Mordell-Weil group as Galois modules

TL;DR

This paper investigates how the Galois group acts on the points of an elliptic curve over number fields, aiming to understand the structure of the Mordell-Weil group as a Galois module using local and base field data.

Contribution

It provides a method to determine the $p$-adic structure of the Mordell-Weil group as a Galois module from base field and local information, for odd primes.

Findings

Characterizes the Galois module structure of $E(K)$

Relates the structure to local field data

Advances understanding of Galois actions on elliptic curves

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 $K$.