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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 : satisfying the four properties given below. Here, the notation and refer to the upper half-plane and the complex plane, respectively. The notation is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element is a 2x2 matrix : with ''a'', ''b'', ''c'', ''d'' real numbers, satisfying ''ad''−''bc''=1. An automorphic factor must satisfy: :1. For a fixed , the function is a holomorphic function of . :2. For all and , one has :: :for a fixed real number ''k''. :3. For all and , one has :: :Here, is the fractional linear transform of by . :4.If , then for all and , one has :: :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
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