Compatible Systems of `-adic Representations of Dimension Two

where E`n denotes the group of `-torsion points of E. The Galois group GK = Gal(K̄/K) acts continuously on T` and therefore on V`. In a series of works ([5], [6], [8], [9]), Serre investigated the image, Γ`, of GK in GL(V`). If E has complex multiplication over C, Γ` is contained in a Cartan subgroup of GL(V`) (resp. the normalizer of a Cartan) if K contains (resp. does not contain) the endomorphism ring of E. If E does not admit complex multiplication, Γ` is Zariski-dense in GL(V`) and for all ` 0, Γ` = GL(T`). In [11], an analogous result is proved for compatible systems of 2-dimensional Galois representations arising from elliptic modular forms. Fix a finite set S of primes ofK. For the purposes of this paper, a compatible system of Galois representations, unramified outside S will be a system of continuous representations