Workshop on Special Cycles and Related Topics

Abstract: Abstract: Let $C$ be a curve defined over a finite field, and let $X/C$ be a non-isotrivial family of K3 surfaces. In joint work with Maulik-Tang, under a compactness assumption (an assumption removed in later work by Tayou), we prove that if the K3 surface is generically ordinary, the locus where the Picard rank is greater than the generic Picard rank is infinite. There are examples of non-ordinary families of K3 surfaces where the locus where the Picard rank jumps is empty, and examples where this locus is finite. I will speak on joint work in progress with Ruofan Jiang and Ziquan Yang where we classify exactly when this locus is infinite under the hypothesis that the endomorphism ring of the generic Kuga-Satake abelian variety is determined by the generic Picard rank.