Factor of automorphy

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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.

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

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