Functiones et Approximatio Commentarii Mathematici

A height inequality for rational points on elliptic curves implied by the abc-conjecture

Ulf Kühn and Jan Steffen Müller

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this short note we show that the uniform $abc$-conjecture puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of Vojta's height inequality formulated by Mochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferich primes.

Article information

Source
Funct. Approx. Comment. Math. Volume 52, Number 1 (2015), 127-132.

Dates
First available in Project Euclid: 20 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.facm/1426857040

Digital Object Identifier
doi:10.7169/facm/2015.52.1.10

Mathematical Reviews number (MathSciNet)
MR3326129

Zentralblatt MATH identifier
06425018

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11G50: Heights [See also 14G40, 37P30]

Keywords
elliptic curves abc-conjecture rational points heights Wieferich primes

Citation

Kühn, Ulf; Müller, Jan Steffen. A height inequality for rational points on elliptic curves implied by the abc-conjecture. Funct. Approx. Comment. Math. 52 (2015), no. 1, 127--132. doi:10.7169/facm/2015.52.1.10. https://projecteuclid.org/euclid.facm/1426857040.


Export citation

References

  • E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006.
  • M. Einsiedler, G. Everest, and T. Ward, Primes in elliptic divisibility sequences, LMS J. Comput. Math. 4 (2001), 1–13.
  • G. Everest, J. Reynolds, and S. Stevens, On the denominators of rational points on elliptic curves, Bull. Lond. Math. Soc. 39 (2007), no. 5, 762–770.
  • A. Granville and H.M. Stark, ABC implies no “Siegel zeros” for L-functions of characters with negative discriminant, Inventiones Math. 139 (2000), 509–523.
  • S. Mochizuki, Arithmetic elliptic curves in general position, Math. J. Okayama Univ. 52 (2010), 1–28.
  • S. Mochizuki, Inter-universal Teichmüller theory IV: Log-volume computations and set-theoretic foundations, Preprint (2012).
  • J. Reynolds, Perfect powers in elliptic divisibility sequences, J. Number Theory 132 (2012), no. 5, 998–1015.
  • J.H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.
  • J.H. Silverman, Wieferich's criterion and the $abc$-conjecture, J. Number Theory 30 (1988), no. 2, 226–237.
  • K. Stange, Elliptic nets and elliptic curves, Algebra and Number Theory 5 (2011), no. 2, 197–229.
  • M. Van Frankenhuysen, The $ABC$ conjecture implies Vojta's height inequality for curves, J. Number Theory 95 (2002), no. 2, 289–302.
  • J.F. Voloch, Elliptic Wieferich primes, J. Number Theory 81 (2000), no. 2, 205–209.
  • M. Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31–74.