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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 groups. To prove this, we construct explicit 
$\mu _p$-covers of Jacobians of curves 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.
