  The geometric objects of study in this paper are K3 surfaces which admit a
polarization by the unique even unimodular lattice of signature (1,17). A
standard Hodge-theoretic observation about this special class of K3 surfaces is
that their polarized Hodge structures are identical with the polarized Hodge
structures of abelian surfaces that are cartesian products of elliptic curves.
Earlier work of the first two authors gives an explicit normal form and
construction of the moduli space for these surfaces. In the present work, this
normal form is used to derive Picard-Fuchs differential equations satisfied by
periods of these surfaces. We also investigate the subloci of the moduli space
on which the polarization is enhanced. In these cases, we derive information
about the Picard-Fuchs differential equations satisfied by periods of these
subfamilies, and we relate this information to the theory of genus zero
quotients of the upper half-plane by Moonshine groups. For comparison, we also
examine the analogous theory for elliptic curves in Weierstrass form.
