Multiplicity one at full congruence level

Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_F\to {\mathrm{GL}}_2({\overline{\mathbb{F}}}_p)$ be a modular Galois representation which satisfies the Taylor--Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$-torsion in the mod $p$ cohomology of Shimura curves with full congruence level at $v$ as a ${\mathrm{GL}}_2(k_v)$-representation. In particular, it only depends on $\overline{r}{|}_{I_{F_v}}$ and its Jordan--Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic ${\mathrm{GL}}_2({\mathbb{F}}_q)$-projective envelopes and the multiplicity one results of \cite{EGS}.

Accepted for publication at Journal de l Institut de Mathematiques de Jussieu

References in corpus (1)

• A classification of irreducible admissible mod p representations of p-adic reductive groups

Cited by in corpus (6)

• Diagrams in the mod $p$ cohomology of Shimura curves
• Conjectures and results on modular representations of ${\mathrm{GL}}_n(K)$ for a $p$-adic field $K$
• Serre weight conjectures for $p$-adic unitary groups of rank 2
• Gelfand-Kirillov dimension and mod p cohomology for GL2
• Can we dream of a 1-adic Langlands correspondence?
• On the constituents of the mod $p$ cohomology of Shimura curves