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Mathematics of Computation

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Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

Authors: Harris B. Daniels, Álvaro Lozano-Robledo, Filip Najman and Andrew V. Sutherland
Journal: Math. Comp. 87 (2018), 425-458
MSC (2010): Primary 11G05; Secondary 11R21, 12F10, 14H52
Published electronically: May 5, 2017
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Abstract: Let $ E/\mathbb{Q}$ be an elliptic curve and let $ \mathbb{Q}(3^\infty )$ be the compositum of all cubic extensions of $ \mathbb{Q}$. In this article we show that the torsion subgroup of $ E(\mathbb{Q}(3^\infty ))$ is finite and we determine 20 possibilities for its structure, along with a complete description of the $ \overline {\mathbb{Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $ \overline {\mathbb{Q}}$-isomorphism classes of elliptic curves, and a complete list of $ j$-invariants for each of the 4 that do not.

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Additional Information

Harris B. Daniels
Affiliation: Department of Mathematics and Statistics, Amherst College, Amherst, Massachusetts 01002

Álvaro Lozano-Robledo
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Filip Najman
Affiliation: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia

Andrew V. Sutherland
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): February 23, 2016
Received by editor(s) in revised form: August 24, 2016
Published electronically: May 5, 2017
Additional Notes: The third author acknowledges support from the QuantiXLie Center of Excellence. The fourth author was supported by NSF grants DMS-1115455 and DMS-1522526.
Article copyright: © Copyright 2017 American Mathematical Society

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