The period–index problem in WC-groups I: elliptic curves

Abstract

Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full p-torsion. We show that the order of the p-part of the Shafarevich–Tate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O’Neil's period–index obstruction. We deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the Weil–Châtelet group of E/K of period p and index p2.