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In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group acts on a complex-analytic manifold . Then, also acts on the space of holomorphic functions from to the complex numbers. A function is termed an ''automorphic form'' if the following holds: : where is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of . The ''factor of automorphy'' for the automorphic form is the function . An ''automorphic function'' is an automorphic form for which is the identity. Some facts about factors of automorphy: * Every factor of automorphy is a cocycle for the action of on the multiplicative group of everywhere nonzero holomorphic functions. * The factor of automorphy is a coboundary if and only if it arises from an everywhere nonzero automorphic form. * For a given factor of automorphy, the space of automorphic forms is a vector space. * The pointwise product of two automorphic forms is an automorphic form corresponding to the product of the corresponding factors of automorphy. Relation between factors of automorphy and other notions: * Let be a lattice in a Lie group . Then, a factor of automorphy for corresponds to a line bundle on the quotient group . Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle. The specific case of a subgroup of ''SL''(2, R), acting on the upper half-plane, is treated in the article on automorphic factors. ==References== * ''(The commentary at the end defines automorphic factors in modern geometrical language)'' * 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「factor of automorphy」の詳細全文を読む スポンサード リンク
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