The relationship between face cuboids and elliptic curves

Abstract

A rational face cuboid is a cuboid that all of edges, two of three face diagonals and space diagonal have rational lengths. \[ E_{1,s}: y^2=x(x-(2s)^2)(x+(s^2-1)^2) \] for a rational number $s \neq 0, \pm 1$, and define $\tilde{A}$ consisting of all pairs of a rational number $s$ and a non-torsion rational point $(\alpha, \beta ) \in E_{1,s}(\mathbb{Q})$. We construct a surjective map from $\tilde{A}$ to the set $\mathscr{F}$ of equivalence classes of rational face cuboids, and prove that this map is a $32:1$-map. In this way, we show that the set $\mathscr{F}$ has infinite elements. Also, we prove that there are infinitely many $s \in \mathbb{Q} \setminus \{ 0,\pm 1 \}$ with $\mathrm{rank} E_{1,s} (\mathbb{Q})>0$. In this proof, we construct pairs of $s$ and $(\alpha, \beta) \in E_{1,s} (\mathbb{Q})$ which are not parametric solutions.