The geometric Breuil-Mézard conjecture for two-dimensional potentially Barsotti-Tate Galois representations

We establish a geometrisation of the Breuil-Mézard conjecture for potentially Barsotti-Tate representations, as well as of the weight part of Serre’s conjecture, for moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field.