Let $\phi: E \to E'$ be isogeny between elliptic curves. Let $K$ be a number field. Let $Sha(E/K)$ be a Tate-Shafarevich group of elliptic curve $E/K$.What is the definition of $Sha(E/K)[\phi]$ ?
Silverman's book 'The arithmetic of elliptic curves' deals with this group, but I think there is no definition of $Sha(E/K)[\phi]$.
I think the definition is $Sha(E/K)[\phi]=ker(H^1(G_K,E(K)[\phi])\to \prod H^1(G_{K_v},E(K_v)[\phi]))$.
Is my definition correct ? Thank you for your help.