Automorphic function
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 formula_1 acts on a complex-analytic manifold formula_2. Then, formula_1 also acts on the space of holomorphic functions from formula_2 to the complex numbers. A function formula_5 is termed an "automorphic  form" if the following holds: where formula_7 is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of formula_1. The "factor of automorphy" for the automorphic form formula_5 is the function formula_10. An "automorphic function" is an automorphic form for which formula_10 is the identity. Some facts about factors of automorphy: Relation between factors of automorphy and other notions: The specific case of formula_13 a subgroup of "SL"(2, R), acting on the upper half-plane, is treated in the article on automorphic factors.