Abstract

The field of p-adic numbers is a well-known example of a non-Archimedean field. To try and develop a theory of geometry over such fields, it seems reasonable to take inspiration from classical geometric constructions like real differentiable, or complex analytic, manifolds. However, for topological reasons, this naive approach is problematic and, in most cases, is of limited geometric interest.

A more sophisticated way to study the geometry of non-Archimedean fields is provided by rigid analytic geometry. Affinoid spaces lie at the heart of this theory. These are the rigid analytic analogues of affine schemes from algebraic geometry. After I have introduced affinoid spaces, I will spend most of this talk discussing an appropriate definition of de Rham cohomology in the affinoid setting. I will also attempt to convey why, from the viewpoint of representation theory, affinoid de Rham cohomology is an interesting thing to consider.