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In algebraic geometry, the Mumford–Tate group (or Hodge group) ''MT''(''F'') constructed from a Hodge structure ''F'' is a certain algebraic group ''G''. When ''F'' is given by a rational representation of an algebraic torus, the definition of ''G'' is as the Zariski closure of the image in the representation of the circle group, over the rational numbers. introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. introduced the ''p''-adic analogue of Mumford's construction for Hodge–Tate modules, using the work of on p-divisible groups, and named them Mumford–Tate groups. ==Formulation== The algebraic torus ''T'' used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of ''a''+''bi'' on the basis of the complex numbers C over R: : The circle group inside this group of matrices is the unitary group ''U''(1). Hodge structures arising in geometry, for example on the cohomology groups of Kähler manifolds, have a lattice consisting of the integral cohomology classes. Not quite so much is needed for the definition of the Mumford–Tate group, but it does assume that the vector space ''V'' underlying the Hodge structure has a given rational structure, i.e. is given over the rational numbers ''Q''. For the purposes of the theory the complex vector space ''V''''C'', obtained by extending the scalars of ''V'' from ''Q'' to ''C'', is used. The weight ''k'' of the Hodge structure describes the action of the diagonal matrices of ''T'', and ''V'' is supposed therefore to be homogeneous of weight ''k'', under that action. Under the action of the full group ''V''''C'' breaks up into subspaces ''V''''pq'', complex conjugate in pairs under switching ''p'' and ''q''. Thinking of the matrix in terms of the complex number λ it represents, ''V''''pq'' has the action of λ by the ''p''th power and of the complex conjugate of λ by the ''q''th power. Here necessarily :''p'' + ''q'' = ''k''. In more abstract terms, the torus ''T'' underlying the matrix group is the Weil restriction of the multiplicative group ''GL''(1), from the complex field to the real field, an algebraic torus whose character group consists of the two homomorphisms to ''GL''(1), interchanged by complex conjugation. Once formulated in this fashion, the rational representation ρ of ''T'' on ''V'' setting up the Hodge structure ''F'' determines the image ρ(''U''(1)) in ''GL''(''V''''C''); and ''MT''(''F'') is by definition the Zariski closure, for the ''Q''-Zariski topology on ''GL''(''V''), of this image.〔http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf, pp. 7–9.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mumford–Tate group」の詳細全文を読む スポンサード リンク
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