Let $E_1$ and $E_2$ be elliptic curves over a field $k$, and let $l$ be a prime coprime to the characteristic of $k$ (if $char(k) \ne 0$). Let $\varphi$ denote the canonical map

$Hom(E_1,E_2)\otimes_{\mathbb{Z}} \mathbb{Z}_l \rightarrow Hom_{G_k}(T_lE_1,T_lE_2)$ `.

For any field $k$, $\varphi$ is easily shown to be injective, and (a case of) the Tate conjecture says that if $k$ is a finite field or a number field then $\varphi$ is surjective (though the proof is hard). What can we say if instead $k$ is a local field?

More precisely, my questions are:

1)Is there a reason/counterexample explaining why $\varphi$ will not be surjective for $k$ a local field?

2)If the answer to (1) is `yes', is there a weaker statement along the lines of the above which is (or is expected to be) true for $k$ a local field? (Sorry this is rather vague).

Thanks, David