Failures of the Integral Hasse Principle for Affine Quadric Surfaces

Abstract

Quadric hypersurfaces are well-known to satisfy the Hasse principle. However, this is no longer true in the case of the Hasse principle for integral points, where counter-examples are known to exist in dimension 1 and 2. This work explores the frequency that such counter-examples arise in a family of affine quadric surfaces defined over the integers.

20 pages

References in corpus (2)

• Failures of weak approximation in families
• What is the probability that a random integral quadratic form in $n$ variables has an integral zero?

Cited by in corpus (4)

• A positive proportion of Thue equations fail the integral Hasse principle
• Semi-integral Brauer-Manin obstruction and quadric orbifold pairs
• Integral points on affine quadric surfaces
• Local-global principles for semi-integral points on Markoff orbifold pairs