The Period-index Problem in Wc-groups Iii: Biconic Curves

This note continues our study of the period-index problem in the Weil-Châtelet group of an elliptic curve, begun in [6]. In this paper, we conjectured that for any elliptic curve E/K defined over a number field and any prime number p, there exists an infinite subgroup G of H(K,E), every nonzero element of which has period p and index p. This conjecture was proved when E has full p-torsion defined over K. In [7] we went on to pursue analogous questions in the Weil-Châtelet group of a higher-dimensional abelian variety.