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 $${{\mathbb{Q}(\sqrt{D})}}$$ , where D is a squarefree integer. For this purpose, we give a characterization in terms of $${{\mathbb{Q}(\sqrt{D})}}$$ -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 $${{\mathbb{Q}(\sqrt{D})}}$$ and we give an explicit answer, in terms of D, to the finiteness of the free part of $${E({\mathbb{Q}(\sqrt{D})})}$$ for some cases. We translate this task to computing whether the rank of the quadratic D-twist of the modular curve X 0(36) is zero or not.