Three cubes in arithmetic progression over quadratic fields

TL;DR

This paper investigates the existence of three-term arithmetic progressions of cubes over quadratic fields by analyzing rational points on elliptic curves and the rank of associated modular curve twists.

Contribution

It characterizes arithmetic progressions of cubes over quadratic fields using elliptic curve rational points and computes the Mordell-Weil group over these fields.

Findings

Determined the torsion subgroup of the elliptic curve over quadratic fields.

Provided partial results on the finiteness of the free part of the elliptic curve group.

Connected the problem to the rank of quadratic twists of a modular curve.

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^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.