Towards a theory of local Shimura varieties

Abstract

This is a survey article that advertises the idea that there should exist a theory of p-adic local analogues of Shimura varieties. Prime examples are the towers of rigid-analytic spaces defined by Rapoport–Zink spaces, and we also review their theory in the light of this idea. We also discuss conjectures on the ℓ-adic cohomology of local Shimura varieties