Pell surfaces

DOI10.1007/S10474-019-01008-2arXiv1906.08818MaRDI QIDQ6320808

János Kollár

Publication date: 20 June 2019

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.

Mathematics Subject Classification ID

Rational and ruled surfaces (14J26) Quadratic and bilinear Diophantine equations (11D09) Global ground fields in algebraic geometry (14G25) Affine fibrations (14R25) Multiplicative and norm form equations (11D57)

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