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Two infinite families of elliptic curves with Mordell-Weil rank at least $3$

Pankaj Patel, Debopam Chakraborty, Jaitra Chattopadhyay

TL;DR

The paper constructs two infinite families of elliptic curves over $\mathbb{Q}$, $E_{a,b}$ and $E'_{a,b}$, and proves that under mild coprimality and parity conditions their torsion is trivial while the Mordell–Weil rank is at least $3$ for infinitely many $(a,b)$. The approach combines explicit Néron–Tate height calculations for carefully chosen rational points with height-pairing arguments to establish linear independence, and leverages Pell-type unit arguments in $\mathbb{Z}[\sqrt{3}]$ (via $a^{2}-3b^{2}=1$) to generate infinite families with the desired properties. The results extend prior work by Brown–Myres, Fujita–Nara, and Hatley–Stack, and generalize several earlier rank bounds to wider parameter choices. This provides explicit, infinite families of rank at least $3$ curves and contributes to understanding how rank can be forced by canonical height methods in parametric families.

Abstract

In this paper, we consider two infinite parametric families of elliptic curves defined over $\mathbb{Q}$ given by the equations $E_{a,b} : y^{2} = x^{3} - a^{2}x + b^{2}$ and $E^{\prime}_{a,b} : y^{2} = x^{3} - a^{2}x + b^{6}$, where $a,b \in \mathbb{N}$ satisfy certain mild conditions. We prove that the torsion group of $E_{a,b}(\mathbb{Q})$ is trivial and the Mordell-Weil ranks of both $E_{a,b}(\mathbb{Q})$ and $E^{\prime}_{a,b}(\mathbb{Q})$ are at least $3$ for infinitely many choices of $a$ and $b$ by using the Néron-Tate height of a rational point and by exploiting the unit group of the ring of integers of $\mathbb{Q}(\sqrt{3})$. This is an extension of the results of Brown-Myres and Fujita-Nara where lower bounds of the ranks were provided under the assumption that $a = 1$ or $b = 1$. Also, our families of elliptic curves vastly generalize the curves recently investigated by Hatley and Stack.

Two infinite families of elliptic curves with Mordell-Weil rank at least $3$

TL;DR

The paper constructs two infinite families of elliptic curves over , and , and proves that under mild coprimality and parity conditions their torsion is trivial while the Mordell–Weil rank is at least for infinitely many . The approach combines explicit Néron–Tate height calculations for carefully chosen rational points with height-pairing arguments to establish linear independence, and leverages Pell-type unit arguments in (via ) to generate infinite families with the desired properties. The results extend prior work by Brown–Myres, Fujita–Nara, and Hatley–Stack, and generalize several earlier rank bounds to wider parameter choices. This provides explicit, infinite families of rank at least curves and contributes to understanding how rank can be forced by canonical height methods in parametric families.

Abstract

In this paper, we consider two infinite parametric families of elliptic curves defined over given by the equations and , where satisfy certain mild conditions. We prove that the torsion group of is trivial and the Mordell-Weil ranks of both and are at least for infinitely many choices of and by using the Néron-Tate height of a rational point and by exploiting the unit group of the ring of integers of . This is an extension of the results of Brown-Myres and Fujita-Nara where lower bounds of the ranks were provided under the assumption that or . Also, our families of elliptic curves vastly generalize the curves recently investigated by Hatley and Stack.
Paper Structure (3 sections, 3 theorems, 18 equations)

This paper contains 3 sections, 3 theorems, 18 equations.

Table of Contents

  1. introduction
  2. Néron-Tate height and the Proof of Theorem \ref{['1st']}
  3. Proof of Theorem \ref{['3rd']}

Key Result

Theorem 1

Let $a$ and $b$ be positive integers with $\gcd(a,b) = 1$ and $b$ is an odd integer. If $3 \mid b$ and $4 \nmid a$, then the torsion group of $E_{a,b} : y^{2} = x^{3} - a^{2}x + b^{2}$ is trivial. Moreover, there exist infinitely many pairs of positive integers $(a,b)$ with $a^{2} = 3b^{2} + 1$ such

Theorems & Definitions (5)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Proposition 2.1