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

We establish a geometrization 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. These results are first proved for the stacks of our earlier papers, and then transferred to the stacks of Emerton and Gee by means of a comparison of versal rings.