We will discuss joint work with Bhargav Bhatt and Matthew Morrow which proves a comparison between integral p-adic etale cohomology and integral crystalline cohomology, using a geometric construction of Breuil-Kisin modules.

Hida proved in 1981 that if a prime p divides the algebraic part of the value at s = 1 of the adjoint L-function of a holomorphic cusp form f, then there is another cusp form g such that f is congruent to g modulo the prime p. This result was generalized to Hilbert modular forms by Eknath Ghate and Mladen Dimitrov, and to certain cusp forms on GL(2) over an imaginary quadratic field by Eric Urban, and recently to cusp forms on GL(2) over any number field by Namikawa. In this talk, I will discuss further generalizations of this phenomenon to the context adjoint L-values for cohomological cuspidal automorphic representations of GL(n) over any number field. This is joint work with Baskar Balasubramanyam

Questions about decomposition of representations of groups over local fields to subgroups, such as from SO(n) to SO(n-1), has been of  considerable interest in the recent past with impressive works of Waldspurger, Moeglin-Waldspurger, and Raphael Beuzart-Plessis. Corresponding global question for unitary groups has been partly settled by Wei Zhang. 

These questions have so far been considered only for tempered representations (of the group and the subgroup). In this lecture, I will discuss the situation with non-tempered representations.  This is a report on work done with B. Gross and Wee Teck Gan.
 

Given a reductive group G over the p-adic numbers I will explain a classification result for G-bundles over the curve defined in my joint work with Fontaine. I will then state and explain a conjecture linked to this classification

In this talk, I will present some results concerning the classification of the cuspidal automorphic representations of GL(n) over Q whose archimedean component is `` algebraic ‘’ and whose non archimedean components are all unramified.  As an application, I will sketch a proof of the following result : there are exactly 23 Siegel modular cusp forms of weight 12, for the full Siegel modular group, in genus at most 12.

A new method introduced by Treumann, Venkatesh and the speaker makes it possible to prove cases of Langlands functoriality for cohomology classes mod p. I will show how this implies, in this context, automorphic induction for radicial extensions E=F(x^(1/p)) of number fields. I will also discuss the application to endoscopy for unitary groups.

Bluegrass performance in Simons Auditorium.