2019/06/20 by Kollár, János
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1906.08818
In 1826 Abel started the study of the polynomial Pell equation x2-g(u)y2=1. Its solvability in polynomials x(u), y(u) depends on a certain torsion point on the Jacobian of the hyperelliptic curve v2=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.