Abstract Juan Esteban Rodriguez Camargo

Juan Esteban Rodriguez Camargo will speak on

Locally analytic completed cohomology of Shimura varieties

The goal of this talk is to relate the locally analytic vectors of the completed cohomology of a Shimura variety with the locally analytic structural sheaf at infinite level, generalizing the work of Lue Pan for the modular curve. As a consequence, one can deduce a rational version of the Calegari-Emerton conjectures. More precisely, we will sketch the construction of the geometric Sen operator of a rigid analytic space, and explain how it is used to calculate proétale cohomology. Then, we show that the geometric Sen operator of a Shimura variety is obtained as the pullback of a G-equivariant vector bundle of the flag variety via the Hodge-Tate period map. As a consequence, we will be able to compute (Hodge-Tate) proétale cohomology as Lie algebra cohomology of certain D-modules over the flag variety.