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On the periods of abelian varieties. (English) Zbl 1475.11121

Summary: In this expository paper, we review the formula of Chowla and Selberg for the periods of elliptic curves with complex multiplication, and discuss two methods of proof. One uses Kronecker’s limit formula and the other uses the geometry of a family of abelian varieties. We discuss a generalization of this formula, which was proposed by Colmez, as well as some explicit Hodge cycles which appear in the geometric proof.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
11G15 Complex multiplication and moduli of abelian varieties
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14K10 Algebraic moduli of abelian varieties, classification
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