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 r : Gal(F /F ) → GL 2 (Fp) 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 GL 2 (Fv) occurring in the corresponding Hecke eigenspaces of the mod p cohomology of Shimura varieties associated to D have Gelfand-Kirillov dimension [Fv : Qp]. We also prove that any such representation can be generated as a GL 2 (Fv)-representation by its subspace of invariants under the first principal congruence subgroup. If moreover [Fv : Qp] = 2, we prove that such representations have length 3, confirming a speculation of Breuil and Paškūnas.

Factorization de la cohomologie étale p-adique de la tour de Drinfeld

“…son noyau et son conoyau sont de type fini sur O L . L'argument qui suit est inspiré des lemmes 5.4 et 5.11 de [39], et demande quelques préliminaires.…”

On some mod $p$ representations of quaternion algebra over $\mathbb{Q}_p$

“…Up to present, the (mod p) correspondence in the case of GL 2 (Q p ) has been well-understood in various aspects, by the work of [Bre03], [Col10], [Eme11], and [Paš13]. Recently, there have been significant progress towards a Langlands correspondence for GL 2 (L), when L is a finite unramified extension of Q p ([BHH + 20], [HW20], [BHH + 21]). However, a mod p Jacquet-Langlands correspondence is still largely unknown, even in the case of GL 2 (Q p ).…”

“…Remark 7.5. Under some (stronger) genericity condition on r| GF v , the assumption on dim K (π B ′ (r)) of Proposition 7.4 is verified in [BHH + 20], [HW20].…”