Applications

We apply the above results to prove that Conjecture 4.1 implies the existence of elements of Shafarevich-Tate groups of twisted powers of elliptic curves of every prime order. We also construct an abelian variety $A$ over  $\mathbb{Q}$ such that the Birch and Swinnerton-Dyer conjecture predicts that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/\mathbb{Q})[3]=\mathbb{Z}/3\mathbb{Z}$ and that $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/\mathbb{Q})$ is not a square or twice a square.