Quadratic integral solutions to double Pell equations

TL;DR

This paper investigates quadratic integral points on curves defined by Pell equations in projective three-space, providing explicit descriptions of solution families and bounds on finite solution sets, using geometric methods adapted from Vojta's theorem.

Contribution

It introduces a novel geometric approach to bound and describe quadratic integral solutions on Pell-related curves, extending Vojta's theorem to this context.

Findings

Explicit description of solution families

Finite set bounds based on field and equations

Adaptation of Vojta's theorem for Pell curves

Abstract

We study the quadratic integral points-that is, (S-)integral points defined over any extension of degree two of the base field-on a curve defined in P_3 by a system of two Pell equations. Such points belong to three families explicitly described, or belong to a finite set whose cardinality may be explicitly bounded in terms of the base field, the equations defining the curve and the set S. We exploit the peculiar geometry of the curve to adapt the proof of a theorem of Vojta, which in this case does not apply.