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On the characterization of complex Shimura varieties. (English) Zbl 1036.11028
Shimura varieties generalize classical modular curves, and over $$\mathbb C$$ they are finite disjoint unions of arithmetic quotients of Hermitian symmetric domains. One approach to the study of Shimura varieties was introduced by Kazhdan, who showed that the conjugate of a Shimura variety is again a Shimura variety. Kazhdan’s result can be used to prove the existence of the canonical models for all Shimura varieties. In this paper the author recalls basic properties of complex Shimura varieties and proves that they actually characterize Shimura varieties. This characterization implies the explicit form of Kazhdan’s result. In the appendix the author also provides a modern formulation and a proof of Weil’s descent theorem.

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