Dmitri Whitmore (University of Cambridge)

Abstract:
The Langlands programme predicts a correspondence between automorphic forms and Galois representations. A key expectation is local-global compatibility, which relates this global correspondence to the local Langlands correspondence. For regular algebraic cuspidal automorphic representations over CM fields, associated p-adic Galois representations have been constructed, but establishing this compatibility at primes above p has remained an important open problem.

In this talk, I will discuss joint work with Lambert A’Campo, Bence Hevesi, and Jack Thorne establishing local-global compatibility at p up to semisimplification. As a consequence, the associated Galois representations are geometric. I will discuss applications to new cases of the adjoint Bloch–Kato Selmer vanishing conjecture and outline how we remove the hypotheses on the mod p Galois representations required by previous approaches to local-global compatibility.