In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. The term "Shimura variety" applies to the higher-dimensional case, in the case of one-dimensional varieties one speaks of Shimura curves. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties.
Special instances of Shimura varieties were originally introduced by Goro Shimura in the course of his generalization of the complex multiplication theory. Shimura showed that while initially defined analytically, they are arithmetic objects, in the sense that they admit models defined over a number field, the reflex field of the Shimura variety. In the 1970s, Pierre Deligne created an axiomatic framework for the work of Shimura. Around the same time Robert Langlands remarked that Shimura varieties form a natural realm of examples for which equivalence between motivic and automorphic L-functions postulated in the Langlands program can be tested. Automorphic forms realized in the cohomology of a Shimura variety are more amenable to study than general automorphic forms; in particular, there is a construction attaching Galois representations to them.
Other articles related to "shimura variety, shimura":
... Shimura varieties play an outstanding role in the Langlands program ... The prototypical theorem, the Eichler–Shimura congruence relation, implies that the Hasse-Weil zeta function of a modular curve is a product of L-functions associated to explicitly determined modular forms of weight 2 ... process of generalization of this theorem that Goro Shimura introduced his varieties and proved his reciprocity law ...
... Shimura was a colleague and a friend of Yutaka Taniyama ... Shimura then wrote a long series of major papers, extending the phenomena found in the theory of complex multiplication and modular forms to higher dimensions (amo ... It equally brought out the concept, in general, of Shimura variety which is the higher-dimensional equivalent of modular curve ...
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