  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).
