Multiplicity one for the $\mathrm{mod} \: p$ cohomology of Shimura curves: the tame case

Abstract

Let F be a totally real field, p an unramified place of F dividing p and r : Gal(F /F ) → GL 2 (Fp) a continuous irreducible modular representation. The work of Buzzard, Diamond and Jarvis [9] associates to r an admissible smooth representation of GL 2 (Fp) on the mod p cohomology of Shimura curves attached to indefinite division algebras which split at p. When r| Gal(Fp/Fp) is tamely ramified and generic (and under some additional technical assumptions), we determine the subspace of invariants of this representation under the principal congruence subgroup of level p. In particular, the subspace depends only on r| Gal(Fp/Fp) and satisfies a multiplicity one property.Theorem 1.1. Assume that F v is unramified over Q p and ρ is tamely ramified and generic. Under certain assumptions (see Cor. 3.9), we have πRemark 1.2. After the first version of the paper was written, we are informed that Le, Morra and Schraen obtain a similar result independently [19]. Both the proofs use results of [13] as a global input, however, the local representation theory part is quite different.We also prove a similar result when D is a definite quaternion algebra unramified at all places over p. In [13], the subspace of pro-p-Iwahori fixed vectors of π D v (r) is determined, for ρ generic, but could be reducible non-split. The proof of our theorem uses the construction of [13] as a main tool. We intend to extend the result to reducible non-split ρ in future work.The organization of the paper is as follows. In Section 2, we give all the local results that we need, especially the local criterion Corollary 2.29. Note that the criterion does not apply to ρ reducible non-split (see Remark 2.27). In Section 3, we deduce our main theorem from our local results and results of [13].

An application of a theorem of Emerton to mod p representations of GL 2

“…Remark Although the decomposition (hence the existence of ) is still conjectural, under some extra assumptions on and (see [, § 3.3]) Breuil and Diamond have constructed a certain candidate for , which we denote by , showing that if holds (at least in the situation of Theorem ). In fact, the results of are proved for the representation . The proof of Theorem works equally for this .…”

“…A conjecture of Buzzard, Diamond and Jarvis says that the space decomposes as a restricted tensor product where each factor is an admissible smooth representation of and depends only on the restriction of at . Note that, when , the local factor is expected to be the right representation in the mod local Langlands (or Jacquet–Langlands) program, and many important properties about it have been proved, see for example .…”

Multiplicity One at Full Congruence Level

“…Relating the two approaches led to this collaboration. After our paper had been written, we were notified that Hu and Wang also obtained a similar result independently [16].…”

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

“…For this reason, in the following we write (by abuse of notation) π(ρ) def = π D v (r). There has been a lot of works studying the representation-theoretic properties of π(ρ), see [BDJ10], [Gee11], [GK14], [Bre14], [BD14], [EGS15], [Hu17], [HW18], [LMS], [Le19], [DL19], etc. These works often have the common aim to determine certain invariants attached to the restriction of π(ρ) to K def = GL 2 (O Fv ), like the socle, the subspace of invariants under the first principal subgroup K 1 def = 1 + pM 2 (O Fv ) or the pro-p Iwahori subgroup I 1 , and also some local-global compatibility related to these subspaces.…”

“…These works often have the common aim to determine certain invariants attached to the restriction of π(ρ) to K def = GL 2 (O Fv ), like the socle, the subspace of invariants under the first principal subgroup K 1 def = 1 + pM 2 (O Fv ) or the pro-p Iwahori subgroup I 1 , and also some local-global compatibility related to these subspaces. For example, it is known that (under various mild assumptions) (i) soc K π(ρ) ∼ = ⊕ σ∈D(ρ) σ, where D(ρ) is an explicit set of Serre weights associated to ρ in [BP12,§9], see [GK14], [EGS15]; (ii) π(ρ) K1 ∼ = D 0 (ρ), where D 0 (ρ) is a representation of GL 2 (F p f ) constructed in [BP12,§13], see [HW18], [LMS], [Le19].…”