Let $L$ be a finite extension of number field $K$. Let $G=Gal(L/K)$. Let $E/L$ be an elliptic curve defined over $L$. Let $E(L)$ be $L $ rational point of $E$.
Then, why is Galois cohomology $H^1(G,E(L))$ finite group ?
If $E/L$ has rank $0$, both $G$ and $E(L)$ is finite, so the claim is obvious. But in the case $rank(E/L)≧1$, $E(L)$ is infinite set so I think there is no reason $H^1(G,E(L))$ to be finite.