Complex multiplication and brauer groups of K3 surfaces

One may ask how much of the classical theory of complex multiplication translates to K3 surfaces. This question looks natural and it is justified by the deep similarities between K3 surfaces and Abelian varieties, that are geometric (they are the only Calabi-Yau surfaces) or motivic (in some appropriate category, the motive of every K3 is Abelian) or moduli-space theoretical, since both objects are parametrised by Shimura varieties. The aim of this thesis is to assemble all these similarities to obtain a theory for CM K3 surfaces which bears many resemblances and yet many interesting differences to the classical one of Abelian varieties. Since our original motivation was to understand the Brauer groups of CM K3 surfaces, the results obtained will also have practical applications in this direction. In particular, given a number field $K$ and a CM number field $E$, we are able to write down the finitely many Brauer groups $\text{Br}(\overline{X})^{G_K}$ of any K3 surface $X/K$ with CM by $E$. A second question we are going to be interested in regards fields of definition. It was known since Piatetski-Shapiro and Shafarevich that every complex K3 surface with CM can be descended over $\overbar{\Q}$, so one would like to know if there is a natural choice for a field of definition, as it happens for elliptic curves. We show that this is true under some mild condition on the quadratic form associated to the transcendental lattice $T(X)$, and this allows us to give an elementary proof of a finiteness theorem only recently proved by Orr and Skorobogatov.Open Acces