Brauer-Manin obstruction for Markoff surfaces

Abstract

Ghosh and Sarnak have studied integral points on surfaces defined by an equation x^2+y^2+z^2-xyz= m over the integers. For these affine surfaces, we systematically study the Brauer group and the Brauer-Manin obstruction to the integral Hasse principle. We prove that strong approximation for integral points on any such surface, away from any finite set of places, fails, and that, for m\neq 0, 4, the Brauer group does not control strong approximation.

This is the final version, to appear in Annali della Scuola Normale di Pisa, classe di Scienze, issue 3 of the XXI/2021 volume, series V, due to be published at the end of September 2021