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.
Categories: math.NT math.RT