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Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer (BSD) 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 $\text{ord}_{s=1}L(E, s)\le 1$.

My question is: what is known in the case $\text{rank} E(\mathbb{Q})\le 1$? Has the BSD conjecture been proved under this assumption?

Thank you!

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Yes, but only with the additional assumption that $ Ш(E/\mathbf{Q})$ is finite (one actually only needs that $Ш(E/\mathbf{Q})[p^\infty]$ is finite for almost any prime $p$). This then implies that $\mathrm{Sel}_{p^\infty}(E/\mathbf{Q})$ has corank 1, and then it follows from work of Skinner, Skinner-Zhang, and Zhang.

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