Some new cases of the Breuil-Mézard conjecture via degenerations in the affine grassmannian

Abstract: The Breuil-Mézard conjecture relates the mod $p$ geometry of moduli spaces of $n$-dimensional potentially crystalline (or semi-stable) Galois representations in terms of the mod $p$ representation theory of $\mathrm{GL}_n$.

In this talk I will explain a proof of this result for two dimensional crystalline representations with sufficiently small Hodge-Tate weights (roughly $\leq p/e$ for $e$ the ramification degree). The main idea is to relate the geometry of these moduli spaces to degenerations of products of flag varieties in an affine grassmannian, and to prove a version of Breuil-Mézard for these degenerations.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)