  The zeta function of a K3 surface over a finite field satisfies a number of
obvious (archimedean and l-adic) and a number of less obvious (p-adic)
constraints. We consider the converse question, in the style of Honda-Tate:
given a function Z satisfying all these constraints, does there exist a K3
surface whose zeta-function equals Z? Assuming semi-stable reduction, we show
that the answer is yes if we allow a finite extension of the finite field. An
important ingredient in the proof is the construction of complex projective K3
surfaces with complex multiplication by a given CM field.
