Two-dimensional locally $\mathbb{K}$-Nash groups

TL;DR

This paper classifies two-dimensional connected abelian locally Nash groups over real and complex fields, using Painlevé's algebraic addition theorem and analyzing meromorphic maps.

Contribution

It provides a comprehensive classification of 2D abelian locally Nash groups over real and complex fields, extending previous work with new algebraic and analytic techniques.

Findings

Complete classification of 2D connected abelian locally Nash groups over complex numbers.

Derived real classification from complex case.

Utilized Painlevé's algebraic addition theorem in the analysis.

Abstract

We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlev\'e's description of meromorphic maps admitting an Algebraic Addition Theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification.