The sixth Painleve transcendent and uniformization of algebraic curves

Details The sixth Painleve transcendent and uniformization of algebraic curves The sixth Painleve transcendent and uniformization of algebraic curves

2010-11-07

arxiv.org/abs/1011.1645v2

Mathematics

Classical Analysis and ODEs

(Major changes, completely reconstructed and rewritten); English, 47 pages, 1 table, no figures, LaTeX

Scientific paper

We exhibit a remarkable connection between sixth equation of Painleve 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 Apery used to prove the irrationality of Riemann's zeta(3).

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