Rational p-adic Hodge theory for non-proper rigid-analytic varieties

Abstract: The goal of this talk will be to discuss the rational p-adic Hodge theory of general smooth rigid-analytic varieties. The study of this subject for varieties that are not necessarily proper (e.g. Stein) is motivated in part by the desire of finding a geometric incarnation of the p-adic Langlands correspondence in the cohomology of local Shimura varieties. In this context, one difficulty is that the relevant cohomology groups (such as the p-adic (pro-)étale, and de Rham ones) are usually infinite-dimensional, and, to study them, it becomes important to exploit the topological structure that they carry. But, in doing so, one quickly runs into several topological issues: for example, the de Rham cohomology groups of a smooth affinoid space are, in general, not Hausdorff. We will explain how to overcome these issues, using the condensed and solid formalisms recently developed by Clausen and Scholze, and we will report on a comparison theorem describing the geometric p-adic (pro-)étale cohomology in terms of de Rham data, for a large class of smooth rigid-analytic varieties defined over a p-adic field. In particular, we recover results of Colmez, Dospinescu, and Nizioł.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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