On meromorphic mappings admitting an Algebraic Addition Theorem

A proper or singular abelian mapping from $C^n$ to ${\bar{\text{C}}}^n$ is parametrized by $n$ meromorphic functions with at most $2n$ 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.