Elliptic Curves with Surjective Adelic Galois Representations

Let K be a number field. The Gal((K) over bar /K)-action on the torsion of an elliptic curve E/K gives rise to an adelic representation rho(E): Gal((K) over bar /K)-> GL(2)((Z) over cap). From an analysis of maximal closed subgroups of GL(2)((Z) over cap) we derive useful necessary and sufficient conditions for rho(E) to be surjective. Using these conditions, we compute an example of a number field K and an elliptic curve E/K that admits a surjective adelic Galois representation.