CoCalc Shared Fileswww / papers / nonsquaresha / removed
Author: William A. Stein
1\comment{
2Elliptic curves~$E$ over~$\Q$ such that $\Sha(E/\Q)[p]$ is nontrivial are
3known by explicit computation for a finite list of small values
4of~$p$.  It remains an open problem to prove that there are infinitely
5many~$p$ such that $\Sha(E/\Q)[p]$ is nonzero.  In this paper we study a
6related problem.
7
8A {\em twisted power} of an elliptic curve~$E$ is an abelian variety
9over~$\Q$ that is a twist of $E^{\times n}=E\cross\cdots \cross E$ for
10some positive integer~$n$.  Proposition~\ref{prop:all_prime_orders}
11asserts that Conjecture~\ref{conj:nonvanishing} about nonvanishing of
12certain twisted $L$-functions implies that, for every prime~$p$, there
13is a twisted power~$A$ of some elliptic curve such that
14$\Sha(A/\Q)[p]\neq \{0\}$.
15
16Robert Pollack has informed the author that he can probably prove that
17there exist nontrivial $\Sha(A/\Q)[p]$ using his results and recent
18results of Perron-Riou on the size of the~$p$-primary part of
19$\Sha(E/L)$, for~$E$ an elliptic curve with supersingular reduction
20at~$p$ and~$L$ varying over subfields of the cyclotomic
21$\Z_p$-extension of~$\Q$.  We have not investigated Pollack's approach
22further, because our true interest in this problem is to link
23Mordell-Weil and Shafarevich-Tate groups of abelian varieties, in
24order to obtain new results toward the conjecture of Birch and
25Swinnerton-Dyer (see Section~\ref{sec:bsd}).  In fact,
26Theorem~\ref{thm:main} asserts, again assuming the nonvanishing
27conjecture, that for all sufficiently large primes~$p$, there is a
28twisted power~$A$ of~$E$ such that
29$$E(\Q)/p E(\Q)\subset \Sha(A/\Q).$$
30}
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