  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^3-27. We compute the torsion subgroup of
the Mordell-Weil 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 D-twist of the
modular curve X_0(36) is zero or not.
