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 (padically) completed cohomology of modular curves and prove a padic analogue of this result. As applications, we are able to prove a classicality result for overconvergent eigenforms and give a new proof of FontaineMazur 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.