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Towards a theory of local Shimura varieties. (English) Zbl 1378.11070
Summary: 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 $$\ell$$-adic cohomology of local Shimura varieties.

MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 14G22 Rigid analytic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology 11-02 Research exposition (monographs, survey articles) pertaining to number theory
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