View source for Factor of automorphy

In [[Mathematics|mathematics]], the notion of '''factor of automorphy''' arises for a [[Group (mathematics)|group]] [[Group action (mathematics)|acting]] on a complex-analytic manifold. Suppose a group <math>G</math> acts on a complex-analytic manifold <math>X</math>. Then, <math>G</math> also acts on the space of [[Holomorphic function|holomorphic function]]s from <math>X</math> to the complex numbers. A function <math>f</math> is termed an ''[[Automorphic form|automorphic form]]'' if the following holds:
: <math>f(g.x) = j_g(x)f(x)</math>

where <math>j_g(x)</math> is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of <math>G</math>.

The ''factor of automorphy'' for the automorphic form <math>f</math> is the function <math>j</math>. An ''automorphic function'' is an automorphic form for which <math>j</math> is the identity.

Some facts about factors of automorphy:

* Every factor of automorphy is a cocycle for the action of <math>G</math> 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 <math>\Gamma</math> be a lattice in a Lie group <math>G</math>. Then, a factor of automorphy for <math>\Gamma</math> corresponds to a [[Line bundle|line bundle]] on the quotient group <math>G/\Gamma</math>. Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.

The specific case of <math>\Gamma</math> a subgroup of ''SL''(2,&nbsp;'''R'''), acting on the [[Upper half-plane|upper half-plane]], is treated in the article on [[Automorphic factor|automorphic factor]]s.

==References==
*{{springer|id=a/a014170|author=A.N. Andrianov, A.N. Parshin|title=Automorphic Function}} ''(The commentary at the end defines automorphic factors in modern geometrical language)''
*{{springer|id=a/a014160|author=A.N. Parshin|title=Automorphic Form}}

[[Category:Automorphic forms]]

{{Sourceattribution|Factor of automorphy|1}}