Article ID Journal Published Year Pages File Type
4609624 Journal of Differential Equations 2016 50 Pages PDF
Abstract

We exhibit a remarkable connection between sixth equation of Painlevé list and infinite families of explicitly uniformizable algebraic curves. Fuchsian equations, congruences for group transformations, differential calculus of functions and differentials on corresponding Riemann surfaces, Abelian integrals, analytic connections (generalizations of Chazy's equations), and other attributes of uniformization can be obtained for these curves. As byproducts of the theory, we establish relations between Picard–Hitchin's curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous differential equation which Apéry used to prove the irrationality of Riemann's ζ(3)ζ(3).

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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