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

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 Paškūnas.

Final version

References in corpus (1)

• Multiplicity one for the mod $p$ cohomology of Shimura curves: the tame case