On the locally analytic vectors of the completed cohomology of modular curves

On the locally analytic vectors of the completed cohomology of modular curves

-
Lue Pan, University of Chicago

Zoom link:  https://princeton.zoom.us/j/97126136441

Password: the three digit integer that is the cube of the sum of its digits

A classical result identifies holomorphic modular forms with highest weight vectors of certain representations of SL_2(\mathbb{R}). We study locally analytic vectors of the (p-adically) completed cohomology of modular curves and prove a p-adic analogue of this result. As applications, we are able to prove a classicality result for overconvergent eigenforms 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.