Finite Length for Unramified $GL_2$: Beyond Multiplicity One

TL;DR

This paper proves that certain smooth mod p representations of GL_2 over unramified extensions of Q_p, arising from Shimura curve cohomology, have finite length without assuming multiplicity one, extending prior results.

Contribution

It establishes finite length of these representations in a broader setting, removing the multiplicity one assumption at tame level.

Findings

Representations are of finite length.

Extends previous results to cases without multiplicity one.

Applicable to cohomology of Shimura curves.

Abstract

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. Building on recent work of Breuil, Herzig, Hu, Morra and Schraen, we study the smooth mod $p$ representations of $\mathrm{GL}_2(K)$ appearing in a tower of mod $p$ Hecke eigenspaces of the cohomology of Shimura curves, under mild genericity assumptions but notably no multiplicity one assumption at tame level, and prove that these representations are of finite length, thereby extending a previous result of the aforementioned authors.