On meromorphic mappings admitting an Algebraic Addition Theorem

Abstract: A proper or singular abelian mapping from

C^{n}

to

is parametrized by

n

meromorphic functions with at most

2 n

periods. We develop the existence and structure theorems of the classical theory of an abelian mapping purely on the basis of its defining functional equation, the so-called algebraic addition theorem (AAT), with no appeal to any representation as quotients of theta functions. We offer two new proofs of the periodicity of a nonrational mapping admitting an AAT. We also prove by new arguments the existence of a rational group law on an associated algebraic variety, and that all proper and singular abelian mappings do admit an AAT.

Mathematics Subject Classification ID

Analytic theory of abelian varieties; abelian integrals and differentials (14K20) Meromorphic mappings in several complex variables (32H04) Algebraic dependence theorems (32J10)

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