{{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 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.
==Factor of automorphy== In mathematics, the notion of '''factor of automorphy''' arises for a group 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 functions 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 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 case of <math>G</math> a subgroup of <math>SL_2(\mathbb{R})</math>, acting on the upper half-plane, is treated in the article on automorphic factors. In particular, automorphic functions for the modular group <math>G=SL_2(\mathbb{Z})</math> are called modular functions.
==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