We show how the Galois representation of an elliptic curve over a number field can be used to determine the structure of the (topological) group of adelic points  of that elliptic curve.

As a consequence, we find that for "almost all" elliptic curves over a number field K,  the adelic point group is a universal topological group depending only on the degree  of K. Still, we can construct infinitely many pairwise non-isomorphic elliptic curves  over K that have an adelic point group not isomorphic to this universal group.

This generalizes work of my student Athanasios Angelakis (PhD Leiden, 2015).