Hilbert's 10th Problem Over Rings of Integers

We show that for every 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. By work of Shlapentokh, this implies that Hilbert's tenth problem over the ring of integers R of any number field has a negative solution. That is, there does not exist an algorithm to determine whether a polynomial equation in several variables over R has solutions in R. This is joint work with Levent Alpöge, Manjul Bhargava, and Wei Ho.