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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Torsion subgroups of rational elliptic curves over the compositum of all cubic fields
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by Harris B. Daniels, Álvaro Lozano-Robledo, Filip Najman and Andrew V. Sutherland PDF
Math. Comp. 87 (2018), 425-458 Request permission

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
  • MR Author ID: 1105200
  • Email: hdaniels@amherst.edu
  • Álvaro Lozano-Robledo
  • Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
  • Email: alvaro.lozano-robledo@uconn.edu
  • Filip Najman
  • Affiliation: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
  • MR Author ID: 886852
  • Email: fnajman@math.hr
  • Andrew V. Sutherland
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • MR Author ID: 852273
  • ORCID: 0000-0001-7739-2792
  • Email: drew@math.mit.edu
  • 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.
  • © Copyright 2017 American Mathematical Society
  • Journal: Math. Comp. 87 (2018), 425-458
  • MSC (2010): Primary 11G05; Secondary 11R21, 12F10, 14H52
  • DOI: https://doi.org/10.1090/mcom/3213
  • MathSciNet review: 3716201