Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? Here, $()^{ab}$ denotes the maximal Abelian extension of a field, $K_{\nu}$ is the localization of a field $K$ at a finite prime $\nu$, $\mu_{\nu}:Gal(K_{\nu}^{ab}/K_{\nu})\rightarrow Gal(K^{ab}/K)$ is induced by the injection $K\rightarrow K_{\nu}$ and $E_{p^n}$ are the $p^n$-torsion points of a (non-singular) elliptic curve $E$, defined over $K$.
I am not an expert on the subject, so please have mercy on my soul.