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Introduction to Shimura varieties. (English) Zbl 1148.14011
Arthur, James (ed.) et al., Harmonic analysis, the trace formula, and Shimura varieties. Proceedings of the Clay Mathematics Institute 2003 summer school, Toronto, Canada, June 2–27, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3844-X/pbk). Clay Mathematics Proceedings 4, 265-378 (2005).
Shimura varieties are quotients of a bounded symmetric domain by an arithmetic group, and they are closely linked to the theory of automorphic forms. The modern theory of Shimura varieties emerged with the study of abelian varieties with complex multiplication by Shimura, Taniyama and Weil as well as with Shimura’s proof of the existence of canonical models for certain types of Shimura varieties. Deligne introduced a more systematic treatment of Shimura varieties using the language of abstract reductive groups and extended Shimura’s results on canonical models. On the other hand, Langlands made Shimura varieties an essential part of his program both as a source of Galois group representations and as tests for the conjecture that all motivic \(L\)-functions are automorphic.
In this article the author provides a comprehensive survey of the theory of Shimura varieties from the point of view of Deligne and Langlands. In particular, he discusses Siegel modular varieties, PEL Shimura varieties, Shimura varieties of Hodge type and complex multiplication along with the existence of canonical models for general Shimura varieties. He also describes the good reduction of Shimura varieties and gives a formula for the number of points on Shimura varieties over a finite field.
For the entire collection see [Zbl 1083.11002].

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