Automorphic function

From Quire, the open reference·Mathematical function on a space that is invariant under the action of some group·2 min read

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#edit

In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group ${\displaystyle G}$ acts on a complex-analytic manifold ${\displaystyle X}$ . Then, ${\displaystyle G}$ also acts on the space of holomorphic functions from ${\displaystyle X}$ to the complex numbers. A function ${\displaystyle f}$ is termed an automorphic form if the following holds:

${\displaystyle f(g.x)=j_{g}(x)f(x)}$

where ${\displaystyle j_{g}(x)}$ is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of ${\displaystyle G}$ .

The factor of automorphy for the automorphic form ${\displaystyle f}$ is the function ${\displaystyle j}$ . An automorphic function is an automorphic form for which ${\displaystyle j}$ is the identity.

Some facts about factors of automorphy:

Every factor of automorphy is a cocycle for the action of ${\displaystyle G}$ 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 ${\displaystyle \Gamma }$ be a lattice in a Lie group ${\displaystyle G}$ . Then, a factor of automorphy for ${\displaystyle \Gamma }$ corresponds to a line bundle on the quotient group ${\displaystyle G/\Gamma }$ . Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.

The case of ${\displaystyle G}$ a subgroup of ${\displaystyle SL_{2}(\mathbb {R} )}$ , acting on the upper half-plane, is treated in the article on automorphic factors. In particular, automorphic functions for the modular group ${\displaystyle G=SL_{2}(\mathbb {Z} )}$ are called modular functions.

References#edit

A.N. Parshin (2001) [1994], "Automorphic Form", Encyclopedia of Mathematics, EMS Press
Andrianov, A.N.; Parshin, A.N. (2001) [1994], "Automorphic Function", Encyclopedia of Mathematics, EMS Press
Ford, Lester R. (1929), Automorphic functions, New York: McGraw-Hill, JFM 55.0810.04
Fricke, Robert; Klein, Felix (1897), Vorlesungen über die Theorie der automorphen Functionen (in German), vol. I. Die gruppentheoretischen Grundlagen., Leipzig: B. G. Teubner, JFM 28.0334.01
Fricke, Robert; Klein, Felix (1912), Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen. (in German), Leipzig: B. G. Teubner., JFM 32.0430.01

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