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Number of integral points on quadratic twists of elliptic curves

Seokhyun Choi

Abstract

We consider the integral points on the quadratic twists $E_D : y^2 = x^3+D^2Ax+D^3B$ of the elliptic curve $E : y^2 = x^3+Ax+B$ over $\mathbb{Q}$. For sufficiently large values of $D$, we prove that the number of integral points on $E_D$ admits the upper bound $\ll 4^r$, where $r$ denotes the Mordell-Weil rank of $E_D$.

Number of integral points on quadratic twists of elliptic curves

Abstract

We consider the integral points on the quadratic twists of the elliptic curve over . For sufficiently large values of , we prove that the number of integral points on admits the upper bound , where denotes the Mordell-Weil rank of .

Paper Structure

This paper contains 12 sections, 28 theorems, 212 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminary lemmas
  4. Spherical codes
  5. Bound for small points
  6. Bound for medium-small points
  7. Bound for medium-large points
  8. Diophantine approximations
  9. Quantitative Roth's theorem
  10. Bound for large points
  11. Calculations for Lemma \ref{['P+Q_positive_bound']} and Lemma \ref{['P+Q_negative_bound']}
  12. Table for Proposition \ref{['MS_bound']}

Key Result

Theorem 1.1

Let $E/\mathbb{Q}$ be an elliptic curve defined by the Weierstrass equation For any squarefree positive integer $D$, let $E_D$ be the quadratic twist of $E$ by $D$, defined by the Weierstrass equation Then for sufficiently large $\lvert D \rvert$ (depending on $A,B$), where $r$ is the rank of $E_D/\mathbb{Q}$.

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 48 more