Towards a theory of local Shimura varieties

TL;DR

This survey explores the concept of local Shimura varieties as p-adic analogues of classical Shimura varieties, discussing examples like Rapoport-Zink spaces and conjectures on their cohomology.

Contribution

It proposes a unifying perspective on local Shimura varieties and reviews their theory and related conjectures, highlighting their significance in p-adic geometry.

Findings

Rapoport-Zink spaces as key examples of local Shimura varieties

Discussion of conjectures on the $$-adic cohomology of these varieties

Proposal of a broader theory connecting local and global Shimura varieties

Abstract

This is a survey article that advertizes 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 $\ell$-adic cohomology of local Shimura varieties.