Finite length for unramified GL₂: beyond multiplicity one, non-semisimple case

2026/07/25 by Lucrezia Bertoletti
Mathematics · #math.NT

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Abstract

Let p be a prime number and K a finite unramified extension of ℚ_p. 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 they are of finite length, thereby extending some recent results of Breuil, Herzig, Hu, Morra and Schraen to higher multiplicity. In a previous companion article we investigated the case where the local Galois representation attached to the Hecke eigensystem is semisimple; this article treats the non-semisimple case.

Related

• Finite Length for Unramified GL₂: Beyond Multiplicity One
• Finite length for unramified GL₂
• On the constituents of the mod p cohomology of Shimura curves
• An application of a theorem of Emerton to mod p representations of GL₂
• Multiplicity one for the mod p cohomology of Shimura curves: the tame case