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 $K/F$ K / F , there exists an abelian variety $A/F$ A / F of positive rank whose rank does not grow upon base change to K 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 O K of integers of any number field K K , there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over OK O K has solutions in OK O K .