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: It follows from these axioms that X has a unique structure of a complex manifold (possibly, disconnected) such that for every representation ρ: GR → GL(V), the family (V, ρ ⋅ h) is a holomorphic family of Hodge structures; moreover, it forms a variation of Hodge structure, and X is a finite disjoint union of hermitian symmetric domains.
For every sufficiently small compact open subgroup K of G(Aƒ), the double coset space is a finite disjoint union of locally symmetric varieties of the form
Shimura's approach, later presented in his monograph, was largely phenomenological, pursuing the widest generalizations of the reciprocity law formulation of complex multiplication theory.
In Deligne's formulation, Shimura varieties are parameter spaces of certain types of Hodge structures.
Let F be a totally real number field and D a quaternion division algebra over F. The multiplicative group D× gives rise to a canonical Shimura variety.
The qualitative nature of the Zariski closure of sets of special points on a Shimura variety is described by the André–Oort conjecture.
Indeed, it was in the process of generalization of this theorem that Goro Shimura introduced his varieties and proved his reciprocity law.
On the basis of their results, Robert Langlands made a prediction that the Hasse-Weil zeta function of any algebraic variety W defined over a number field would be a product of positive and negative powers of automorphic L-functions, i.e. it should arise from a collection of automorphic representations.
[1] However philosophically natural it may be to expect such a description, statements of this type have only been proved when W is a Shimura variety.