Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T22:03:08.127Z Has data issue: false hasContentIssue false
  • English
  • Français

Height estimates on cubic twists of the Fermat elliptic curve

Published online by Cambridge University Press:  17 April 2009

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

Extract

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.

We give bounds for the canonical height of rational and integral points on cubic twists of the Fermat elliptic curve. As a corollary we prove that there is no integral arithmetic progression on certain curves in this family.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 2 , October 2005 , pp. 177 - 186
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Bennet, M.A., ‘On the representation of unity by binary cubic forms’, (preprint).Google Scholar
[2]Bremner, A., Silverman, J.H. and Tzanakis, N., ‘Intergral points in arithmetic progression on y 2 = x(x 2 − n 2)’, J. Number Theory 80 (2000), 187–208.CrossRefGoogle Scholar
[3]Cremona, J.E., http://www.maths.nott.ac.uk/personal/jec/ftp/progs.Google Scholar
[4]Hindry, M. and Silverman, J.H., ‘The cannonical heights and integral points on elliptic curves’, Invent. Math. 93 (1998), 419–450.Google Scholar
[5]Lang, S., Elliptic curves: Diophantine analysis, Grundlehren der Math. Wiss (Springer-Verlag, Berlin, New York, 1978).Google Scholar
[6]Silverman, J.H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Math. 151 (Springer-Verlag, New York, 1994).CrossRefGoogle Scholar
[7]Tate, J., ‘Algorithm for determining the type of a singular fiber in an elliptic pencil’, in Modular Functions of One Variable IV, Lecture Notes in Math. 476 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar