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11G15 Number theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Complex multiplication and moduli of abelian varieties

14D07 Algebraic geometry -- Families, fibrations -- Variation of Hodge structures

14G35 Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Modular and Shimura varieties

14H30 Algebraic geometry -- Curves -- Coverings, fundamental group En-ligne: Springerlink | Zentralblatt

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The lecture notes under review is concerned about the construction of families of Calabi–Yau 3-manifolds with complex multiplication (CMCY for short). Such families of CMCY 3-manifolds are constructed using curves with dense sets of fibers with complex multiplication.

The theory of complex multiplication for abelian varieties is applied to construct curves with complex multiplication. The first six chapters are devoted to this task. Since curves can be determined by their Hodge structures. In this note, Hodge structures and variations of Hodge structures are studied in detail. The main tool for the study of Hodge structures is the generic Mumford-Tate groups. The Hodge groups are discussed and in particular, the Hodge structure is computed for a universal family of hyperelliptic curves. The necessary theoretical background materials, such as Shimura varietes, Hodge theory, are also collected in these early chapters.

The last five chapters describe the construction of Calabi–Yau 3-folds with complex multiplication (CYCM). ... (Zentralblatt)

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