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A [[Meromorphic function|meromorphic function]] on a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141601.png" /> of the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141602.png" /> that, for some [[Discrete group of transformations|discrete group of transformations]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141603.png" /> operating on this domain, satisfies an equation:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141604.png" /></td> </tr></table>
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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141605.png" /> is the Jacobian of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141607.png" /> is an integer known as the weight of the automorphic form. If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141608.png" /> acts fixed-point free, then automorphic forms define differential forms on the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a0141609.png" /> and vice versa. Automorphic forms may be used in the construction of non-trivial automorphic functions (cf. [[Automorphic function|Automorphic function]]). It has been shown that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a01416010.png" /> is a function that is holomorphic and bounded on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a01416011.png" />, then the series
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A [[Meromorphic function|meromorphic function]] on a bounded domain  $  D $
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of the complex space  $  \mathbf C  ^ {n} $
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that, for some [[Discrete group of transformations|discrete group of transformations]] $  \Gamma $
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operating on this domain, satisfies an equation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a01416012.png" /></td> </tr></table>
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$$
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f ( \gamma ( x ) )  = j _  \gamma  ^ {-m} ( x ) f ( x ) ,
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\  x \in D , \gamma \in \Gamma ,
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$$
  
converges for large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a01416013.png" />, thus representing a non-trivial automorphic function of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a014/a014160/a01416014.png" />. These series are called Poincaré theta-series.
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Here  $  j _  \gamma  (x) $
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is the Jacobian of the mapping  $  x \rightarrow \gamma (x) $
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and  $  m $
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is an integer known as the weight of the automorphic form. If the group  $  \Gamma $
 +
acts fixed-point free, then automorphic forms define differential forms on the quotient space  $  D / \Gamma $
 +
and vice versa. Automorphic forms may be used in the construction of non-trivial automorphic functions (cf. [[Automorphic function|Automorphic function]]). It has been shown that if  $  g(x) $
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is a function that is holomorphic and bounded on a domain  $  D $,
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then the series
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$$
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\sum _ {\gamma \in \Gamma }
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g( \gamma (x)) j _  \gamma  ^ {m} (x)
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$$
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 +
converges for large values of $  m $,  
 +
thus representing a non-trivial automorphic function of weight $  m $.  
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These series are called Poincaré theta-series.
  
 
The classical definition of automorphic forms, given above, has recently served as the starting point of a far-reaching generalization in the theory of discrete subgroups of Lie groups and adèle groups.
 
The classical definition of automorphic forms, given above, has recently served as the starting point of a far-reaching generalization in the theory of discrete subgroups of Lie groups and adèle groups.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  , ''Oeuvres de H. Poincaré'' , Gauthier-Villars  (1916–1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.L. Siegel,  "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen  (1955)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  , ''Oeuvres de H. Poincaré'' , Gauthier-Villars  (1916–1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.L. Siegel,  "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen  (1955)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 10:01, 25 April 2020


A meromorphic function on a bounded domain $ D $ of the complex space $ \mathbf C ^ {n} $ that, for some discrete group of transformations $ \Gamma $ operating on this domain, satisfies an equation:

$$ f ( \gamma ( x ) ) = j _ \gamma ^ {-m} ( x ) f ( x ) , \ x \in D , \gamma \in \Gamma , $$

Here $ j _ \gamma (x) $ is the Jacobian of the mapping $ x \rightarrow \gamma (x) $ and $ m $ is an integer known as the weight of the automorphic form. If the group $ \Gamma $ acts fixed-point free, then automorphic forms define differential forms on the quotient space $ D / \Gamma $ and vice versa. Automorphic forms may be used in the construction of non-trivial automorphic functions (cf. Automorphic function). It has been shown that if $ g(x) $ is a function that is holomorphic and bounded on a domain $ D $, then the series

$$ \sum _ {\gamma \in \Gamma } g( \gamma (x)) j _ \gamma ^ {m} (x) $$

converges for large values of $ m $, thus representing a non-trivial automorphic function of weight $ m $. These series are called Poincaré theta-series.

The classical definition of automorphic forms, given above, has recently served as the starting point of a far-reaching generalization in the theory of discrete subgroups of Lie groups and adèle groups.

References

[1] H. Poincaré, , Oeuvres de H. Poincaré , Gauthier-Villars (1916–1965)
[2] C.L. Siegel, "Automorphe Funktionen in mehrerer Variablen" , Math. Inst. Göttingen (1955)

Comments

References [a2] and [a3] can serve to get some idea of modern developments and topics in the theory of automorphic forms and its relations with other parts of mathematics. (Cf. the comments to the article Automorphic function for a more general notion).

References

[a1] W.L. Baily jr., "Introductory lectures on automorphic forms" , Iwanami Shoten & Princeton Univ. Press (1973)
[a2] A. Borel (ed.) W. Casselman (ed.) , Automorphic forms, representations and -functions , Proc. Symp. Pure Math. , 33:1–2 , Amer. Math. Soc. (1979)
[a3] S.S. Gelbart, "Automorphic forms on adèle groups" , Princeton Univ. Press (1975)
How to Cite This Entry:
Automorphic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Automorphic_form&oldid=45527
This article was adapted from an original article by A.N. Parshin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article