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

TL;DR

This paper explores the structure of mod p representations of GL_2 over a p-adic field, using Emerton's theorem to analyze cohomological aspects related to Shimura curves and division algebras.

Contribution

It applies Emerton's theorem to study the ordinary parts of mod p representations of GL_2, providing new insights into their structure in the context of Shimura curves.

Findings

Identification of ordinary parts of mod p cohomology representations

Application of Emerton's theorem to p-adic representation analysis

Insights into the structure of representations arising from Shimura curves

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}.