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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 lists of N for which J0(N), J1(N), and J1(2,2N) have 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 Galois-orbits of cusps of X1(N) for N ≤55, N ̸=54.
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This content will become publicly available on April 1, 2026
Ogg’s torsion conjecture: Fifty years later
Andrew Ogg’s mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us. Ogg’s celebrated torsion conjecture—as it relates to modular curves—can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. We give a survey of Ogg’s torsion conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years.
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- Award ID(s):
- 1945452
- PAR ID:
- 10656984
- Publisher / Repository:
- AMS
- Date Published:
- Journal Name:
- Bulletin of the American Mathematical Society
- Volume:
- 62
- Issue:
- 2
- ISSN:
- 0273-0979
- Page Range / eLocation ID:
- 235 to 268
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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