Three cubes in arithmetic progression over quadratic fields
Abstract
We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields Q(sqrt(D)), where D is a squarefree integer. For this purpose, we give a characterization in terms of Q(sqrt(D))rational points on the elliptic curve E:y^2=x^327. We compute the torsion subgroup of the MordellWeil group of this elliptic curve over Q(sqrt(D)) and we give partial answers to the finiteness of the free part of E(Q(sqrt(D))). This last task will be translated to compute if the rank of the quadratic Dtwist of the modular curve X_0(36) is zero or not.
 Publication:

arXiv eprints
 Pub Date:
 September 2009
 arXiv:
 arXiv:0909.0227
 Bibcode:
 2009arXiv0909.0227G
 Keywords:

 Mathematics  Number Theory;
 11B25;
 14H52
 EPrint:
 Archiv der Mathematik Vol. 95, no. 3, 233241 (2010)