Cambridge Journal of Mathematics

Volume 10 (2022)

Number 2

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

Pages: 261 – 431

DOI: https://dx.doi.org/10.4310/CJM.2022.v10.n2.a1

Authors

Yongquan Hu (Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; and University of the Chinese Academy of Sciences, Beijing, China)

Haoran Wang (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

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 $\overline{r} : \mathrm{Gal}(\overline{F} / F) \to \mathrm{GL}_2 (\overline{\mathbb{F}}_p)$ be a continuous irreducible representation which, when restricted to a fixed place $v \vert p$, 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 $\operatorname{mod} p$ 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.

Keywords

$\operatorname{mod}p$ Langlands program, Gelfand–Kirillov dimension

2010 Mathematics Subject Classification

11F70, 22E50

Received 9 July 2021

Published 23 June 2022