Difference between revisions of "NTS/Abstracts"

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== Anton Gershaschenko  ==
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== Aug 28 ==
  
 
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| bgcolor="#DDDDDD" align="center"| Title: Moduli of Representations of Unipotent Groups
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Lemke Oliver'''
 
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| bgcolor="#BCD2EE"  align="center" | ''The distribution of 2-Selmer groups of elliptic curves with two-torsion''
Abstract: Representations of reductive groups are discretely parameterized, but unipotent groups can have non-trivial families of representations, so it makes sense try to construct and understand a moduli stack (or space) of representations of a given unipotent group. If you restrict to certain kinds of representations, it is possible to actually get your hands on the moduli stack and to construct a moduli space. I'll summarize the few things I know about the general case and then give you a tour of some interesting features that appear in small examples.
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Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over Q, when curves are ordered by height, is exactly 3, and Bhargava and Ho have shown that, in the family of curves with a marked point, the average is exactly 6.  In stark contrast to these results, we show that the average size in the family of elliptic curves with a two-torsion point is unbounded. This follows from an understanding of the Tamagawa ratio associated to such elliptic curves, which we prove is "normally distributed with infinite variance". This work is joint with Zev Klagsbrun.
 
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== Keerthi Madapusi ==
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== Sep 04 ==
  
 
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| bgcolor="#DDDDDD" align="center"| Title: A rationality property of Hodge cycles on abelian varieties, with an application to arithmetic compactifications of Shimura varieties
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen'''
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|-
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| bgcolor="#BCD2EE"  align="center" | ''Unramified deformation rings''
 
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Abstract: Let A be an abelian variety over a number field E, and let v be a finite place of E where A has bad, split, semi-stable reduction. Then the toric part of the reduction of A at v gives rise to a partial integral structure on the (l-adic, p-adic, deRham, log crystalline) cohomology group H^1(A_{E_v} ), arising essentially from the character group of the torus. Let η be a Hodge cycle on A; then one can ask if η is rational (in a precise sense) with respect to this new integral structure on the cohomology. This question was first considered by Andre ́.
+
Class field theory allows one to precisely understand ramification in abelian extensions of number fields. A consequence is that infinite pro-p abelian extensions of a number field are infinitely ramified above p. Boston conjectured a nonabelian analogue of this fact, predicting that certain universal p-adic representations that are unramified at p act via a finite quotient, and this conjecture strengthens the unramified version of the Fontaine-Mazur conjecture. We show in many cases that one can deduce Boston's conjecture from the unramified Fontaine-Mazur conjecture, which allows us to deduce (unconditionally) Boston's conjecture in many two-dimensional cases. This is joint work with F. Calegari.
Using the theory of Shimura varieties and the Faltings-Chai toroidal compactifications of the moduli of principally polarized abelian varieties, we convert this question into one of deciding if a certain sub-scheme of a torus embedding is again a torus embedding. In the situation where the Mumford-Tate group of A has a reductive model over Zp, for v|p (this is the unramified situation), we employ a generalization of the methods introduced by Faltings and Kisin–initially used to construct smooth integral models of Shimura varieties–to answer this question positively.
 
A by-product of this rationality result is the construction of good toroidal compactifications of the integral models of Shimura varieties mentioned above. This was in fact the main motivation for considering, in the first place, the possibility of such a result. A formal consequence of the existence of these compactifications is the following result, which is a slightly weakened version of a conjecture of Yasuo Morita: Suppose the Mumford-Tate group G of A is anisotropic mod center, then, for any prime p such that G admits a reductive model over Zp, A has potentially good reduction at all finite places v|p of E.
 
The first part of the talk will be expository: we will introduce Hodge cycles on abelian varieties and their properties, so that we can state the problem at hand. Then, we will switch track and talk about the question on toric embeddings referenced above. After this, we will focus on the case where the reduction of A at v is a split torus. In this case, the theory is more combinatorial, but the key ideas for the general case are already visible. We will quickly sketch the properties that we need of Shimura varieties and the Faltings-Chai compactification, and see how they can be used to reduce the problem to the one about toric embeddings. If time remains, we will say something about how to solve this latter problem using the Faltings-Kisin method.
 
 
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== Bei Zhang  ==
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== Sep 11 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: p-adic L-function of automorphic form of GL(2)
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood'''
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|-
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| bgcolor="#BCD2EE"  align="center" | The distribution of sandpile groups of random graphs &#42;&#42;&#42;
 
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Abstract: Modular symbol is used to construct p-adic L-functions
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The sandpile group is an abelian group associated to a graph, given as the cokernel of the graph Laplacian.  An Erd&#337;s–R&#233;nyi random graph then gives some distribution of random abelian groups.  We will give an introduction to various models of random finite abelian groups arising in number theory and the connections to the distribution conjectured by Payne et. al. for sandpile groups. We will talk about the moments of random finite abelian groups, and how in practice these are often more accessible than the distributions themselves, but frustratingly are not a priori guaranteed to determine the distribution.  In this case however, we have found the moments of the sandpile groups of random graphs, and proved they determine the measure, and have proven Payne's conjecture.
associated to a modular form. In this talk, I will explain how to
+
 
generalize this powerful tool to the construction of p-adic L-functions
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&#42;&#42;&#42; ''This is officially a '''probability seminar''', but will occur in the usual NTS room B105 at a slightly '''earlier time''', 2:25 PM.''
attached to an automorphic representation on GL_{2}(A) where A is the ring
 
of adeles over a number field. This is a joint work with Matthew Emerton.
 
 
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== David Brown ==
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== Sep 18 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Explicit modular approaches to generalized Fermat equations
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Takehiko Yasuda'''
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|-
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| bgcolor="#BCD2EE"  align="center" | ''Distributions of rational points and number fields, and height zeta functions''
 
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Abstract: TBA
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In this talk, I will talk about my attempt to relate Malle's conjecture on the distribution of number fields with Batyrev and Tschinkel's generalization of Manin's conjecture on the distribution of rational points on singular Fano varieties. The main tool for relating these is the height zeta function.
 
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== Sep 25 ==
 +
 +
<center>
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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|-
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Ramin Takloo-Bigash'''
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|-
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| bgcolor="#BCD2EE"  align="center" | ''Counting orders in number fields and p-adic integrals''
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|-
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| bgcolor="#BCD2EE"  | 
 +
In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded
 +
discriminant in a given quintic number field emphasizing the role played by p-adic (and motivic) integration.  This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (Caltech).
 +
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 +
</center>
  
 
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<br>
  
== Tony Várilly-Alvarado ==
+
== Oct 02 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: TBA  
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Pham Huu Tiep'''
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|-
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| bgcolor="#BCD2EE" align="center" | ''Nilpotent Hall and abelian Hall subgroups''
 
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|-
| bgcolor="#DDDDDD"|   
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| bgcolor="#BCD2EE" |   
Abstract: TBA
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To which extent can one generalize the Sylow theorems? One possible direction is to assume the existence of a nilpotent subgroup whose order and index are coprime. We will discuss recent
 +
joint work with various collaborators that gives a criterion to detect the existence of such subgroups in any finite group.
 
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</center>
 
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<br>
== Wei Ho ==
+
 
 +
== Oct 09 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: TBA  
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Woodbury'''
 +
|-
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| bgcolor="#BCD2EE" align="center" | ''An Adelic Kuznetsov Trace Formula for GL(4)''
 
|-
 
|-
| bgcolor="#DDDDDD"|  
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| bgcolor="#BCD2EE" |
Abstract: TBA
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An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s trace formula.  Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations of this to GL(3) have been given which are useful for number theoretic applications.  In my talk I will discuss joint work with Dorian Goldfeld in which we further generalize the said GL(3) results to GL(4).  I will discuss some of the new features and complications which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical Sato-Tate theorem.
 
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</center>
 
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<br>
 
<br>
== Rob Rhoades ==
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 +
== Oct 16 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: TBA  
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Grizzard'''
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|-
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| bgcolor="#BCD2EE" align="center" | ''Small points and free abelian groups''
 
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|-
| bgcolor="#DDDDDD"|   
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| bgcolor="#BCD2EE" |   
Abstract: TBA
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Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F* modulo torsion, respectively on E(F) modulo torsion. The groups F* and E(F) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values.  We prove the failure of the converse to this statement by explicitly constructing counterexamples.  This is joint work with Philipp Habegger and Lukas Pottmeyer.
 
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</center>
 
</center>
  
 
<br>
 
<br>
== TBA ==
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 +
== Oct 23 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: TBA 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang'''
 
|-
 
|-
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| bgcolor="#BCD2EE" align="center" | ''A conjecture of Colmez''
Abstract: TBA
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| bgcolor="#BCD2EE"  |
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In his seminal work on the Mordell conjecture, Faltings introduces and studies the (semistable)  height of an Abelian variety. When the Abelian veriety is a CM elliptic curve, its Faltings height is essentially the local derivative (at the critical point s=1) of the Dirichlet L-series associated to the imaginary quadratic field by the famous Chowla-Selberg formula.  In 1990s, Colmez gave a precise conjectural formula to compute the Faltings height of a CM abelian variety of CM type (E, &Phi;) in terms of the log derivative at s=1 of some `Artin' L-function   associated to the CM type &Phi;. He proved the conjecture when the CM number field when E is abelian, refining Gross and Anderson's work on periods. Around 2007, I proved the first non-abelian case of the Colmez conjecture using a totally different method--arithmetic intersection and Borcherds product. In this talk, I will talk about its generalization to a new family of CM type, which is an ongoing joint work with Bruinier, Howard, Kudla, and Rapoport.
 
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</center>
 
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<br>
== Chris Davis ==
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 +
== Oct 30 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: TBA  
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Laura DeMarco'''
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|-
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| bgcolor="#BCD2EE" align="center" | ''Elliptic curves and complex dynamics''
 
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|-
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| bgcolor="#BCD2EE" |   
Abstract: TBA
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I will discuss relations between the dynamics of complex rational functions and the arithmetic of elliptic curves. My goal is to present some new work (in progress) that reproves/generalizes a 1959 result of Lang and Neron about rational points on elliptic curves over function fields.  On the dynamical side, the same ideas lead to a characterization of stability for families of rational maps on P<sup>1</sup>.
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</center>
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<br>
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 +
== Nov 06 ==
 +
 
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<center>
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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|-
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Magee'''
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|-
 +
| bgcolor="#BCD2EE"  align="center" | ''Zero sets of Hecke polynomials on the sphere''
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|-
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| bgcolor="#BCD2EE"  | 
 +
The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators that arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'.
 
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</center>
 
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<br>
 
<br>
== Andrew Obus ==
+
 
 +
== Nov 13 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Cyclic Extensions and the Local Lifting Problem 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yiwei She'''
 
|-
 
|-
| bgcolor="#DDDDDD"|  
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| bgcolor="#BCD2EE" align="center" | ''The Shafarevich conjecture for K3 surfaces''
Abstract: The "local lifting problem" asks: given a G-Galois extension A/k[[t]], where k is algebraically closed of characteristic p, does there exist a G-Galois extension A_R/R[[t]] that reduces to A/k[[t]], where R is a characteristic zero DVR with residue field k? (here a Galois extension is an extension of integrally closed rings that gives a Galois extension on fraction fields.) The Oort conjecture states that the local lifting problem should always have a solution for G cyclicThis is basic Kummer theory when p does not divide |G|, and has been proven when v_p(|G|) = 1 (Oort, Sekiguchi, Suwa) and when v_p(|G|) = 2 (Green, Matignon). We will first motivate the local lifting problem from geometry, and then we will show that it has a solution for a large family of cyclic extensions. This family includes all extensions where v_p(|G|) = 3 and many extensions where v_p(|G|) is arbitrarily high. This is joint work with Stefan Wewers.
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 +
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Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfacesBuilding on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces. I will also explain the connections between the Shafarevich conjecture and the Tate conjecture.
 
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== Bianca Viray ==
+
== Nov 20 ==
  
 
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<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Descent on elliptic surfaces and transcendental Brauer element 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall'''
 
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|-
| bgcolor="#DDDDDD"|  
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| bgcolor="#BCD2EE"  align="center" | ''Endoscopy and cohomology of unitary groups''
Abstract: Elements of the Brauer group of a variety X are hard to
+
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compute. Transcendental elements, i.e. those that are not in the
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| bgcolor="#BCD2EE"  |
kernel of the natural map Br X -->  Br \overline{X}, are notoriously
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We will give a rough outline of the endoscopic classification of representations of quasi-split unitary groups carried out by Mok, following Arthur and others. We will show how this can be used to prove asymptotics for the L<sup>2</sup> Betti numbers of families of locally symmetric spaces.
difficult. Wittenberg and Ieronymou have developed methods to find
 
explicit representatives of transcendental elements of an elliptic
 
surface, in the case that the Jacobian fibration has rational
 
2-torsion. We use ideas from descent to develop techniques for general
 
elliptic surfaces.
 
 
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== Frank Thorne ==
+
== Dec 04 ==
  
 
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<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: TBA  
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Joel Specter'''
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|-
 +
| bgcolor="#BCD2EE" align="center" | ''Commuting Endomorphisms of the p-adic Unit Disk''
 
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|-
| bgcolor="#DDDDDD"|   
+
| bgcolor="#BCD2EE" |   
Abstract: TBA
+
When can a pair of endomorphisms of <math>\mathbf{Z}_p[[X]]/\mathbf{Z}_p</math> commute? Approaching this problem from the vantage point of dynamics on the p-adic unit disk, Lubin proved that whenever a non-invertible endomorphism f commutes with a non-torsion automorphism u, the pair f and u exhibit many of the same properties as endomorphisms of a formal group over <math>\mathbf{Z}_p</math>. Because of this, he posited that for such a pair of endomorphisms to exist, there in fact had to be a formal group 'somehow in the background.' In this talk, I will discuss how some of the dynamical systems of Lubin occur naturally as the restriction of the Galois action on certain Fontaine period rings. Using this observation, I will construct, in some cases, the formal groups conjectured by Lubin.
 
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== Rafe Jones ==
+
 
 +
== Dec 11 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Galois theory of iterated quadratic rational functions 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Ila Varma'''
 
|-
 
|-
| bgcolor="#DDDDDD"|   
+
| bgcolor="#BCD2EE" align="center" | ''The mean number of 3-torsion elements in ray class groups of quadratic fields''
 
+
|-
Abstract: I'll describe recent work investigating the arboreal Galoisrepresentation attached to a degree-2 rational function, focusing on the case where the function commuteswith a non-trivial Mobius transformation. In a sense this is a dynamical systems analogue to the p-adic Galois representation attached to an elliptic curve, with particular attention to the CM case.  In joint work with Michelle Manes, we give criteria for the image of this representation to be large, and show thatthese criteria are often satisfied in the "CM" case.
+
| bgcolor="#BCD2EE"  |   
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In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the
 +
class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over <math>\mathbb{Q}</math>. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
Line 191: Line 245:
 
<br>
 
<br>
  
== Jonathan Blackhurst  ==
+
 
 +
<!--
 +
== October 3 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Polynomials of the Bifurcation Points of the Logistic Map 
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrew Bridy''' (Madison)
 
|-
 
|-
| bgcolor="#DDDDDD"|   
+
| bgcolor="#BCD2EE"  align="center" | Title: The Artin–Mazur zeta function of a Lattes map in positive characteristic
Abstract: The logistic map f(r,x)=rx(1-x) was originally studied by ecologists modeling the population of a species from one generation to the next. Here, the population of the next generation f(r,x) depends on the population x of the current generation and a parameter r called the probiotic potential. To find the long-term behavior of the population for a fixed r, we look at iterates of the critical point c=1/2 under the map f. If 0<r<2 the iterates approach zero, so the population goes extinct. If 2<r<3, the iterates approach a single non-zero value, and the population is in equilibrium. If 3<r<1+sqrt(6), the iterates oscillate between two values (even iterates approach one value, while odd iterates approach another) in a boom-and-bust cycle. As r increases, the population begins oscillating between four values, then eight, then sixteen, and this period-doubling continues until an accumulation point (r=3.57, approximately) where the behavior of the population is no longer periodic. Even after the accumulation point, iterates of the critical point may begin again to exhibit periodic behavior. For example, at r=1+sqrt(8), they oscillate between three values. These values of r—2, 3, 1+sqrt(6), and 1+sqrt(8)—where qualitative behavior of the model changes are called bifurcation points. Little has been known about the polynomials these algebraic numbers satisfy. We find their degrees, show that their roots come in pairs whose mean is 1, put constraints on the size and prime factors of their constant coefficients, and record the number of real roots.
+
|-
 +
| bgcolor="#BCD2EE"  |   
 +
Abstract: The Artin–Mazur zeta function of a dynamical system is a generating function that captures information about its periodic points. In characteristic zero, the zeta function of a rational map from '''P'''<sup>1</sup> to '''P'''<sup>1</sup> is known to always be a rational function. In positive characteristic, the situation is much less clear. Lattes maps are rational maps on '''P'''<sup>1</sup> that are finite quotients of endomorphisms of elliptic curves, and they have many interesting dynamical properties related to the geometry and arithmetic of elliptic curves. I show that the zeta function of a separable Lattes map in positive characteristic is a transcendental function.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== Liang Xiao ==
+
<br>
 +
 
 +
== October 10 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Computing log-characteristic cycles using ramification theory
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Bogdan Petrenko''' (Eastern Illinois University)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Generating an algebra from the probabilistic standpoint
 
|-
 
|-
| bgcolor="#DDDDDD"|   
+
| bgcolor="#BCD2EE" |   
Abstract: There is an analogy among vector bundles with integrable
+
Abstract: Let ''A'' be a ring whose additive group is free Abelian of finite
connections, overconvergent F-isocrystals, and lisse l-adic sheaves.
+
rank. The topic of this talk is the following question: what is the
Given one of the objects, the property of being clean says that the
+
probability that several random elements of ''A'' generate it as a ring? After
ramification is controlled by the ramification along all generic
+
making this question precise, I will show that it has an interesting
points of the ramified divisors. In this case, one expects that the
+
answer which can be interpreted as a local-global principle. Some
Euler characteristics may be expressed in terms of (subsidiary) Swan
+
applications will be discussed. This talk will be based on my joint work
conductors; and (in first two cases) the log-characteristic cycles may
+
with Rostyslav Kravchenko (University of Chicago) and Marcin Mazur
be described in terms of refined Swan conductors. I will explain the
+
(Binghamton University).
proof of this in the vector bundle case and report on the recent
+
 
progress on the overconvergent F-isocrystal case if time is permitted.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== Winnie Li ==
+
<br>
 +
 
 +
== October 17 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Modularity of Low Degree Scholl Representations
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Anthony Várilly-Alvarado''' (Rice)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups
 
|-
 
|-
| bgcolor="#DDDDDD"|  To the space of d-dimensional cusp forms of weight k > 2 for
+
| bgcolor="#BCD2EE" |   
a noncongruence
+
Abstract: Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space.  They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface.   I will explain how to construct, for every non-trivial, non-constant element A of Br X, a rational genus-one fibration X -> P^1 such that A is "vertical" for this map. This implies, for example, that if there is a cohomological obstruction to the existence of a point on X, then there is a genus-one fibration X -> P^1 where none of the fibers are locally soluble, giving a concrete, geometric way of "seeing" a Brauer-Manin obstruction. The construction also gives a fast, practical algorithm for computing the Brauer group of X.  Conjecturally, this gives a mechanical way of testing for the existence of rational points on these surfaces. This is joint work with Bianca Viray.
subgroup of SL(2, Z), Scholl has attached a family of 2d-dimensional
 
compatible l-adic
 
representations of the Galois group over Q. Since his construction is
 
motivic, the associated
 
L-functions of these representations are expected to agree with
 
certain automorphic L-functions
 
according to Langlands' philosophy. In this talk we shall survey
 
recent progress on this topic.
 
More precisely, we'll see that this is indeed the case when d=1. This
 
also holds true when d=2,
 
provided that the representation space admits quaternion
 
multiplications. This is a joint work
 
with Atkin, Liu and Long.
 
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
 +
<br>
 +
 +
== October 24 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Garrett''' (Minnesota)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Spectra of pseudo-Laplacians on spaces of automorphic forms
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: Faddeev–Pavlov and Lax–Phillips observed that certain
 +
restrictions of the Laplacian to parts of automorphic continuous
 +
spectrum have discrete spectrum. Colin de Verdiere used this to prove
 +
meromorphic continuation of Eisenstein series, and proposed ways to
 +
exploit this idea to construct self-adjoint operators with spectra
 +
related to zeros of ''L''-functions. We show that simple forms of this
 +
construction produce at most very sparse spectra, due to
 +
incompatibility with pair correlations for zeros. Ways around some of
 +
the obstacles are sketched. (Joint with E. Bombieri.)
 +
|}                                                                       
 +
</center>
  
 +
<br>
  
== Avraham Eizenbud ==
+
== October 31 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#DDDDDD" align="center"| Title: Multiplicity One Theorems - a Uniform Proof 
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jerry Wang''' (Princeton)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Pencils of quadrics and the arithmetic of hyperelliptic curves
 
|-
 
|-
| bgcolor="#DDDDDD"|   
+
| bgcolor="#BCD2EE" |   
 +
Abstract: In recent joint works with Manjul Bhargava and Benedict Gross, we showed that a positive proportion of hyperelliptic curves over '''Q''' of genus ''g'' have no points over any odd degree extension of '''Q'''. This is done by computing certain 2-Selmer averages and applying a result of Dokchitser–Dokchitser on the parity of the rank of the 2-Selmer groups in biquadratic twists. In this talk, we will see how arithmetic invariant theory and the geometric theory of pencils of quadrics are used to obtain the 2-Selmer averages.
 +
|}                                                                       
 +
</center>
  
Abstract: Let F be a local field of characteristic 0.
+
<br>
We consider distributions on GL(n+1,F) which are invariant under the adjoint action of
 
GL(n,F). We prove that such distributions are invariant under
 
transposition. This implies that an irreducible representation of
 
GL(n+1,F), when restricted to GL(n,F) "decomposes" with multiplicity one.
 
  
 +
== November 7 ==
  
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''who?''' (where?)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: tba
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: tba
 +
|}                                                                       
 +
</center>
  
Such property of a group and a subgroup is called strong Gelfand property.
+
<br>
It is used in representation theory and automorphic forms. This property
 
was introduced by Gelfand in the 50s for compact groups. However, for
 
non-compact groups it is much more difficult to establish.
 
  
For our pair (GL(n+1,F),GL(n,F)) it was proven in 2007 in [AGRS] for
+
== November 12 ==
non-Archimedean F, and in 2008 in [AG] and [SZ] for Archimedean F. In this
 
lecture we will present a uniform for both cases.
 
This proof is based on the above papers and an additional new tool. If time
 
permits we will discuss similar theorems that hold for orthogonal and
 
unitary groups.
 
  
[AG] A. AizenbudD. Gourevitch, Multiplicity one theorem for (GL(n+1,R),GL(n,R))", arXiv:0808.2729v1 [math.RT]
+
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Counting extensions of number fields of given degree, bounded (rho)-discriminant, and specified Galois closure
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: A very basic question in algebraic number theory is: how many number fields are there? A natural way to order the fields of a fixed degree n is by discriminant, and classical results of Minkowski then assure us that there are only finitely many fields with a given discriminant. We are also often interested in counting number fields, or relative extensions, with other properties, such as having a particular Galois closure. A folk conjecture sometimes attributed to Linnik states that the number of extensions of degree n and absolute discriminant less than X is on the order of X. A great deal of recent and ongoing work has been focused towards achieving upper bounds on counts of this nature (quite successfully, in degree 5 and lower), but there is comparatively little known in higher degrees, for relative extensions, or for sufficiently complicated Galois closures: the primary results are those of Schmidt and Ellenberg-Venkatesh. I will discuss these results and my thesis work, in which I generalize several of their results and introduce another counting metric, the "rho-discriminant".
 +
|}                                                                       
 +
</center>
  
[AGRS] A. Aizenbud,  D. Gourevitch, S. Rallis, G. Schiffmann, Multiplicity One Theorems, arXiv:0709.4215v1 [math.RT], to appear in the Annals of Mathematics.
+
<br>
  
 +
== November 21 ==
  
[SZ] B. Sun and C.-B. Zhu, Multiplicity one theorems: the Archimedean case, preprint available at http://www.math.nus.edu.sg/~matzhucb/Multiplicity_One.pdf
+
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Lipnowski''' (Duke)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: tba
 +
|-
 +
| bgcolor="#BCD2EE"  |
 +
Abstract: tba
 +
|}                                                                       
 +
</center>
 +
 
 +
<br>
 +
 
 +
== November 26 ==
  
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Kane''' (Stanford)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Diffuse decompositions of polynomials
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: We study some problems relating to polynomials evaluated
 +
either at random Gaussian or random Bernoulli inputs.  We present some
 +
new work on a structure theorem for degree-''d'' polynomials with Gaussian
 +
inputs.  In particular, if ''p'' is a given degree-''d'' polynomial, then ''p''
 +
can be written in terms of some bounded number of other polynomials
 +
''q''<sub>1</sub>, ..., ''q''<sub>''m''</sub> so that the joint probability density function of
 +
''q''<sub>1</sub>(''G''), ..., ''q''<sub>''m''</sub>(''G'') is close to being bounded.  This says essentially
 +
that any abnormalities in the distribution of ''p''(''G'') can be explained by
 +
the way in which ''p'' decomposes into the ''q''<sub>''i''</sub>.  We then present some
 +
applications of this result.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
 +
<br>
  
 +
== December 5 ==
 +
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jennifer Park''' (MIT)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: tba
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: tba
 +
 +
|}                                                                       
 +
</center>
  
 
<br>
 
<br>
  
== Organizer contact information ==
+
== December 12 ==
[http://www.math.wisc.edu/~brownda/ David Brown:]
 
  
[http://www.math.wisc.edu/~cais/ Bryden Cais:]
+
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Vivek Shende''' (Berkeley)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Equidistribution on the space of rank two vector bundles over the projective line
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: I will discuss how the algebraic geometry of hyperelliptic curves gives an approach to a function field analogue of the 'mixing conjecture' of Michel and Venkatesh.  (For a rather longer abstract, see the [http://arxiv.org/abs/1307.8237 arxiv posting] of the same name as the talk). This talk presents joint work with Jacob Tsimerman.  
 +
|}                                                                       
 +
</center>
  
 
<br>
 
<br>
  
 +
-->
 +
 +
== Organizer contact information ==
  
 +
Sean Rostami (srostami@math.wisc.edu)
  
 
----
 
----

Latest revision as of 13:53, 28 November 2014

Aug 28

Robert Lemke Oliver
The distribution of 2-Selmer groups of elliptic curves with two-torsion

Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over Q, when curves are ordered by height, is exactly 3, and Bhargava and Ho have shown that, in the family of curves with a marked point, the average is exactly 6. In stark contrast to these results, we show that the average size in the family of elliptic curves with a two-torsion point is unbounded. This follows from an understanding of the Tamagawa ratio associated to such elliptic curves, which we prove is "normally distributed with infinite variance". This work is joint with Zev Klagsbrun.


Sep 04

Patrick Allen
Unramified deformation rings

Class field theory allows one to precisely understand ramification in abelian extensions of number fields. A consequence is that infinite pro-p abelian extensions of a number field are infinitely ramified above p. Boston conjectured a nonabelian analogue of this fact, predicting that certain universal p-adic representations that are unramified at p act via a finite quotient, and this conjecture strengthens the unramified version of the Fontaine-Mazur conjecture. We show in many cases that one can deduce Boston's conjecture from the unramified Fontaine-Mazur conjecture, which allows us to deduce (unconditionally) Boston's conjecture in many two-dimensional cases. This is joint work with F. Calegari.



Sep 11

Melanie Matchett Wood
The distribution of sandpile groups of random graphs ***

The sandpile group is an abelian group associated to a graph, given as the cokernel of the graph Laplacian. An Erdős–Rényi random graph then gives some distribution of random abelian groups. We will give an introduction to various models of random finite abelian groups arising in number theory and the connections to the distribution conjectured by Payne et. al. for sandpile groups. We will talk about the moments of random finite abelian groups, and how in practice these are often more accessible than the distributions themselves, but frustratingly are not a priori guaranteed to determine the distribution. In this case however, we have found the moments of the sandpile groups of random graphs, and proved they determine the measure, and have proven Payne's conjecture.

*** This is officially a probability seminar, but will occur in the usual NTS room B105 at a slightly earlier time, 2:25 PM.



Sep 18

Takehiko Yasuda
Distributions of rational points and number fields, and height zeta functions

In this talk, I will talk about my attempt to relate Malle's conjecture on the distribution of number fields with Batyrev and Tschinkel's generalization of Manin's conjecture on the distribution of rational points on singular Fano varieties. The main tool for relating these is the height zeta function.


Sep 25

Ramin Takloo-Bigash
Counting orders in number fields and p-adic integrals

In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded discriminant in a given quintic number field emphasizing the role played by p-adic (and motivic) integration. This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (Caltech).


Oct 02

Pham Huu Tiep
Nilpotent Hall and abelian Hall subgroups

To which extent can one generalize the Sylow theorems? One possible direction is to assume the existence of a nilpotent subgroup whose order and index are coprime. We will discuss recent joint work with various collaborators that gives a criterion to detect the existence of such subgroups in any finite group.


Oct 09

Michael Woodbury
An Adelic Kuznetsov Trace Formula for GL(4)

An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s trace formula. Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations of this to GL(3) have been given which are useful for number theoretic applications. In my talk I will discuss joint work with Dorian Goldfeld in which we further generalize the said GL(3) results to GL(4). I will discuss some of the new features and complications which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical Sato-Tate theorem.


Oct 16

Robert Grizzard
Small points and free abelian groups

Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F* modulo torsion, respectively on E(F) modulo torsion. The groups F* and E(F) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. We prove the failure of the converse to this statement by explicitly constructing counterexamples. This is joint work with Philipp Habegger and Lukas Pottmeyer.


Oct 23

Tonghai Yang
A conjecture of Colmez

In his seminal work on the Mordell conjecture, Faltings introduces and studies the (semistable) height of an Abelian variety. When the Abelian veriety is a CM elliptic curve, its Faltings height is essentially the local derivative (at the critical point s=1) of the Dirichlet L-series associated to the imaginary quadratic field by the famous Chowla-Selberg formula. In 1990s, Colmez gave a precise conjectural formula to compute the Faltings height of a CM abelian variety of CM type (E, Φ) in terms of the log derivative at s=1 of some `Artin' L-function associated to the CM type Φ. He proved the conjecture when the CM number field when E is abelian, refining Gross and Anderson's work on periods. Around 2007, I proved the first non-abelian case of the Colmez conjecture using a totally different method--arithmetic intersection and Borcherds product. In this talk, I will talk about its generalization to a new family of CM type, which is an ongoing joint work with Bruinier, Howard, Kudla, and Rapoport.


Oct 30

Laura DeMarco
Elliptic curves and complex dynamics

I will discuss relations between the dynamics of complex rational functions and the arithmetic of elliptic curves. My goal is to present some new work (in progress) that reproves/generalizes a 1959 result of Lang and Neron about rational points on elliptic curves over function fields. On the dynamical side, the same ideas lead to a characterization of stability for families of rational maps on P1.


Nov 06

Michael Magee
Zero sets of Hecke polynomials on the sphere

The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators that arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'.


Nov 13

Yiwei She
The Shafarevich conjecture for K3 surfaces

Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces. Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces. I will also explain the connections between the Shafarevich conjecture and the Tate conjecture.


Nov 20

Simon Marshall
Endoscopy and cohomology of unitary groups

We will give a rough outline of the endoscopic classification of representations of quasi-split unitary groups carried out by Mok, following Arthur and others. We will show how this can be used to prove asymptotics for the L2 Betti numbers of families of locally symmetric spaces.


Dec 04

Joel Specter
Commuting Endomorphisms of the p-adic Unit Disk

When can a pair of endomorphisms of [math]\displaystyle{ \mathbf{Z}_p[[X]]/\mathbf{Z}_p }[/math] commute? Approaching this problem from the vantage point of dynamics on the p-adic unit disk, Lubin proved that whenever a non-invertible endomorphism f commutes with a non-torsion automorphism u, the pair f and u exhibit many of the same properties as endomorphisms of a formal group over [math]\displaystyle{ \mathbf{Z}_p }[/math]. Because of this, he posited that for such a pair of endomorphisms to exist, there in fact had to be a formal group 'somehow in the background.' In this talk, I will discuss how some of the dynamical systems of Lubin occur naturally as the restriction of the Galois action on certain Fontaine period rings. Using this observation, I will construct, in some cases, the formal groups conjectured by Lubin.


Dec 11

Ila Varma
The mean number of 3-torsion elements in ray class groups of quadratic fields

In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over [math]\displaystyle{ \mathbb{Q} }[/math]. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava.



Organizer contact information

Sean Rostami (srostami@math.wisc.edu)


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