Papers › A Classification of Genus 0 Modular Curves with Rational Points
A Classification of Genus 0 Modular Curves with Rational Points
Rakvi
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Let E be a non-CM elliptic curve defined over ℚ. Fix an algebraic closure ℚ of ℚ. We get a Galois representation ρ_E Gal(ℚ/ℚ) →GL₂(ℤ̂) associated to E by choosing a compatible bases for the N-torsion subgroups of E(ℚ). Associated to an open subgroup G of GL₂(ℤ̂) satisfying -I ∈G and det(G)=ℤ̂^×, we have the modular curve (X_G,π_G) over ℚ which loosely parametrises elliptic curves E such that the image of ρ_E is conjugate to a subgroup of Gᵗ. 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.
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