Elliptic Curves with Surjective Adelic Galois Representations

Abstract

Let $K$ be a number field. The $\operatorname{Gal}(\bar{K}/K)$-action on the torsion of an elliptic curve $E/K$ gives rise to an adelic representation $ρ_E: \operatorname{Gal}(\bar{K}/K) \to \mathrm{GL}_2(\hat{\mathbb{Z}})$. From an analysis of maximal closed subgroups of $\mathrm{GL}_2(\hat{\mathbb{Z}})$ we derive useful necessary and sufficient conditions for $ρ_E$ to be surjective. Using these conditions, we compute an example of a number field $K$ and an elliptic curve $E/K$ that admits a surjective adelic Galois representation.