Locally analytic vectors in the completed cohomology of unitary Shimura curves

2025/05/15 by Qiu Tian, Qiu, Tian, Buchin Su +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.10290

openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

We use the methods introduced by Lue Pan to study the locally analytic vectors in the completed cohomology of unitary Shimura curves. As an application, we prove a classicality result on two-dimensional regular σ-de Rham representations of Gal( L/L) appearing in the locally σ-analytic vectors of the completed cohomology, where L is a finite extension of ℚ_p and σ:L\hookrightarrow E is an embedding of L into a sufficiently large finite extension E of ℚ_p. We also prove that if a two-dimensional representation of Gal( L/L) appears in the locally σ-algebraic vectors of the completed cohomology then it is σ-de Rham. Finally, we give a geometric realization of some locally σ-analytic representations of GL₂(L). This realization has some applications to the p-adic local Langlands program, including a locality theorem for Galois representations arising from classical automorphic forms, an admissibility result for coherent cohomology of Drinfeld curves, and some special cases of the Breuil's locally analytic Ext¹-conjecture for GL₂(L).

Cited by

• Locally analytic completed cohomology of Shimura varieties of Hodge type

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