K3 surfaces with Picard rank 20

TL;DR

This paper classifies all complex K3 surfaces with the maximum Picard rank of 20 over Q, using advanced tools like modularity, the Artin-Tate conjecture, and class group theory.

Contribution

It provides a complete classification of K3 surfaces with Picard rank 20 over Q, extending previous results with new techniques.

Findings

All such K3 surfaces over Q are classified.

The Neron-Severi group is generated by divisors defined over Q.

The methods apply to general singular K3 surfaces with Picard rank 20.

Abstract

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.