Factor of automorphy


In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group acts on a complex-analytic manifold. Then, also acts on the space of holomorphic functions from to the complex numbers. A function is termed an automorphic form if the following holds:
where is an everywhere 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.
Some facts about factors of automorphy:
Relation between factors of automorphy and other notions:
The specific case of a subgroup of SL, acting on the upper half-plane, is treated in the article on automorphic factors.