Crystalline liftings and weight part of Serre's conjecture

TL;DR

This paper advances the understanding of Serre's conjecture by proving new cases related to the weight part for mod p Galois representations, using crystalline lift techniques on unitary groups.

Contribution

It generalizes previous methods to establish new cases of the weight part of Serre's conjecture for automorphic Galois representations on unitary groups.

Findings

Proved new cases of the weight part of Serre's conjecture.

Developed methods for studying crystalline lifts of Galois representations.

Extended previous work to higher-dimensional unitary groups.

Abstract

We prove some new cases of weight part of Serre's conjectures for mod $p$ Galois representations associated to automorphic representations on unitary groups $U(d)$. The approach is a generalization of the work of Gee-Liu-Savitt, namely, we study reductions of certain crystalline representations, as well as crystalline lifts of these reductions.