On locally analytic vectors of the completed cohomology of modular curves II

2022/09/14 by Lue Pan, Pan, Lue
Mathematics · #11F77 (Primary) 11F33 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.06366

openalex publication_date 2022/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This is a continuation of our previous work on the locally analytic vectors of the completed cohomology of modular curves. We construct differential operators on modular curves with infinite level at p in both "holomorphic" and "anti-holomorphic" directions. As applications, we reprove a classicality result of Emerton which says that every absolutely irreducible two dimensional Galois representation which is regular de Rham at p and appears in the completed cohomology of modular curves comes from an eigenform. Moreover we give a geometric description of the locally analytic representations of GL₂(ℚ_p) attached to such a Galois representation in the completed cohomology.

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