Cyclic coverings, Calabi-Yau manifolds and Complex multiplication

We construct families of Calabi-Yau manifolds with dense set of complex multiplication fibers in an arbitrary dimension. We will also give explicite examples of complex multiplication fibers. For this construction we use families of curves with dense set of complex multiplication fibers. In addition, we give examples of such families for each genus less or equal 7 and we study the generic Hodge groups of families of cyclic covers of the projective line.