Degenerating products of flag varieties and applications to the Breuil--Mezard conjecture

TL;DR

This paper studies degenerations of products of flag varieties and connects their geometry to Galois representations, leading to new results on the Breuil--Mézard conjecture in dimension two.

Contribution

It introduces a geometric model for crystalline Galois deformation spaces and proves new cases of the Breuil--Mézard conjecture using these degenerations.

Findings

Degenerations model the geometry of crystalline subspaces in Galois stacks.

Established relations between cycle classes and Galois representation theory.

Proved new cases of the Breuil--Mézard conjecture in dimension two.

Abstract

We consider closed subschemes in the affine grassmannian obtained by degenerating $e$-fold products of flag varieties, embedded via a tuple of dominant cocharacters. For $G= \operatorname{GL}_2$, and cocharacters small relative to the characteristic, we relate the cycles of these degenerations to the representation theory of $G$. We then show that these degenerations smoothly model the geometry of (the special fibre of) low weight crystalline subspaces inside the Emerton--Gee stack classifying $p$-adic representations of the Galois group of a finite extension of $\mathbb{Q}_p$. As an application we prove new cases of the Breuil--M\'ezard conjecture in dimension two.