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A guide to the reduction modulo $$p$$ of Shimura varieties. (English) Zbl 1084.11029
Tilouine, Jacques (ed.) et al., Automorphic forms (I). Proceedings of the Semester of the Émile Borel Center, Paris, France, February 17–July 11, 2000. Paris: Société Mathématique de France (ISBN 2-85629-172-4/pbk). Astérisque 298, 271-318 (2005).
In these notes the author concentrates on the reduction modulo $$p$$ of Shimura varieties for a parahoric level structure, specifically on those aspects which have a group-theoretic interpretation. He puts some of the existing literature in its context and points out unsolved questions. As motivation he starts with the elliptic modular curve. The first part of the article is concerned with the local theory in which the following themes are treated: parahoric subgroups, $$\mu$$-admissible and $$\mu$$-permissible set, affine Deligne-Lusztig varieties, relations to local models. The second part is concerned with the global theory on the themes: geometry of the reduction of a Shimura variety, pseudomotivic and quasi-pseudomotivic Galois gerbes, description of the point set in the reduction, the semisimple zeta function.
For the entire collection see [Zbl 1063.11002].

MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties
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