Automorphic function

Short description: Mathematical function on a space that is invariant under the action of some group

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 $G$ acts on a complex-analytic manifold $X$. Then, $G$ also acts on the space of holomorphic functions from $X$ to the complex numbers. A function $f$ is termed an automorphic form 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 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 $\mathrm{\Gamma }$ be a lattice in a Lie group $G$. Then, a factor of automorphy for $\mathrm{\Gamma }$ corresponds to a line bundle on the quotient group $G/\mathrm{\Gamma }$. Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.

The specific case of $\mathrm{\Gamma }$ 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