Potentially crystalline deformation rings and Serre weight conjectures

TL;DR

This paper proves the weight part of Serre's conjecture for certain automorphic forms on U(3), establishes automorphy lifting theorems in dimension three, and confirms the Breuil-Mézard conjecture for specific deformation rings.

Contribution

It provides the first proof of the weight part of Serre's conjecture for U(3) in generic cases and introduces explicit descriptions of deformation rings with new patching techniques.

Findings

Confirmed the geometric Breuil-Mézard conjecture for specific deformation rings.

Proved automorphy lifting theorems in dimension three.

Validated the weight part of Serre's conjecture for forms of U(3).

Abstract

We prove the weight part of Serre's conjecture in generic situations for forms of $U(3)$ which are compact at infinity and split at places dividing $p$ as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights $(2,1,0)$ for $K/\mathbb{Q}_p$ unramified combined with patching techniques. Our results show that the (geometric) Breuil-M\'ezard conjectures hold for these deformation rings.