Rank stability makes rings of integers diophantine

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