In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group.

Factor of automorphy

In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group G acts on a complex-analytic manifold X. Then, G also acts on the space of holomorphic functions from X to the complex numbers. A function f is termed an automorphic form if the following holds: where j_g(x) is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of G. The factor of automorphy for the automorphic form f is the function j. An automorphic function is an automorphic form for which j is the identity. Some facts about factors of automorphy: Relation between factors of automorphy and other notions: The specific case of \Gamma a subgroup of SL(2, R), acting on the upper half-plane, is treated in the article on automorphic factors.

Examples