The relationship between face cuboids and elliptic curves

TL;DR

This paper explores the connection between rational face cuboids and elliptic curves, constructing a surjective map to show infinitely many such cuboids exist and analyzing the rank of associated elliptic curves.

Contribution

It establishes a surjective 32:1 map from pairs of rational points on elliptic curves to equivalence classes of rational face cuboids, proving their infinitude.

Findings

Infinite set of rational face cuboids established.

Constructed pairs of parameters and points not derived from parametric solutions.

Proved existence of infinitely many elliptic curves with positive rank.

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.