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Picard-Fuchs equations of families of QM abelian surfaces. (English) Zbl 1420.14101

For a family of complex algebraic varieties, the periods of the fibers are integrals of holomorphically varying differential forms against a topologically constant family of homology classes, satisfying a certain differential equation, called the Picard-Fuchs equation (PFE). The authors describe an algorithm for computing the PFE for a family of twists of a fixed elliptic surface. The algorithm is then applied to obtain the PFEs for several examples, which come from families of Kummer surfaces over Shimura curves, as previously studied by the authors. This is used to find correspondences between the parameter spaces of these families and Shimura curves.

MSC:

14K10 Algebraic moduli of abelian varieties, classification
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14G35 Modular and Shimura varieties
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