Papers › Explicit open images for elliptic curves over ℚ
Explicit open images for elliptic curves over ℚ
David Zywina
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For a non-CM elliptic curve E defined over ℚ, the Galois action on its torsion points gives rise to a Galois representation ρ_E: Gal(ℚ/ℚ)→GL₂(ℤ) that is unique up to isomorphism. A renowned theorem of Serre says that the image of ρ_E is an open, and hence finite index, subgroup of GL₂(ℤ). We describe an algorithm that computes the image of ρ_E up to conjugacy in GL₂(ℤ); this algorithm is practical and has been implemented. Up to a positive answer to a uniformity question of Serre and finding all the rational points on a finite number of explicit modular curves of genus at least 2, we give a complete classification of the groups ρ_E(Gal(ℚ/ℚ))∩SL₂(ℤ) and the indices [GL₂(ℤ):ρ_E(Gal(ℚ/ℚ))] for non-CM elliptic curves E/ℚ. Much of the paper is dedicated to the efficient computation of modular curves via modular forms expressed in terms of Eisenstein series.
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