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arXiv · 2510.19553

Rank stability makes rings of integers diophantine

Bjorn Poonen
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Abstract

The recent negative answer to Hilbert's tenth problem over rings of integers relies on a theorem that for every extension of number fields $L/K$, if there is an abelian variety $A$ over $K$ such that $0 < \operatorname{rank} A(K) = \operatorname{rank} A(L)$, then $\mathcal{O}_K$ is $\mathcal{O}_L$-diophantine. We present an alternative proof of this theorem and review how it is used.

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Bjorn Poonen. 2025-10-22. Rank stability makes rings of integers diophantine. https://arxiv.org/abs/2510.19553

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