Pell surfaces

TL;DR

This paper investigates affine lines on Pell surfaces defined by polynomial Pell equations, providing a comprehensive description for even-degree cases and exploring properties of certain curves, while leaving odd-degree cases open.

Contribution

It offers a near-complete classification of affine lines on Pell surfaces for even degrees and analyzes curves with one place at infinity, advancing understanding of polynomial solutions.

Findings

Complete description of affine lines for even-degree Pell surfaces

Identification of curves with only 1 place at infinity on these surfaces

Odd-degree cases remain unresolved

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.