Resolutions of spaces of crystalline representations and modularity

We introduce a new partial resolution of crystalline spaces of Galois representations when the gaps in Hodge–Tate weights are smaller than $p$, with no bound on ramification. Furthermore, when $n=3$ in the case of minimal regular weight, we are able to show that the resolution is normal (assuming the ramification index is divisible by 3). Employing base change techniques and further analysis of the resolution, we are able to show that all the components of the crystalline deformation rings are potentially diagonalizable. As a consequence, we deduce automorphy lifting, the weight part of Serre’s conjecture, and the Breuil-Mézard conjecture in dimension three for minimal regular weight.