On locally analytic vectors of the completed cohomology of modular curves

We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of. 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 and prove a converse result.