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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Torsion subgroups of elliptic curves in short Weierstrass form
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by Michael A. Bennett and Patrick Ingram PDF
Trans. Amer. Math. Soc. 357 (2005), 3325-3337 Request permission

Abstract:

In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves $E/\mathbb {Q}$ in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any $\varepsilon >0$, all but finitely many curves \[ E_{A,B} \; : \; y^2 = x^3 + A x + B, \] where $A$ and $B$ are integers satisfying $A>|B|^{1+\varepsilon }>0$, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to $|A|>|B|^{2+\varepsilon }>0$, then the $E_{A,B}$ now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.
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Additional Information
  • Michael A. Bennett
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
  • MR Author ID: 339361
  • Patrick Ingram
  • Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4
  • MR Author ID: 759982
  • Received by editor(s): December 20, 2003
  • Received by editor(s) in revised form: February 15, 2004
  • Published electronically: March 10, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 357 (2005), 3325-3337
  • MSC (2000): Primary 11G05, 11J68
  • DOI: https://doi.org/10.1090/S0002-9947-05-03629-9
  • MathSciNet review: 2135750