Sporadic Cubic Torsion [article]

Maarten Derickx, Anastassia Etropolski, Mark van Hoeij, Jackson S. Morrow, David Zureick-Brown
2020 arXiv   pre-print
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 X_1(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
more » ... lete list of N for which J_0(N) (resp., J_1(N), resp., J_1(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 J_1(N)(Q) is generated by Gal(Q̅/Q)-orbits of cusps of X_1(N)_Q̅ for N≤ 55, N ≠ 54.
arXiv:2007.13929v1 fatcat:mz4tjwlwvvbuhjsrllt2a3e2li