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.
- A.N. Andrianov, A.N. Parshin (2001), "Automorphic Function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 (The commentary at the end defines automorphic factors in modern geometrical language)
- A.N. Parshin (2001), "Automorphic Form", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4