An application of a theorem of Emerton to mod $p$ representations of $\mathrm{GL}_2$

Abstract

Let $p$ be a prime and $L$ be a finite extension of $\mathbb{Q}_p$. We study the ordinary parts of $\mathrm{GL}_2(L)$-representations arised in the mod $p$ cohomology of Shimura curves attached to indefinite division algebras which splits at a finite place above $p$. The main tool of the proof is a theorem of Emerton \cite{Em3}.