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

Ananth N. Shankar, Arul Shankar, Yunqing Tang, Salim Tayou

Published 2019-09-16Version 1

Given a K3 surface $X$ over a number field $K$, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite, assuming that $X$ has everywhere potentially good reduction. The result is a special case of a more general one on exceptional classes for K3 type motives associated to GSpin Shimura varieties and several other applications are given. As a corollary, we give a new proof of the fact that $X_{\overline{K}}$ has infinitely many rational curves.