A classification of genus 0 modular curves with rational points

Let E E be a non-CM elliptic curve defined over Q \mathbb {Q} . Fix an algebraic closure Q ¯ {\overline {\mathbb Q}} of Q \mathbb {Q} . We get a Galois representation \[ ρ E : G a l ( Q ¯ / Q ) → G L 2 ( Z ^ ) \rho _E \colon {Gal}({\overline {\mathbb Q}}/\mathbb {Q})\to GL_2({\widehat {\mathbb {Z}}}) \] associated to E E by choosing a system of compatible bases for the N N -torsion subgroups of E ( Q ¯ ) . E({\overline {\mathbb Q}}). Associated to an open subgroup G G of G L 2 ( Z ^ ) GL_2({\widehat {\mathbb {Z}}}) satisfying − I ∈ G -I \in G and det ( G ) = Z ^ × \det (G)={\widehat {\mathbb {Z}}}^{\times } , we have the modular curve ( X G , π G ) (X_G,\pi _G) over Q \mathbb {Q} which loosely parametrises elliptic curves E E such that the image of ρ E \rho _E is conjugate to a subgroup of G t . G^t. In this article we give a complete classification of all such genus 0 0 modular curves that have a rational point. This classification is given in finitely many families. Moreover, for each such modular curve morphism π G : X G → X G L 2 ( Z ^ ) \pi _G \colon X_G \to X_{GL_2({\widehat {\mathbb {Z}}})} can be explicitly computed.