Nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over quadratic number fields

TL;DR

The paper links solvability of the Fermat-type equation $x^3+y^3=kz^3$ over the quadratic field $K=\mathbb{Q}(\sqrt{d})$ to rational points on the elliptic curve $E: y^2=x^3-432d^3k^2$ by proving that a non-torsion $E(\mathbb{Q})$-point yields a nontrivial $K$-solution. Under the hypotheses that $d$ is squarefree, $k>0$ cubefree, and BSD holds for rank $>1$, it derives computable criteria based on root numbers and signatures to determine the existence of such solutions, including a detailed treatment for $k=1$ with explicit congruence conditions on $|d|$ modulo $9$. The paper also provides an explicit mapping from $E(\mathbb{Q})$-points to $K$-solutions and develops a comprehensive framework (via functional equation signs and local root numbers) to assess rank positivity. Together, these results yield practical, arithmetic criteria for detecting nontrivial Fermat-type solutions over quadratic fields, anchored in the rank of specific elliptic curves.

Abstract

We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over $\mathbb{Q}(\sqrt{d})$ by constructing a relationship with the points on the elliptic curve $y^2=x^3-432d^3k^2$ over $\mathbb{Q}$ for certain $k\in\mathbb{N}$.

Theorems & Definitions (9)

• Theorem 1.1
• Corollary 1.3
• proof
• Proposition 3.1
• Theorem 3.2: D
• Definition 3.3
• Theorem 4.1
• proof
• proof