Studying Hilbert's 10th problem via explicit elliptic curves

TL;DR

This paper advances the understanding of Hilbert's 10th problem by improving results on its unsolvability in certain number fields using multiple elliptic curves and a more direct approach.

Contribution

It extends previous work by increasing the proportion of primes for which Hilbert's 10th problem is unsolvable and broadens the class of number fields considered, employing multiple elliptic curves and a new method.

Findings

Increased the proportion of primes for which the problem is unsolvable.

Extended results to new classes of number fields involving square-free integers.

Replaced Iwasawa theory with a more direct method.

Abstract

N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. We improve their proportions and extend their results to the case of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{Dq})$, where $D$ belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.