Automorphic function

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[edit]

In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group  G {\displaystyle G} acts on a complex-analytic manifold  X {\displaystyle X} . Then,  G {\displaystyle G} also acts on the space of holomorphic functions from  X {\displaystyle X} to the complex numbers. A function  f {\displaystyle f} is termed an automorphic form if the following holds:

f ( g . x ) = j g ( x ) f ( x ) {\displaystyle f(g.x)=j_{g}(x)f(x)}

where  j g ( x ) {\displaystyle j_{g}(x)} is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of  G {\displaystyle G} .

The factor of automorphy for the automorphic form  f {\displaystyle f} is the function  j {\displaystyle j} . An automorphic function is an automorphic form for which  j {\displaystyle j} is the identity.

Some facts about factors of automorphy:

* Every factor of automorphy is a cocycle for the action of  G {\displaystyle G} 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  Γ {\displaystyle \Gamma } be a lattice in a Lie group  G {\displaystyle G} . Then, a factor of automorphy for  Γ {\displaystyle \Gamma } corresponds to a line bundle on the quotient group  G / Γ {\displaystyle G/\Gamma } . Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.

The specific case of  Γ {\displaystyle \Gamma } a subgroup of SL(2, R), acting on the upper half-plane, is treated in the article on automorphic factors.

## Examples[edit]

* Kleinian group
* Elliptic modular function
* Modular function
* Complex torus

## References[edit]

* A.N. Parshin (2001) [1994], "Automorphic Form", Encyclopedia of Mathematics, EMS Press
* Andrianov, A.N.; Parshin, A.N. (2001) [1994], "Automorphic Function", Encyclopedia of Mathematics, EMS Press
* Ford, Lester R. (1929), Automorphic functions, New York, McGraw-Hill, ISBN 978-0-8218-3741-2, JFM 55.0810.04
* Fricke, Robert; Klein, Felix (1897), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen. (in German), Leipzig: B. G. Teubner, ISBN 978-1-4297-0551-6, JFM 28.0334.01
* Fricke, Robert; Klein, Felix (1912), Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen. (in German), Leipzig: B. G. Teubner., ISBN 978-1-4297-0552-3, JFM 32.0430.01