Mirror symmetry and the Breuil—Mezard Conjecture: an update

Abstract: The Breuil—Mezard Conjecture predicts a precise indexing of cycles in moduli spaces of local Galois representations by modular representations of finite groups of Lie type. A couple years ago, Bao Le Hung and I introduced a new approach to the Breuil—Mezard Conjecture based on a connection to an instance of mirror symmetry, which in that instance predicts a precise indexing of Lagrangians in a symplectic variety by representations of a quantum group. Recently, we used this to prove the Breuil—Mezard Conjecture in the generic range for arbitrary unramified groups, including exceptional groups. My intent is to review this and also work-in-progress with Le Hung and Zhongyipan Lin, which aims to extend the result to ramified groups. The key new aspect of the ramified case is a nascent theory of "Spectral Langlands functoriality", an analogue of Langlands functoriality for the spectral (i.e., "Galois") side of the Langlands correspondence.

number theory

Audience: researchers in the topic

BC-MIT number theory seminar