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Let E$E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the p$p$-torsion of the Tate-ShafarevichTate–Shafarevich group in this case? In particular, I would like to know if (or if known under which additional assumption) it vanishes.

Let E be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the p-torsion of the Tate-Shafarevich group in this case? In particular, I would like to know if (or if known under which additional assumption) it vanishes.

Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known under which additional assumption) it vanishes.

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p-torsion in the Tate-Shafarevich group of supersingular elliptic curves

Let E be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the p-torsion of the Tate-Shafarevich group in this case? In particular, I would like to know if (or if known under which additional assumption) it vanishes.