# Adelic point groups of elliptic curves

## TL;DR

This paper characterizes the adelic point groups of elliptic curves over number fields, providing explicit descriptions and showing the diversity of these groups among elliptic curves.

## Contribution

It offers an explicit description of adelic point groups of elliptic curves and demonstrates the variation among curves over a fixed number field.

## Key findings

- Almost all elliptic curves over a number field have a universal adelic point group.
- Existence of infinitely many elliptic curves with distinct adelic point groups.
- The structure of E(A) relates to the Galois representation of torsion points.

## Abstract

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points of E. An explicit description of E(A) is given, and we prove that for K of degree n, almost all elliptic curves over K have an adelic point group topologically isomorphic to a universal group depending on n. We also show that there exist infinitely many elliptic curves over K having a different adelic point group.