Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

TL;DR

This paper proves that for certain K3 surfaces over number fields, the Picard rank jumps infinitely often at primes, implying either many rational curves or unirational specializations, with broader implications for motives and abelian surfaces.

Contribution

It establishes the infinitude of primes where the Picard rank jumps for K3 surfaces with potentially good reduction, linking to broader results on K3 motives and applications to abelian surfaces.

Findings

Infinitely many primes where Picard rank jumps for K3 surfaces.

Existence of infinitely many rational curves or unirational specializations.

Applications to abelian surfaces with potentially good reduction.

Abstract

Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline{K}}$ has infinitely many rational curves or $X$ has infinitely many unirational specializations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field $K$ with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of $K$.