View source for Automorphic function

{{Short description|Mathematical function on a space that is invariant under the action of some 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==

*{{springer|id=a/a014160|author=A.N. Parshin|title=Automorphic Form}}
*{{eom|id=a/a014170|first=A.N. |last=Andrianov|first2= A.N. |last2=Parshin|title=Automorphic Function}}
*{{Citation | last1=Ford | first1=Lester R. |authorlink=Lester R. Ford| title=Automorphic functions | url=https://books.google.com/books?id=aqPvo173YIIC | publisher=New York, McGraw-Hill  | isbn=978-0-8218-3741-2 | jfm=55.0810.04 | year=1929}}
*{{Citation | last1=Fricke | first1=Robert | last2=Klein | first2=Felix |authorlink1=Robert Fricke|authorlink2= Felix Klein| title=Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen. | url=https://archive.org/details/vorlesungenber01fricuoft | publisher=Leipzig: B. G. Teubner | language=German | isbn=978-1-4297-0551-6 | jfm=28.0334.01 | year=1897}}
*{{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 | publisher=Leipzig: B. G. Teubner.  | language=German | isbn=978-1-4297-0552-3 | jfm=32.0430.01 | year=1912}}

[[Category:Automorphic forms]]
[[Category:Discrete groups]]
[[Category:Types of functions]]
[[Category:Complex manifolds]]