Squares in arithmetic progression over certain non-primitive quartic number fields

TL;DR

This work extends known results on arithmetic progressions of squares from quadratic fields to quadratic extensions of $\mathbb{Q}(\sqrt{D})$ for square-free $D$, under explicit rank and class-number hypotheses. By encoding five-square progressions via the genus-5 curve $V$ and a G(t)-parametrization, the authors reduce the problem to the arithmetic of twists $E_{0}^{D}$ and $E_{1}^{\pm D}$ and the torsion-growth behavior of related curves over quadratic fields. They prove a dichotomy: if $\operatorname{rank}_{\mathbb{Z}}E_{0}^{D}(\mathbb{Q})=0$ (and $D\neq \pm 2$), there are no non-constant five-square progressions over any quadratic extension of $\mathbb{Q}(\sqrt{D})$; if the rank is nonzero and the class number is 1, progressions exist and are, up to equivalence, of the form $(a^{2},b^{2},c^{2},\alpha d^{2},e^{2})$ with $\alpha$ non-square. The paper also treats special small-$D$ cases ($D=-1,\pm2,3$) yielding explicit unique examples in certain biquadratic fields, determines when six-term progressions can occur, and provides a detailed parametrization in the $D=-2$ and $D=2$ settings. Overall, the work combines Mordell-style explicit parametrization with torsion-growth data to classify five-square progressions in a broad family of quadratic- and biquadratic-number-field contexts, offering concrete arithmetic descriptions and infinite families in several cases.

Abstract

Let $D$ be a square-free integer. Under certain conditions on $D$, we characterize non-constant arithmetic progressions of squares over quadratic extensions of $\mathbb{Q}(\sqrt{D})$.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Remark
• Theorem 3
• Remark
• Remark
• Conjecture 4
• Lemma 5
• proof
• Proposition 6