Date

Speaker

Topic

April 2

Amir Mohammadi

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

KaiWen 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 lmodular JacquetLanglands correspondence
Let F be a nonArchimedean 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 JacquetLanglands correspondence modulo l. This is a joint work with Vincent Sécherre.

April 23

Wei Zhang (Harvard Univ.)

Relative trace formula and GrossZagier formula
In this talk I will present a relative trace formula approach to
the GrossZagier formula and its high dimensional generalization (a
derivative version of the global GrossPrasad 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

Moshe Adrian (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 nonarchimedean
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, HarishChandra 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 HarishChandra'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 padic interpolation of their arithmetic data, e.g. their padic Lfunctions. Thanks to the work of many mathematicians, we have a detailed understanding of many parts of this picture. But Galoistheoretic aspects in the nonordinary 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 padic Hodge theory and Galois cohomology.

May 14

Bei Zhang (Northwestern Univ.)

FourierJacobi expansion of Eisenstein series on U(3, 1) and the
application
In order to apply congruence among modular forms to study
oneside divisibility towards the main conjecture of GL_{2} × K^{×}, a
pintegral 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 FourierJacobi 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

padic 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 padic period domains for padic Hodge structures.

May 28

Julee Kim (MIT)

Generalized CasselmanShalika formula on GL_{n}.
We calculate Whittaker functions of generalized principal series of GL_{n},
using Hecke algebra isomorphisms.

June 4

Tong Liu (Purdue Univ.)

Automorphy of Galois representation of Gal(Q/Q) to GO_{4}(Q_{p}).
This is a joint work with JiuKang Yu. Let ρ: Gal(Q/Q)→
GO_{4}(Q_{p}) 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 HodgeTate 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.)

padic differential operators on automorphic forms and applications
At certain special points, the values of the Riemann zeta function and many other Lfunctions are algebraic, up to a welldetermined 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 Lfunctions.
Building on work of N. Katz, we introduce a padic analogue of these differential operators, and we discuss an application.
