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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 Automorphic factor ： ウィキペディア英語版 Automorphic factor In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article 'factor of automorphy'. ==Definition== An ''automorphic factor of weight k'' is a function :$\backslash nu:\backslash Gamma\; \backslash times\; \backslash mathbb\; \backslash to\; \backslash mathbb$ satisfying the four properties given below. Here, the notation $\backslash mathbb$ and $\backslash mathbb$ refer to the upper half-plane and the complex plane, respectively. The notation $\backslash Gamma$ is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element $\backslash gamma\backslash in\backslash Gamma$ is a 2x2 matrix : with ''a'', ''b'', ''c'', ''d'' real numbers, satisfying ''ad''−''bc''=1. An automorphic factor must satisfy: :1. For a fixed $\backslash gamma\backslash in\backslash Gamma$, the function $\backslash nu(\backslash gamma,z)$ is a holomorphic function of $z\backslash in\backslash mathbb$. :2. For all $z\backslash in\backslash mathbb$ and $\backslash gamma\backslash in\backslash Gamma$, one has ::$\backslash vert\backslash nu(\backslash gamma,z)\backslash vert=\backslash vert\; cz+d\backslash vert^k$ :for a fixed real number ''k''. :3. For all $z\backslash in\backslash mathbb$ and $\backslash gamma,\backslash delta\backslash in\backslash Gamma$, one has ::$\backslash nu(\backslash gamma\backslash delta,z)=\backslash nu(\backslash gamma,\backslash delta\; z)\backslash nu(\backslash delta,z)$ :Here, $\backslash delta\; z$ is the fractional linear transform of $z$ by $\backslash delta$. :4.If $-I\backslash in\backslash Gamma$, then for all $z\backslash in\backslash mathbb$ and $\backslash gamma\backslash in\backslash Gamma$, one has ::$\backslash nu(-\backslash gamma,z)=\backslash nu(\backslash gamma,z)$ :Here, ''I'' denotes the identity matrix. 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「Automorphic factor」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.