Let $L/K$ be a unramified quadratic extension of local fields. Let $E/K$ be an elliptic curve over $K$.
We can consider reduction map $E(L)\to E(l)$ where $l$ is a residue field of $L$. Suppose characteristic of $k$ is odd. Let $E^1(L)$ be kernel of reduction map.
I want to prove $Gal(L/K)$ acts trivially on $E^1(L)$. How can I prove this ?
Suppose $\sigma \in Gal(L/K)$ acts on $P\in E^1(L)$. $\tilde{\sigma(P)}=0$, thus $\sigma({P})\in E^1(L)$.Thus, $G$ acts on $E^1(L)$, but I cannot derive any infomation on this action.
Hints only is also appreciated.