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Automorphic function
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.

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

f ( g . x ) = j g ( x ) f ( x ) {\displaystyle f(g.x)=j_{g}(x)f(x)}

where j g ( x ) {\displaystyle 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 {\displaystyle G} .

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

Some facts about factors of automorphy:

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

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

Examples

  • Kleinian group – Discrete group of Möbius transformations
  • Elliptic modular function – Modular function in mathematicsPages displaying short descriptions of redirect targets
  • Modular function – Analytic function on the upper half-plane with a certain behavior under the modular groupPages displaying short descriptions of redirect targets
  • Complex torus