Assume $K$ is an imaginary quadratic extension of $\mathbb{Q}$, and $E$ an elliptic curve defined over $\mathbb{Q}$.
Let $p\neq l$ be primes in $\mathbb{Q}$ where $E$ has good reduction. Assume $p$ splits in $K/\mathbb{Q}$ and $l$ stays inert.
Denote by $K\_{l}$ the localization of $K$ at the place $l$.
Is there always an isomorphism


$$H^1(K\_{l},E)\_{p^n}\rightarrow E(K\_l)/p^nE(K\_l),$$
at least for large enough $n$?



Cohomology of elliptic curves
More generally let k be a local field of residual characteristic $\ell$ and $E/k$ an elliptic curve with good reduction. Then for any prime $p\neq\ell$, there is a perfect duality between $E(k)/p^n$ and $H^1(k,E)[p^n]$ discovered by Tate. As both are finite abelian groups, they are indeed isomorphic. Knowing the structure of elliptic curves over local fields, it is easy to see that $E(k)/p^n$ is isomorphic to the same group in the reduction $\tilde E(\tilde k)/p^n$. So it is clear when these groups are non-trivial.


Yet, there is, in general, no canonical choice of an isomorphism. In some places like when using Euler systems, and it may be that you are referring to this, it is useful to fix an isomorphism between them so that one can compare classes. Typically one wants even a complement in $H^1(k,E[p^n])$ to the image of the Kummer map. 


