Arbitrarily large 2-torsion in Tate-Shafarevich groups of Abelian varieties

We show that, for any d, the 2-torsion of Tate-Shafarevich groups of absolutely simple abelian varieties of dimension d over Q can be arbitrarily large. This involves the use of an approach, which we shall describe, for demonstrating arbitrarily large Tate-Shafarevich groups which does not require entire Selmer groups to be found.