On Sufficient Conditions for Rank Zero in Families of Elliptic Curves

In this article, we study the Mordell Weil ranks of elliptic curves in the family ED:y²=x³+Dx, where D\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 ED. Under our congruence hypotheses, the ranks of these matrices give sufficient conditions for ED to have rank zero. We give explicit sufficient conditions for the curves ED to have Mordell Weil rank zero over 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.