Rank stability in quadratic extensions and Hilbert’s tenth problem for the ring of integers of a number field

We show that for any quadratic extension of number fields K/F, there exists an abelian variety A/F of positive rank whose rank does not grow upon base change to K. This result implies that Hilbert’s tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring OK of integers of any number field K, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over OK has solutions in OK.