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 gives conditions for an L-function to come from a representation of a group over a local field or an automorphic form of a group over the adeles.
Weil's converse theorem
The first higher-dimensional converse theorem was proved by Hecke (1936) for level 1, 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. Weil (1967) found an extension to modular forms of higher level, which was described by Ogg (1969, chapter V). Weil's extension has the condition that not only the Dirichlet series, but also its twists by some Dirichlet characters, should satisfy functional equations.
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.
References
- Hecke, E. (1936), "Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung", Mathematische Annalen, 112 (1): 664–699, doi:10.1007/BF01565437, ISSN 0025-5831
- Ogg, Andrew (1969), Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, MR0256993
- Weil, André (1967), "Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Mathematische Annalen, 168: 149–156, doi:10.1007/BF01361551, ISSN 0025-5831, MR0207658