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

Abstract

AbstractGiven a K3 surfaceXover a number fieldKwith potentially good reduction everywhere, we prove that the set of primes ofKwhere the geometric Picard rank jumps is infinite. As a corollary, we prove that either$X_{\overline {K}}$has infinitely many rational curves orXhas infinitely many unirational specialisations.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 fieldKwith potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes ofK.