# Automorphic function

This term is related to: action on complex manifolds
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## 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.