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 over R
has solutions in R. This is joint work with Levent Alpöge, Manjul
Bhargava, and Wei Ho.