Tangent Bundles in p-adic Geometry

Because of the existence of approximate p-power roots, a perfectoid algebra over Q_p admits no continuous derivations, and thus the natural Kahler tangent space of a perfectoid space over Q_p is identically zero. However, it turns out that many perfectoid spaces (or more general diamonds) arising from constructions involving rigid analytic varieties and their cohomology can be equipped with the extra structure of a Banach-Colmez Tangent Bundle, and using these Tangent Bundles natural period maps can be differentiated. As a first example, Tangent Bundles for infinite level basic EL local Shimura varieties have been computed by Ivanov and Weinstein using a heuristic construction due to Fargues and Scholze. However, in this talk, we are interested primarily in examples that go beyond the purview of the Fargues-Scholze heuristic. We will focus on the first non-trivial case of the infinite level modular curve, where we can describe the main points without assuming too much background with perfectoid rings or p-adic geometry. In this case, the existence of the Tangent Bundle is a consequence of an explicit quotient presentation of a closely related space derived from (a simple case of) the p-adic Simpson correspondence. Using this presentation, we will explain how to differentiate certain functions along Vector Fields in order to recover the annihilation property of Pan’s geometric Sen morphism, how to differentiate the Hodge-Tate period map, and how to relate this Tangent Bundle to a local construction via the Fargues-Scholze heuristic after restriction to good reduction residue disks. Time permitting, we will finish with a brief description of how the theory can be extended beyond infinite level modular curves.