Open image computations for elliptic curves over number fields

For a non-CM elliptic curve E defined over a number field K, the Galois action on its torsion points gives rise to a Galois representation ρE: Gal(K̄/K)→ GL₂(Z) that is unique up to isomorphism. A renowned theorem of Serre says that the image of ρE is an open, and hence finite index, subgroup of GL₂(Z). In an earlier work of the author, an algorithm was given, and implemented, that computed the image of ρE up to conjugacy in GL₂(Z) in the special case K=Q. A fundamental ingredient of this earlier work was the Kronecker-Weber theorem whose conclusion fails for number fields K≠ Q. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre's theorem for a typical elliptic curve over a fixed number field.