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Almost all quadratic twists of an elliptic curve have no integral points

Tim Browning, Stephanie Chan

TL;DR

The paper proves that almost all quadratic twists $E_D$ of a fixed elliptic curve $E: y^2=x^3+Ax+B$ have no integral points, when $D$ runs over square-free integers, with a conditional upper bound in the presence of partial $2$-torsion (weak Hall–Lang conjecture) and an unconditional bound when no rational $2$-torsion exists. The authors develop a comprehensive strategy combining Mordell’s construction of binary quartic forms from integral points with Cremona’s reduction theory to translate the problem into counting rational points on a singular cubic surface, and they control the counts via detailed Heath–Brown type character-sum estimates. They also derive higher-moments bounds for the number of integral points across the twists, using Smith’s Selmer-rank results and Szpiro ratio arguments to bound exponential ranks and Selmer-related quantities. The work handles both full and partial $2$-torsion cases, providing a robust framework that links geometric reduction, arithmetic of quartic forms, and analytic number theory to establish density-zero results with quantitative bounds, and thereby informs the distribution of integral points and Selmer elements in quadratic twist families.

Abstract

For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown's analysis of Selmer group statistics for the congruent number curve.

Almost all quadratic twists of an elliptic curve have no integral points

TL;DR

The paper proves that almost all quadratic twists of a fixed elliptic curve have no integral points, when runs over square-free integers, with a conditional upper bound in the presence of partial -torsion (weak Hall–Lang conjecture) and an unconditional bound when no rational -torsion exists. The authors develop a comprehensive strategy combining Mordell’s construction of binary quartic forms from integral points with Cremona’s reduction theory to translate the problem into counting rational points on a singular cubic surface, and they control the counts via detailed Heath–Brown type character-sum estimates. They also derive higher-moments bounds for the number of integral points across the twists, using Smith’s Selmer-rank results and Szpiro ratio arguments to bound exponential ranks and Selmer-related quantities. The work handles both full and partial -torsion cases, providing a robust framework that links geometric reduction, arithmetic of quartic forms, and analytic number theory to establish density-zero results with quantitative bounds, and thereby informs the distribution of integral points and Selmer elements in quadratic twist families.

Abstract

For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown's analysis of Selmer group statistics for the congruent number curve.
Paper Structure (30 sections, 43 theorems, 275 equations)

This paper contains 30 sections, 43 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Main results and moments
  3. Counting points on a cubic surface
  4. Preliminaries
  5. Proof of the upper bound in Theorem \ref{['theorem:main']}
  6. Proof of the lower bound in Theorem \ref{['theorem:main']}
  7. Integral points on quadratic twists with large $\gcd(x,D)$
  8. Integral points on quadratic twists with small $\gcd(x,D)$
  9. Preliminaries
  10. Quartic forms associated to integral points
  11. Bounding the number of equivalence classes of quartic forms
  12. Completing the proof of Theorem \ref{['theorem:maintwists']}
  13. Character sum input
  14. Preliminaries
  15. Dissection
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $\varepsilon>0$. Let $A,B\in \mathbb{Z}$ such that $4A^3+27B^2\neq 0$, and let $E_D$ be given by eq:jive. Assume that Conjecture con holds. Then where the implied constants depend at most on $A, B$ and $\varepsilon$. Moreover, Conjecture con is only required in the proof of the upper bound, and only then when $x^3+Ax+B$ has precisely one root over $\mathbb{Q}$.

Theorems & Definitions (76)

  • Conjecture 1.1: weak Hall--Lang
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5: Smith
  • Theorem 2.6
  • ...and 66 more