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

TL;DR

This paper proves a multiplicity one property for the mod p cohomology of Shimura curves associated with certain Galois representations over totally real fields, under tame ramification and genericity conditions.

Contribution

It establishes that the invariants of the associated smooth representation depend solely on the local Galois representation and confirms a multiplicity one result in this setting.

Findings

Invariants depend only on local Galois representation

Verifies multiplicity one property for the representation

Applicable under tame ramification and genericity assumptions

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.