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September 30, 2025Open Access

Finite Length for Unramified GL₂: Beyond Multiplicity One

Key Points

These smooth mod p representations of GL2(K) are established to be of finite length.
The study extends earlier findings on representations without the multiplicity one assumption in tame level settings.
A careful analysis of Hecke eigenspaces reveals their critical roles in the cohomology of Shimura curves.
This work builds on foundational research by Breuil, Herzig, Hu, Morra, and Schraen in the field of p-adic representations.

Abstract

Let p be a prime number and K a finite unramified extension of Qₚ. Building on recent work of Breuil, Herzig, Hu, Morra and Schraen, we study the smooth mod p representations of GL₂ (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.

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Lucrezia Bertoletti (Mon,) studied this question.

synapsesocial.com/papers/68dc12c58a7d58c25ebb0a9e — DOI: https://doi.org/10.48550/arxiv.2505.20134

Finite Length for Unramified GL₂: Beyond Multiplicity One

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