Two infinite families of elliptic curves with Mordell-Weil rank at least $3$
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| Format: | Preprint |
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2026
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| _version_ | 1866914251772264448 |
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| author | Patel, Pankaj Chakraborty, Debopam Chattopadhyay, Jaitra |
| author_facet | Patel, Pankaj Chakraborty, Debopam Chattopadhyay, Jaitra |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_08570 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Two infinite families of elliptic curves with Mordell-Weil rank at least $3$ Patel, Pankaj Chakraborty, Debopam Chattopadhyay, Jaitra Number Theory 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. |
| title | Two infinite families of elliptic curves with Mordell-Weil rank at least $3$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.08570 |