• Corpus ID: 221818751

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

@article{Hu2020OnTM,
  title={On the mod \$p\$ cohomology for \$\mathrm\{GL\}\_2\$: the non-semisimple case},
  author={Yong Hu and Haoran Wang},
  journal={arXiv: Number Theory},
  year={2020}
}

• Yong Hu, Haoran Wang
• Published 21 September 2020
• Mathematics
• arXiv: Number Theory

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 $\overline{r}:\mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ 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.