Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T02:58:04.970Z Has data issue: false hasContentIssue false
  • English
  • Français

Families of elliptic curves with trivial Mordell-Weil group

Published online by Cambridge University Press:  17 April 2009

Andrzej Dabrowski  and
Małgorzata Wieczorek
Andrzej Dabrowski
Affiliation:
University of Szczecin, Institute of Mathematics, ul. Wielkopolska 15, 70–451 Szczecin, Poland, e-mail: [email protected]
Małgorzata Wieczorek
Affiliation:
University of Szczecin, Institute of Mathematics, ul. Wielkopolska 15, 70–451 Szczecin, Poland, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Fix and elliptic curve y2 = x3 + Ax + B, satisfying A, B ∈ ℤ A ≥ |B| > 0. We prove that the associated quadratic family contains infinitely many elliptic curves with trivial Mordell-Weil group.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 2 , October 2000 , pp. 303 - 306
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Breuil, C., Conrad, B., Diamond, F. and Taylor, R., ‘On the modularity conjecture for all elliptic curves’, (article in preparation).Google Scholar
[2]Bump, D., Friedberg, S. and Hoffstein, H., ‘On some applications of automorphic forms to number theory’, Bull. Amer. Math. Soc. (New Series) 33 (1996), 157–175.CrossRefGoogle Scholar
[3]Kolyvagin, V.A., ‘Finitness of E (ℚ) and III(E, ℚ) for a subclass of Weil curves’, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 522–540.Google Scholar
[4]Kubert, D.S., ‘Universal bounds on the torsion of elliptic curves’, Proc. London Math. Soc. 33 (1976), 193–237.CrossRefGoogle Scholar
[5]Mazur, B., ‘Rational isogenies of prime degree’, Invent. Math. 44 (1978), 129–162.CrossRefGoogle Scholar
[6]Nakagawa, J. and Horie, K., ‘Elliptic curves with no rational points’, Proc. Amer. Math. Soc. 104 (1988), 20–24.CrossRefGoogle Scholar
[7]Olson, L.D., ‘Torsion points on elliptic curves with given j-invariant’, Manuscripta Math. 16 (1975), 145–150.CrossRefGoogle Scholar
[8]Olson, L.D., ‘Points of finite order on elliptic curves with complex multiplication’, Manuscripta Math. 14 (1974), 195–205.CrossRefGoogle Scholar
[9]Silverman, J., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 (Springer-Verlag, Berlin, Heidelberg, New York, 1986).CrossRefGoogle Scholar