Computation of highly ramified coverings

TL;DR

This paper explicitly constructs three almost Belyi coverings of degrees 11, 12, and 20, and demonstrates their use in computing algebraic solutions to the sixth Painleve equation.

Contribution

It provides explicit examples of almost Belyi coverings of specific degrees and applies them to solve algebraic cases of the sixth Painleve equation.

Findings

Explicit constructions of three almost Belyi coverings

Application to algebraic solutions of Painleve VI

Illustration of the use of coverings in complex differential equations

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.