# Automorphic function

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This term is related to: action on complex manifolds

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## Definition

Let be a group acting on a complex manifold . An **automorphic function** for this action is a map from to the space of holomorphic functions from to satisfying any of the following equivalent conditions:

- There exists a function such that:

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

- The divisor of (which stores the zeroes and poles with their multiplicities) is invariant under the action of .

## 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.