Periods of Drinfeld modules and local shtukas with complex multiplication

Colmez [Col93] 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. Mathematics Subject Classification (2000): 11G09, (11R42, 11R58, 14L05)