Let $K$ be a number field and $E/K$ be an elliptic curve. Let $G_K=Gal(\overline{K}/K)$. Let $M_K$ be a set of all places of $K$.
Tate-Shafarevich group is usually defined as $$Sha(E/K)\stackrel{\mathrm{def}}{=} Ker(H^1(G_K,E) \to \prod_{v\in M_K} {H^1(G_{K_v},E)})$$
My question is, what the group
$$Ker(H^1(G_K,E)\to \bigoplus_{v\in M_K} {{H^1}(G_{K_v},E)})$$ looks like.
Is this group worth defining ?