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Rational and integral points on Markoff-type K3 surfaces

Quang-Duc Dao

TL;DR

This work analyzes rational and integral points on Markoff-type K3 (MK3) surfaces, a symmetry-rich subclass of Wehler K3 surfaces. It constructs a concrete MK3 family with Zariski-dense rational points that nevertheless experiences an integral Brauer–Manin obstruction to the integral Hasse principle, and it provides counting results for Hasse failures. The authors determine the geometric Picard group and the algebraic Brauer group for a representative MK3 family, giving explicit quaternion algebra generators and relating them to integral obstructions. They further prove rational points can be Zariski-dense, derive explicit instances of Brauer–Manin obstructions from quaternion algebras, and establish asymptotic counting for admissible and obstructed $k$-values, while discussing implications for strong approximation and broader Brauer–Picard phenomena on MK3 surfaces.

Abstract

Following recent works by E. Fuchs et al. and by the author, we study rational and integral points on Markoff-type K3 (MK3) surfaces, i.e., Wehler K3 surfaces of Markoff type. In particular, we construct a family of MK3 surfaces which have a Zariski dense set of rational points but fail the integral Hasse principle due to the Brauer-Manin obstruction and provide some counting results for this family. We also give some remarks on Brauer groups, Picard groups, and failure of strong approximation on MK3 surfaces.

Rational and integral points on Markoff-type K3 surfaces

TL;DR

This work analyzes rational and integral points on Markoff-type K3 (MK3) surfaces, a symmetry-rich subclass of Wehler K3 surfaces. It constructs a concrete MK3 family with Zariski-dense rational points that nevertheless experiences an integral Brauer–Manin obstruction to the integral Hasse principle, and it provides counting results for Hasse failures. The authors determine the geometric Picard group and the algebraic Brauer group for a representative MK3 family, giving explicit quaternion algebra generators and relating them to integral obstructions. They further prove rational points can be Zariski-dense, derive explicit instances of Brauer–Manin obstructions from quaternion algebras, and establish asymptotic counting for admissible and obstructed -values, while discussing implications for strong approximation and broader Brauer–Picard phenomena on MK3 surfaces.

Abstract

Following recent works by E. Fuchs et al. and by the author, we study rational and integral points on Markoff-type K3 (MK3) surfaces, i.e., Wehler K3 surfaces of Markoff type. In particular, we construct a family of MK3 surfaces which have a Zariski dense set of rational points but fail the integral Hasse principle due to the Brauer-Manin obstruction and provide some counting results for this family. We also give some remarks on Brauer groups, Picard groups, and failure of strong approximation on MK3 surfaces.

Paper Structure

This paper contains 19 sections, 18 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Wehler K3 surfaces
  4. Markoff-type K3 surfaces
  5. Brauer groups and the Brauer--Manin obstruction
  6. Brauer groups
  7. The Brauer--Manin obstruction
  8. Rational points on MK3 surfaces
  9. Brauer groups of Markoff-type K3 surfaces
  10. Geometry of Wehler K3 surfaces
  11. The geometric Picard group
  12. The algebraic Brauer group
  13. Brauer--Manin obstruction from quaternion algebras
  14. Existence of local integral points
  15. Integral Brauer--Manin obstruction
  16. ...and 4 more sections

Key Result

Theorem 1.1

For $k \in \mathbb{Z}$, let $W_{k} \subset \mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}$ be the MK3 surface defined over $\mathbb{Q}$ by the $(2,2,2)$-form Let $\mathcal{U}_{k}$ be the integral model of $U_{k}$ defined over $\mathbb{Z}$ by the same equation. If $k$ satisfies the condition: then $\mathcal{U}_{k}(\textbf{A}_{\mathbb{Z}}) \not= \emptyset$ and there is an algebraic Bra

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.4
  • ...and 28 more