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 Id : 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 ?

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 ?