Automorphic function: Difference between revisions

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In{{Short mathematics, an '''automorphic function''' is adescription|Mathematical function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. }}

In mathematics, an '''automorphic function''' is a function on a space that is invariant under the [[Group action (mathematics)|action]] of some [[group (mathematics)|group]], in other words a function on the [[Quotient space (topology)|quotient space]]. Often the space is a [[complex manifold]] and the group is a [[discrete group]].

==Factor of automorphy==

In [[mathematics]], the notion of '''factor of automorphy''' arises for a [[group (mathematics)|group]] [[Group action (mathematics)|acting]] on a [[complex-analytic manifold]]. Suppose a group <math>G</math> acts on a complex-analytic manifold <math>X</math>. Then, <math>G</math> also acts on the space of [[holomorphic function]]s from <math>X</math> to the complex numbers. A function <math>f</math> is termed an ''[[automorphic form]]'' if the following holds:

: <math>f(g.x) = j_g(x)f(x)</math>

where <math>j_g(x)</math> is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of <math>G</math>.

The ''factor of automorphy'' for the automorphic form <math>f</math> is the function <math>j</math>. An ''automorphic function'' is an automorphic form for which <math>j</math> is the identity.

Some facts about factors of automorphy:

* Every factor of automorphy is a [[Cocycle (algebraic topology)|cocycle]] for the action of <math>G</math> on the multiplicative group of everywhere nonzero holomorphic functions.

* The factor of automorphy is a [[coboundary]] if and only if it arises from an everywhere nonzero automorphic form.

* For a given factor of automorphy, the space of automorphic forms is a vector space.

* The pointwise product of two automorphic forms is an automorphic form corresponding to the product of the corresponding factors of automorphy.

Relation between factors of automorphy and other notions:

* Let <math>\Gamma</math> be a lattice in a Lie group <math>G</math>. Then, a factor of automorphy for <math>\Gamma</math> corresponds to a [[line bundle]] on the quotient group <math>G/\Gamma</math>. Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.

The specific case of <math>\Gamma</math> a subgroup of ''SL''(2,&nbsp;'''R'''), acting on the [[upper half-plane]], is treated in the article on [[automorphic factor]]s.

==Examples==

*[[{{annotated link|Kleinian group]]}}

*[[{{annotated link|Elliptic modular function]]}}

*[[{{annotated link|Modular function]]}}

*{{annotated link|Complex torus}}

==References==

*{{eomspringer|id=a/a014170a014160|firstauthor=A.N. |last=Andrianov|first2= A.N. |last2=Parshin|title=Automorphic FunctionForm}}

*{{eom|id=a/a014170|first=A.N. |last=Andrianov|first2= A.N. |last2=Parshin|title=Automorphic Function}}

*{{Citation | last1=Ford | first1=Lester R. | title=Automorphic functions | url=http://books.google.com/books?id=aqPvo173YIIC | publisher=New York, McGraw-Hill | isbn=978-0-8218-3741-2 | id={{JFM|55.0810.04}} | year=1929}}

*{{Citation | last1=FrickeFord | first1=RobertLester |R. last2|authorlink=KleinLester | first2=FelixR. Ford| title=VorlesungenAutomorphic über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen.functions | url=httphttps://wwwbooks.archivegoogle.orgcom/details/vorlesungenber01fricuoftbooks?id=aqPvo173YIIC | publisher___location=Leipzig: B. G. TeubnerNew York| languagepublisher=German | isbn=978McGraw-1-4297-0551-6Hill | idjfm={{JFM|2855.03340810.01}}04 | year=18971929}}

*{{Citation | last1=Fricke | first1=Robert | last2=Klein | first2=Felix |authorlink1=Robert Fricke|authorlink2= Felix Klein| title=Vorlesungen über die Theorie der automorphen Functionen.|volume Zweiter= Band:I. Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphengruppentheoretischen FunktionenGrundlagen. | url=httphttps://www.archive.org/details/vorlesungenber02fricuoftvorlesungenber01fricuoft | ___location=Leipzig|publisher=Leipzig: B. G. Teubner. | language=German | isbnjfm=978-1-4297-0552-3 | id={{JFM|3228.04300334.01}} | year=19121897}}

*{{Citation | last1=Fricke | first1=Robert | last2=Klein | first2=Felix | title=Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen. | url=https://archive.org/details/vorlesungenber02fricuoft | ___location=Leipzig|publisher= B. G. Teubner. | language=German | jfm=32.0430.01 | year=1912}}

[[Category:Automorphic forms]]

[[Category:Discrete groups]]

[[Category:Types of functions]]

[[Category:Complex manifolds]]