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

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

but also its twists

${\displaystyle L_{\chi }(s)=\sum {\frac {\chi (n)a_{n}}{n^{s}}}}$

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 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, 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