  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.
