Adelic point groups of elliptic curves

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.