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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Torsion subgroups of elliptic curves in short Weierstrass form


Authors: Michael A. Bennett and Patrick Ingram
Journal: Trans. Amer. Math. Soc. 357 (2005), 3325-3337
MSC (2000): Primary 11G05, 11J68
Published electronically: March 10, 2005
MathSciNet review: 2135750
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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

\begin{displaymath}E_{A,B} \; : \; y^2 = x^3 + A x + B, \end{displaymath}

where $A$ and $B$ are integers satisfying $A>\vert B\vert^{1+\varepsilon}>0$, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to $\vert A\vert>\vert B\vert^{2+\varepsilon}>0$, 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

Patrick Ingram
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03629-9
PII: S 0002-9947(05)03629-9
Received by editor(s): December 20, 2003
Received by editor(s) in revised form: February 15, 2004
Published electronically: March 10, 2005
Article copyright: © Copyright 2005 American Mathematical Society