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arXiv · 2503.04561

On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves

Pankaj Patel·Debopam Chakraborty·Jaitra Chattopadhyay
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Abstract

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.

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Pankaj Patel, Debopam Chakraborty, Jaitra Chattopadhyay. 2025-03-06. On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves. https://arxiv.org/abs/2503.04561

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