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We study the locally analytic vectors in the completed cohomology of modularcurves and determine the eigenvectors of a rational Borel subalgebra of$\mathfrak{gl}_2(\mathbb{Q}_p)$. As applications, we prove a classicalityresult for overconvergent eigenforms of weight one and give a new proof of theFontaine-Mazur conjecture in the irregular case under some mild hypotheses. Foran overconvergent eigenform of weight $k$, we show its corresponding Galoisrepresentation has Hodge-Tate-Sen weights $0,k-1$ and prove a converse result.

On locally analytic vectors of the completed cohomology of modular curves