Joint IAS/PU Arithmetic Geometry

If X is a smooth proper rigid variety over C_p and L is a Z_p-local system on X, the cohomology groups H*(X,L) are finitely generated Z_p-modules by a basic result of Scholze. If L is merely a Q_p-local system, its cohomology groups are still finite dimensional, but in a much subtler sense: they are finite-dimensional Banach-Colmez spaces. This is a very recent result of Anschutz-Le Bras-Mann, with another approach in progress due to Li-Niziol-Reinecke-Zavyalov. I will give some context for these results, explain some interesting examples, and formulate some new conjectures.