Galois Representations and Automorphic Forms

I will outline the proof of various cases of the local-global compatibility statement alluded to in the title, and also explain its applications to the Fontaine—Mazur conjecture, and to a conjecture of Kisin.

Let F be a locally compact non-Archimedean field, p its residue characteristic and G a connected reductive algebraic group over F . The classical Satake isomorphism describes the Hecke algebra (over the field of complex numbers) of double classes in G with respect to a special maximal compact subgroup K of G . In our setting K is a slightly smaller special parahoric subgroup, we introduce an absolutely irreducible smooth representation of K on a vector field V over a field of characteristic p , and we get a description of an analoguous Hecke algebra with respect to V .

Let p and l be two distinct prime numbers, and fix a positive integer d . I will explain how the F_l-cohomology complex of the Lubin-Tate tower of height d of a p-adic field K realizes mod l versions of both the semi-simple Langlands correspondence for GL_d(K) and the "Langlands-Jacquet" transfer from GL_d(K) to the central division K-algebra of invariant 1/d . Then I will give an explicit description of the supercuspidal part of the integral l-adic cohomology of this LT tower in terms of certain universal deformations.

<a href="http://math.ias.edu/files/seminars/FontaineAbst.pdf">http://math.ias.edu/files/seminars/FontaineAbst.pdf</a>