Shafarevich-Tate Groups of Twisted Powers of Elliptic Curves

William A. Stein




Abstract

Let E be an elliptic curve over 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 Ex(p-1) such that E(Q)/p E(Q) is naturally contained in a subgroup of Sha(A/Q). We also give an example of an abelian variety A over Q such that the Birch and Swinnerton-Dyer conjecture would, if true, imply that Sha(A/Q)[3] has order 3.



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