There is the well known conjecture of Colmez, which describes the logarithmic derivative of the $L$-function of a character via the periods of CM-abelian varieties. Equivalently it describes the Faltings heights of an abelian variety with complex multiplication in terms of the logarithmic derivatives of $L$-functions of characters.
We know the conjecture in the abelian case, in some cases in degree four and "on average" which is a powerful enough result to show the André-Oort conjecture.
I'm a bit uncertain however what the conjecture itself is telling me "philosophically/morally". Colmez's explenation that it comes from a product formula (suitably made sensible) for the periods of CM abelian varieties is cool, yet I'm not sure which side is the "easy" one giving informations about the "hard" side (typically the logarithmic derivative should be "easy" giving us access to arithmetic information, say in the class number formula).
Does Colmez's conjecture help us understand more general abelian varieties or what would be applications of it? Ultimately why would we like to bound the Faltings height of CM abelian varieties (I'm sure we want to, I'm just ignorant)?
