# 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.