# Elliptic Curves

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.

• Take a look at this – Stahl Oct 28 '14 at 8:11
• 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 mathoverflow.net/questions/124202/… The converse is not known to be true up to now. – Ferra Oct 28 '14 at 15:34
• The converse is known to be true under the assumption that the Tate-Shafarevich group is finite. See arxiv.org/abs/1405.7294 – Brandon Carter Nov 26 '14 at 22:59