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Author: William A. Stein
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Elliptic curves~$E$ over~$\Q$ such that $\Sha(E/\Q)[p]$ is nontrivial are
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known by explicit computation for a finite list of small values
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of~$p$. It remains an open problem to prove that there are infinitely
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many~$p$ such that $\Sha(E/\Q)[p]$ is nonzero. In this paper we study a
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related problem.
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A {\em twisted power} of an elliptic curve~$E$ is an abelian variety
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over~$\Q$ that is a twist of $E^{\times n}=E\cross\cdots \cross E$ for
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some positive integer~$n$. Proposition~\ref{prop:all_prime_orders}
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asserts that Conjecture~\ref{conj:nonvanishing} about nonvanishing of
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certain twisted $L$-functions implies that, for every prime~$p$, there
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is a twisted power~$A$ of some elliptic curve such that
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$\Sha(A/\Q)[p]\neq \{0\}$.
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Robert Pollack has informed the author that he can probably prove that
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there exist nontrivial $\Sha(A/\Q)[p]$ using his results and recent
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results of Perron-Riou on the size of the~$p$-primary part of
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$\Sha(E/L)$, for~$E$ an elliptic curve with supersingular reduction
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at~$p$ and~$L$ varying over subfields of the cyclotomic
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$\Z_p$-extension of~$\Q$. We have not investigated Pollack's approach
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further, because our true interest in this problem is to link
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Mordell-Weil and Shafarevich-Tate groups of abelian varieties, in
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order to obtain new results toward the conjecture of Birch and
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Swinnerton-Dyer (see Section~\ref{sec:bsd}). In fact,
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Theorem~\ref{thm:main} asserts, again assuming the nonvanishing
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conjecture, that for all sufficiently large primes~$p$, there is a
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twisted power~$A$ of~$E$ such that
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$$E(\Q)/p E(\Q)\subset \Sha(A/\Q).$$
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}
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