Mirror symmetry and the Breuil-Mézard Conjecture

The Breuil-Mézard Conjecture predicts the existence of hypothetical “Breuil-Mezard cycles” in the moduli space of mod $p$ Galois representations of $\mathrm{G}\mathrm{a}\mathrm{l}({\bar{\mathbb{Q}}}_q\mathrm{/}{\mathbb{Q}}_q)$ 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.