Multiplicity one at full congruence level

TL;DR

This paper proves that the mod p cohomology of Shimura curves at full congruence level exhibits multiplicity one for Jordan-Hölder factors, depending only on the restriction of the Galois representation to inertia.

Contribution

It provides a detailed description of the $ar{m}$-torsion in the cohomology, showing it depends solely on the restriction of the Galois representation and establishing multiplicity one results.

Findings

The $ar{m}$-torsion is determined by $ar{r}|_{I_{F_v}}$.

Jordan-Hölder factors appear with multiplicity one.

The structure relies on generic $ ext{GL}_2( extbf{F}_q)$-projective envelopes.

Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ 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--H\"{o}lder 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}.