The sixth Painlevé transcendent and uniformization of algebraic curves

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).