Using elliptic curves of rank one towards the undecidability of   Hilbert's Tenth Problem over rings of algebraic integers
  Let F and K be number fields, with F contained in K. and let O_F and O_K be their rings of integers. If there exists an elliptic curve E over F such that E(F) and E(K) have rank 1, then there exists a diophantine definition of O_F over O_K. 
Bjorn Poonen