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In the mathematical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem states that a representation of an algebraic group over the adeles is automorphic whenever the L-functions of various twists of it are well behaved. ==Weil's converse theorem== The first converse theorems were proved by who characterized the Riemann zeta function by its functional equation, and by who showed that if a Dirichlet series satisfied a certain functional equation and some growth conditions then it was the Mellin transform of a modular form of level 1. found an extension to modular forms of higher level, which was described by . Weil's extension states that if not only the Dirichlet series : but also its twists : by some Dirichlet characters χ, satisfy suitable functional equations relating values at ''s'' and 1−''s'', then the Dirichlet series is essentially the Mellin transform of a modular form of some level. 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「Converse theorem」の詳細全文を読む スポンサード リンク
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