Hashing to elliptic curves of $j=0$ and Mordell--Weil groups

Abstract

Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 - b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_{\!q}$ such that $\sqrt[3]{b} \notin \mathbb{F}_{\!q}$. This article tries to resolve the problem of constructing a rational $\mathbb{F}_{\!q}$-curve on the Kummer surface of the direct product $E_b \!\times\! E_b^\prime$, where $E_b^\prime$ is the quadratic $\mathbb{F}_{\!q}$-twist of $E_b$. More precisely, we propose to search such a curve among infinite order $\mathbb{F}_{\!q}$-sections of some elliptic surface of $j=0$, analyzing its Mordell--Weil group. Unfortunately, we prove that it is just isomorphic to $\mathbb{Z}/3$.