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# Mathematical Research Letters

## Volume 25 (2018)

### Number 3

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

Pages: 843 – 873

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n3.a6

#### Authors

#### Abstract

Let $F$ be a totally real field, $\mathfrak{p}$ an unramified place of $F$ dividing $p$ and $\overline{r} : \mathrm{Gal}(\overline{F} / F) \to \mathrm{GL}_2 (\overline{\mathbb{F}}_p)$ a continuous irreducible modular representation. The work of Buzzard, Diamond and Jarvis (Duke Math Journal, 2010) associates to $\overline{r}$ an admissible smooth representation of $\mathrm{GL}_2 (F_{\mathfrak{p}})$ on the $\mathrm{mod} \: p$ étale cohomology of Shimura curves attached to indefinite division algebras which split at $\mathfrak{p}$. When $\overline{r}$ satisfies the Taylor–Wiles hypotheses and ${\overline{r}\vert}_{\mathrm{Gal}(\overline{F}_{\mathfrak{p}} / F_{\mathfrak{p}})}$ is tamely ramified and generic, we determine the subspace of invariants of this representation under the principal congruence subgroup of level $\mathfrak{p}$. As a consequence, this subspace depends only on ${\overline{r}\vert}_{\mathrm{Gal}(\overline{F}_{\mathfrak{p}} / F_{\mathfrak{p}})}$ and satisfies a multiplicity one property.

Received 14 September 2016

Accepted 27 February 2017

Published 3 August 2018