Three cubes in arithmetic progression over quadratic fields

We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields Q( √ D), where D is a squarefree integer. For this purpose, we give a characterization in terms of Q( √ D)-rational points on the elliptic curve E : y = x − 27. We compute the torsion subgroup of the Mordell–Weil group of this elliptic curve over Q( √ D) and we give an explicit answer, in terms of D, to the finiteness of the free part of E(Q( √ D)) for some cases. We translate this task to computing whether the rank of the quadratic D-twist of the modular curve X0(36) is zero or not. Mathematics Subject Classification (2010). 11B25, 14H52.