Two infinite families of elliptic curves with Mordell-Weil rank at least 3

In this paper, we consider two infinite parametric families of elliptic curves defined over Q given by the equations Ea,b : y² = x³ - a²x + b² and E^a,b : y² = x³ - a²x + b⁶, where a,b ∈ N satisfy certain mild conditions. We prove that the torsion group of Ea,b(Q) is trivial and the Mordell-Weil ranks of both Ea,b(Q) and E^a,b(Q) are at least $3$ for infinitely many choices of a and b by using the Néron-Tate height of a rational point and by exploiting the unit group of the ring of integers of Q(√3). This is an extension of the results of Brown-Myres and Fujita-Nara where lower bounds of the ranks were provided under the assumption that $a = 1$ or $b = 1$. Also, our families of elliptic curves vastly generalize the curves recently investigated by Hatley and Stack.