In 1826 Abel started the study of the polynomial Pell equation x 2 − g ( u ) y 2 = 1. Its solvability in polynomials x ( u ), y ( u ) depends on a certain torsion point on the Jacobian of the hyperelliptic curve v 2 = g ( u ). In this paper we study the affine surfaces defined by the Pell equations in 3-space with coordinates x, y, u , and aim to describe all affine lines on it. These are polynomial solutions of the equation x ( t ) 2 − g ( u ( t )) y ( t ) 2 = 1. Our results are rather complete when the degree of g is even but the odd degree cases are left completely open. For even degrees we also describe all curves on these Pell surfaces that have only 1 place at infinity.

中文翻译：

---

佩尔表面

1826 年，Abel 开始研究多项式 Pell 方程 x 2 − g ( u ) y 2 = 1。它在多项式 x ( u ), y ( u ) 中的可解性取决于超椭圆曲线 v 的雅可比矩阵上的某个扭点2 = 克（你）。在本文中，我们研究了由 Pell 方程在坐标为 x、y、u 的 3 空间中定义的仿射曲面，并旨在描述其上的所有仿射线。这些是方程 x ( t ) 2 − g ( u ( t )) y ( t ) 2 = 1 的多项式解。当 g 的次数为偶数时，我们的结果相当完整，但奇数次的情况完全开放。对于偶数度，我们还描述了这些 Pell 曲面上在无穷远处只有 1 个位置的所有曲线。