K3 surfaces with Picard rank 20

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 apply our methods to general singular K3 surfaces, that is, those with Neron-Severi group of rank 20, but not necessarily generated by divisors over Q.