Integral points on Markoff type cubic surfaces

Abstract

For integers $k$, we consider the affine cubic surface $V_k$ given by $M(\mathbf{x})=x_1^2+x_2^2+x_3^2-x_1{x}_2{x}_3=k$. We show that for almost all $k$ the Hasse Principle holds, namely that $V_k(\mathbb{Z})$ is non-empty if $V_k({\mathbb{Z}}_p)$ is non-empty for all primes $p$, and that there are infinitely many $k$'s for which it fails. The Markoff morphisms act on $V_k(\mathbb{Z})$ with finitely many orbits and a numerical study points to some basic conjectures about these "class numbers" and Hasse failures. Some of the analysis may be extended to less special affine cubic surfaces.

This is the final version of this paper. The published version of this paper contains an abridged Sec. 10 on computations. We make the full version accessible here. There are various updates, in particular references to the papers [CTWX20], [LM20] and [GMS22] on failures to the Hasse Principle. The published version has some additional differences

References in corpus (3)

• Markoff Surfaces and Strong Approximation: 1
• Integral Hasse principle and strong approximation for Markoff surfaces
• Commutators in ${\mathrm{SL}}_2$ and Markoff surfaces I

Cited by in corpus (5)

• Integral Hasse principle and strong approximation for Markoff surfaces
• Commutators in ${\mathrm{SL}}_2$ and Markoff surfaces I
• Brauer-Manin obstruction for integral points on Markoff-type cubic surfaces
• Brauer-Manin obstruction for Wehler K3 surfaces of Markoff type
• Local-global principles for semi-integral points on Markoff orbifold pairs