Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Brauer-Manin obstruction for Wehler K3 surfaces of Markoff type

Quang-Duc Dao

TL;DR

This work extends the study of Brauer–Manin obstructions to integral points from Markoff-type cubic surfaces to Wehler K3 surfaces of Markoff type (MK3), by computing the geometric Picard group and the algebraic Brauer group for explicit MK3 families and constructing explicit quaternion algebras generating the obstruction. The authors prove integral Hasse principle failures explained by the obstruction in three concrete MK3 families, and provide counting results showing a nontrivial lower bound, $\gg M^{1/2}/\log M$, for the number of such obstructions as $|k|\le M$. They also analyze the obstruction’s impact on strong approximation and examine rational points on affine MK3 surfaces, revealing both obstructions and examples with rational points despite the lack of integral points. Overall, the paper advances understanding of integral points on higher-dimensional arithmetic surfaces by connecting Picard, Brauer, and adelic methods in a K3 setting with Markoff-type symmetry, and it highlights the nuanced arithmetic between integral, rational, and local solubility properties.

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-Thélène-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.

Brauer-Manin obstruction for Wehler K3 surfaces of Markoff type

TL;DR

This work extends the study of Brauer–Manin obstructions to integral points from Markoff-type cubic surfaces to Wehler K3 surfaces of Markoff type (MK3), by computing the geometric Picard group and the algebraic Brauer group for explicit MK3 families and constructing explicit quaternion algebras generating the obstruction. The authors prove integral Hasse principle failures explained by the obstruction in three concrete MK3 families, and provide counting results showing a nontrivial lower bound, , for the number of such obstructions as . They also analyze the obstruction’s impact on strong approximation and examine rational points on affine MK3 surfaces, revealing both obstructions and examples with rational points despite the lack of integral points. Overall, the paper advances understanding of integral points on higher-dimensional arithmetic surfaces by connecting Picard, Brauer, and adelic methods in a K3 setting with Markoff-type symmetry, and it highlights the nuanced arithmetic between integral, rational, and local solubility properties.

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-Thélène-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.
Paper Structure (18 sections, 28 theorems, 133 equations)

This paper contains 18 sections, 28 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Wehler K3 surfaces
  4. Markoff-type K3 surfaces
  5. The Brauer group of Markoff-type K3 surfaces
  6. Geometry of K3 surfaces
  7. The geometric Picard group
  8. The algebraic Brauer group
  9. The Brauer--Manin obstruction
  10. Review of the Brauer--Manin obstruction
  11. Brauer--Manin obstruction from quaternion algebras
  12. Existence of local points
  13. Integral Brauer--Manin obstructions
  14. Counting the Hasse failures
  15. Further remarks
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $\mathcal{U}$ be the affine scheme over $\mathbb{Z}$ defined by where such that its natural compactification $X$ is smooth over $\mathbb{Q}$. Then we have as $M \rightarrow +\infty$.

Theorems & Definitions (61)

  • Conjecture 1.1: Conjecture 10.2 in GS22
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.4
  • ...and 51 more