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 ?

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