On the computation of the Picard group for $K3$ surfaces

TL;DR

This paper develops a refined method to compute the Picard group of K3 surfaces, enabling the construction of examples with Picard rank 1 more efficiently by analyzing Galois module structures on étale cohomology.

Contribution

It introduces a Galois module analysis approach that overcomes previous computational limitations, allowing for more efficient construction of K3 surfaces with Picard rank 1.

Findings

Constructed examples of K3 surfaces with Picard rank 1

Reduced computational time for examples requiring less calculation

Extended the method beyond the previous Picard rank 2 restriction

Abstract

We construct examples of $K3$ surfaces of geometric Picard rank $1$. Our method is a refinement of that of R. van Luijk. It is based on an analysis of the Galois module structure on \'etale cohomology. This allows to abandon the original limitation to cases of Picard rank $2$ after reduction modulo $p$. Furthermore, the use of Galois data enables us to construct examples which require significantly less computation time.