Picard rank jumps for K3 surfaces with bad reduction

Picard rank jumps for K3 surfaces with bad reduction Salim TayouLet 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 and Tang to the case where X has bad reduction.We prove a similar result for generically ordinary nonisotrivial families of K3 surfaces over curves over p which extends previous work of Maulik, Shankar and Tang.As a consequence, we give a new proof of the ordinary Hecke orbit conjecture for orthogonal and unitary Shimura varieties.1. Introduction 77 2. GSpin Shimura varieties: integral models and their compactifications 80 3. Arithmetic intersection theory and modularity 92 4. The main estimates and proof of the main theorems 94 5. Bounding the contribution from bad reduction places 104 6. Applications