On the uniformization of algebraic curves

Details On the uniformization of algebraic curves On the uniformization of algebraic curves

2002-05-24

arxiv.org/abs/math/0205261v6

Moscow Math. Journ. (2008), v.8(2), 233-271

Mathematics

Classical Analysis and ODEs

Final version; 36 pages, 4 figures

Scientific paper

Based 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.

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