Algebro-Geometric Approach to an Okamoto Transformation, the Painlevé VI and Schlesinger Equations

A new method of constructing algebro-geometric solutions of rank two four-point Schlesinger system is presented. For an elliptic curve represented as a ramified double covering of CP1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {CP}^1$$\end{document}, a meromorphic differential is constructed with the following property: The common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of a Painlevé VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painlevé VI equations. The corresponding solution of the rank two Schlesinger system associated with a family of elliptic curves is constructed in terms of this differential. The initial data for construction of the meromorphic differential include a point in the Jacobian of the curve, under the assumption that this point has non-variable coordinates with respect to the lattice of the Jacobian while the branch points vary. It appears that the cases where the coordinates of the point are rational correspond to the Poncelet polygons inscribed and circumscribed in a pair of conics. Thus, this is a generalization of a situation studied by Hitchin, who related algebraic solutions of a Painlevé VI equation with the Poncelet polygons.