An infinite family of elliptic curves and Galois module structure

Abstract

Let E be an elliptic curve dened over a number eld F with everywhere good reduction. By dividing F -rational torsion points with respect to the group law of E M. Taylor dened certain Kummer orders and studied their Galois module structure. His results led to the conjecture that these Kummer orders are free over an explicitly given Hopf order. In this paper we prove that the conjecture does not hold for innitely many elliptic curves which are dened over quadratic imaginary number elds k and endowed with a k-rational 2torsion point.