In mathematics, an automorphic function is a function on a space that is invariant under the

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

of some

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

, in other words a function on the quotient space. Often the space is a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

and the group is a

discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...

Factor of automorphy

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the notion of factor of automorphy arises for a

acting Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad r ...

on a

complex-analytic manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

. Suppose a group

G

acts on a complex-analytic manifold

X

. Then,

G

also acts on the space of

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...

s from

X

to the complex numbers. A function

f

is termed an ''

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

'' if the following holds: :

f(g.x) = j_g(x)f(x)

where

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

. The ''factor of automorphy'' for the automorphic form

f

is the function

j

. An ''automorphic function'' is an automorphic form for which

j

is the identity. Some facts about factors of automorphy: * Every factor of automorphy is a cocycle for the action of

G

on the multiplicative group of everywhere nonzero holomorphic functions. * The factor of automorphy is a

coboundary In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...

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

\Gamma

be a lattice in a Lie group

G

. Then, a factor of automorphy for

\Gamma

corresponds to a

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

on the quotient group

G/\Gamma

. Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle. The specific case of

\Gamma

a subgroup of ''SL''(2, R), acting on the

upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...

, is treated in the article on

automorphic factor In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article 'factor of automorphy'. De ...

Examples

Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...

Elliptic modular function In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is hol ...

Modular function In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...

Complex torus In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...

References

* * * * *{{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 Automorphic forms Discrete groups Types of functions Complex manifolds