1998/06/13 by Mark B. Villarino, Villarino, Mark B.
Mathematics · #14K20 (Secondary) #32H04 (Primary) #32J10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #math.CV #msc:14K20 #msc:32H04 #msc:32J10
paper · pdf · doi:10.48550/arxiv.math/9806078
20 pages, Latex2e
arxiv created 1998/06/13 · openalex publication_date 1998/06/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A proper or singular abelian mapping from Cn to \Cn 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.