On the solutions to $Ax^p+By^p+Cz^p=0$ over quadratic fields

TL;DR

This paper investigates the Diophantine equation $A x^p + B y^p + C z^p = 0$ over quadratic fields $K = \\mathbb{Q}(\\sqrt{d})$ with pairwise coprime $p$th-power-free integers $A,B,C$ and prime $p>3$. It develops a hyperelliptic-curve framework via the curve $Y^2 = X^p + \frac{A^2(BC)^{p-1}}{4}$, linking $K$-rational solutions to rational or quadratic-field points and using this to deduce strong nonexistence and structure results. The main findings are: (i) there are no solutions in $K\\setminus\\mathbb{Q}$ when $BC eq \\pm 1$; (ii) when $BC = \\pm 1$ and $A eq \\pm 2$, possible solutions lie in $K\\setminus\\mathbb{Q}$ with explicit constraints; and (iii) in the special case $ABC = \\pm 1$, the curve simplifies and provides an explicit correspondence from rational points to $K$-solutions, yielding a complete description in quadratic fields. These results give a precise, curve-theoretic handle on the arithmetic of $Ax^p+By^p+Cz^p=0$ over quadratic extensions.

Abstract

We provide the necessary conditions for the existence of solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ over any quadratic number field $K$ with $A,B,C$ pth powerfree integer numbers. We determine when $x$, $y$ and $z$ are rational numbers for pairwise coprime integers $A$, $B$ and $C$. Moreover, we prove that $x$, $y$ and $z$ are in $K\setminus\mathbb{Q}$ when $BC=\pm 1$ and $A\neq \pm 2$. Finally, we prove that no solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ exist in $K\setminus\mathbb{Q}$ when $BC\neq \pm 1$.

Theorems & Definitions (39)

• Theorem 1.1
• Corollary 1.2
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5