Brauer-Manin obstruction for integral points on Markoff-type cubic surfaces

TL;DR

This paper investigates the Brauer-Manin obstruction on Markoff-type cubic surfaces, providing counterexamples to strong approximation and the integral Hasse principle, with new counting results and explanations.

Contribution

It constructs new counterexamples to strong approximation explained by the Brauer-Manin obstruction and identifies counterexamples to the integral Hasse principle not explained by it.

Findings

Counterexamples to strong approximation explained by Brauer-Manin obstruction.

Counterexamples to the integral Hasse principle not explained by Brauer-Manin obstruction.

Counting results related to integral points on Markoff-type cubic surfaces.

Abstract

Following [GS22], [LM20] and [CWX20], we study the Brauer-Manin obstruction for integral points on similar Markoff-type cubic surfaces. In particular, we construct a family of counterexamples to strong approximation which can be explained by the Brauer-Manin obstruction with some counting results of similar nature to those in [LM20] and [CWX20]. We also give some counterexamples to the integral Hasse principle which cannot be explained by the (algebraic) Brauer-Manin obstruction.