In joint work with Barry Mazur, we show that over every number field there are many elliptic curves of rank zero, and (assuming the finiteness of Shafarevich-Tate groups) many elliptic curves of rank one.

Combining our results about ranks of twists with ideas of Poonen and Shlapentokh, we show that if one assumes the finiteness of Shafarevich-Tate groups of elliptic curves, then Hilbert's Tenth Problem is undecidable (i.e., has a negative answer) over the ring of integers of every number field.