Variation of Neron-Severi ranks of reductions of K3 surfaces

TL;DR

This paper investigates how the geometric Picard ranks of K3 surfaces over the rationals change when reduced modulo various primes, providing computational data and statistical analysis for smooth quartic examples.

Contribution

It offers the first extensive computational study of Picard rank variations of K3 surfaces under reduction, including data for all primes less than 2^16.

Findings

Picard ranks vary with prime reduction

Statistical patterns observed in rank distribution

Computational methods for Picard rank determination

Abstract

We study the behavior of geometric Picard ranks of K3 surfaces over the rationals under reduction modulo primes. We compute these ranks for reductions of smooth quartic surfaces modulo all primes $p<2^{16}$ in several representative examples and investigate the resulting statistics.