Towards Hilbert's Tenth Problem for rings of integers through Iwasawa theory and Heegner points

TL;DR

This paper demonstrates that for many primes, the ring of integers in certain cubic and quadratic extensions has an unsolvable Hilbert's tenth problem, using advanced techniques from Iwasawa theory and Heegner points.

Contribution

It introduces a new explicit family of number fields where Hilbert's tenth problem is undecidable, employing novel applications of Iwasawa theory and Heegner point congruences.

Findings

Proves Z is Diophantine in rings of integers of specific number fields.

Establishes infinite families of fields with undecidable Hilbert's tenth problem.

Uses elliptic curve rank stability via Iwasawa theory and Heegner points.

Abstract

For a positive proportion of primes $p$ and $q$, we prove that $\mathbb{Z}$ is Diophantine in the ring of integers of $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$. This provides a new and explicit infinite family of number fields $K$ such that Hilbert's tenth problem for $O_K$ is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.