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Quantitative strong approximation for quaternary quadratic forms

Zhizhong Huang, Damaris Schindler, Alec Shute

TL;DR

The paper develops a quantitative framework for quaternary and non-split quadric forms, turning Heath–Brown’s delta-method into a tool for precise equidistribution of rational points on smooth non-split quadrics and for the explicit growth of integral points on the punctured affine cone. By combining real and p-adic (congruence) conditions with a refined double Kloosterman analysis, the authors derive sharp asymptotics with optimal real-place error terms and reveal how Brauer–Manin obstructions enter the circle-method through explicit leading constants tied to Tamagawa measures and Peyre’s constants. A key novelty is a quantitative encoding of Brauer–Manin obstructions within circle-method counts (via a 𝔎-term and density Ξ_{W_i}), and a demonstration that the resulting asymptotics satisfy a relative Hardy–Littlewood principle for the punctured affine cone. The work thus unifies delta-methods, oscillatory integrals, and arithmetic geometry (Tamagawa numbers, Artin L-functions, and Brauer groups) to yield precise counts for rational and integral points on quadrics with non-split discriminants, including new instances where Brauer–Manin obstructions are detected nontrivially by the circle method. These results have implications for strong approximation, arithmetic purity, and the explicit evaluation of leading constants predicted by Peyre.

Abstract

The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real place. On the other hand, we also study the growth of integral points on the three-dimensional punctured affine cone, as a quantitative version of strong approximation with Brauer--Manin obstruction for this quasi-affine variety.

Quantitative strong approximation for quaternary quadratic forms

TL;DR

The paper develops a quantitative framework for quaternary and non-split quadric forms, turning Heath–Brown’s delta-method into a tool for precise equidistribution of rational points on smooth non-split quadrics and for the explicit growth of integral points on the punctured affine cone. By combining real and p-adic (congruence) conditions with a refined double Kloosterman analysis, the authors derive sharp asymptotics with optimal real-place error terms and reveal how Brauer–Manin obstructions enter the circle-method through explicit leading constants tied to Tamagawa measures and Peyre’s constants. A key novelty is a quantitative encoding of Brauer–Manin obstructions within circle-method counts (via a 𝔎-term and density Ξ_{W_i}), and a demonstration that the resulting asymptotics satisfy a relative Hardy–Littlewood principle for the punctured affine cone. The work thus unifies delta-methods, oscillatory integrals, and arithmetic geometry (Tamagawa numbers, Artin L-functions, and Brauer groups) to yield precise counts for rational and integral points on quadrics with non-split discriminants, including new instances where Brauer–Manin obstructions are detected nontrivially by the circle method. These results have implications for strong approximation, arithmetic purity, and the explicit evaluation of leading constants predicted by Peyre.

Abstract

The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real place. On the other hand, we also study the growth of integral points on the three-dimensional punctured affine cone, as a quantitative version of strong approximation with Brauer--Manin obstruction for this quasi-affine variety.

Paper Structure

This paper contains 23 sections, 25 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Effective equidistribution on the non-split projective quadratic surface
  3. Counting integral points on the punctured affine cone
  4. Methods and further comments
  5. Notation and conventions
  6. Set-up
  7. The congruence condition and Poisson summation
  8. The real condition and the oscillatory integrals
  9. The quadratic exponential sums
  10. The contribution from $\boldsymbol{c}=\boldsymbol{0}$
  11. The Tamagawa number $\mathfrak{S}_{L,\boldsymbol{\Lambda}}({\mathscr {V}})$
  12. Counting points with real conditions
  13. More on $S_q(\boldsymbol{c})$
  14. Proof of Theorem \ref{['thm:realzoom']}
  15. Proof of Theorem \ref{['thm:realzoommain']}
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume eq:symmetric, $\Delta_F\neq\square$, and Assume that $V$ has local points at all places of $\mathbb{Q}$. Then where ${\mathscr {I}}(w_R)$ is the "weighted singular integral" (see eq:weightedsingint for its definition) and $\mathfrak{S}(V)$ is the finite Tamagawa measure of ${\mathscr {V}}(\widehat{{\mathbb {Z}}})$ (à la Peyre Peyre, with respect to the naive metric).

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1: H-Bdelta Lemma 16
  • Lemma 2.2: H-Bdelta Lemma 18
  • Lemma 2.3: H-Bdelta Lemma 22
  • proof : Sketch of Proof
  • Proposition 2.4
  • proof
  • ...and 41 more