Automorphic function

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.

In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group ${\displaystyle G}$ acts on a complex-analytic manifold ${\displaystyle X}$. Then, ${\displaystyle G}$ also acts on the space of holomorphic functions from ${\displaystyle X}$ to the complex numbers. A function ${\displaystyle f}$ is termed an automorphic form if the following holds:

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

where ${\displaystyle j_{g}(x)}$ is an everywhere nonzero holomorphic function. Equivalently, an automorphic form is a function whose divisor is invariant under the action of ${\displaystyle G}$.

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

Some facts about factors of automorphy:

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

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