Automorphic function

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title: "Automorphic function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["automorphic-forms", "discrete-groups", "types-of-functions", "complex-manifolds"] description: "Mathematical function on a space that is invariant under the action of some group" topic_path: "general/automorphic-forms" source: "https://en.wikipedia.org/wiki/Automorphic_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical function on a space that is invariant under the action of some group ::

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. Suppose a group G acts on a complex-analytic manifold X. Then, G also acts on the space of holomorphic functions from X to the complex numbers. A function f is termed an automorphic form if the following holds:

: f(g.x) = j_g(x)f(x)

where 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.

The factor of automorphy for the automorphic form f is the function j. An automorphic function is an automorphic form for which j is the identity.

Some facts about factors of automorphy:

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

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

Examples

References

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