Geodesic
On locally analytic vectors of the completed cohomology of modular curves
Forum of Mathematics, Pi, Tome 10 (2022)

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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. 
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Lue Pan. On locally analytic vectors of the completed cohomology of modular curves. Forum of Mathematics, Pi, Tome 10 (2022). doi: 10.1017/fmp.2022.1

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