## Torsion subgroups of elliptic curves in short Weierstrass form

HTML articles powered by AMS MathViewer

- 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.## References

- 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**, DOI 10.4064/aa-74-3-273-292 - Andrzej Da̧browski and Małgorzata Wieczorek,
*Families of elliptic curves with trivial Mordell-Weil group*, Bull. Austral. Math. Soc.**62**(2000), no. 2, 303–306. MR**1786212**, DOI 10.1017/S0004972700018773 - 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** - Daniel Sion Kubert,
*Universal bounds on the torsion of elliptic curves*, Proc. London Math. Soc. (3)**33**(1976), no. 2, 193–237. MR**434947**, DOI 10.1112/plms/s3-33.2.193 - B. Mazur,
*Modular curves and the Eisenstein ideal*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR**488287**, DOI 10.1007/BF02684339 - K. F. Roth,
*Rational approximations to algebraic numbers*, Mathematika**2**(1955), 1–20; corrigendum, 168. MR**72182**, DOI 10.1112/S0025579300000644 - Joseph H. Silverman,
*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210**, DOI 10.1007/978-1-4757-1920-8 - H. M. Stark,
*Effective estimates of solutions of some Diophantine equations*, Acta Arith.**24**(1973), 251–259. MR**340175**, DOI 10.4064/aa-24-3-251-259 - Małgorzata Wieczorek,
*Torsion points on certain families of elliptic curves*, Canad. Math. Bull.**46**(2003), no. 1, 157–160. MR**1955623**, DOI 10.4153/CMB-2003-016-6

## 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