Multiplicity one for the mod $p$ cohomology of Shimura curves: the tame case

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 associates to $\overline{r}$ an admissible smooth representation of ${\mathrm{GL}}_2(F_{\mathfrak{p}})$ on the mod $p$ cohomology of Shimura curves attached to indefinite division algebras which split at $\mathfrak{p}$. When $\overline{r}{|}_{\mathrm{Gal}(\overline{F_{\mathfrak{p}}}/F_{\mathfrak{p}})}$ is tamely ramified and generic (and under some technical assumptions), we determine the subspace of invariants of this representation under the principal congruence subgroup of level $\mathfrak{p}$. In particular, it depends only on $\overline{r}{|}_{\mathrm{Gal}(\overline{F_{\mathfrak{p}}}/F_{\mathfrak{p}})}$ and verifies a multiplicity one property.

21 pages. Title changed; revised version