Hyperkähler varieties and their relation to Shimura stacks

This thesis consists of three parts. The first part is an exposition of the well known fact that certain Shimura varieties are moduli spaces of abelian motives. This is known for all Shimura varieties of abelian type, but we restrict ourselves to those of Hodge and orthogonal type. Using this and the global Torelli theorem, it has been shown that the period map for polarized K3 surfaces descends to an open embedding of the moduli space of polarized K3 surfaces over Q into a Shimura stack of orthogonal type. The second part of the thesis generalizes this to hyperkahler varieties of higher dimension. We focus in particular on hyperkahler varieties of K3^[n] type, for which we show an analogue over Q of a result of Markman on their monodromy over C. In the final part we use these results to explicitly compute the spinor norm of monodromy operators on K3 surfaces in mixed characteristic. This gives rise to a necessary condition for a lattice to be the Neron-Severi lattice of a K3 surface over a non-closed field. It also allows us to compute the value at -1 of the characteristic polynomial of the Frobenius on the second cohomology of certain K3 surfaces over a finite field.