Bounds for Serre's open image theorem for elliptic curves over number fields

For an elliptic curve E∕K without potential complex multiplication we bound the index of the image of Gal(K̅/K) in GL₂(ˆℤ), the representation being given by the action on the Tate modules of E at the various primes. The bound is explicit and only depends on [K:Q] and on the stable Faltings height of E. We also prove a result relating the structure of closed subgroups of GL₂(Zℓ) to certain Lie algebras naturally attached to them.