Papers › Sporadic Cubic Torsion
Sporadic Cubic Torsion
Maarten Derickx, Anastassia Etropolski, Mark van Hoeij, Jackson S. Morrow, David Zureick-Brown
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Let K be a number field, and let E/K be an elliptic curve over K. The Mordell--Weil theorem asserts that the K-rational points E(K) of E form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of E(K) for K a cubic number field. To do so, we determine the cubic points on the modular curves X₁(N) for N = 21, 22, 24, 25, 26, 28, 30, 32, 33, 35, 36, 39, 45, 65, 121. As part of our analysis, we determine the complete list of N for which J₀(N) (resp., J₁(N), resp., J₁(2,2N)) has rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on J₁(N)(ℚ) is generated by Gal(ℚ̅/ℚ)-orbits of cusps of X₁(N)_(ℚ̅) for N≤55, N ≠54.
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