For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\operatorname{ Gal}(\bar{ K}/K)$ in $\operatorname{ GL}_2(\hat{ \mathbb{ Z}})$, the representation being given by the action on the Tate modules of $E$ at the various primes. The bound is effective and only depends on $[K:\mathbb{ Q}]$ and on the stable Faltings height of $E$. We also prove a result relating the structure of subgroups of $\operatorname{ GL}_2(\mathbb{ Z}_\ell)$ to certain Lie algebras naturally attached to them.
Categories: math.NT math.AG