Computation of highly ramified coverings

Abstract

An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painleve equation.

26 pages

References in corpus (7)

• Transformations ${RS}_4^2(3)$ of the Ranks $\le 4$ and Algebraic Solutions of the Sixth Painlevé Equation
• Transformations of some Gauss hypergeometric functions
• Some explicit solutions to the Riemann-Hilbert problem
• Special Functions of the Isomonodromy Type, Rational Transformations of Spectral Parameter, and Algebraic Solutions of the Sixth Painlevé Equation
• Some Examples of RS^2_3(3)-Transformations of Ranks 5 and 6 as the Higher Order Transformations for the Hypergeometric Function
• Galois Action on Diameter Four Trees
• Computation of RS-pullback transformations for algebraic Painleve VI solutions

Cited by in corpus (8)

• Calabi-Yau Varieties: from Quiver Representations to Dessins d'Enfants
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• Algorithms and differential relations for Belyi functions
• Composite Genus One Belyi Maps
• Classification of algebraic solutions of irregular Garnier systems
• Schlesinger transformations for algebraic Painleve VI solutions
• Computation of RS-pullback transformations for algebraic Painleve VI solutions
• Genus One Belyi Maps by Quadratic Correspondences