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

The specific case of $$\Gamma$$ 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