Tate-Shafarevich groups and Hasse principle (reference)

Question

I'm looking for a proof of the fact that the Hasse local-global principle holds for an elliptic curve $E$ defined over $Q$ if and only if the Tate-Shafarevich group of $E$ vanishes. I just need to know where i can find this, book/article and page should be enough. Thank you

Answer 1

The Hasse principle automatically holds for an elliptic curve $E$ over $\mathbf{Q}$. By definition, $E$ has a marked point defined over $\mathbf{Q}$.

The correct statement is that the Hasse principle holds for a genus one curve $C$ defined over $\mathbf{Q}$ if and only if $C$ represents the trivial class in the Tate-Shafarevich group of its Jacobian. In particular, if $Ш(Jac(C)/\mathbf{Q})$ is trivial, then the Hasse principle holds for $C/\mathbf{Q}$. However, it is important to note that this is latter statement is not an if and only if. In particular, an elliptic curve with nontrivial Tate-Shafarevich group still satisfies the Hasse principle for the reason mentioned above.

Barry Mazur wrote an excellent expository paper all about local-global principles in algebraic geometry, which you can read here. 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.