Arithmetic occult period maps

Abstract

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.