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.
Read more about Shimura Variety: Examples, Canonical Models and Special Points, Role in The Langlands Program
Famous quotes containing the word variety:
“To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain proofs that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth.”
—David Hume (17111776)