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 $\rho_E \colon {\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to {\rm GL}_2(\hat{\mathbb{Z}})$. In this article, we give possibilities for groups generated by $\rho_E({\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}))$ and $-I$ as $E$ varies over non CM elliptic curves defined over $\mathbb{Q}$ such that the group $\pm \rho_E({\rm Gal}(\bar{\mathbb{Q}}/\mathbb{Q}))$ has genus 0. However, our list is not minimal.