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Bounds for Serre's open image theorem for elliptic curves over number fields (1403.3813v5)
Published 15 Mar 2014 in math.NT and math.AG
Abstract: 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.