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

Abstract

For an elliptic curve $E∕K$ without potential complex multiplication we bound the index of the image of Gal$\left(\overline{K}∕K\right)$ in GL${}_2\left(\hat{\mathbb{Z}}\right)$, 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 $\left[K:\mathbb{Q}\right]$ and on the stable Faltings height of $E$. We also prove a result relating the structure of closed subgroups of GL${}_2\left({\mathbb{Z}}_{\ell }\right)$ to certain Lie algebras naturally attached to them.