Let A"ƒ" be the ring of finite adeles of Q. For any sufficiently small compact open subgroup "K" of "G"(A"ƒ"), the double coset space  is a finite disjoint union of locally symmetric varieties of the form "Γ" \ "X"+, where the plus superscript indicates a connected component. The varieties "Sh""K"("G","X") are complex algebraic varieties and they form an inverse system over all sufficiently small compact open subgroups "K". This inverse system  admits a natural right action of "G"(A"ƒ"). It is called the Shimura variety associated with the Shimura datum ("G", "X") and denoted "Sh"("G", "X"). For special types of hermitian symmetric domains and congruence subgroups "Γ", algebraic varieties of the form "Γ" \ "X" = "Sh""K"("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. 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. Its dimension "d" is the number of infinite places over which "D" splits. In particular, if "d" = 1 (for example, if "F" = Q and "D" ⊗ R ≅ M2(R)), fixing a sufficiently small arithmetic subgroup of "D"×, one gets a Shimura curve, and curves arising from this construction are already compact (i.e. projective). Some examples of Shimura curves with explicitly known equations are given by the Hurwitz curves of low genus:   and by the Fermat curve of degree 7.