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.

References

*springer|id=a/a014170|author=A.N. Andrianov,A.N. Parshin|title=Automorphic Function "(The commentary at the end defines automorphic factors in modern geometrical language)"
*springer|id=a/a014160|author=A.N. Parshin|title=Automorphic Form