On the mod $p$ cohomology for $\mathrm{GL}_2$: the non-semisimple case

TL;DR

This paper investigates the structure of mod p cohomology for certain Shimura varieties, focusing on non-semisimple Galois representations and establishing properties like Gelfand-Kirillov dimension and representation length.

Contribution

It proves that the representations in question have Gelfand-Kirillov dimension equal to the degree of the local field extension and confirms the length of these representations in quadratic cases.

Findings

Representations have Gelfand-Kirillov dimension equal to [F_v:Q_p].

Any such representation is generated by its invariants under the first principal congruence subgroup.

In degree 2 extensions, representations have length 3, confirming a conjecture.

Abstract

Let $F$ be a totally real field unramified at all places above $p$ and $D$ be a quaternion algebra which splits at either none, or exactly one, of the infinite places. Let $\bar{r}:\mathrm{Gal}(\bar{F}/F)\to \mathrm{GL}_2(\bar{\mathbb{F}}_p)$ be a continuous irreducible representation which, when restricted to a fixed place $v|p$, is non-semisimple and sufficiently generic. Under some mild assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(F_v)$ occurring in the corresponding Hecke eigenspaces of the mod $p$ cohomology of Shimura varieties associated to $D$ have Gelfand-Kirillov dimension $[F_v:\mathbb{Q}_p]$. We also prove that any such representation can be generated as a $\mathrm{GL}_2(F_v)$-representation by its subspace of invariants under the first principal congruence subgroup. If moreover $[F_v:\mathbb{Q}_p]=2$, we prove that such representations have length $3$, confirming a speculation of Breuil and Pa\v{s}k\=unas.