title: Automorphic function
text: # 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.
  
  ## Factor of automorphy
  
  In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): 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.
  
  ## Examples
  
  * Kleinian group
  * Elliptic modular function
  * Modular function
  * Complex torus
  