Abstract

For a positive proportion of primes p and q, we prove that Z is Diophantine in the ring of integers of Q( 3 v p, v -q). This provides a new and explicit infinite family of number fields K such that Hilbert's tenth problem for OK is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.