NTS/Abstracts
Aug 28
| Robert Lemke Oliver |
| The distribution of 2-Selmer groups of elliptic curves with two-torsion |
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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. |
Sep 04
| Patrick Allen |
| Unramified deformation rings |
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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 *** |
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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 |
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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. |
Sep 25
| Ramin Takloo-Bigash |
| Counting orders in number fields and p-adic integrals |
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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 |
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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. |
Oct 09
| Michael Woodbury |
| Coming soon... |
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Coming soon... |
Oct 16
| Robert Grizzard |
| Small points and free abelian groups |
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Coming soon... |
Oct 23
| Simon Marshall |
| Endoscopy and cohomology of unitary groups |
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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^2 Betti numbers of families of locally symmetric spaces. |
Oct 30
| Laura DeMarco |
| Elliptic curves and complex dynamics |
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Coming soon... |
Nov 06
| Yueke Hu |
| Coming soon... |
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Coming soon... |
Nov 13
| Yiwei She |
| Coming soon... |
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Coming soon... |
Nov 20
| Tonghai Yang |
| CM values and central derivatives of L-functions |
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In this talk, I will describe how the CM values of some automorphic Green functions (in a Shimura variety) are directly related to the central derivatives of some Rankin-Selberg L-functions. In the special case where the L-function vanishes entirely, the formula is a vast generalization of the Gross-Zagier factorization formula for singular moduli. In the general case, one can try to prove a higher-dimensional analogue of the Gross-Zagier formula. |
Dec 04
| Joel Specter |
| Coming soon... |
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Coming soon... |
Dec 11
| Ila Varma |
| Coming soon... |
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Coming soon... |
Organizer contact information
Sean Rostami (srostami@math.wisc.edu)
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