Resolutions of spaces of crystalline representations and modularity

TL;DR

The paper develops a new partial resolution for crystalline Galois representation spaces, proving normality and potential diagonalizability, leading to automorphy and Serre's conjecture results in dimension three.

Contribution

Introduces a novel partial resolution of crystalline spaces with small Hodge--Tate gaps, establishing normality and potential diagonalizability in dimension three.

Findings

Resolution is normal for n=3 in minimal regular weight.

All components of crystalline deformation rings are potentially diagonalizable.

Automorphy lifting and Serre's conjecture are confirmed in dimension three.

Abstract

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-M\'ezard conjecture in dimension three for minimal regular weight.