## University of Chicago Number Theory Seminar

### Spring 2010: Friday 3:30-5:00pm, room E207

This is the home page of Number Theory Seminar at University of Chicago. To get on (or off) the seminar mailing list, please contact Sug Woo Shin or Liang Xiao.

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

One may also want to check out Geometric Langlands Seminar and Algebraic Geometry Seminar.

Click here to see the schedule of other quarters: Winter 2010 / Spring 2010 / Fall 2010 / Winter 2011 / Spring 2011 / Fall 2011 / Winter 2012 / Spring 2012 / Fall 2012..

## Schedule

Date

Speaker

Topic

April 2

Quadratic forms and dynamics on homogeneous space
Over the past 40 years or so unipotent flow on homogeneous spaces and its applications to number theory especially to Diophantine approximation has attracted considerable attention. The aim of this talk is to draw this connection and address some techniques and major results in the subject. If time allows we will address some recent works in this direction as well.

April 9

Kai-Wen Lan
(Princeton Univ.)

Vanishing theorems for torsion automorphic sheaves
In this talk, I will explain my joint work with Junecue Suh on when and why the cohomology of Shimura varieties (with nontrivial integral coefficients) has no torsion, based on certain vanishing theorems we have proved recently. (All conditions involved will be explicit, independent of level, and effectively computable.)

April 16

Alberto Minguez
(Jussieu)

Towards an l-modular Jacquet-Langlands correspondence
Let F be a non-Archimedean locally compact field of residue characteristic p, and let G be an inner form of GL(n,F), that is a group of the form GL(m,D) where D is a division algebra of centre F. Given R an algebraically closed field of characteristic different from p, I will explain how to classify the irreducible smooth representations of G with coefficients in R, in terms of parameters involving the supercuspidal representations of the Levi subgroups of G. We will discuss about the possibility of having a Jacquet-Langlands correspondence modulo l. This is a joint work with Vincent Sécherre.

April 23

Wei Zhang
(Harvard Univ.)

Relative trace formula and Gross-Zagier formula
In this talk I will present a relative trace formula approach to the Gross-Zagier formula and its high dimensional generalization (a derivative version of the global Gross-Prasad Conjecture) for unitary Shimura variety. In particular, a conjectural Arithmetic Fundamental Lemma (AFL) is proposed. Some results proved recently will be presented, including the AFL for the unitary group in three variables.

April 30

(Univ. of Maryland)

A New Construction of the Tame Local Langlands Correspondence for GL(n,F), n a prime
In my thesis, I give a new construction of the tame local Langlands correspondence for $GL(n,F)$, $n$ a prime, where $F$ is a non-archimedean local field of characteristic zero. The Local Langlands Correspondence for $GL(n,F)$ has been proven recently by Henniart, Harris/Taylor. In the tame case, supercuspidal representations correspond to characters of elliptic tori, but the local Langlands correspondence is unnatural because it involves a twist by some character of the torus. Taking the cue from the theory of real groups, supercuspidal representations should instead be parameterized by characters of covers of tori. Stephen DeBacker has calculated the distribution characters of supercuspidal representations for $GL(n,F)$, $n$ prime, and they are written in terms of functions on elliptic tori. Over the reals, Harish-Chandra parameterized discrete series representations of real groups by describing their distribution characters restricted to compact tori. Those distribution characters are written down in terms of functions on a canonical double cover of real tori. We show that if one writes down a natural analogue of Harish-Chandra's distribution character for $p$-adic groups, then it is the distribution character of a unique supercuspidal representation of $GL(n,F)$, where $n$ is prime, away from the local character expansion. These results pave the way for a natural construction of the local Langlands correspondence for $GL(n,F)$, $n$ a prime. In particular, there is no need to introduce any character twists.

May 7

Jay Pottharst
(Boston College)

Iwasawa theory of modular forms at nonordinary primes
The Iwasawa theory of modular forms concerns the p-adic interpolation of their arithmetic data, e.g. their p-adic L-functions. Thanks to the work of many mathematicians, we have a detailed understanding of many parts of this picture. But Galois-theoretic aspects in the non-ordinary case have been particularly resistant to analysis, and understanding this case is a major goal of current research. We will explain a new method for bringing most nonordinary primes onto an equal footing with the ordinary ones, in such a way that much of our intuition generalizes. We make use of recent improvements in p-adic Hodge theory and Galois cohomology.

May 14

Bei Zhang
(Northwestern Univ.)

Fourier-Jacobi expansion of Eisenstein series on U(3, 1) and the application
In order to apply congruence among modular forms to study one-side divisibility towards the main conjecture of GL2 × K×, a p-integral Eisenstein series on U(3, 1) needs to be constructed so that it does not vanish modulo p. I will introduce the calculation result about the Fourier-Jacobi coefficient of this Eisenstein series, and then explain a strategy about how such calculation can help to argue the nonvanishing modulo p of this Eisenstein series.

May 21

Kazuya Kato

p-adic period domains and toroidal partial compactifications
S. Usui, C. Nakayama, and I constructed toroidal partial compactifications of period domains for Hodge structures. In this talk, I explain that similarly we can construct toroidal partial compactifications of p-adic period domains for p-adic Hodge structures.

May 28

Julee Kim
(MIT)

Generalized Casselman-Shalika formula on GLn.
We calculate Whittaker functions of generalized principal series of GLn, using Hecke algebra isomorphisms.

June 4

Tong Liu
(Purdue Univ.)

Automorphy of Galois representation of Gal(Q/Q) to GO4(Qp).
This is a joint work with Jiu-Kang Yu. Let ρ: Gal(Q/Q)→ GO4(Qp) be a continuous representation. We study the automorphy of ρ, that is, when ρ arises from an automorphic form. Under the assumptions that \rho is unramified almost everywhere, crystalline at p with Hodge-Tate weights {0, 0, r, r} such that 2r < p and the eigenvalues of the complex conjugate is 1, 1, -1 , -1, we prove ρ is indeed automorphic. We apply this result to the representations of Scholl's motive which relates to noncongruence modular forms and show that if the dimension of the representaion is 4 and the weight of noncongruence modular form is odd then the representation is automorphic.

June 11

Ellen Eischen
(Northwestern Univ.)

p-adic differential operators on automorphic forms and applications
At certain special points, the values of the Riemann zeta function and many other L-functions are algebraic, up to a well-determined transcendental factor. G. Shimura, H. Maass, and M. Harris extensively studied a class of differential operators on automorphic forms; the action of these operators on Eisenstein series plays an important role in proofs of algebraicity properties of many L-functions.
Building on work of N. Katz, we introduce a p-adic analogue of these differential operators, and we discuss an application.

This page is maintained by Liang Xiao; it was shamelessly copied from Kiran Kedlaya, 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.