Pell surfaces

Abstract

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.