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

action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...

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 iden ...

, 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 a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

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 ...

Factor of automorphy

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad range of sk ...

on a

complex-analytic manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas of charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are holomorp ...

. 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 de ...

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 o ...

'' 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 In mathematics a cocycle is a closed cochain (algebraic topology), cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in gr ...

for the action of

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

\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 organis ...

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, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...

, is treated in the article on automorphic factors.

Examples

* * * *

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 , location=Leipzig, publisher= B. G. Teubner. , language=German , jfm=32.0430.01 , year=1912 Automorphic forms Discrete groups Types of functions Complex manifolds