The Breuil-M\'ezard conjecture for potentially Barsotti-Tate representations

TL;DR

This paper proves the Breuil-Mézard conjecture for 2-dimensional potentially Barsotti-Tate Galois representations over finite extensions of Q_p for p>2, and applies these results to advance the understanding of Serre's conjecture.

Contribution

It establishes the Breuil-Mézard conjecture for a broad class of Galois representations and determines most multiplicities in the unramified case, impacting Serre's conjecture.

Findings

Proved the Breuil-Mézard conjecture for 2-dimensional potentially Barsotti-Tate representations for p>2.

Determined most multiplicities in the unramified case.

Applied results to confirm cases of Serre's conjecture.

Abstract

We prove the Breuil-M\'ezard conjecture for 2-dimensional potentially Barsotti-Tate representations of the absolute Galois group G_K, K a finite extension of Q_p, for any p>2 (up to the question of determining precise values for the multiplicities that occur). In the case that K/Q_p is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre's conjecture, proving a variety of results including the Buzzard-Diamond-Jarvis conjecture.