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Cubic surfaces failing the integral Hasse principle

Julian Lyczak, Vladimir Mitankin, H. Uppal

TL;DR

This work proves the integral Hasse principle can fail for affine diagonal cubic surfaces by computing the integral Brauer--Manin obstruction for $U$ and constructing two infinite families of counterexamples. It provides a complete description of the Brauer group $\\mathrm{Br}\,U$ in terms of $\\mathrm{Br}\,X$ and a potential $\\mathbb{Z}/2\\mathbb{Z}$-factor, with explicit generators in special subfamilies. The authors also develop a general framework for computing invariant maps via Hilbert symbols, and they quantify how often integral Hasse failures and integral strong approximation off $\\infty$ occur across natural counting problems, obtaining matching upper and lower bounds up to polylogarithmic factors. Additionally, they present concrete examples where the obstruction occurs without requiring explicit Brauer representatives, highlighting the practical utility of the integral Brauer--Manin approach for log K3-type surfaces. Overall, the paper advances both the theoretical understanding of integral obstructions on diagonal affine cubic surfaces and the quantitative study of their distribution in families.

Abstract

We study the integral Brauer--Manin obstruction for affine diagonal cubic surfaces, which we employ to construct the first counterexamples to the integral Hasse principle in this setting. We then count in three natural ways how such counterexamples are distributed across the family of affine diagonal cubic surfaces and how often such surfaces satisfy integral strong approximation off $\infty$.

Cubic surfaces failing the integral Hasse principle

TL;DR

This work proves the integral Hasse principle can fail for affine diagonal cubic surfaces by computing the integral Brauer--Manin obstruction for and constructing two infinite families of counterexamples. It provides a complete description of the Brauer group in terms of and a potential -factor, with explicit generators in special subfamilies. The authors also develop a general framework for computing invariant maps via Hilbert symbols, and they quantify how often integral Hasse failures and integral strong approximation off occur across natural counting problems, obtaining matching upper and lower bounds up to polylogarithmic factors. Additionally, they present concrete examples where the obstruction occurs without requiring explicit Brauer representatives, highlighting the practical utility of the integral Brauer--Manin approach for log K3-type surfaces. Overall, the paper advances both the theoretical understanding of integral obstructions on diagonal affine cubic surfaces and the quantitative study of their distribution in families.

Abstract

We study the integral Brauer--Manin obstruction for affine diagonal cubic surfaces, which we employ to construct the first counterexamples to the integral Hasse principle in this setting. We then count in three natural ways how such counterexamples are distributed across the family of affine diagonal cubic surfaces and how often such surfaces satisfy integral strong approximation off .
Paper Structure (20 sections, 37 theorems, 133 equations, 2 tables)

This paper contains 20 sections, 37 theorems, 133 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Integral Brauer--Manin obstruction
  3. Brauer groups of diagonal affine cubic surfaces
  4. Algebraic Brauer group
  5. Transcendental Brauer group
  6. Jacobian of diagonal cubic curves
  7. Proof of Theorem \ref{['thm: Brauer group thm']}
  8. Explicit generators
  9. Hilbert symbols
  10. Construction of the Hilbert symbol
  11. Conditions for a Brauer--Manin obstruction
  12. Computing invariant maps for generic families
  13. A three coefficient family
  14. Lower bound
  15. Establishing the lower bound
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume that $a_0, a_1, a_2, a_3 \in \mathbb{Q}^{\ast}$. Then the algebraic part of $\mathop{\mathrm{Br}}\nolimits U$ is isomorphic to $\mathop{\mathrm{Br}}\nolimits X$. Moreover,

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Brauer--Grothendieck group
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 3.1
  • ...and 75 more