# 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:

- Every factor of automorphy is a cocycle for the action of 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:

- Let be a lattice in a Lie group . Then, a factor of automorphy for corresponds to a line bundle on the quotient group . Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.

The specific case of a subgroup of *SL*(2, **R**), acting on the upper half-plane, is treated in the article on automorphic factors.

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation
}} *(The commentary at the end defines automorphic factors in modern geometrical language)*

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}