  We determine all complex K3 surfaces with Picard rank 20 over Q. Here the
Neron-Severi group has rank 20 and is generated by divisors which are defined
over Q. Our proof uses modularity, the Artin-Tate conjecture and class group
theory. With different techniques, the result has been established by Elkies to
show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface.
We then apply our methods to general singular K3 surfaces, i.e. with
Neron-Severi group of rank 20, but not necessarily generated by divisors over
Q.
