Picard ranks of reductions of K3 surfaces over global fields

For a K3 surface X over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of X has larger geometric Picard rank than that of the generic fiber X. A similar statement still holds true for ordinary K3 surfaces over global function fields. In this talk, I will present the proofs via the intersection theory on GSpin Shimura varieties and also discuss various applications. These results are joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.