Rank of the family of elliptic curves $y^2 = x^3- 5px$
Arkabrata Ghosh
TL;DR
This work analyzes the Mordell-Weil rank of the elliptic curve family $E_p: y^2 = x^3 - 5 p x$ for odd primes $p$, establishing explicit rank criteria based on congruence classes modulo $40$ and certain square conditions. The author employs a 2-descent framework, using a degree-2 isogeny and descent maps $\alpha$, $\overline{\alpha}$ and associated torsors, to bound the rank via $2^r = (1/4) |\alpha(\Gamma)| \cdot |\overline{\alpha}(\overline{\Gamma})|$, while also leveraging twists to relate ranks over quadratic fields, particularly $\mathbb{Q}(i)$. The main results classify $r(E_p/\mathbb{Q})$ as $0$, $1$, or at least $1$ under three families of congruence/square conditions, and derive corresponding ranks over $\mathbb{Q}(i)$; a novelty is achieving these results without BSD or parity conjectures. The paper also includes computational verifications supporting the theoretical criteria and discusses potential extensions to higher ranks.
Abstract
This article considers the family of elliptic curves given by $E_{p}: y^2=x^3-5px$ and certain conditions on an odd prime $p$. More specifically, we have shown that if $p \equiv 7, 23 \pmod {40}$, then the rank of $E_{p}$ is zero for both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the prime $ p $ is of the form $ 40k_1 + 3 $ or $ 40k_2 + 27$, where $k_1, k_2 \in \mathbb{Z}$ such that $(5k_1+1)$ or $(5k_2 +4)$ are perfect squares, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Moreover, if the prime $ p $ is of the form $ 40k_3 + 11 $ or $ 40k_4 + 19$ where $k_3 ~\text{and}~ k_4 \in \mathbb{Z}$ such that $(160k_3+49)$ or $(160k_4 + 81) $ are perfect squares, then the given family of elliptic curves has rank at least one over $\mathbb{Q}$ and rank at least two over $\mathbb{Q}(i)$.