Two-dimensional locally Nash groups

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. As a consequence, we obtain a classification of two-dimensional abelian irreducible algebraic groups defined over R.