Multiplicities in the ordinary part of mod $p$ cohomology for $\mathrm{GL}_n(\mathbb{Q}_p)$

TL;DR

This paper investigates the multiplicities of certain ordinary parts of mod p cohomology for GL_n(Q_p), demonstrating uniform occurrence of indecomposable components in automorphic cohomology via Taylor-Wiles patching.

Contribution

It proves that all indecomposable components of the ordinary mod p cohomology representation occur with equal multiplicity at specific levels, extending understanding of mod p automorphic forms.

Findings

Indecomposable components occur with equal multiplicity.

Application of Taylor-Wiles patching to automorphic cohomology.

Uniform multiplicity result for ordinary mod p cohomology.

Abstract

Given a continuous ordinary Galois representation $\bar{\rho}:G_{\mathbb{Q}_p}\rightarrow\mathrm{GL}_n(\overline{\mathbb{F}}_p)$, Breuil and Herzig constructed an admissible smooth $\overline{\mathbb{F}}_p$-representation $\Pi(\bar{\rho})^{\mathrm{ord}}$ of $\mathrm{GL}_n(\mathbb{Q}_p)$ and showed that it occurs in certain globally defined mod $p$ cohomology spaces. By applying Taylor-Wiles patching to spaces of ordinary automorphic representations we prove that the indecomposable pieces of $\Pi(\bar{\rho})^{\mathrm{ord}}$ each occur with the same multiplicity at a well-chosen tame level.