Introduction

Let $E$ be an elliptic curve over  $\mathbb{Q}$. We prove that a very plausible conjecture about nonvanishing of prime-degree twists of $L(E,s)$ implies that for all but finitely many primes $p$ there is a twist $A$ of $E^{\times (p-1)}$ such that $E(\mathbb{Q})/p E(\mathbb{Q})$ is ``visible'' as a subgroup of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/\mathbb{Q})$. When $E$ is the elliptic curve of conductor $43$, our construction yields a twist of $E\times E$ such that the Birch and Swinnerton-Dyer conjecture predicts that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A/\mathbb{Q})[3]$ has order $3$.

This paper is organized out as follows. In Section 1 we define twisted powers, Tamagawa numbers, and rigid primes. We recall in Section 2 the definition of the restriction of scalars of an elliptic curve and prove a proposition about a map induced by trace. In Section 3 we recall a construction of the author and Amod Agashe of visible subgroups of Shafarevich-Tate groups. We state a conjecture about nonvanishing of twists of prime degree in Section 4, and give computational evidence for this conjecture. In Section 5 we prove triviality of the $p$-torsion of several abelian groups attached to twisted powers of an elliptic curve. The heart of the paper is Section 6, which uses the above results to construct subgroups of Shafarevich-Tate groups of twisted powers. Section 7 pulls together the results of the previous sections; there we prove that the conjecture of Section 4 implies the existence of elements of Shafarevich-Tate groups of every prime order, we discuss the extent to which the order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$ can fail to be square, and describe a connection with the Birch and Swinnerton-Dyer conjecture.

Acknowledgement: It is a pleasure to thank Gautam Chinta, Benedict Gross, Emanuel Kowalski, Barry Mazur, Bjorn Poonen, David Rohrlich, and Michael Stoll for helpful comments and conversations.