A geometric perspective on the Breuil-M\'ezard conjecture

TL;DR

This paper provides a geometric refinement of the Breuil-Mézard conjecture for 2-dimensional mod p Galois representations, extends it to higher dimensions, and proves special cases under certain hypotheses.

Contribution

It introduces a geometric perspective on the Breuil-Mézard conjecture, proposes a conjectural generalization to n-dimensional cases, and offers a conditional proof based on R = T theorems.

Findings

Proved a geometric refinement for 2-dimensional cases.

Formulated a conjecture for higher-dimensional representations.

Established an unconditional result for two-dimensional potentially Barsotti-Tate representations.

Abstract

Let p > 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil-M\'ezard conjecture for 2-dimensional mod p representations of the absolute Galois group of Qp. We also state a conjectural generalisation to n-dimensional representations of the absolute Galois group of an arbitrary finite extension of Qp, and give a conditional proof of this conjecture, subject to a certain R = T-type theorem together with a strong version of the weight part of Serre's conjecture for rank n unitary groups. We deduce an unconditional result in the case of two-dimensional potentially Barsotti-Tate representations.