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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Factor of automorphy ： ウィキペディア英語版 Factor of automorphy In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group $G$ acts on a complex-analytic manifold $X$. Then, $G$ also acts on the space of holomorphic functions from $X$ to the complex numbers. A function $f$ is termed an ''automorphic form'' if the following holds: : $f(g.x)\; =\; j\_g(x)f(x)$ where $j\_g(x)$ is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of $G$. The ''factor of automorphy'' for the automorphic form $f$ is the function $j$. An ''automorphic function'' is an automorphic form for which $j$ is the identity. Some facts about factors of automorphy: * Every factor of automorphy is a cocycle for the action of $G$ 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 $\backslash Gamma$ be a lattice in a Lie group $G$. Then, a factor of automorphy for $\backslash Gamma$ corresponds to a line bundle on the quotient group $G/\backslash Gamma$. Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle. The specific case of $\backslash Gamma$ 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 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.