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