Automorphic function: Difference between revisions

Revision as of 04:43, 14 May 2007

This term is related to: action on complex manifolds
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Definition

Let $G$ be a group acting on a complex manifold $X$ . An automorphic function for this action is a map $f$ from $G$ to the space of holomorphic functions from $G$ to $C$ satisfying any of the following equivalent conditions:

There exists a function $γ : G \times X \to C^{*}$ such that:

$f (g^{- 1} . x) = γ (g, x) f (x)$

Such a $γ$ is automatically a factor of automorphy and we say that $f$ is an automorphic function corresponding to $γ$ .

The divisor of $f$ (which stores the zeroes and poles with their multiplicities) is invariant under the action of $G$ .

Generalization

The notion of automorphic function can be generalized from complex numbers to arbitrary fields, if we suitably generalize or relax the assumptions of holomorphicity.

@@ Line 6: / Line 6: @@
 * There exists a function <math>\gamma:G \times X \to \mathbb{C}^*</math> such that:
+<math>f(g^{-1}.x) = \gamma(g,x)f(x)</math>
+Such a <math>\gamma</math> is automatically a [[factor of automorphy]] and we say that <math>f</math> is an automorphic function corresponding to <math>\gamma</math>.
+* The divisor of <math>f</math> (which stores the zeroes and poles with their multiplicities) is invariant under the action of <math>G</math>.
+==Generalization==
+The notion of automorphic function can be generalized from complex numbers to arbitrary fields, if we suitably generalize or relax the assumptions of holomorphicity.