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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: tjedr@sus.univ.szczecin.pl
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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