Английская Википедия:Converse theorem

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Шаблон:Short description Шаблон:For 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 Шаблон:Harvs who characterized the Riemann zeta function by its functional equation, and by Шаблон:Harvtxt 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. Шаблон:Harvtxt found an extension to modular forms of higher level, which was described by Шаблон:Harvtxt. Weil's extension states that if not only the Dirichlet series

<math>L(s)=\sum\frac{a_n}{n^s}</math>

but also its twists

<math>L_\chi(s)=\sum\frac{\chi(n)a_n}{n^s}</math>

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.

Higher dimensions

J. W. Cogdell, H. Jacquet, I. I. Piatetski-Shapiro and J. Shalika have extended the converse theorem to automorphic forms on some higher-dimensional groups, in particular GLn and GLm×GLn, in a long series of papers.

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