Integral points on affine quadric surfaces

TL;DR

This paper studies the frequency of Brauer-Manin obstructions to integral points on certain affine quadric surfaces, specifically analyzing the family defined by $ax^2 + by^2 + cz^2 = n$, and improves previous bounds.

Contribution

It advances understanding of when Brauer-Manin obstructions occur for these surfaces and refines bounds established in prior research.

Findings

Identifies conditions under which Brauer-Manin obstruction explains failures of the Hasse principle.

Provides improved bounds on the frequency of Brauer-Manin obstructions.

Enhances theoretical understanding of integral points on affine quadrics.

Abstract

It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely explained by the Brauer-Manin obstruction. We investigate how often the family of quadric hypersurfaces $ax^2 + by^2 +cz^2 = n$ has a Brauer-Manin obstruction. We improve previous bounds of Mitankin.