University of Chicago Number Theory Seminar

Spring 2014: Tuesday 1:30-2:50pm, E308


This is the homepage of the Number Theory Seminar at the University of Chicago. To get on or off the mailing list, you can either go to lists.uchicago.edu or contact Davide Reduzzi.

NOTE: the room for this quarter is Eckhart 308.

Click here to see the location of Eckhart Hall, and here for directions to the University of Chicago.

Previous quarter: Winter 2014 • Next quarter: Fall 2014

Spring 2014 Schedule

Date

Speaker

Title

April 8

Mike Roth
(Queen's University)

Roth's theorem for arbitrary varieties
If X is a variety of general type defined over a number field k, then the Bombieri-Lang conjecture predicts that the k-rational points of X are not Zariski dense. The conjecture is a prediction that a global condition on the canonical bundle (that it is ''generically positive'') implies a global condition about rational points...(Show Abstract)

April 15

Stefano Morra
(Fields Inst./ Univ. of Toronto)

On mod p local-global compatibility for GL3 in the ordinary case
Let ρ:GQpGL3(Fp) be a maximally nonsplit, ordinary Galois representation. If ρ is Fontaine-Lafaille and sufficiently generic, the φ-action on the associated Fontaine-Lafaille module lets us detect a local Galois invariant.
On the other hand, let π be a smooth GL3(Qp)-representation (over Fp). If π verifies appropriate conditions...
(Show Abstract)

April 22

Arul Shankar
(Harvard)

The average 5-Selmer rank of elliptic curves
We use geometry-of-numbers techniques to show that the average size of the 5-Selmer group of elliptic curves is equal to 6. From this, we deduce an upper bound on the average rank of elliptic curves. Then, by constructing families of elliptic curves with equidistributed root number we show that the average rank is less than 1. This is joint work with Manjul Bhargava.

April 29

William Casselman
(Univ. British Columbia)

Symmetric powers and the Satake transform
A classic formula of Tamagawa, following work of Hecke, relates certain Langlands L-functions associated to GL(n,k) groups (k p-adic) and some simple properties of the Satake transform for these groups, but applied to functions in the Schwartz space of the matrix algebra M(n,k).
Recent conjectures of Ngo and others suggest...
(Show Abstract)

May 6

Wei Zhang
(Columbia)

Selmer groups and the divisibility of Heegner points
This talk is about a proof of Kolyvagin's conjecture in 1991 on p-indivisibility of (derived) Heegner points for an ordinary prime p > 3 with some ramification conditions, with some application to the arithmetic of elliptic curves.

May 13

Matthew Young
(Texas A&M)

Equidistribution of Eisenstein series
The behavior of eigenfunctions of the Laplacian is a well-studied topic in analysis, geometry, math physics, etc. In some arithmetical settings, such as for the modular surface, then there are also interesting connections to number theory. In particular, the behavior of the Eisenstein series is related to properties of the Riemann zeta function. I will discuss some recent work showing that the Eisenstein series equidistributes on "thin" sets, with various notions of thin.

May 20

Daniel Le
(UChicago)

Lattices in the algebraic vectors in the cohomology of U(3) Shimura varieties
Little is known about the p-adic Langlands correspondence outside of GL1 and GL2(Qp). Assuming that it satisfies p-adic local-global compatibility, one would expect the correspondence to occur in the completed cohomology of Shimura varieties. Though it is not known that the result is independent of the global situation, we show that under hypotheses the lattice in the algebraic vectors of the completed cohomology of U(3) Shimura varieties depends only on the local Galois representation at p.

May 27

Marc-Hubert Nicole
(Marseille/UCLA)

An l-adic Jacquet-Langlands correspondence for paramodular Siegel threefolds
T. Ibukiyama proposed, in the early ‘80s, a conjecture à la Jacquet-Langlands-Eichler-Shimizu for spaces of Siegel modular forms for the paramodular group K(p), where p is a prime. A concrete geometric consequence of Ibukiyama’s conjecture is that: 1 + the dimension of the space S3(K(p)) of Siegel modular forms of weight three is equal to the number of irreducible components of the supersingular locus of the moduli space A2,1 of principally polarized abelian surfaces mod p....(Show Abstract)

June 3

Masoud Kamgarpour
(Univ. of Queensland)

Preservation of depth in local geometric Langlands program
Local Langlands program aims to establish a relationship between representations of the Galois group of a local field and irreducible representations of the dual group. It is expected that, under mild conditions, this correspondence preserves depths of representations. In this talk, I will explain the geometric analogue of this expectation, in the framework of Frenkel-Gaitsgory's local geometric Langlands correspondence....(Show Abstract)

One may also want to check out:

  • Geometric Langlands Seminar Monday and Thursday 4:30pm;
  • Algebraic Geometry Seminar biweekly on Wednesday 4:30pm-6pm;
  • Northwestern University Number Theory Seminar Monday 4pm;
  • UIC Number Theory Seminar Tuesday 1pm.

    This page is maintained by Davide Reduzzi; it was shamelessly copied from Liang Xiao's page, which was shamelessly copied from Kiran Kedlaya's page, which in turn was shamelessly copied from Jason Starr's page, which in turn was shamelessly copied from Ravi Vakil's page, which in turn was shamelessly copied from Pasha Belorousski's page at the University of Michigan. For more sites with a similar pedigree, see Michael Thaddeus's list or Jim Bryan's list.