# Integral points on Markoff type cubic surfaces

## TL;DR

This paper investigates the distribution of integer solutions on Markoff type cubic surfaces, showing that the Hasse Principle generally holds but fails infinitely often, and explores the structure of solutions under Markoff morphisms.

## Contribution

It proves the Hasse Principle holds for almost all parameters and identifies the existence of infinitely many exceptions, also analyzing the orbit structure of solutions.

## Key findings

- Hasse Principle holds for almost all k
- Infinitely many k where Hasse Principle fails
- Finite orbits under Markoff morphisms

## Abstract

For integers $k$, we consider the affine cubic surface $V_{k}$ given by $M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. 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.