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 {displaystyle G} acts on a complex-analytic manifold X {displaystyle X} . Then, G {displaystyle G} also acts on the space of holomorphic functions from X {displaystyle X} to the complex numbers. A function f {displaystyle f} is termed an automorphic form if the following holds: In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group G {displaystyle G} acts on a complex-analytic manifold X {displaystyle X} . Then, G {displaystyle G} also acts on the space of holomorphic functions from X {displaystyle X} to the complex numbers. A function f {displaystyle f} is termed an automorphic form if the following holds: where j g ( x ) {displaystyle 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 {displaystyle G} . The factor of automorphy for the automorphic form f {displaystyle f} is the function j {displaystyle j} . An automorphic function is an automorphic form for which j {displaystyle j} is the identity.