Rank stability makes rings of integers diophantine

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.