Automorphic function
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. Often the space is a complex manifold and the group is a discrete group.
Factor of automorphy
In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group [math]\displaystyle{ G }[/math] acts on a complex-analytic manifold [math]\displaystyle{ X }[/math]. Then, [math]\displaystyle{ G }[/math] also acts on the space of holomorphic functions from [math]\displaystyle{ X }[/math] to the complex numbers. A function [math]\displaystyle{ f }[/math] is termed an automorphic form if the following holds:
- [math]\displaystyle{ f(g.x) = j_g(x)f(x) }[/math]
where [math]\displaystyle{ 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]\displaystyle{ G }[/math].
The factor of automorphy for the automorphic form [math]\displaystyle{ f }[/math] is the function [math]\displaystyle{ j }[/math]. An automorphic function is an automorphic form for which [math]\displaystyle{ j }[/math] is the identity.
Some facts about factors of automorphy:
- Every factor of automorphy is a cocycle for the action of [math]\displaystyle{ 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]\displaystyle{ \Gamma }[/math] be a lattice in a Lie group [math]\displaystyle{ G }[/math]. Then, a factor of automorphy for [math]\displaystyle{ \Gamma }[/math] corresponds to a line bundle on the quotient group [math]\displaystyle{ 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]\displaystyle{ \Gamma }[/math] 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
- Hazewinkel, Michiel, ed. (2001), "Automorphic Form", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=a/a014160
- Hazewinkel, Michiel, ed. (2001), "Automorphic Function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=a/a014170
- Ford, Lester R. (1929), Automorphic functions, New York, McGraw-Hill, ISBN 978-0-8218-3741-2, https://books.google.com/books?id=aqPvo173YIIC
- Fricke, Robert; Klein, Felix (1897) (in German), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen., Leipzig: B. G. Teubner, ISBN 978-1-4297-0551-6, https://archive.org/details/vorlesungenber01fricuoft
- Fricke, Robert; Klein, Felix (1912) (in German), Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen., Leipzig: B. G. Teubner., ISBN 978-1-4297-0552-3, https://archive.org/details/vorlesungenber02fricuoft
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