Serre weights for three-dimensional wildly ramified Galois representations

TL;DR

This paper proves the weight part of Serre's conjecture for three-dimensional mod p Galois representations with wild ramification, under genericity conditions, advancing understanding of Galois representations and automorphic forms.

Contribution

It extends previous results by removing tame ramification restrictions and establishes new links between Galois representations, Emerton--Gee stacks, and local models.

Findings

Proved the weight part of Serre's conjecture for wild ramification cases.

Validated a version of Breuil's lattice conjecture.

Established a mod p multiplicity one result for U(3)-arithmetic manifold cohomology.

Abstract

We formulate and prove the weight part of Serre's conjecture for three-dimensional mod $p$ Galois representations under a genericity condition when the field is unramified at $p$. This removes the assumption in \cite{arXiv:1512.06380}, \cite{arXiv:1608.06570} that the representation be tamely ramified at $p$. We also prove a version of Breuil's lattice conjecture and a mod $p$ multiplicity one result for the cohomology of $U(3)$-arithmetic manifolds. The key input is a study of the geometry of the Emerton--Gee stacks \cite{arXiv:2012.12719} using the local models introduced in \cite{arXiv:2007.05398}.