Converse theorem

In the matheamtical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to come from an automorphic form.

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 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.

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