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Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that $$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$ Thanks to the work of Gross-Zagier and Kolyvagin, we know that this conjecture is true if $\text{ord}_{s=1}L(E, s)\le 1$.

What is known in the case $\text{rank} E(\mathbb{Q})\le 1$? Is it known that if $\text{rank} E(\mathbb{Q})=1$, then $L'(E, 1)\neq 0$?

Thank you!

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    $\begingroup$ In fairness, it's probably worth mentioning that in addition to the work of G-Z and Kolyvagin, one also needs the theorem of Wiles et al that every elliptic curve over $\mathbb Q$ is modular, since [G-Z] and [K] start with the assumption that their elliptic curve is modular. $\endgroup$ – Joe Silverman Aug 25 '18 at 11:35
  • $\begingroup$ ...and a non-vanishing result for special values of L functions. $\endgroup$ – Pasten Aug 26 '18 at 0:29
  • $\begingroup$ Cross-posted: math.stackexchange.com/questions/2893669 $\endgroup$ – Watson Oct 20 '18 at 12:00
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The following theorem is due to Chris Skinner, in this 2014 paper.

Let E/Q be an elliptic curve such that rank E(Q) = 1 and the Tate-Shafarevich group Sha(E / Q) is finite, and some other technical assumptions hold. Then $ord_{s = 1} L(E, s) = 1$, and in particular $L'(E, 1) \ne 0$.

This is, as far as I know, the best one can do at the moment; if you don't know that Sha (or at least its p-primary part for some p) is finite, then you're stuck.

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