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 ...
and the group is a
discrete group.
Factor of automorphy
In
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 bro ...
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 de ...
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 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
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 ...
, is treated in the article on
automorphic factors.
Examples
*
Kleinian group
*
Elliptic modular function
*
Modular function
*
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'', w ...
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