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Integral Hasse principle and strong approximation for Markoff surfaces

Daniel Loughran, Vladimir Mitankin

Abstract

We study the failure of the integral Hasse principle and strong approximation for Markoff surfaces, as studied by Ghosh and Sarnak, using the Brauer-Manin obstruction.

Integral Hasse principle and strong approximation for Markoff surfaces

Abstract

We study the failure of the integral Hasse principle and strong approximation for Markoff surfaces, as studied by Ghosh and Sarnak, using the Brauer-Manin obstruction.

Paper Structure

This paper contains 22 sections, 35 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Geometry of affine cubic surfaces
  3. Basic geometry
  4. Cubic surfaces with $3$ coplanar lines
  5. The conic bundle
  6. Brauer group
  7. Geometry of projective Markoff surfaces
  8. Brauer group of affine Markoff surfaces
  9. Transcendental Brauer group
  10. Algebraic Brauer group
  11. The Brauer--Manin obstruction
  12. Set-up
  13. Assumptions of Corollary \ref{['cor:generic']}
  14. Review of the Brauer--Manin obstruction
  15. Calculating the local invariants
  16. ...and 7 more sections

Key Result

Theorem 1.1

We have

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Gysin sequence
  • ...and 61 more