Arbitrarily large $p$-torsion in Tate-Shafarevich groups

We show that, for any prime $p$, there exist absolutely simple abelian varieties over $\mathbb{Q}$ with arbitrarily large $p$-torsion in their Tate-Shafarevich group. To prove this, we construct explicit ${\mu }_p$-covers of Jacobians of the form $y^p=x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.