On locally analytic vectors of the completed cohomology of modular curves

Abstract

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 one 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.

arXiv comment88 pages; v3: revised version. We incorporate the recent result of Paskunas-Tung (arXiv:2104.08948)