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arXiv · 1909.01434

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

Natalia Garcia-Fritz·Hector Pasten
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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.

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BibTeXRIS

Natalia Garcia-Fritz, Hector Pasten. 2019-09-03. Towards Hilbert's Tenth Problem for rings of integers through Iwasawa theory and Heegner points. https://arxiv.org/abs/1909.01434

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