This term is related to: action on complex manifolds
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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 .
The notion of automorphic function can be generalized from complex numbers to arbitrary fields, if we suitably generalize or relax the assumptions of holomorphicity.