In this thesis, we study the Hodge-Tate structure of the pro´etale cohomology of Shimura varieties. This document is divided in four main issues.
First, we construct an integral model of the perfectoid modular curve. Using this formal scheme, we prove some vanishing results for the coherent cohomology of the perfectoid modular curve, we also provide a description of the dual completed cohomology as an inverse limit of integral modular forms of weight 2 by normalized traces.
Secondly, we construct theoverconvergent Eichler-Shimura map for the first coherent cohomology group, complementing the work of Andreatta-Iovita-Stevens. More precisely, we construct a map from the overconvergent cohomology with compact support of Boxer-Pilloni to the locally analytic modular symbols of Ash-Stevens. We reinterpret the construction of these maps in terms of the Hodge-Tate period map and the perfectoid modular curve. We also reprove the classical Faltings’s Eichler-Shimura decomposition using the Hodge-Tate period map, and the dual BGG resolution of irreducible representations of GL2. We show that the overconvergent Eichler-Shimura maps are compatible with the Up-operator, and that their small slope vectors interpolate the classical Eichler-Shimura maps.
Thirdly, in a joint work with Joaqu´ın Rodrigues Jacinto, we develop the classical theory of locally analytic representations of p-adic Lie groups in the context of condensed mathematics. Inspired from foundational works of Lazard, Schneider-Teitelbaum and Emerton, we define a notion of solid locally analytic representation for a compact p-adic Lie group. We prove that the category of solid locally analytic representations can be described as modules over algebras of analytic distributions. As an application, we prove a cohomological comparison theorem between solid group cohomology, solid group cohomology of the (derived) locally analytic vectors, and Lie algebra cohomology.
Finally, we generalize the work of Lue Pan to arbitrary Shimura varieties. We construct a geometric Sen operator for a class of proetale ô- modules F which we call relative locally analytic. We prove that this Sen operator is related with the p-adic Simpson correspondence, and that it computes the pro´etale cohomology of F. We apply this theory to Shimura varieties, obtaining that the computation of pro´etale cohomology can be translated in terms of Lie algebra cohomology over the flag variety via the Hodge-Tate period map. In particular, we prove that the Cp-extension of scalars of the locally analytic completed cohomology can be described as the analytic cohomology of the infinite-at-p level Shimura variety, of the locally analytic sections of the structural sheaf. This implies a rational version of the Calegari-Emerton conjectures for any Shimura variety without the hypothesis of the infinite-at-p level Shimura variety to be perfectoid.