Local to global principle for the moduli space of K3 surfaces

Abstract

Recently S. Patrikis, J.F. Voloch and Y. Zarhin have proven, assuming several well known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, under some technical restrictions. This is possible since abelian varieties and K3s are quite well described by `Hodge-theoretical' results. In particular the theorem we present can be interpreted as follows: a family of $\ell$-adic representations that looks like the one induced by the transcendental part of the $\ell$-adic cohomology of a K3 surface (defined over a number field) determines a Hodge structure which in turn determines a K3 surface (which may be defined over a number field).