Exceptional jumps of Picard rank of K3 surfaces over number fields

Given a K3 surface XX over a number field KK, we prove that the set of primes of KK where the geometric Picard rank jumps is infinite, assuming that XX has everywhere potentially good reduction. This result is formulated in the general framework of GSpin Shimura varieties and I will explain other applications to abelian surfaces. I will also discuss applications to the existence of rational curves on K3 surfaces. The results in this talk are joint work with Ananth Shankar, Arul Shankar and Yunqing Tang.