Modularity and the construction of singular K3 surfaces

In the first half of the talk I want to give a brief introduction to some notions surrounding the concept of modularity of an algebraic variety. The goal is to make sense of the statement ``a singular K3 surface over Q is modular''.

In the second half of the talk I want to show how this property of singular K3 surfaces can be used to find specializations with maximal Picard rank 20 in a family of K3 surfaces over Q.