An infinite family of elliptic curves and Galois module structure

Abstract

Let E be an elliptic curve defined over a number field F with everywhere good reduction. By dividing F -rational torsion points with respect to the group law of E M. Taylor defined 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 infinitely many elliptic curves which are defined over quadratic imaginary number fields k and endowed with a k-rational 2-torsion point.