The geometric Breuil-M\'ezard conjecture for two-dimensional potentially Barsotti-Tate Galois representations

TL;DR

This paper geometrically proves the Breuil-Mézard conjecture and the weight part of Serre's conjecture for two-dimensional mod p Galois representations over p-adic fields, advancing understanding of their structure.

Contribution

It provides a geometric framework for the Breuil-Mézard and Serre's conjectures in the context of two-dimensional mod p Galois representations.

Findings

Established a geometric version of the Breuil-Mézard conjecture.

Connected the conjecture to the structure of moduli stacks of Galois representations.

Enhanced understanding of the weight part of Serre's conjecture.

Abstract

We establish a geometrisation of the Breuil-M\'ezard conjecture for potentially Barsotti-Tate representations, as well as of the weight part of Serre's conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field.