Shimura variety

In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura curves are the one-dimensional Shimura varieties. 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. In 1979, 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. Let S = ResC/R Gm be the Weil restriction of the multiplicative group from complex numbers to real numbers. It is a real algebraic group, whose group of R-points, S(R), is C* and group of C-points is C×C. A Shimura datum is a pair (G, X) consisting of a (connected) reductive algebraic group G defined over the field Q of rational numbers and a G(R)-conjugacy class X of homomorphisms h: S → GR satisfying the following axioms: For any h in X, only weights (0,0), (1,−1), (−1,1) may occur in gC, i.e. the complexified Lie algebra of G decomposes into a direct sum where for any z ∈ S, h(z) acts trivially on the first summand and via (respectively, ) on the second (respectively, third) summand.