On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves
Pankaj Patel, Debopam Chakraborty, Jaitra Chattopadhyay
TL;DR
The paper analyzes a specially crafted parametric family of elliptic curves $E_m$ with $m$ even and twin-prime constraints, aiming to bound the Mordell-Weil rank and the $2$-Selmer rank. It proves the torsion subgroup is $\\mathbb{Z}/2\\mathbb{Z} \\oplus \\mathbb{Z}/2\\mathbb{Z}$ and establishes a rank lower bound $r(E_m) \\\ge 2$ using canonical height pairings for two independent points. A detailed $2$-descent analysis yields a lower bound $s_2(E_m) \\\ge w$, where $w$ counts specific prime divisors of $m^4-1$ meeting Legendre-symbol conditions relative to the prime factors of $m^4-1\\-4m^2$ and $m^4-1\\+4m^2$. Theorem \\text{mainthm} consolidates these results, and the corollary shows that under stronger primality hypotheses one gets $s_2(E_m) = w+1$ and, assuming the parity conjecture, $r(E_m)$ is odd and at least $3$. These findings provide concrete families of elliptic curves with rank at least $2$ and illuminate the structure of their $2$-Selmer groups and descent data.
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.