Shafarevich-Tate Groups of Twisted Powers of Elliptic Curves
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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