It is known that if the Tate-Shafarevich group of an elliptic curve is trivial, then Hasse local-global principle holds for every homogeneuos space over the elliptic curve. I read here and here (at 2.2.3 page 24), that we can get additional information by the Tate-Shafarevich group of the jacobian variety of the elliptic curve. More specifically, here Brandon Carter says that "Hasse principle holds for a genus one curve $C$ defined over $\Bbb{Q}$ if and only if $C$ represents the trivial class in the Tate-Shafarevich group of its Jacobian" and then says "To get the precise statement I mentioned above, it is enough to read Chapter X, Sections 3 and 4 of Silverman's Arithmetic of Elliptic Curves book."
But Silverman mentions the Tate-Shafarevich group of the Jacobian exectly 0 times (I've already read sections 3 and 4 in chapter X).
Where can I find a reference to Brandon Carter's statement?
Thanks in advance.