On Locally Analytic Vectors of the Completed Cohomology of Modular Curves

AbstractWe 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 weightk, we show its corresponding Galois representation has Hodge–Tate–Sen weights$0,k-1$and prove a converse result.