On classification of genus 0 modular curves with a rational point

Let E be an elliptic curve defined over ℚ. Fix ℚ̅ as an algebraic closure of ℚ. We get a Galois representation

ρ_E: Gal(ℚ) → GL₂(ℤ̂)

by choosing a compatible bases for the N-torsion subgroups of E(ℚ̅). Let G be an open subgroup of GL₂(ℤ̂) such that det(G)=ℤ̂ and -I ∈ G. Associated to G we have the modular curve X_G which loosely parametrises elliptic curves E such that im(ρ_E) ⊆ G. In this talk I will start with the description of modular curves and then discuss some techniques to classify genus 0 modular curves with a rational point.