Shimura Variety: Modular Curve, Hermitian Symmetric Space, Congruence Subgroup

9786131209987: Shimura Variety: Modular Curve, Hermitian Symmetric Space, Congruence Subgroup
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a Shimura variety is an analogue of a modular curve, and is (roughly) a quotient of an Hermitian symmetric space by a congruence subgroup of an algebraic group. The simplest example is the quotient of the upper half-plane by SL2(Z). The term Shimura variety applies to the higher-dimensional case, in the case of one-dimensional varieties one speaks of Shimura curves. Such algebraic varieties, formed by compactification of selected quotients of that type, were introduced in a series of papers of Goro Shimura during the 1960s. Shimura''s approach was largely phenomenological, pursuing the widest generalizations of the reciprocity law formulation of complex multiplication theory, in his book (see references). In retrospect, the name Shimura variety was introduced, to recognise that these varieties form the appropriate higher-dimensional class of complex manifolds building on the idea of modular curve. Abstract characterizations were introduced, to the effect that Shimura varieties are parameter spaces of certain types of Hodge structures.

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