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.
Definition
Shimura datum 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
g ⊗ C = k ⊕ p + ⊕ p − , {\displaystyle {\mathfrak {g}}\otimes \mathbb {C} ={\mathfrak {k}}\oplus {\mathfrak {p}}^{+}\oplus {\mathfrak {p}}^{-},}
where for any z ∈ S, h(z) acts trivially on the first summand and via z / z ¯ {\displaystyle z/{\bar {z}}} (respectively, z ¯ / z {\displaystyle {\bar {z}}/z} ) on the second (respectively, third) summand. The adjoint action of h(i) induces a Cartan involution on the adjoint group of GR. The adjoint group of GR does not admit a factor H defined over Q such that the projection of h on H is trivial. 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.
Shimura variety Let Aƒ be the ring of finite adeles of Q. For every sufficiently small compact open subgroup K of G(Aƒ), the double coset space
Sh K ( G , X ) = G ( Q ) ∖ X × G ( A f ) / K {\displaystyle \operatorname {Sh} _{K}(G,X)=G(\mathbb {Q} )\backslash X\times G(\mathbb {A} _{f})/K}
is a finite disjoint union of locally symmetric varieties of the form Γ i ∖ X + {\displaystyle \Gamma _{i}\backslash X^{+}} , where the plus superscript indicates a connected component. The varieties ShK(G,X) are complex algebraic varieties and they form an inverse system over all sufficiently small compact open subgroups K. This inverse system
( Sh K ( G , X ) ) K {\displaystyle (\operatorname {Sh} _{K}(G,X))_{K}}
admits a natural right action of G(Aƒ). It is called the Shimura variety associated with the Shimura datum (G, X) and denoted Sh(G, X).
History For special types of hermitian symmetric domains and congruence subgroups Γ, algebraic varieties of the form Γ \ X = ShK(G,X) and their compactifications were introduced in a series of papers of Goro Shimura during the 1960s. 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 retrospect, the name "Shimura variety" was introduced by Deligne, who proceeded to isolate the abstract features that played a role in Shimura's theory. In Deligne's formulation, Shimura varieties are parameter spaces of certain types of Hodge structures. Thus they form a natural higher-dimensional generalization of modular curves viewed as moduli spaces of elliptic curves with level structure. In many cases, the moduli problems to which Shimura varieties are solutions have been likewise identified.
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