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 also acts on the space of holomorphic functions is termed an automorphic form if the following holds:

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The factor of automorphy. An automorphic function is the identity.

Some facts about factors of automorphy:

• Every factor of automorphy is a cocycle on the multiplicative group of everywhere nonzero holomorphic functions.

• The factor of automorphy is a coboundary if and only if it arises from an everywhere nonzero automorphic form.

• For a given factor of automorphy, the space of automorphic forms is a vector space.

• The pointwise product of two automorphic forms is an automorphic form corresponding to the product of the corresponding factors of automorphy.

Relation between factors of automorphy and other notions:

• corresponds to a line bundle. Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.

, acting on the upper half-plane, is treated in the article on automorphic factors. In particular, automorphic functions for the modular group are called modular functions.

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