The sixth Painleve transcendent and uniformization of algebraic curves

TL;DR

This paper explores the deep connection between the sixth Painleve equation and the uniformization of algebraic curves, revealing new relationships and methods in complex analysis and algebraic geometry.

Contribution

It establishes a novel link between Painleve VI and explicit uniformization of algebraic curves, integrating various mathematical tools and uncovering new relations among classical curves and equations.

Findings

Connection between Painleve VI and algebraic curve uniformization

Relations between Picard-Hitchin's curves, hyperelliptic curves, and tori

Link to Apery's proof of zeta(3) irrationality

Abstract

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).