Rank stability makes rings of integers diophantine

TL;DR

This paper explores how rank stability of abelian varieties over number field extensions influences the diophantine definability of rings of integers, providing an alternative proof of a key theorem related to Hilbert's tenth problem.

Contribution

It offers a new proof of a theorem linking rank stability to diophantine properties of rings of integers, impacting the understanding of Hilbert's tenth problem over number fields.

Findings

Alternative proof of the key theorem on rank stability and diophantine definability.

Clarification of the theorem's role in the negative solution to Hilbert's tenth problem.

Review of applications of the theorem in number theory and logic.

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.