A generalization of a result of Hurwitz and Mordell on the torsion subgroups of certain elliptic curves
Let \(k\) be an algebraic number field and let \(a,b,c,d\) be elements of \(k\) such that \(abc(d^ 3-27abc)\neq 0\). The author considers the elliptic curve given by \(E:aX^ 3+bY^ 3+cZ^ 3=dXYZ\) or rather the torsion subgroup of the group \(E(k)\) of points with coordinates in \(k\) (assuming such points exist). In case \(k=\mathbb{Q}\) there are classical results of Hurwitz and of Mordell. The author defines a certain map, resembling a height function, from \(E(k)\) into the set of integrals ideals of \(k\), which enables him to count the number of torsion points on the curve. Rather general results are obtained.
scientific article; zbMATH DE number 61919