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 $\overline{r}: \mathrm{Gal}(\overline{F}/F)\rightarrow\mathrm{GL}_2(\overline{\mathbb{F}}_p)$ 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.