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 Hamburger (1921) who characterized the Riemann zeta function by its functional equation, and by Hecke (1936) 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 states that if not only the Dirichlet series
L ( s ) = ∑ a n n s {\displaystyle L(s)=\sum {\frac {a_{n}}{n^{s}}}}
but also its twists
L χ ( s ) = ∑ χ ( n ) a n n s {\displaystyle 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−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.
References Cogdell, James W.; Piatetski-Shapiro, I. I. (1994), "Converse theorems for GLn", Publications Mathématiques de l'IHÉS, 79 (79): 157–214, doi:10.1007/BF02698889, ISSN 1618-1913, MR 1307299 Cogdell, James W.; Piatetski-Shapiro, I. I. (1999), "Converse theorems for GLn. II", Journal für die reine und angewandte Mathematik, 507 (507): 165–188, doi:10.1515/crll.1999.507.165, ISSN 0075-4102, MR 1670207 Cogdell, James W.; Piatetski-Shapiro, I. I. (2002), "Converse theorems, functoriality, and applications to number theory", in Li, Tatsien (ed.), Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Beijing: Higher Ed. Press, pp. 119–128, arXiv:math/0304230, Bibcode:2003math......4230C, ISBN 978-7-04-008690-4, MR 1957026, archived from the original on 2011-08-20, retrieved 2011-06-18 Cogdell, James W. (2007), "L-functions and converse theorems for GLn", in Sarnak, Peter; Shahidi, Freydoon (eds.), Automorphic forms and applications, IAS/Park City Math. Ser., vol. 12, Providence, R.I.: American Mathematical Society, pp. 97–177, ISBN 978-0-8218-2873-1, MR 2331345 Hamburger, Hans (1921), "Über die Riemannsche Funktionalgleichung der ζ-Funktion", Mathematische Zeitschrift, 10 (3): 240–254, doi:10.1007/BF01211612, ISSN 0025-5874 Hecke, Erich (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, MR 0256993 Weil, André (1967), "Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Mathematische Annalen, 168: 149–156, doi:10.1007/BF01361551, ISSN 0025-5831, MR 0207658
External links Cogdell's papers on converse theorems