Integral Hasse principle for Markoff type cubic surfaces

TL;DR

This paper provides new upper bounds on the failure rate of the integral Hasse principle for Markoff type cubic surfaces, showing it holds for almost all such surfaces in certain sequences.

Contribution

It introduces improved bounds on failures of the integral Hasse principle for a family of cubic surfaces, advancing understanding of their arithmetic properties.

Findings

Bound on failures improves previous results

Hasse principle holds for density 1 in certain sequences

Results contribute to understanding of rational points on cubic surfaces

Abstract

We establish new upper bounds on the number of failures of the integral Hasse principle within the family of Markoff type cubic surfaces $x^2+ y^2+ z^2- xyz= a$ with $|a|\leq A$ as $A\to \infty$. Our bound improves upon existing work of Ghosh and Sarnak. As a result, we demonstrate that the integral Hasse principle holds for a density $1$ of surfaces in certain sparse sequences.