Explicit estimates on a theorem of Shioda concerning the ranks of curves given by $y^2=x^3-a^2x+m^2$

Numerous papers in the literature contain results on the ranks of elliptic curves which generalize a theorem of Brown and Myers on $y^2=x^3-x+m^2$ in various ways. We have recently proven an effective version of a generalization which subsumes most of what has appeared in the literature on this topic, although the bounds have not been worked out explicitly. We consider here the subfamily $y^2=x^3-a^2x+m^2$, give explicit lower bounds for $m$ in terms of $a$ for such a curve to have rank at least $2$, and show how these lower bounds are influenced by the $abc$ conjecture.