Density of arithmetic Hodge loci

I will explain a conjecture on density of arithmetic Hodge loci which includes and generalizes several recent density results of these loci in arithmetic geometry. This conjecture has also analogues over functions fields that I will survey. As a particular instance, I will outline the proof of the following result: a K3 surface over a number field admits infinitely many specializations where its Picard rank jumps. This last result is joint work with Ananth Shankar, Arul Shankar and Yunqing Tang.