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
Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular:
*Quotient space (topology), in case of topological spaces
* Quotient space (linear algebra), in case of vector spaces
*Quotient ...
. 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.
Factor of automorphy
In
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
group 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. Suppose a group
acts on a complex-analytic manifold
. Then,
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
to the complex numbers. A function
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:
:
where
is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of
.
The ''factor of automorphy'' for the automorphic form
is the function
. An ''automorphic function'' is an automorphic form for which
is the identity.
Some facts about factors of automorphy:
* Every factor of automorphy is a
cocycle
In mathematics a cocycle is a closed 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 group cohomology. In autonomous d ...
for the action of
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
be a lattice in a Lie group
. Then, a factor of automorphy for
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
. Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.
The specific case of
a subgroup of ''SL''(2, R), acting on the
upper half-plane, is treated in the article on
automorphic factors.
Examples
*
Kleinian group
*
Elliptic modular function
*
Modular function
*
Complex torus
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