  We study the action of the Galois group $G$ of a finite extension $K/k$ of
number fields on the points on an elliptic curve $E$. For an odd prime $p$, we
aim to determine the structure of the $p$-adic completion of the Mordell-Weil
group $E(K)$ as a $\mathbb{Z}_p[G]$-module only using information of $E$ over
$k$ and the completions of $K$.
