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Automorphic function

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

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Source updated May 21, 2026

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.

01Factor 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 case of $G$ a subgroup of $SL_{2}(\mathbb {R} )$ , acting on the upper half-plane, is treated in the article on automorphic factors. In particular, automorphic functions for the modular group $G=SL_{2}(\mathbb {Z} )$ are called modular functions.

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This article is adapted from the Wikipedia article “Automorphic function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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