Automorphic function

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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.