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Let A be an elliptic curve over a number field K, let p 7 be a prime and let } be a prime ideal of K lying above p, such that the j-invariant of A is non-integral at }. We extend a result of Rohrlich to show that a certain deformation of the Galois representation attached to the Tate module of A is surjective, under some restrictions involving the ramification index of }, p and j(A). 1. Surjectivity of a Galois Representation Let K be a number field, fix K, an algebraic closure of K, and let j be transcendental over K. Let E be an elliptic curve defined over the field K(j) such that j(E) = j. Given a prime number p 7, the natural action of Gal(K(j)/K(j)) on the group of p-torsion points of E induces a repre- sentation f E: Gal(K(j)/K(j)) ! SL(2,Fp). The universal deformation of f E, with respect to certain ramification conditions (see (Roh), (Roh04)), is an epimorphism E: Gal(K(j)/K(j)) ! SL(2,Zp((X))).

On the surjectivity of Galois representations attached to elliptic curves over number fields