On the distribution of the Picard ranks of the reductions of a $K3$ surface

TL;DR

This paper investigates how the geometric Picard ranks of K3 surfaces change when reduced modulo primes, introducing a quadratic jump character to describe rank increases at certain primes.

Contribution

It introduces the jump character, a quadratic character that predicts rank jumps of K3 surfaces upon reduction at good primes.

Findings

The jump character determines when Picard rank increases.

Rank increases occur at primes where the jump character evaluates to -1.

The distribution of Picard ranks is linked to the properties of the jump character.

Abstract

We report on our results concerning the distribution of the geometric Picard ranks of $K3$ surfaces under reduction modulo various primes. In the situation that $\rk \Pic S_{\overline{K}}$ is even, we introduce a quadratic character, called the jump character, such that $\rk \Pic S_{\overline\bbF_{\!\frakp}} > \rk \Pic S_{\overline{K}}$ for all good primes, at which the character evaluates to $(-1)$.