  Several natural complex configuration spaces admit surprising uniformizations
as arithmetic ball quotients, by identifying each parametrized object with the
periods of some auxiliary object. In each case, the theory of canonical models
of Shimura varieties gives the ball quotient the structure of a variety over
the ring of integers of a cyclotomic field. We show that the
(transcendentally-defined) period map actually respects these algebraic
structures, and thus that occult period maps are arithmetic. As an intermediate
tool, we develop an arithmetic theory of lattice-polarized K3 surfaces.
