Automorphic function

Definition
Let $$G$$ be a group acting on a complex manifold $$X$$. An automorphic function for this action is a map $$f$$ from $$G$$ to the space of holomorphic functions from $$G$$ to $$\mathbb{C}$$ satisfying any of the following equivalent conditions:


 * There exists a function $$\gamma:G \times X \to \mathbb{C}^*$$ such that:

$$f(g^{-1}.x) = \gamma(g,x)f(x)$$

Such a $$\gamma$$ is automatically a factor of automorphy and we say that $$f$$ is an automorphic function corresponding to $$\gamma$$.


 * The divisor of $$f$$ (which stores the zeroes and poles with their multiplicities) is invariant under the action of $$G$$.

Generalization
The notion of automorphic function can be generalized from complex numbers to arbitrary fields, if we suitably generalize or relax the assumptions of holomorphicity.