Quadratic Integral Solutions to Double Pell Equations

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 ${\mathbb{P}}_3$ 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.