Torsion groups of elliptic curves with integral j-invariant over pure cubic fields

Abstract

We determine all possible torsion groups of elliptic curves E with integral j-invariant over pure cubic number fields K. Except for the groups Z/2 Z, Z/3 Z and Z/2 Z ⊕ Z/2 Z, there exist only finitely many curves E and pure cubic fields K such that E over K has a given torsion group E TOR ( K), and they are all calculated here. The curves E over K with torsion group E TOR(K) ≅ Z/2 Z ⊕ Z/2 Z have j-invariants belonging to a finite set. They are also calculated. A preliminary report on the results obtained was given by H. H. Müller, H. Ströher, and H. G. Zimmer ( in “Proceedings, Intern. Numb. Th. Conference at Laval University, Québec, Canada, 1987” (J.-M. DeKoninck and C. Levesque, Eds.), pp. 671–698, de Gruyter, Berlin/New York, 1989) .