On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908756142456832 |
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| author | Patel, Pankaj Chakraborty, Debopam Chattopadhyay, Jaitra |
| author_facet | Patel, Pankaj Chakraborty, Debopam Chattopadhyay, Jaitra |
| contents | We consider the parametric family of elliptic curves over $\mathbb{Q}$ of the form $E_{m} : y^{2} = x(x - n_{1})(x - n_{2}) + t^{2}$, where $n_{1}$, $n_{2}$ and $t$ are particular polynomial expressions in an integral variable $m$. In this paper, we investigate the torsion group $E_{m}(\mathbb{Q})_{\rm{tors}}$, a lower bound for the Mordell-Weil rank $r({E_{m}})$ and the $2$-Selmer group ${\rm{Sel}}_{2}(E_{m})$ under certain conditions on $m$. This extends the previous works done in this direction, which are mostly concerned with the Mordell-Weil ranks of various parametric families of elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04561 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves Patel, Pankaj Chakraborty, Debopam Chattopadhyay, Jaitra Number Theory We consider the parametric family of elliptic curves over $\mathbb{Q}$ of the form $E_{m} : y^{2} = x(x - n_{1})(x - n_{2}) + t^{2}$, where $n_{1}$, $n_{2}$ and $t$ are particular polynomial expressions in an integral variable $m$. In this paper, we investigate the torsion group $E_{m}(\mathbb{Q})_{\rm{tors}}$, a lower bound for the Mordell-Weil rank $r({E_{m}})$ and the $2$-Selmer group ${\rm{Sel}}_{2}(E_{m})$ under certain conditions on $m$. This extends the previous works done in this direction, which are mostly concerned with the Mordell-Weil ranks of various parametric families of elliptic curves. |
| title | On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.04561 |