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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.
| | I am Rogelio from Chicago. I love to play Clarinet. Other hobbies are Cricket.<br><br>Here is my webpage: [http://www.areadevelopment.com/newsItems/2-15-2013/awesome-products-locates-facility-airy-north-carolina437894.shtml LD Hardas] |
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| ==Weil's converse theorem==
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| 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 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
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| :<math>L(s)=\sum\frac{a_n}{n^s}</math>
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| but also its twists
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| :<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−''s'', then the Dirichlet series is essentially the Mellin transform of a modular form of some level.
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| ==Higher dimensions==
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| 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>×GL<sub>''n''</sub>, in a long series of papers.
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| ==References==
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| *{{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}}
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| *{{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}}
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| *{{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}}
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| ==External links==
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| * [http://www.math.osu.edu/~cogdell/ Cogdell's papers on converse theorems]
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| [[Category:Automorphic forms]]
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I am Rogelio from Chicago. I love to play Clarinet. Other hobbies are Cricket.
Here is my webpage: LD Hardas