# Automorphic function

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{{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 [action](/source/Group_action_(mathematics)) of some [group](/source/group_(mathematics)), in other words a function on the [quotient space](/source/Quotient_space_(topology)).  Often the space is a [complex manifold](/source/complex_manifold) and the group is a [discrete group](/source/discrete_group).

==Factor of automorphy==
In [mathematics](/source/mathematics), the notion of '''factor of automorphy''' arises for a [group](/source/group_(mathematics)) [acting](/source/Group_action_(mathematics)) on a [complex-analytic manifold](/source/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](/source/holomorphic_function)s from <math>X</math> to the complex numbers. A function <math>f</math> is termed an ''[automorphic form](/source/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](/source/Cocycle_(algebraic_topology)) for the action of <math>G</math> on the multiplicative group of everywhere nonzero holomorphic functions.
* The factor of automorphy is a [coboundary](/source/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](/source/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 case of <math>G</math> a subgroup of <math>SL_2(\mathbb{R})</math>, acting on the [upper half-plane](/source/upper_half-plane), is treated in the article on [automorphic factor](/source/automorphic_factor)s. In particular, automorphic functions for the [modular group](/source/modular_group) <math>G=SL_2(\mathbb{Z})</math> are called [modular function](/source/modular_function)s.

==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 | location=New York|publisher= McGraw-Hill | 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|volume = I. Die gruppentheoretischen Grundlagen. | url=https://archive.org/details/vorlesungenber01fricuoft | location=Leipzig|publisher= B. G. Teubner | language=German | 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 | 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

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Adapted from the Wikipedia article [Automorphic function](https://en.wikipedia.org/wiki/Automorphic_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Automorphic_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.