Algebraic numbers associated to certain punctured spheres (Q1087912)

The paper concerns possible algebraic implications of the uniformization theorem for \(P^ 1_{{\mathbb{C}}}\setminus S\) where S is a set of points of cardinality \(\geq 3\). When card S\(=3\), full results are known and classical and associated with the theory of Dedekind's \(\eta\)-function. The author studies the covering map in the more general case, and by combining properties of the Schwarzian derivative with a recent result of Belyi is led to a conjecture concerning the algebraic nature of certain Laurent series coefficients. The result of Belyi concerning the existence of certain functions on sets of algebraic numbers used is reproved in Section 5 of the paper, while Section 4 treats the case card S\(=3\) in the context of the paper's considerations.