On Possible Genus 0 adelic Galois Images of non CM Elliptic Curves over $\mathbb{Q}$

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$/ Associated to $E$, there is an adelic Galois representation $\ddot{I}{}_E:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to {\mathrm{GL}}_2(\hat{\mathbb{Z}})$. In this article, we give possibilities for groups generated by $\ddot{I}{}_E(\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}))$ and $-I$ as $E$ varies over non CM elliptic curves defined over $\mathbb{Q}$ such that the group $\pm \ddot{I}{}_E(\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}))$ has genus 0. However, our list is not minimal.

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