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Product formulas for periods of CM abelian varieties and the function field analog. (English) Zbl 1469.11179

Summary: We survey Colmez’s theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and \(A\)-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and \(A\)-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G15 Complex multiplication and moduli of abelian varieties
11R42 Zeta functions and \(L\)-functions of number fields
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