Mirror symmetry and the Breuil-Mézard Conjecture

Abstract

The Breuil-Mézard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" in the moduli space of mod $p$ Galois representations of $\mathrm{Gal}({\overline{\mathbb{Q}}}_q/{\mathbb{Q}}_q)$ that should govern congruences between mod $p$ automorphic forms. For generic parameters, we propose a construction of Breuil-Mézard cycles in arbitrary rank, and verify that they satisfy the Breuil-Mézard Conjecture for all sufficiently generic tame types and small Hodge-Tate weights. Our method is purely local and group-theoretic, and completely distinct from previous approaches to the Breuil-Mézard Conjecture. In particular, we leverage new connections between the Breuil-Mézard Conjecture and phenomena occurring in homological mirror symmetry and geometric representation theory.

This version improves significantly over the previous one, generalizing the results to essentially arbitrary unramified groups (see Section 1.5 for discussion of changes). Includes a new appendix written jointly with Zhongyipan Lin