A Classification of Genus 0 Modular Curves with Rational Points

2021/05/30 by Rakvi · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2105.14623

Abstract

Let E be a non-CM elliptic curve defined over \mathbb Q. Fix an algebraic closure \mathbb Q of \mathbb Q. We get a Galois representation ρ_E \colon Gal(\mathbb Q/\mathbb Q) → GL₂(\mathbb Z) associated to E by choosing a compatible bases for the N-torsion subgroups of E(\mathbb Q). Associated to an open subgroup G of GL₂(\mathbb Z) satisfying -I ∈ G and det(G)=\mathbb Z^×, we have the modular curve (X_G,π_G) over \mathbb Q which loosely parametrises elliptic curves E such that the image of ρ_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.

Cited by

• Modular curves of prime-power level with infinitely many quadratic points

Related

• Explicit open images for elliptic curves over ℚ
• Genus 0 Modular curves of prime power level with a point defined over number fields other than \Q
• On the possible images of the mod ell representations associated to\n elliptic curves over Q
• Rational curves on elliptic surfaces
• Families Of Elliptic Curves With The Same Mod 8 Representations