Let $E/K$ be an elliptic curve over a number field $K$. Let $\widehat{E(K)}$ be profinite completion of $E(K)$.//
Let $v$ be a place of $K$. Let $K_v$ be completion of $K$ at $v$.
Is there an injection from $\widehat{E(K)}$ to $E(K_v)$ ? How can we construct it ?
Background of this question:
I encountered a beautiful exact sequence of Tate-Shafarevich group $Sha(E/K)$, $$0\to Sha(E/K)\to H^1(G_K,E) \to \bigoplus_{v} {H^1(G_{K_v},E)}\to {\widehat{E(K)}}^*\to0$$ where $*$ is Pontryagin dual.
However, I couldn't understand what the last map $\bigoplus_{v} {H^1(G_{K_v},E)}\to {\widehat{E(K)}}^*$ is. If there is a map $\widehat{E(K)}\to \prod_v E(K_v)$, we can take its Pontryagin dual and that is the map we wanted. However, what is the map $\widehat{E(K)}\to \prod_v E(K_v)$ ? If I can embed $\widehat{E(K)}$ to $E(K_v)$, that's ok, so this question arised. Another direct approach to this background is also appreciated.