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In 2014, Bhargava and other authors proved that more than 66 % of all elliptic curves defined over $ \mathbb{Q} $ satisfy the Birch and Swinnerton-Dyer conjecture. Tonight I stumbled upon an article of Kozuma Morita (arXiv:1803.11074) claiming to prove that all elliptic curves with complex multiplication satisfy this conjecture, entailing a solution of the congruent number problem through Tunnell's theorem.

Assuming such a result holds, which lower bound to the proportion of rational elliptic curves satisfying BSD conjecture could be reached ?

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    $\begingroup$ The work of Morita has been discussed here before mathoverflow.net/a/270820 mathoverflow.net/a/244478 $\endgroup$ – j.c. Jun 17 '18 at 20:17
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    $\begingroup$ Very few curves over $\mathbb{Q}$ have complex multiplication. This would not change the % anyway. $\endgroup$ – Chris Wuthrich Jun 17 '18 at 20:18
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    $\begingroup$ Various refinements since the Bhargava-Skinner-Zhang paper means that it is now possible to improve the percentage... at a talk in 2016 Bhargava mentioned that (as of September 2016) it should be possible to push the percentage above the 80s. Since this is quite tedious and somewhat unrewarding work, it appears that it has not been written down yet. $\endgroup$ – Stanley Yao Xiao Jun 17 '18 at 20:24
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    $\begingroup$ If correct, this would be a huge breakthrough. Note that even the statement "100% of $E/\mathbb Q$ satisfy BSD" would likely involve first proving that 100% of the curves have rank at most 1. But Morita claims to prove BSD for all $E/\mathbb Q$ having CM, which would include large numbers of curves with rank $\ge2$. I am very curious where he's finding the points of infinite order in the case that $L$ vanishes to order $r\ge2$. It's known that the Heegner point construction fails in that case. $\endgroup$ – Joe Silverman Jun 17 '18 at 20:43
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    $\begingroup$ @StanleyYaoXiao That answer in its entirety addresses an old paper of Morita. A completely new one was posted in March (and updated a week ago) on arXiv. $\endgroup$ – Wojowu Jun 19 '18 at 16:58

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