Mordell-Weil group as Galois modules

Abstract

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 $K$.