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

Abstract

Given a continuous ordinary Galois representation $\bar{\rho }:G_{{\mathbb{Q}}_p}\to {\mathrm{GL}}_n({\overline{\mathbb{F}}}_p)$, 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.

25 pages

References in corpus (2)

• On the automorphy of $l$-adic Galois representations with small residual image
• The classification of irreducible admissible mod p representations of a p-adic GL_n