Open image computations for elliptic curves over number fields

Abstract

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 $\rho_E: Gal(\overline{K}/K)\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A renowned theorem of Serre says that the image of $\rho_E$ is an open, and hence finite index, subgroup of $GL_2(\widehat{\mathbb{Z}})$. In an earlier work of the author, an algorithm was given, and implemented, that computed the image of $\rho_E$ up to conjugacy in $GL_2(\widehat{\mathbb{Z}})$ in the special case $K=\mathbb{Q}$. A fundamental ingredient of this earlier work was the Kronecker-Weber theorem whose conclusion fails for number fields $K\neq \mathbb{Q}$. 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.