Converse theorem

For converse of a theorem, see Converse (logic).

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

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

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−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 GL_n and GL_m×GL_n, in a long series of papers.

References

• Cogdell, James W.; Piatetski-Shapiro, I. I. (1994), "Converse theorems for GL_n", 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 GL_n. 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 GL_n", 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, 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, 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