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Failures of the Integral Hasse Principle for Affine Quadric Surfaces

Vladimir Mitankin

TL;DR

The paper investigates the frequency of counter-examples to the integral Hasse principle within the family of affine diagonal quadrics $a x^2 + b y^2 + c z^2 = n$. It leverages the integral Brauer–Manin obstruction as the main tool to detect failures of integral solubility and provides precise asymptotics for when ad\`eles are locally nonempty, and bounds for the occurrence of Brauer–Manin obstructions. The principal contributions are (i) an explicit product formula for the density of locally soluble surfaces, with $\\sigma_{\\infty}=1$ and computable $\\sigma_p$ (Theorem Th0); (ii) a sharp upper bound $N_{\\mathrm{Br}}(B) \\ leq_n C B^{3/2}\\log B^3$ showing that Brauer–Manin obstructions are rare (Theorem Th1); and (iii) a nontrivial lower bound in a structured subfamily, giving $N'_{\\mathrm{Br}}(B) \\ approx C B^{3/2}/\\sqrt{\\log B}$ and hence that the upper bound is close to optimal (Theorem Th2). Collectively, the results quantify how often the integral Hasse principle fails in this natural family and highlight the dominant role of local densities and the Brauer–Manin obstruction in the integral setting.

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.

Failures of the Integral Hasse Principle for Affine Quadric Surfaces

TL;DR

The paper investigates the frequency of counter-examples to the integral Hasse principle within the family of affine diagonal quadrics . It leverages the integral Brauer–Manin obstruction as the main tool to detect failures of integral solubility and provides precise asymptotics for when ad\`eles are locally nonempty, and bounds for the occurrence of Brauer–Manin obstructions. The principal contributions are (i) an explicit product formula for the density of locally soluble surfaces, with and computable (Theorem Th0); (ii) a sharp upper bound showing that Brauer–Manin obstructions are rare (Theorem Th1); and (iii) a nontrivial lower bound in a structured subfamily, giving and hence that the upper bound is close to optimal (Theorem Th2). Collectively, the results quantify how often the integral Hasse principle fails in this natural family and highlight the dominant role of local densities and the Brauer–Manin obstruction in the integral setting.

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.

Paper Structure

This paper contains 5 sections, 8 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The integral Brauer--Manin obstruction
  3. Proof of Theorem \ref{['Th0']}
  4. Proof of Theorem \ref{['Th1']}
  5. Proof of Theorem \ref{['Th2']}

Key Result

Theorem 1.1

For any non-zero $n \in \mathbb{Z}$ the limit $\lim_{B \to \infty} B^{-3} N_{\mathop{\mathrm{loc}}\nolimits}(B)$ exists, it is non-zero and it is given as a product of local densities where $\sigma_{\infty}$ and $\sigma_p$ are defined in Omega and localDensDef. Furthermore, $\sigma_{\infty} = 1$ and for any prime $p \nmid n$ we have

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Lemma 5.1
  • proof
  • ...and 2 more