Title: On the locally analytic vectors of the completed cohomology of modular curves I: Main results Speaker: Lue Pan (University of Chicago) Time: 2020-9-10, 9:30-10:30 Place: ZOOM;ID:4663562952;Pswd:mcm1234 Abstract: We study locally analytic vectors of the completed cohomology of modular curves and determine eigenvectors of a rational Borel subalgebra of gl_2(Q_p). As applications, we are able to prove a classicality result for overconvergent eigenform of weight one 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. Attachment: