Families of Prym varieties of abelian coverings and shimura varieties

Under the condition that the Prym map is injective in characteristic p, we prove that the special subvarieties in the moduli space of abelian varieties of dimension l and polarization type D, Al,D, arising from families of abelian covers of P1 are of a very restrictive nature. In other words, if the family is one-dimensional or if it contains an eigenspace of certain type for the group action on the cohomology of fibers, then the Shimura varieties arising from such families can only be constructed by the group action of the family.