2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/168641Based on Burnside's parametrization of the algebraic curve $y^2=x^5-x$ we provide remaining attributes of its uniformization: Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons, etc. As a generalization, we construct the zero genus uniformization of arbitrary curves. For hyperelliptic curves all the objects of the theory are explicitly described. We consider a large number of examples and, briefly, applications: Abelian integrals, metrics of Poincare, differential equations of the Jacobi--Chazy and Picard--Fuchs type, and others.Final version; 36 pages, 4 figuresClassical Analysis and ODEsHigh Energy Physics - TheoryMathematical Physics30F10; 30F35text