On locally analytic vectors of the completed cohomology of modular curves

By Lue Pan. Forum of Mathematics, Pi, 2020

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Abstract We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak {gl}_2(\mathbb {Q}_p)$ . As applications, we prove a classicality result for overconvergent eigenforms of weight 1 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/68b016a8b7d8bf40422450e06e66748b00be113a