Sporadic cubic torsion

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(Q̄/Q)-orbits of cusps of X₁(N)Q̄ for N≤ 55, N ≠ 54.