Mirror symmetry and the Breuil-M\'ezard Conjecture

Abstract

The Breuil-M\'{e}zard 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-M\'{e}zard cycles in arbitrary rank, and verify that they satisfy the Breuil-M\'{e}zard 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-M\'ezard Conjecture. In particular, we leverage new connections between the Breuil-M\'ezard Conjecture and phenomena occurring in homological mirror symmetry and geometric representation theory.