Hilbert's 10th Problem via Mordell curves

TL;DR

The paper establishes new undecidability results for Hilbert’s 10th problem over rings of integers in number fields formed by cubic extensions and their composites with CM Mordell curves. It links the cube-sum problem to ranks of cubic twists of Mordell curves ${\mathsf E}_a: y^2=x^3+a$, using CM structure to translate cube-sum properties into positive rank criteria, and then applies Shlapentokh’s rank-implication framework to deduce Diophantine undecidability in extensions like $\mathbb Q(\zeta_3,\sqrt[3]{p})$ for a positive density of primes $p$, and in infinite families $L_{D,p}=\mathbb Q(\zeta_3,\sqrt{D},\sqrt[3]{p})$. The core results show unsolvability for about $5/6$ of primes $p$ in the first class and extend to degree-12 extensions under explicit density statements, with explicit cube-sum data guiding the rank behavior of the associated Mordell curves. The methods differ from quadratic-twist approaches by focusing on cubic twists and Mordell curves, and yield infinitely many new number fields where Hilbert’s 10th problem remains unsolvable, contributing to the landscape of Diophantine undecidability in algebraic number theory.

Abstract

We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(ζ_3, \sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields $\mathbb{Q}(ζ_3, \sqrt{D}, \sqrt[3]{p})$ for every $D \in S$ and every prime $p \equiv 2,5 \pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof.

Theorems & Definitions (29)

• Theorem A
• Proposition A
• Theorem 1.1
• Theorem 1.2: A. Shlapentokh shlapentokh2008elliptic
• Theorem 1.3
• Lemma 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Example 2.5