I want some clarification regarding some concept in elliptic curves. In many papers I have seen that, let $E:y^2=x^3+Ax+B $ be an elliptic curve if $L(E,1) $ (corresponding L-function at s=1) is nonzero and Tate Shafarevich group is also nontrivial then MWrank of $E $ is zero. Is it true for all elliptic curves? What is the reason? Is converse also true.
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Take a look at this – Stahl Oct 28 '14 at 08:11
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Yes, it is true for every elliptic curve (over $\mathbb Q$), and it follows from Gross-Zagier and Kolyvagin's work. See here for example http://mathoverflow.net/questions/124202/deducing-bsd-from-gross-zagier-and-kolyvagin The converse is not known to be true up to now. – Ferra Oct 28 '14 at 15:34
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The converse is known to be true under the assumption that the Tate-Shafarevich group is finite. See http://arxiv.org/abs/1405.7294 – bzc Nov 26 '14 at 22:59