论文标题
使用Kajiwara-noumi-Yamada的椭圆曲线代表求解Quispel-Roberts-Thompson地图
Solving the Quispel-Roberts-Thompson maps using Kajiwara-Noumi-Yamada's representation of elliptic curves
论文作者
论文摘要
众所周知，由Quispel-Roberts-Thompson地图（QRT地图）确定的动力系统将二级多项式曲线的铅笔保留在$ {\ Mathbb {cp}}}^1 \ times {\ times {\ times {\ mathbb {cp}}}^1 $上。在大多数情况下，该铅笔是椭圆形的，即其通用成员是一属的平滑代数曲线，并且可以将系统作为椭圆纤维上最初点所属的椭圆纤维的翻译求解。但是，该过程非常复杂，尤其是在标准化过程中。在本文中，对于不变椭圆曲线上的给定初始点，我们提出了一种使用Kajiwara-noumi-Yamada的椭圆形曲线的参数表示，直接根据Weierstrass Sigma函数构建解决方案的方法。
It is well known that the dynamical system determined by a Quispel-Roberts-Thompson map (a QRT map) preserves a pencil of biquadratic polynomial curves on ${\mathbb{CP}}^1 \times {\mathbb{CP}}^1$. In most cases this pencil is elliptic, i.e. its generic member is a smooth algebraic curve of genus one, and the system can be solved as a translation on the elliptic fiber to which the initial point belongs. However, this procedure is rather complicated to handle, especially in the normalization process. In this paper, for a given initial point on an invariant elliptic curve, we present a method to construct the solution directly in terms of the Weierstrass sigma function, using Kajiwara-Noumi-Yamada's parametric representation of elliptic curves.
