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



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)


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.


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

2010 Mathematics Subject Classification

11F70, 22E50

Yongquan Hu is partially supported by the National Key R&D Program of China 2020YFA0712600, by the National Natural Science Foundation of China Grants 12288201 and 11971028; and by the National Center for Mathematics and Interdisciplinary Sciences and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences.

Haoran Wang is partially supported by National Natural Science Foundation of China Grants 11971028 and 11901331, and by the Beijing Natural Science Foundation (1204032).

Received 9 July 2021

Published 23 June 2022