A geometric perspective on the Breuil–Mézard conjecture

Source abstract

Abstract Let $p\gt 2$ be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard conjecture for two-dimensional mod $p$ representations of the absolute Galois group of ${ \mathbb{Q} }_{p}$ . We also state a conjectural generalization to $n$ -dimensional representations of the absolute Galois group of an arbitrary finite extension of ${ \mathbb{Q} }_{p}$ , and give a conditional proof of this conjecture, subject to a certain $R= \mathbb{T}$ -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.