (Madison)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_19 Proportion of ordinary curves in some families]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 26<br />
| bgcolor="#F0B0B0" align="center" | [https://www.math.wisc.edu/wiki/index.php/Colloquia Special Colloquium Lecture]<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 3<br />
| bgcolor="#F0B0B0" align="center" | Patrick Allen (UIUC)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_3 On the modularity of elliptic curves over imaginary quadratic fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 10<br />
| bgcolor="#F0B0B0" align="center" | Borys Kadets (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_10 Sectional monodromy groups of projective curves]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 17<br />
| bgcolor="#F0B0B0" align="center" | Yousheng Shi (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_17 Generalized special cycles and theta series]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Oct 24<br />
| bgcolor="#F0B0B0" align="center" | Simon Marshall (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_24 Counting cohomological automorphic forms on $GL_3$ ]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 7<br />
| bgcolor="#F0B0B0" align="center" | Asif Zaman (Toronto)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_7 A zero density estimate for Dedekind zeta functions ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 14<br />
| bgcolor="#F0B0B0" align="center" | Liyang Yang (Caltech)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_14 Holomorphic Continuation of Certain $L$-functions via Trace Formula] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 21<br />
| bgcolor="#F0B0B0" align="center" | Tony Feng (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_21 Steenrod operations and the Artin-Tate pairing]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 26 (Note different day)<br />
| bgcolor="#F0B0B0" align="center" | Brandon Alberts<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_26 Counting Towers of Number Fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Dec 5<br />
| bgcolor="#F0B0B0" align="center" | Benjamin Breen <br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Dec_05 Unit signatures and narrow class groups of odd abelian number fields]<br />
|-<br />
<br />
<br />
<br />
<br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18492NTS ABSTRACTFall20192019-11-25T22:09:06Z<p>Shusterman: /* Dec 05 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields<br />
|-<br />
| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \triangleleft G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Dec 05 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Benjamin Breen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On unit signatures and narrow class groups of odd abelian number fields: structure and heuristics<br />
|-<br />
| bgcolor="#BCD2EE" | What is the probability that the ring of integers in a number field contains a unit of mixed signature? In this talk, we present Cohen-Lenstra style heuristics for unit signatures and narrow class groups of odd abelian number fields. In addition, we analyze the equation $x^3 - ax^2 + bx - 1 = 0$ to prove that there are infinitely many cyclic cubic number fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18491NTS ABSTRACTFall20192019-11-25T22:08:58Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields<br />
|-<br />
| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \triangleleft G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Dec 05 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Benjamin Breen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On unit signatures and narrow class groups of odd abelian number fields: structure and heuristics<br />
|-<br />
| bgcolor="#BCD2EE" | What is the probability that the ring of integers in a number field contains a unit of mixed signature? In this talk, we present Cohen-Lenstra style heuristics for unit signatures and narrow class groups of odd abelian number fields. In addition, we analyze the equation $x^3 - ax^2 + bx - 1 = 0$ to prove that there are infinitely many cyclic cubic number fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18481NTS ABSTRACTFall20192019-11-22T23:26:19Z<p>Shusterman: /* Nov 26 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields<br />
|-<br />
| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \triangleleft G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18480NTS ABSTRACTFall20192019-11-22T23:26:05Z<p>Shusterman: /* Nov 26 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields<br />
|-<br />
| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18479NTS ABSTRACTFall20192019-11-22T23:25:32Z<p>Shusterman: /* Nov 26 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields<br />
|-<br />
| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18478NTS ABSTRACTFall20192019-11-22T23:25:02Z<p>Shusterman: /* Nov 26 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields<br />
|-<br />
| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\rm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18477NTS ABSTRACTFall20192019-11-22T23:24:50Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brandon Alberts'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting Towers of Number Fields<br />
|-<br />
| bgcolor="#BCD2EE" | Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of G-extensions of number fields $F/K$ with discriminant bounded above by X. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T\trianglelefteq G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\rm Gal}(F/L) \cong T$ and ${\rm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18476NTS2019-11-22T23:22:16Z<p>Shusterman: /* Fall 2019 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Fall_2019 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our next semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2020 Spring 2020]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Fall 2019 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Sep 5<br />
| bgcolor="#F0B0B0" align="center" | Will Sawin (Columbia)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_5 The sup-norm problem for automorphic forms over function fields and geometry]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 12<br />
| bgcolor="#F0B0B0" align="center" | Yingkun Li (Darmstadt)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_12 CM values of modular functions and factorization]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 19<br />
| bgcolor="#F0B0B0" align="center" | Soumya Sankar (Madison)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_19 Proportion of ordinary curves in some families]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 26<br />
| bgcolor="#F0B0B0" align="center" | [https://www.math.wisc.edu/wiki/index.php/Colloquia Special Colloquium Lecture]<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 3<br />
| bgcolor="#F0B0B0" align="center" | Patrick Allen (UIUC)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_3 On the modularity of elliptic curves over imaginary quadratic fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 10<br />
| bgcolor="#F0B0B0" align="center" | Borys Kadets (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_10 Sectional monodromy groups of projective curves]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 17<br />
| bgcolor="#F0B0B0" align="center" | Yousheng Shi (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_17 Generalized special cycles and theta series]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Oct 24<br />
| bgcolor="#F0B0B0" align="center" | Simon Marshall (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_24 Counting cohomological automorphic forms on $GL_3$ ]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 7<br />
| bgcolor="#F0B0B0" align="center" | Asif Zaman (Toronto)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_7 A zero density estimate for Dedekind zeta functions ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 14<br />
| bgcolor="#F0B0B0" align="center" | Liyang Yang (Caltech)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_14 Holomorphic Continuation of Certain $L$-functions via Trace Formula] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 21<br />
| bgcolor="#F0B0B0" align="center" | Tony Feng (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_21 Steenrod operations and the Artin-Tate pairing]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 26 (Note different day)<br />
| bgcolor="#F0B0B0" align="center" | Brandon Alberts<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_26 Counting Towers of Number Fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Dec 5<br />
| bgcolor="#F0B0B0" align="center" | Benjamin Breen <br />
| bgcolor="#BCE2FE"|<br />
|-<br />
<br />
<br />
<br />
<br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18475NTS2019-11-22T23:21:47Z<p>Shusterman: /* Fall 2019 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Fall_2019 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our next semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2020 Spring 2020]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Fall 2019 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Sep 5<br />
| bgcolor="#F0B0B0" align="center" | Will Sawin (Columbia)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_5 The sup-norm problem for automorphic forms over function fields and geometry]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 12<br />
| bgcolor="#F0B0B0" align="center" | Yingkun Li (Darmstadt)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_12 CM values of modular functions and factorization]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 19<br />
| bgcolor="#F0B0B0" align="center" | Soumya Sankar (Madison)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_19 Proportion of ordinary curves in some families]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 26<br />
| bgcolor="#F0B0B0" align="center" | [https://www.math.wisc.edu/wiki/index.php/Colloquia Special Colloquium Lecture]<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 3<br />
| bgcolor="#F0B0B0" align="center" | Patrick Allen (UIUC)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_3 On the modularity of elliptic curves over imaginary quadratic fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 10<br />
| bgcolor="#F0B0B0" align="center" | Borys Kadets (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_10 Sectional monodromy groups of projective curves]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 17<br />
| bgcolor="#F0B0B0" align="center" | Yousheng Shi (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_17 Generalized special cycles and theta series]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Oct 24<br />
| bgcolor="#F0B0B0" align="center" | Simon Marshall (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_24 Counting cohomological automorphic forms on $GL_3$ ]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 7<br />
| bgcolor="#F0B0B0" align="center" | Asif Zaman (Toronto)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_7 A zero density estimate for Dedekind zeta functions ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 14<br />
| bgcolor="#F0B0B0" align="center" | Liyang Yang (Caltech)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_14 Holomorphic Continuation of Certain $L$-functions via Trace Formula ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 21<br />
| bgcolor="#F0B0B0" align="center" | Tony Feng (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_21 Steenrod operations and the Artin-Tate pairing]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 26 (Note different day)<br />
| bgcolor="#F0B0B0" align="center" | Brandon Alberts<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_26 Counting Towers of Number Fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Dec 5<br />
| bgcolor="#F0B0B0" align="center" | Benjamin Breen <br />
| bgcolor="#BCE2FE"|<br />
|-<br />
<br />
<br />
<br />
<br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18474NTS2019-11-22T23:21:22Z<p>Shusterman: /* Fall 2019 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Fall_2019 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our next semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2020 Spring 2020]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Fall 2019 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Sep 5<br />
| bgcolor="#F0B0B0" align="center" | Will Sawin (Columbia)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_5 The sup-norm problem for automorphic forms over function fields and geometry]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 12<br />
| bgcolor="#F0B0B0" align="center" | Yingkun Li (Darmstadt)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_12 CM values of modular functions and factorization]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 19<br />
| bgcolor="#F0B0B0" align="center" | Soumya Sankar (Madison)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_19 Proportion of ordinary curves in some families]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 26<br />
| bgcolor="#F0B0B0" align="center" | [https://www.math.wisc.edu/wiki/index.php/Colloquia Special Colloquium Lecture]<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 3<br />
| bgcolor="#F0B0B0" align="center" | Patrick Allen (UIUC)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_3 On the modularity of elliptic curves over imaginary quadratic fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 10<br />
| bgcolor="#F0B0B0" align="center" | Borys Kadets (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_10 Sectional monodromy groups of projective curves]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 17<br />
| bgcolor="#F0B0B0" align="center" | Yousheng Shi (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_17 Generalized special cycles and theta series]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Oct 24<br />
| bgcolor="#F0B0B0" align="center" | Simon Marshall (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_24 Counting cohomological automorphic forms on $GL_3$ ]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 7<br />
| bgcolor="#F0B0B0" align="center" | Asif Zaman (Toronto)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_7 A zero density estimate for Dedekind zeta functions ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 14<br />
| bgcolor="#F0B0B0" align="center" | Liyang Yang (Caltech)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_14 Holomorphic Continuation of Certain $L$-functions via Trace Formula ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 21<br />
| bgcolor="#F0B0B0" align="center" | Tony Feng (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_21 Steenrod operations and the Artin-Tate pairing]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 26 (Note different day)<br />
| bgcolor="#F0B0B0" align="center" | Brandon Alberts<br />
| bgcolor="#BCE2FE"| [Counting Towers of Number Fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Dec 5<br />
| bgcolor="#F0B0B0" align="center" | Benjamin Breen <br />
| bgcolor="#BCE2FE"|<br />
|-<br />
<br />
<br />
<br />
<br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18442NTS2019-11-17T18:31:10Z<p>Shusterman: /* Fall 2019 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Fall_2019 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our next semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2020 Spring 2020]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Fall 2019 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Sep 5<br />
| bgcolor="#F0B0B0" align="center" | Will Sawin (Columbia)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_5 The sup-norm problem for automorphic forms over function fields and geometry]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 12<br />
| bgcolor="#F0B0B0" align="center" | Yingkun Li (Darmstadt)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_12 CM values of modular functions and factorization]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 19<br />
| bgcolor="#F0B0B0" align="center" | Soumya Sankar (Madison)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_19 Proportion of ordinary curves in some families]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 26<br />
| bgcolor="#F0B0B0" align="center" | [https://www.math.wisc.edu/wiki/index.php/Colloquia Special Colloquium Lecture]<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 3<br />
| bgcolor="#F0B0B0" align="center" | Patrick Allen (UIUC)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_3 On the modularity of elliptic curves over imaginary quadratic fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 10<br />
| bgcolor="#F0B0B0" align="center" | Borys Kadets (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_10 Sectional monodromy groups of projective curves]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 17<br />
| bgcolor="#F0B0B0" align="center" | Yousheng Shi (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_17 Generalized special cycles and theta series]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Oct 24<br />
| bgcolor="#F0B0B0" align="center" | Simon Marshall (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_24 Counting cohomological automorphic forms on $GL_3$ ]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 7<br />
| bgcolor="#F0B0B0" align="center" | Asif Zaman (Toronto)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_7 A zero density estimate for Dedekind zeta functions ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 14<br />
| bgcolor="#F0B0B0" align="center" | Liyang Yang (Caltech)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_14 Holomorphic Continuation of Certain $L$-functions via Trace Formula ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 21<br />
| bgcolor="#F0B0B0" align="center" | Tony Feng (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_21 Steenrod operations and the Artin-Tate pairing]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 26 (Note different day)<br />
| bgcolor="#F0B0B0" align="center" | Brandon Alberts<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Dec 5<br />
| bgcolor="#F0B0B0" align="center" | Benjamin Breen <br />
| bgcolor="#BCE2FE"|<br />
|-<br />
<br />
<br />
<br />
<br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18441NTS2019-11-17T18:30:26Z<p>Shusterman: /* Fall 2019 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Fall_2019 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our next semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2020 Spring 2020]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Fall 2019 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Sep 5<br />
| bgcolor="#F0B0B0" align="center" | Will Sawin (Columbia)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_5 The sup-norm problem for automorphic forms over function fields and geometry]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 12<br />
| bgcolor="#F0B0B0" align="center" | Yingkun Li (Darmstadt)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_12 CM values of modular functions and factorization]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 19<br />
| bgcolor="#F0B0B0" align="center" | Soumya Sankar (Madison)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_19 Proportion of ordinary curves in some families]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 26<br />
| bgcolor="#F0B0B0" align="center" | [https://www.math.wisc.edu/wiki/index.php/Colloquia Special Colloquium Lecture]<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 3<br />
| bgcolor="#F0B0B0" align="center" | Patrick Allen (UIUC)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_3 On the modularity of elliptic curves over imaginary quadratic fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 10<br />
| bgcolor="#F0B0B0" align="center" | Borys Kadets (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_10 Sectional monodromy groups of projective curves]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 17<br />
| bgcolor="#F0B0B0" align="center" | Yousheng Shi (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_17 Generalized special cycles and theta series]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Oct 24<br />
| bgcolor="#F0B0B0" align="center" | Simon Marshall (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_24 Counting cohomological automorphic forms on $GL_3$ ]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 7<br />
| bgcolor="#F0B0B0" align="center" | Asif Zaman (Toronto)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_7 A zero density estimate for Dedekind zeta functions ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 14<br />
| bgcolor="#F0B0B0" align="center" | Liyang Yang (Caltech)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_14 Holomorphic Continuation of Certain $L$-functions via Trace Formula ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 21<br />
| bgcolor="#F0B0B0" align="center" | Tony Feng (MIT)<br />
| bgcolor="#BCE2FE"| Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 26 (Note different day)<br />
| bgcolor="#F0B0B0" align="center" | Brandon Alberts<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Dec 5<br />
| bgcolor="#F0B0B0" align="center" | Benjamin Breen <br />
| bgcolor="#BCE2FE"|<br />
|-<br />
<br />
<br />
<br />
<br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18440NTS ABSTRACTFall20192019-11-17T18:28:36Z<p>Shusterman: /* Sep 19 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18439NTS ABSTRACTFall20192019-11-17T18:28:24Z<p>Shusterman: /* Oct 3 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18438NTS ABSTRACTFall20192019-11-17T18:28:09Z<p>Shusterman: /* Oct 10 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18437NTS ABSTRACTFall20192019-11-17T18:27:56Z<p>Shusterman: /* Oct 17 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18436NTS ABSTRACTFall20192019-11-17T18:27:35Z<p>Shusterman: /* Oct 24 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18435NTS ABSTRACTFall20192019-11-17T18:27:11Z<p>Shusterman: /* Nov 14 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18434NTS ABSTRACTFall20192019-11-17T18:26:54Z<p>Shusterman: /* Nov 21 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18433NTS ABSTRACTFall20192019-11-17T18:26:27Z<p>Shusterman: </p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
== Nov 21 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tony Feng'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Steenrod operations and the Artin-Tate pairing<br />
|-<br />
| bgcolor="#BCD2EE" | In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations. <br />
<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_Semester_2020&diff=18429NTS Spring Semester 20202019-11-15T22:53:09Z<p>Shusterman: /* Schedule */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2019 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our last semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS Fall 2019].<br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_2018_Semester Spring 2018].<br />
<br />
<br />
= Schedule =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Jan 23<br />
| bgcolor="#F0B0B0" align="center" | Rahul Krishna<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Jan 30<br />
| bgcolor="#F0B0B0" align="center" | Eric Stubley<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 6<br />
| bgcolor="#F0B0B0" align="center" | Brian Smithling<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 13<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 20<br />
| bgcolor="#F0B0B0" align="center" | Shai Evra<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Feb 27<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 5<br />
| bgcolor="#F0B0B0" align="center" | Jessica Fintzen<br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 12<br />
| bgcolor="#F0B0B0" align="center" | Andrea Dotto<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 19<br />
| bgcolor="#F0B0B0" align="center" | Spring Break<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | March 26<br />
| bgcolor="#F0B0B0" align="center" | Daniel Litt<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 2<br />
| bgcolor="#F0B0B0" align="center" | Caroline Turnage-Butterbaugh<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 9<br />
| bgcolor="#F0B0B0" align="center" | Rahul Dalal<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 16 <br />
| bgcolor="#F0B0B0" align="center" | Aaron Pollack<br />
| bgcolor="#BCE2FE"| <br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 23<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | April 30<br />
| bgcolor="#F0B0B0" align="center" | <br />
| bgcolor="#BCE2FE"|<br />
|- <br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18377NTS ABSTRACTFall20192019-11-08T23:58:35Z<p>Shusterman: /* Nov 7 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
This is a joint work with Jesse Thorner.<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS&diff=18370NTS2019-11-08T16:43:40Z<p>Shusterman: /* Fall 2019 Semester */</p>
<hr />
<div>= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30 PM – 3:30 PM<br />
*'''Where:''' Van Vleck B321<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Fall_2019 graduate-level seminar], which meets on Tuesdays.<br><br />
<br />
You can find our next semester speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2020 Spring 2020]. <br />
<br><br />
You can find our Spring 2019 speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Spring_Semester_2019 Spring 2019]. <br />
<br><br />
You can find our previous speakers in [https://www.math.wisc.edu/wiki/index.php/NTS_Fall_2018_Semester Fall 2018].<br />
<br />
= Fall 2019 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Sep 5<br />
| bgcolor="#F0B0B0" align="center" | Will Sawin (Columbia)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_5 The sup-norm problem for automorphic forms over function fields and geometry]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 12<br />
| bgcolor="#F0B0B0" align="center" | Yingkun Li (Darmstadt)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_12 CM values of modular functions and factorization]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 19<br />
| bgcolor="#F0B0B0" align="center" | Soumya Sankar (Madison)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Sep_19 Proportion of ordinary curves in some families]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Sep 26<br />
| bgcolor="#F0B0B0" align="center" | [https://www.math.wisc.edu/wiki/index.php/Colloquia Special Colloquium Lecture]<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 3<br />
| bgcolor="#F0B0B0" align="center" | Patrick Allen (UIUC)<br />
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_3 On the modularity of elliptic curves over imaginary quadratic fields]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 10<br />
| bgcolor="#F0B0B0" align="center" | Borys Kadets (MIT)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_10 Sectional monodromy groups of projective curves]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Oct 17<br />
| bgcolor="#F0B0B0" align="center" | Yousheng Shi (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_17 Generalized special cycles and theta series]<br />
|- <br />
| bgcolor="#E0E0E0" align="center" | Oct 24<br />
| bgcolor="#F0B0B0" align="center" | Simon Marshall (Madison)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Oct_24 Counting cohomological automorphic forms on $GL_3$ ]<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 7<br />
| bgcolor="#F0B0B0" align="center" | Asif Zaman (Toronto)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_7 A zero density estimate for Dedekind zeta functions ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 14<br />
| bgcolor="#F0B0B0" align="center" | Liyang Yang (Caltech)<br />
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTFall2019#Nov_14 Holomorphic Continuation of Certain $L$-functions via Trace Formula ] <br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 21<br />
| bgcolor="#F0B0B0" align="center" | Tony Feng (MIT)<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Nov 26 (Note different day)<br />
| bgcolor="#F0B0B0" align="center" | Brandon Alberts<br />
| bgcolor="#BCE2FE"|<br />
|-<br />
| bgcolor="#E0E0E0" align="center" | Dec 5<br />
| bgcolor="#F0B0B0" align="center" | Benjamin Breen <br />
| bgcolor="#BCE2FE"|<br />
|-<br />
<br />
<br />
<br />
<br />
|}<br />
</center><br />
<br />
<br><br />
<br />
*to be confirmed<br />
<br />
= Organizer contact information =<br />
<br />
[http://www.math.wisc.edu/~shusterman/ Mark Shusterman]<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18369NTS ABSTRACTFall20192019-11-08T16:41:35Z<p>Shusterman: /* Nov 14 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shustermanhttps://wiki.math.wisc.edu/index.php?title=NTS_ABSTRACTFall2019&diff=18368NTS ABSTRACTFall20192019-11-08T16:40:57Z<p>Shusterman: /* Nov 14 */</p>
<hr />
<div>Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]<br />
<br />
<br />
== Sep 5 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Will Sawin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | The sup-norm problem for automorphic forms over function fields and geometry<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The sup-norm problem is a purely analytic question about <br />
automorphic forms, which asks for bounds on their largest value (when <br />
viewed as a function on a modular curve or similar space). We describe <br />
a new approach to this problem in the function field setting, which we <br />
carry through to provide new bounds for forms in GL_2 stronger than <br />
what can be proved for the analogous question about classical modular <br />
forms. This approach proceeds by viewing the automorphic form as a <br />
geometric object, following Drinfeld. It should be possible to prove <br />
bounds in greater generality by this approach in the future.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yingkun Li'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | CM values of modular functions and factorization<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Sep 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Soumya Sankar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Proportion of ordinary curves in some families <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
An abelian variety in characteristic <math>p</math> is said to be ordinary if its <math>p</math> torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the <math>p</math> -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so. <br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Oct 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | On the modularity of elliptic curves over imaginary quadratic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Borys Kadets'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Sectional monodromy groups of projective curves<br />
|-<br />
| bgcolor="#BCD2EE" | Let $K$ be a field. Fix a projective curve $X \subset \mathbb{P}^r_K$ of degree $d$. A general hyperplane $H \in \mathbb{P}^{r*}$ intersects $X$ in $d$ points; the monodromy of $X \bigcap H$ as $H$ varies is a subgroup $G_X$ of $S_d$ known as the sectional monodromy group of $X$. When $K=\mathbb{C}$ (or in fact for $\mathrm{char} K = 0$), the equality $G_X=S_d$ was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion $G_X \supset A_d$ holds. On the other hand, for a seemingly simple family of curves $X_{m,n}$ given by the equation $x^n=y^mz^{n-m}$ in $\mathbb{P}^2$ I will completely characterize the possibilities for $G_{X_{n,m}}$; the list of possibilities includes linear groups $\mathrm{AGL}_n(q)$, $\mathrm{PGL}_2(q)$ as well as some sporadic simple groups.<br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Oct 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yousheng Shi'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Generalized special cycles and theta series<br />
|-<br />
| bgcolor="#BCD2EE" | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
== Oct 24 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Counting cohomological automorphic forms on $GL_3$<br />
|-<br />
| bgcolor="#BCD2EE" | I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
<br />
<br />
<br />
== Nov 7 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Zaman'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | A zero density estimate for Dedekind zeta functions<br />
|-<br />
| bgcolor="#BCD2EE" | Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. <br />
<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br><br />
<br />
== Nov 14 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Liyang Yang'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Holomorphic Continuation of Certain $L$-functions via Trace Formula<br />
|-<br />
| bgcolor="#BCD2EE" | In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that these $L$-functions are closely related to each other by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.<br />
<br />
<br />
|} <br />
<br />
</center><br />
<br />
<br></div>Shusterman