On locally analytic vectors of the completed cohomology of modular curves

2020/08/17 by Lue Pan, Pan, Lue · 5 citations
Mathematics · #11F33 #11F77 (Primary) 11F22 #14G22 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2008.07099

openalex publication_date 2020/08/17 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of \mathfrakgl₂(ℚ_p). As applications, we prove a classicality result for overconvergent eigenforms of weight one and give a new proof of the Fontaine-Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight k, we show its corresponding Galois representation has Hodge-Tate-Sen weights 0,k-1 and prove a converse result.

Cited by

• The geometric Sen morphism is the unique lift of the Kodaira--Spencer morphism
• The de Jong fundamental group of ℙ¹_C depends on C and is not always topologically countably generated
• Modularity theorems for abelian surfaces
• Locally analytic completed cohomology of Shimura varieties of Hodge type
• de Rham theory and locally analytic vectors

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