A geometric perspective on the Breuil-Mézard conjecture

Let p> 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard con-jecture for 2-dimensional mod p representations of the absolute Galois group of Qp. We also state a conjectural generalisation to n-dimensional representa-tions 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