A Classification of Genus 0 Modular Curves with Rational Points

Abstract

Let $E$ be a non-CM elliptic curve defined over $\mathbb{Q}$. Fix an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$. We get a Galois representation \[ρ_E \colon Gal(\overline{\mathbb {Q}}/\mathbb {Q}) \to GL_2(\hat{\mathbb {Z}})\] associated to $E$ by choosing a compatible bases for the $N$-torsion subgroups of $E(\overline{\mathbb{Q}}).$ Associated to an open subgroup $G$ of $G{L}_2(\hat{\mathbb{Z}})$ satisfying $-I\in G$ and $det(G)={\hat{\mathbb{Z}}}^{\times }$, we have the modular curve $(X_G,\ddot{I}{€}_G)$ over $\mathbb{Q}$ which loosely parametrises elliptic curves $E$ such that the image of $\ddot{I}{}_E$ is conjugate to a subgroup of $G^t.$ In this article we give a complete classification of all such genus $0$ modular curves that have a rational point. This classification is given in finitely many families. Moreover, each such modular curve can be explicitly computed.

Minor changes in introduction, corrected some typos, made the list more minimal