TITLE: Faltings height of a CM abelian variety
QUESTION [7 upvotes]: Let A be a CM abelian variety, say simple of dimension g, with $End(A) = O_K$, where $K$ is a CM field
of degree $2g$. 
Is there an upper bound for the Faltings height $h(A)$ in terms of the discriminant $d_K$ of $K$?

REPLY [2 votes]: The way your question is stated, the answer is positive by Faltings's finiteness theorem for abelian varieties: there are only finitely many $K$-isom. classes of g-dimensional abelian varieties over $K$ with good reduction over $O_K$. In particular, the Faltings height of an abelian scheme over $O_K$ of relative dimension $g$ is bounded (ineffectively speaking) in terms of $g$ and $K$ (or $g$ and $d_K$ by Hermite-Minkowski). 
Let me say some things about effectivity. I am guessing this is what you are really interested in.
Firstly, there is no effective version (in general) of the above ineffective statement at the moment. You can hope to do some things in particular cases. 
For instance, in the case of dimension one, you can look at Chapter 1 of von Kaenel's thesis:
http://e-collection.library.ethz.ch/eserv/eth:2519/eth-2519-02.pdf
Note that the results in von Kaenel's thesis give you (much) more than just bounds on Faltings heights of elliptic curves with good reduction everywhere.
Moreover, when K is of small discriminant, we know by results of Abrashkin-Fontaine, that there are no non-trivial abelian schemes over O_K, thus you can take the bound to be zero when $d_K$ is at most 8. (But that's cheap...)
Finally, Rafael von Kaenel knows how to bound Faltings heights of certain classes of abelian varieties explicitly in terms of their reduction behaviour, and maybe his methods also work to bound stable Faltings heights of CM abelian varieties. You could email him or me about this if you are interested, as these results are not yet available.