Picard rank jumps for K3 surfaces with bad reduction

Abstract

Let $X$ be a K3 surface over a number field. We prove that $X$ has infinitely many specializations where its Picard rank jumps, hence extending our previous work with Shankar--Shankar--Tang to the case where $X$ might have potentially bad reduction. We prove a similar result for generically ordinary non-isotrivial families of K3 surfaces over curves over ${\overline{\mathbb{F}}}_p$ which extends previous work of Maulik--Shankar--Tang. As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.

Minor revisions. To appear in Algebra & Number Theory

References in corpus (3)

• Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
• Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
• On the ordinary Hecke orbit conjecture