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



Yongquan Hu (Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; and the University of the Chinese Academy of Sciences, Beijing, China)

Haoran Wang (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


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