Studying Hilbert’s 10th problem via explicit elliptic curves

Abstract

N. García-Fritz and H. Pasten showed that Hilbert’s 10th problem is unsolvable in the ring of integers of number fields of the form \(\mathbb {Q}(\root 3 \of {p},\sqrt{-q})\) 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 \(\mathbb {Q}(\root 3 \of {p},\sqrt{Dq})\), 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.