Squares in arithmetic progression over quadratic extensions of number fields

TL;DR

This paper investigates arithmetic progressions of squares over quadratic extensions of number fields, characterizing such progressions via genus 5 curves and elliptic curve properties, establishing bounds on their length.

Contribution

It introduces a novel approach linking quadratic points on genus 5 curves to progressions of squares over quadratic fields, extending Mordell's method.

Findings

Quadratic progressions of five or six squares are very restricted.

No progressions of length greater than six exist under certain conditions.

The method involves analyzing algebraic properties of elliptic curves after base change.

Abstract

We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we determine the set of $K$-quadratic points on this curve under certain conditions on the base field $K$. Our main results rely on the algebraic properties of specific elliptic curves after performing a base change to suitable number fields. As a consequence, we establish that, under appropriate assumptions, any non-elementary arithmetic progression of five or six squares properly defined over a quadratic extension of $K$ must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions.