Introduction

These lecture notes deal with construction methods of Calabi-Yau manifolds with a special arithmetic property. In these methods we use curves with a similar arithmetic property, namely, complex multiplication. In the case of abelian varieties complex multiplication has been well studied by number theorists. The first six chapters describe how this theory for abelian varieties can be applied to the construction of curves with complex multiplication. The remaining five chapters and the appendix are devoted to the construction methods of Calabi-Yau manifolds with a similarly defined arithmetic property.

We give new examples of families of curves with dense sets of complex multiplication fibers and new examples of families of Calabi-Yau manifolds with a dense set of fibers with a similar arithmetic property. Moreover we will acquaint the reader with Mumford-Tate groups, which we use as a main tool for the study of Hodge structures and of variations of Hodge structures. The generic Mumford-Tate groups of families of cyclic covers of the projective line will be computed for a large class of examples.