Abstracts

Matthew Emerton (Northwestern)
An introduction to the mod p and padic Langlands program for GL_2
I will give an introduction to some parts of the mod p and padic Langlands program for GL_2. The focus will be on motivation and simple examples. I will get as far as time permits (which may not be very far).

Pierre Colmez (Ecole Polytechnique)
On the padic Langlands correspondence for GL_2(Q_p)

Ellen Eischen (Northwestern)
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; these differential operators play 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, which should be similarly significant in the study of many padic Lfunctions, in particular padic Lfunctions attached to families of padic automorphic forms on unitary groups.

Teruyoshi Yoshida (Cambridge)
Converse of the weightmonodromy conjecture
This is an expository talk on (classical) global/local
Langlands correspondence between Galois representations and
automorphic/smooth representations of GL(n). The longstanding
weightmonodromy conjecture claims that all geometric local Galois
representations (i.e. appearing in etale cohomology of varieties) are
pure, but we can show that all pure local Galois representations are
found in geometry. It seems that we can only do this via global
argument (HondaTate theory etc), and we introduce Langlands
correspondence in the course of going through several proofs.

Bei Zhang (Northwestern)
padic Lfunction of supersingular Hilbert modular form
Pollack constructed padic Lfunction of modular form f at supersingular primes. Especially, when the Fourier coefficient of f at p is zero, plus and minus padic Lfunctions can be obtained as elements inside Iwasawa algebra. They can serve as one side of Iwasawa main conjecture for an elliptic curve over Q when a_p=0 by Kobayashi's work about +/ Selmer group of this elliptic curve at p. As the generalization of Pollack's work, I will discuss the construction of padic Lfunction for supersingular Hilbert modular form using RankinSelberg method. And will define the +/ padic Lfunctions in this case.

Nick Ramsey (DePaul)
Euclidean Ideal Classes
In the 1970's, Lenstra generalized the notion of a Euclidean ring to that of a ring with a Euclidean ideal. In the context of Dedekind domains, the consequence of the existence of such an ideal is the cyclicity of the class group in much the same way that the consequence of the existence of a Euclidean algorithm is the triviality of the class group. In this talk, I'll discuss Lentra's notion in light of some recent developments of Hester Graves. In particular, I'll discuss a joint result with Graves classifying the quadratic imaginary fields (which play a rather exception role in the theory) that have a Euclidean ideal.

David Savitt (Arizona)
Lifting local Galois representations
I will explain the proof of some cases of the weight part of
Serre's conjecture over totally real fields. One key ingredient is the
construction and study of certain lifts of mod p Galois
representations, and this will be the focus of the talk. This is
joint work with Toby Gee.

Florian Herzig (Northwestern)
The classification of irreducible mod p representations of a padic GL_n
Let F be a finite extension of the padic numbers. We describe the classification of irreducible admissible smooth representations of GL_n(F) over an algebraically closed field of characteristic p, in terms of "supersingular" representations. This generalizes results of BarthelLivne for n = 2. Our motivation is the hypothetical mod p Langlands correspondence for GL_n, which is supposed to relate smooth mod p representations of GL_n(F) to ndimensional mod p Galois representations.

Sug Woo Shin (Chicago)
Plancherel density theorem
Serre asked and answered the following question: Fix a prime p. Consider the eigenvalues of the Hecke
operator T_p on the space of cuspforms of weight k and level N, where (p,N)=1. What is the distribution of the
eigenvalues as k and/or N tend to infinity? The answer turns out to be the Plancherel measure. First I will
review the Plancherel measure in the context of harmonic analysis on padic (and real) Lie groups. Next I will
interpret, reformulate and generalize Serre's question to the case of arbitrary reductive groups over Q. (Serre's
situation is concerned with the case G=GL_2. When G is anisotropic over Q, this result is due to Sauvageot.)

Joël Bellaïche (Brandeis)
Critical padic Lfunctions
For three decades, it was only possible to attach a padic Lfunction to a cuspidal eigenform of weight k whose slope
(that is, pvaluation of the eigenvalue for the operator U_p) was strictly less than k+1. More recently, Stevens and
Pollack explained how to attach, in a natural way, a padic Lfunction for some slope k+1 (or critical slope)
cuspidal eigenforms, namely the cuspidal forms that are not critical.
I shall explain how, using our knowledge of the geometry of the eigencurve, one can extend
their work and define natural padic Lfunctions for the missing cuspidal eigenforms as well as
for the evil Eisenstein series. I shall also show that all those padic Lfunctions, old and new, fit in
a twovariables padic Lfunction on the eigencurve, extending earlier results of Stevens, Panchishkin and Emerton.

Toby Gee (Harvard)
The SatoTate conjecture for Hilbert modular forms
I will discuss the SatoTate conjecture for Hilbert modular forms, which I recently proved in collaboration with Thomas BarnetLamb and David Geraghty. No previous knowledge of Hilbert modular forms or the SatoTate conjecture will be assumed.
