Brauer-Manin obstruction for Wehler K3 surfaces of Markoff type

TL;DR

This paper investigates the Brauer-Manin obstruction on Wehler K3 surfaces of Markoff type, demonstrating how it explains failures of the integral Hasse principle and strong approximation, with new constructions and counting results.

Contribution

It constructs families of Wehler K3 surfaces where the Brauer-Manin obstruction accounts for the failure of the integral Hasse principle and provides new counterexamples to strong approximation.

Findings

Families of surfaces failing the integral Hasse principle explained by Brauer-Manin obstruction

Counterexamples to strong approximation with explanations via Brauer-Manin obstruction

Counting results related to integral points on these surfaces

Abstract

Following recent work by E. Fuchs et al., we study the Brauer-Manin obstruction for integral points on Wehler K3 surfaces of Markoff type. In particular, we construct some families which fail the integral Hasse principle via the Brauer-Manin obstruction with some counting results of similar nature to those in previous works by Ghosh-Sarnak, Loughran-Mitankin, Colliot-Th\'el\`ene-Wei-Xu. We also give some counterexamples to strong approximation (where integral points can exist) which can be explained by the Brauer-Manin obstruction, and study a few aspects of rational points on affine surfaces.