I saw the paper claiming the proof of the BSD conjecture for the CM case. Apart from the truth and falsehood of this paper, I noticed that the author says that the congruent number problem can be solved if the BSD conjecture for CM elliptic curves over an imaginary quadratic field $K$. I think that the congruent number problem is a problem over $\mathbb{Q}$. Is it possible to deduce it as a corollary of true BSD over an imaginary quadratic field $K$? More precisely, my question is $$L(E_K,s)=r(E(K))\Longrightarrow L(E_{\mathbb{Q}},s)=r(E(\mathbb{Q}))?$$
http://kazuma-morita.jimdo.com/app/download/12319896690/Sh.BSD.pdf?t=1452154642
(I know that he withdraw a paper claiming the full BSD from the arxiv at 2013.)