On the Breuil-M\'ezard conjecture

TL;DR

This paper provides a new local proof of the Breuil-Mézard conjecture specifically for two-dimensional Galois representations over p with scalar endomorphisms, for primes p.

Contribution

It introduces a novel local proof of the Breuil-Mézard conjecture for a specific class of two-dimensional Galois representations.

Findings

Proof applies for p with scalar endomorphisms.

Establishes the conjecture in the case p.

Provides new insights into the local properties of Galois representations.

Abstract

We give a new local proof of Breuil-M\'ezard conjecture for two dimensional representations of the absolute Galois group of $\mathbb{Q}_p$, when $p\ge 5$ and the representation has scalar endomorphisms.