Families of elliptic curves of rank $\geq 5$ over $\mathbb Q(t)$

Abstract

We construct two infinite families of curves with high rank. The first is defined by the equation $y^2=x(x-a^2)(x-b^2)+a^2b^2$, where $a,b \in \mathbb {Q}(t)$. The second family arises from a system of rational cuboids, i.e., a rectangular box for which the lengths of the edges and face diagonals are all rational. We create a second family with defining equation $y^2=(x-a^2)(x-b^2)(x-c^2)+a^2b^2c^2$, where $a,b,c \in \mathbb {Q}(t)$ are the edge lengths of a rational cuboid. We show that the rank of both families is $\geq 5$ over $\mathbb Q(t)$. We conclude by studying corresponding families of curves defined by other known solutions to the rational cuboid problem, and find some specific examples of curves from these various families with high rank.