On weight elimination for $\mathrm{GL}_n(\mathbb{Q}_{p^f})$

TL;DR

This paper investigates the properties of modular Serre weights for generic mod p Galois representations over unramified p-adic fields, providing bounds on their genericity and enhancing weight elimination results.

Contribution

It extends previous work to establish bounds on the genericity of modular Serre weights and improves weight elimination theorems for certain Galois representations.

Findings

Modular Serre weights of generic mod p Galois representations are themselves generic.

Provided nearly optimal bounds on the genericity of these weights.

Improved theorems related to weight elimination for unramified p-adic fields.

Abstract

We show that the modular Serre weights of a sufficiently generic mod $p$ Galois representation of an unramified $p$-adic field are themselves generic, and give precise bounds on the genericity, by extending previous work of Emerton, Gee and Herzig. Our bounds are nearly optimal in some cases. We use this to improve recent weight elimination theorems.