Factor of automorphy

In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group [math]\displaystyle{ G }[/math] acts on a complex-analytic manifold [math]\displaystyle{ X }[/math]. Then, [math]\displaystyle{ G }[/math] also acts on the space of holomorphic functions from [math]\displaystyle{ X }[/math] to the complex numbers. A function [math]\displaystyle{ f }[/math] is termed an automorphic form if the following holds:

[math]\displaystyle{ f(g.x) = j_g(x)f(x) }[/math]

where [math]\displaystyle{ j_g(x) }[/math] is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of [math]\displaystyle{ G }[/math].

The factor of automorphy for the automorphic form [math]\displaystyle{ f }[/math] is the function [math]\displaystyle{ j }[/math]. An automorphic function is an automorphic form for which [math]\displaystyle{ j }[/math] is the identity.

Some facts about factors of automorphy:

• Every factor of automorphy is a cocycle for the action of [math]\displaystyle{ G }[/math] 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 [math]\displaystyle{ \Gamma }[/math] be a lattice in a Lie group [math]\displaystyle{ G }[/math]. Then, a factor of automorphy for [math]\displaystyle{ \Gamma }[/math] corresponds to a line bundle on the quotient group [math]\displaystyle{ G/\Gamma }[/math]. Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.

The specific case of [math]\displaystyle{ \Gamma }[/math] a subgroup of SL(2, R), acting on the upper half-plane, is treated in the article on automorphic factors.

References

• Hazewinkel, Michiel, ed. (2001), "Automorphic Function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=a/a014170  (The commentary at the end defines automorphic factors in modern geometrical language)
• Hazewinkel, Michiel, ed. (2001), "Automorphic Form", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=a/a014160