The sixth Painleve transcendent and uniformization of algebraic curves

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:260

Article

Brezhnev, Yurii V.¹ 

[1] Tomsk State Univ, Dept Quantum Field Theory, Tomsk 634050, Russia

关键词: Painleve-6 equation;    Picard-Hitchin solutions;    Algebraic curves;    Theta-functions;    Automorphic functions;    Fuchsian equations;   

DOI  :  10.1016/j.jde.2015.10.009

来源: Elsevier

【 摘 要 】

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). (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free