Joint IAS/Princeton University Number Theory Seminar

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. 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.