The p-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large

Abstract

In this paper it is shown that for every prime p>5 the dimension of the p-torsion in the Tate-Shafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K:Q] bounded by a constant depending only on p. From this we deduce that the dimension of the p-torsion in the Tate-Shafarevich group of A/Q can be arbitrarily large, where A is an abelian variety, with dim A bounded by a constant depending only on p.