Converse theorem

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
 * $$L(s)=\sum\frac{a_n}{n^s}$$

but also its twists
 * $$L_\chi(s)=\sum\frac{\chi(n)a_n}{n^s}$$

by some Dirichlet characters χ, satisfy  suitable functional equations relating values at s and 1&minus;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&times;GLn, in a long series of papers.