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

Abstract

For $E/K$ an elliptic curve without complex multiplication we bound the index of the image of $\mathrm{Gal}(\bar{K}/K)$ in ${\mathrm{GL}}_2(\hat{\mathbb{Z}})$, 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 ${\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$ to certain Lie algebras naturally attached to them.

Final version, accepted for publication in Algebra and Number Theory

References in corpus (2)

• Bounds for Serre's open image theorem
• Un critère d'indépendance pour une famille de représentations l-adiques