On Uniformization of Algebraic Curves

Abstract

Abstract. Based on Burnside’s parametrization of the algebraic curve y 2 = x 5 − x we obtain remaining attributes of its uniformization: associated Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons, etc. As a generalization, we propose a way of uniformization of arbitrary curves by zero genus groups. In the hyperelliptic case all the objects of the theory are explicitly described. We consider a large number of examples and, briefly, applications: Abelian integrals, metrics of Poincaré, differential equations of the Jacobi–Chazy and Picard–Fuchs type, and others. 1.