Sean Howe (University of Utah)

AScholze’s theory of diamonds provides a modern foundation for p-adic geometry that allows us to work with moduli spaces and period maps in p-adic Hodge theory that cannot be constructed in the classical theory of rigid analytic varieties. The theory works by replacing a p-adic analytic space with its functor of points on perfectoid rings, which are, roughly speaking, very big p-adic Banach algebras containing approximate p-power roots of all elements. One downside of this theory is that, because perfectoid rings admit no non-zero continuous derivations, diamonds do not carry an intrinsic theory of tangent spaces. Nevertheless, in many cases that arise in “real life”, the way we construct diamonds secretly carries along some extra differentiable information in the form of a Banach-Colmez Tangent Space.  This applies, in particular, to spaces like the infinite level modular curve and the period domains in p-adic Hodge theory. In this talk, we’ll explain how to compute Tangent Spaces for p-adic Lie torsors like the infinite level modular curve and how to differentiate generalized Hodge-Tate and Grothendieck-Messing period maps.