Torsion of rational elliptic curves over different types of cubic fields

Abstract

Let E be an elliptic curve defined over Q, and let. G be the torsion group E(K)(tors) for some cubic field K which does not occur over Q. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) G can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves E/Q together with cubic fields K so that G = E(K)(tors).