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nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of .

The factor of automorphy for the automorphic form is the function . An automorphic function is an automorphic form for which is the identity.

An automorphic form is a function F on G (with values in some fixed finite-dimensional vector space V, in the vector-valued case), subject to three kinds of conditions:

# to transform under translation by elements

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