K3 surfaces with Picard rank 20

Abstract

We determine all complex K3 surfaces with Picard rank 20 over $\mathbb{Q}$. Here the Néron–Severi group has rank 20 and is generated by divisors which are defined over $\mathbb{Q}$. 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 $\mathbb{Q}$ is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Néron–Severi group of rank 20, but not necessarily generated by divisors over $\mathbb{Q}$.