  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.
