Hilbert's 10th Problem via Mordell curves

TL;DR

This paper demonstrates the unsolvability of Hilbert's 10th Problem over certain number fields related to Mordell curves and primes, using CM elliptic curves and properties of cube sums.

Contribution

It establishes new classes of number fields where Hilbert's 10th Problem is undecidable, linking elliptic curves and prime distributions.

Findings

Hilbert's 10th Problem is unsolvable for 5/6 of all primes p in specific fields.

Infinite set S of square-free integers where the problem remains unsolvable over related fields.

Use of CM elliptic curves associated with the cube sum problem in the proof.

Abstract

We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\mathbb{Q}(\zeta_3, \sqrt[3]{p})$. 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 $\mathbb{Q}(\zeta_3, \sqrt{D}, \sqrt[3]{p})$ 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.