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Squares in arithmetic progression over quadratic extensions of number fields

Enrique González-Jiménez

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.

Squares in arithmetic progression over quadratic extensions of number fields

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 curve. Specifically, we determine the set of -quadratic points on this curve under certain conditions on the base field . 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 must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions.
Paper Structure (8 sections, 7 theorems, 64 equations)

This paper contains 8 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Quadratic points on a genus $5$ curve over number fields
  3. A Result on the Non-Growth of Torsion
  4. Proof of Proposition \ref{['prop']}
  5. Proof of Theorem \ref{['quad_points']}
  6. Proof of Theorem \ref{['main']}
  7. Proof of Theorem \ref{['Qz12']}
  8. Proof of Theorem \ref{['six']}

Key Result

Theorem 1

Let $K$ be a number field with $K\ne\mathbb Q(\zeta_{12})$. Suppose that either $\texttt{cond}_A(K)$ or $\texttt{cond}_B(K)$ holds. Let $L$ be a quadratic extension of $K$. Then

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • Remark 8
  • ...and 1 more