| Résume | 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. |
