Abstract 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 O₊ 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 O₊ O K has solutions in O₊ O K.
Levent Alpöge (Mon,) studied this question.