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首页> 外文期刊>Annales Henri Poincare >Algebro-Geometric Approach to an Okamoto Transformation, the Painleve VI and Schlesinger Equations
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Algebro-Geometric Approach to an Okamoto Transformation, the Painleve VI and Schlesinger Equations

机译：代数 - 几何方法，奥马托转换，痛苦vi和schlesinger方程

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摘要

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, 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 Painleve VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve 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 Painleve VI equation with the Poncelet polygons.
机译：提出了一种构建等级秩两点曲折的秩的代数 - 几何解决方案的新方法。对于表示为CP1的分布的双覆盖的椭圆曲线，用以下性能构造纯差分：其两个零在覆盖物的基部上的共同投影，被认为是覆盖物的唯一移动分支点的函数，是痛苦vi方程的解决方案。这种差异提供了痛苦VI方程的经典Okamoto转换的不变制剂。与椭圆曲线系列相关的等级的相应解决方案是根据这种差异构建的。在分支点变化的情况下，用于构造亚圈子差分的初始数据包括曲线的曲线中的曲线中的一点，同时分支点变化。看来，该点的坐标是理性的情况对应于在一对锥形中铭刻和围绕的锁定多边形。因此，这是Hitchin研究的情况的概括，Who与Poncelet多边形有关止痛VI方程的代数解。

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