Extremal weights and a tameness criterion for mod $p$ Galois representations

TL;DR

This paper advances the understanding of the weight part of Serre's conjecture for mod p Galois representations by generalizing conjectures, introducing extremal weights, and proving modularity in specific cases.

Contribution

It generalizes Herzig's conjecture to ramified fields, introduces extremal weights, and proves their modularity using Levi reduction techniques.

Findings

Proved the weight elimination direction of the generalized conjecture.

Introduced and characterized extremal weights for local mod p representations.

Established modularity of extremal weights in certain cases.

Abstract

We study the weight part of Serre's conjecture for generic $n$-dimensional mod $p$ Galois representations. We first generalize Herzig's conjecture to the case where the field is ramified at $p$ and prove the weight elimination direction of our conjecture. We then introduce a new class of weights associated to $n$-dimensional local mod $p$ representations which we call \emph{extremal weights}. Using a ``Levi reduction" property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre's conjecture for unit groups of some division algebras in generic situations.