On Sufficient Conditions for Rank Zero in Families of Elliptic Curves

Abstract

In this article, we study the Mordell Weil ranks of elliptic curves in the family $E_D:y^2=x^3+Dx$, where $D\in\mathbb{Z}\setminus\{0\}$. Under suitable congruence conditions on $D$ and its prime divisors, we use $2$-descent via a rational $2$-isogeny to study the rank of $E_D$. Under our congruence hypotheses, the ranks of these matrices give sufficient conditions for $E_D$ to have rank zero. We give explicit sufficient conditions for the curves $E_D$ to have Mordell Weil rank zero over $\mathbb{Q}$ in the following cases: (i) $D$ is an odd negative integer, (ii) $D$ is an odd positive integer, and (iii) $D$ is a positive even integer. Also we give a quantitative result describing how many square free parameters satisfy the required matrix-rank condition. In particular, our results show that these rank zero criteria hold for a significant proportion of the admissible parameters.