Faltings height of a CM abelian variety

Question

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 $K$?

Answer 1

Fix an $A$ with CM by $K$, and for each $D$, let $A_D$ be the quadratic twist of $A$ by $K(\sqrt{D})/K$. Also let $$h_{sf}(D)=\min(h(Dd^2):d\in K^*)$$ denote the "square-free height" of $D$. Then $$ h_{\text{Faltings}}(A_D) \gg h_{sf}(D), $$ which shows that there is no upper bound of the sort that you want, since $K$ is fixed, while $h_{sf}(D)$ can be arbitrarily large.

Or do you mean to take the semi-stable Faltings height, i.e., the height obtained after going to a field where $A$ has semi-stable reduction. For CM abelian varieties, this would be a field where $A$ has everywhere good reduction, so the Faltings height comes entirely from the archimedean places. In this case, you can use the fact that the Faltings height is more-or-less equal to the height of the associated point in moduli space. (At least, equal enough to talk about boundedness.) For a principally polarized CM abelian variety, the moduli point is essentially given by the periods, which are more-or-less a basis for $\mathcal{O}_K$ over $\mathbb{Z}$. So it seems that one might well be able to get a bound in terms of $\hbox{Disc}(K)$.

You might try looking first at the case of elliptic curves, where the relation between the Faltings height and the periods is very explicit.

Answer 2

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.