Automorphic function

In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold.

also acts on the space of holomorphic functions from

is termed an automorphic form if the following holds: where

Equivalently, an automorphic form is a function whose divisor is invariant under the action of

The factor of automorphy for the automorphic form

Some facts about factors of automorphy: Relation between factors of automorphy and other notions: The specific case of

a subgroup of SL(2, R), acting on the upper half-plane, is treated in the article on automorphic factors.