  We study whether there exist infinitely many surfaces with given discrete
invariants for which the H^2 is of CM type. This is a surface analogue of a
conjecture of Coleman about curves. We construct a large number of examples of
families of surfaces with p_g = 1 that in the moduli space cut out a special
subvariety; these provide a positive answer to our question. Most of these are
families of K3 surfaces but we also obtain some families of surfaces of general
type. As input for our construction we use the work of Deligne and Mostow.
Finally we prove that a very general K3 surface cannot be dominated by a
product of curves of small genus.
