Prais–Winsten estimation: Difference between revisions

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In the mathematical theory of [[automorphic form]]s, 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.
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==Weil's converse theorem==
 
The first  converse theorems  were proved by {{harvs|txt|last=Hamburger|author-link=Hans Hamburger|year=1921}} who characterized the [[Riemann zeta function]] by its functional equation, and by {{harvtxt|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&nbsp;1. {{harvtxt|Weil|1967}} found an extension to modular forms of higher level, which was  described by {{harvtxt|Ogg|1969|loc=chapter V}}.  Weil's extension states that if  not only the Dirichlet series
:<math>L(s)=\sum\frac{a_n}{n^s}</math>
but also its twists
:<math>L_\chi(s)=\sum\frac{\chi(n)a_n}{n^s}</math>
by some [[Dirichlet character]]s χ, satisfy  suitable functional equations relating values at ''s'' and 1&minus;''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<sub>''n''</sub> and GL<sub>''m''</sub>&times;GL<sub>''n''</sub>, in a long series of papers.
 
==References==
 
*{{Citation | last1=Cogdell | first1=James W. | last2=Piatetski-Shapiro | first2=I. I. | title=Converse theorems for GL<sub>n</sub> | url=http://www.numdam.org/item?id=PMIHES_1994__79__157_0 | mr=1307299 | year=1994 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=79 | pages=157–214}}
*{{Citation | last1=Cogdell | first1=James W. | last2=Piatetski-Shapiro | first2=I. I. | title=Converse theorems for GL<sub>n</sub>. II | doi=10.1515/crll.1999.507.165 | mr=1670207 | year=1999 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=507 | pages=165–188}}
*{{Citation | last1=Cogdell | first1=James W. | last2=Piatetski-Shapiro | first2=I. I. | editor1-last=Li | editor1-first=Tatsien | title=Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) | url=http://mathunion.org/ICM/ICM2002.2/ | publisher=Higher Ed. Press | location=Beijing | isbn= |doi=10.1007/BF01361551 | mr=0207658 | year=1967 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=168 | pages=149–156}}
 
==External links==
* [http://www.math.osu.edu/~cogdell/ Cogdell's papers on converse theorems]
 
[[Category:Automorphic forms]]

Latest revision as of 05:04, 6 December 2014

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