Arbitrarily large p-torsion in Tate-Shafarevich groups - Flynn, E; Shnidman, A; Fisher, T (Cambridge University Press, 2024)
We show that, for any prime p, there exist absolutely simple abelian varieties over Q with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit µp-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.