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Title:
On the locally analytic vectors of the completed cohomology of modular curves
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Abstract:
A classical result identifies holomorphic modular forms with highest weight vectors of certain representations of SL2(ℝ). We study locally analytic vectors of the (p-adically) completed cohomology of modular curves and prove a p-adic analogue of this result. As applications, we are able to prove a classicality result for overconvergent eigenforms and give a new proof of Fontaine-Mazur conjecture in the irregular case under some mild hypothesis. One technical tool is relative Sen theory which allows us to relate infinitesimal group action with Hodge(-Tate) structure.
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