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 | References | Similar Articles | Additional Information

Abstract: In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves 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 , all but finitely many curves

where and are integers satisfying , have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to , then the now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.

**1.**Yann Bugeaud and Kálmán Győry,*Bounds for the solutions of Thue-Mahler equations and norm form equations*, Acta Arith.**74**(1996), no. 3, 273–292. MR**1373714****2.**Andrzej Dabrowski and Małgorzata Wieczorek,*Families of elliptic curves with trivial Mordell-Weil group*, Bull. Austral. Math. Soc.**62**(2000), no. 2, 303–306. MR**1786212**, 10.1017/S0004972700018773**3.**A. Ya. Khinchin,*Continued fractions*, Translated from the third (1961) Russian edition, Dover Publications, Inc., Mineola, NY, 1997. With a preface by B. V. Gnedenko; Reprint of the 1964 translation. MR**1451873****4.**Daniel Sion Kubert,*Universal bounds on the torsion of elliptic curves*, Proc. London Math. Soc. (3)**33**(1976), no. 2, 193–237. MR**0434947****5.**B. Mazur,*Modular curves and the Eisenstein ideal*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 33–186 (1978). MR**488287****6.**K. F. Roth,*Rational approximations to algebraic numbers*, Mathematika**2**(1955), 1–20; corrigendum, 168. MR**0072182****7.**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210****8.**H. M. Stark,*Effective estimates of solutions of some Diophantine equations*, Acta Arith.**24**(1973), 251–259. Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, III. MR**0340175****9.**Małgorzata Wieczorek,*Torsion points on certain families of elliptic curves*, Canad. Math. Bull.**46**(2003), no. 1, 157–160. MR**1955623**, 10.4153/CMB-2003-016-6

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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

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