Adrian Vasiu

We first review the connection between polarized $K3$-surfaces and one particular Shimura variety $Sh(G,X)$, with $G^{ad}=SO(2,19)$. Second we use this to show the existence of moduli schemes of polarized (or just pseudo-polarized) $K3$ surfaces (the degree of the polarization being fixed) over $Spec(\dbZ)$ punctured in some points. Third we get some corollaries, inspired from the well known facts for abelian varieties (like Shafarevich type question, like classufication over finite fields, etc). Fourth we show the connection between the study of the special fibres of these integral models and the study of Hecke orbits of points of the special fibre of different integral canonical models of $Sh(G,X)$. Fifth we state a general criterion of density of Hecke orbits of ordinary points in the general context of Shimura varieties of preabelian type: if one such orbit is dense, then all such orbits are dense; this regains the previously known cases of density of Hecke orbits.