Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \mathbb{Q}_\infty$ as the notation for the cyclotomic $\mathbb{Z}_p$-extension. Let $w|p$ be primes in $L$. We have $L_{\infty,w}/ \mathbb{Q}_{\infty,p}$, a Galois extension and say the Galois group is denoted by $G$.
How does one one compute the Galois cohomology $H^i(G, E(L_{\infty,w})_{p^\infty})$ for $i=1,2$.
My understanding is that using a result of Lang and a deep result of Coates-Greenberg one can say $H^i(G, E(L_{\infty,w}))=0$ for $i=1,2$. Consider the following short exact sequence $$ 0\rightarrow \mathcal{F}(\mathfrak{m}_w) \rightarrow E(L_{\infty,w}) \rightarrow \tilde{E}(\ell_w) \rightarrow 0. $$ Here $\mathcal{F}(\mathfrak{m}_w)$ is the corresponding formal group and $\ell_w$ the residue field. $\tilde{E}$ is reduction modulo $\mathfrak{m}_w$. The result of Lang says $H^i(G, \tilde{E}(\ell_w))=0$ for $i=1,2$ and that of Coates-Greenberg says $H^i(G,\mathcal{F}(\mathfrak{m}_w))=0$. This implies $H^i(G, E(L_{\infty,w}))=0$