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Periods of Drinfeld modules and local shtukas with complex multiplication. (English) Zbl 07146739
Summary: Colmez [Périodes des variétés abéliennes a multiplication complexe, Ann. of Math. (2)138(3) (1993), 625-683; available at http://www.math.jussieu.fr/$$\sim$$ colmez] conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at $$s=0$$ of certain Artin $$L$$-functions. In a series of articles we investigate the analog of Colmez’s theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher-dimensional generalizations, so-called $$A$$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM $$A$$-motive at all finite places in terms of Artin $$L$$-series. The latter is achieved by investigating the local shtukas associated with the $$A$$-motive.

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11R42 Zeta functions and $$L$$-functions of number fields 11R58 Arithmetic theory of algebraic function fields 14L05 Formal groups, $$p$$-divisible groups
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