Brauer groups of generalised Markoff surfaces

Abstract

Let $k$ be a field of characteristic zero with $a,b,c\in k^\times$ and let $m\in k\setminus\{0,4abc\}$. We compute the Brauer group of the generalised Markoff surface $ax^2+by^2+cz^2-xyz=m$. We also construct several counterexamples to the integral Hasse principle, which are explained by the Brauer--Manin obstruction.