# Faltings height of a CM abelian variety

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 $K$?

• (This question is old but it was resurrected, and there’s even a chance the poster was Jacob (I doubt it because he has an MO profile but it would be funny! :) ), but in any case I figured I should note, in case someone stumbles upon this in the future, that you can indeed use results towards Colmez (plus Brauer-Siegel of course) to get a good bound ($d_K^{o(1)}$) on said Faltings height. See arxiv.org/abs/1506.01466 for why.) – alpoge Nov 3 at 3:47

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.

• Many thanks for this! Yes, I did mean the semi-stable Faltings height. I will try to follow your suggestion; – user42721 Dec 9 '13 at 15:52
• Colmez, in `Sur la hauteur de Faltings des varietes abeliennes CM' proves that for a CM elliptic curve $E$, one has $h(E) <<_{\epsilon} d^{\epsilon}$ where $d = |discr(End(E))|$ (remarque (i) after Corrolaire 7). This exactly the sort of bound I would like to have for a general simple CM abelian variety. I think one can probably deduce it from Colmez' conjectural formula for the Faltings height, but maybe it is not hard to prove unconditionally using Joe's suggestion? – user42721 Dec 10 '13 at 13:10
• @user42721: This will give you an explicit bound with $\epsilon = 1/2$, but going below that will involve the Yuan-Zhang and Brauer-Siegel theorems. – Vesselin Dimitrov Mar 27 '16 at 5:53
• @Joe Silverman: I think you meant upper bound in the sentence "which shows that there is no lower bound of the sort that you want"... – Bombyx mori Nov 2 at 23:12
• @Bombyxmori Sure, that's a typo. Feel free to fix it (that's what MO editing is all about). And thanks. – Joe Silverman Nov 3 at 0:41

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.

• This is all very interesting. In particular, bounding the Faltings height of everywhere-good-reduction abelian varieties is a very interesting (and hard) problem. But I think that the CM case is likely to be much easier. – Joe Silverman Dec 10 '13 at 16:35
• Many thanks for this reply!!! What I am really interested in, is proving that in the simple principal CM case, the Faltings height is $<<_{\epsilon} d^{\epsilon}$ where $d$ is the discriminant of the endomorphism ring, which is $O_K$ where $K$ is a CM field of degree $2g$. Of course, $K$ is related to the field of definition of $A$, we know that $A$ is defined over the Hilbert class field of $K$ but it seems very difficult to relate the bounds one can obtain by the methods you mention to $\discr(O_K)$.... – user42721 Dec 10 '13 at 17:20
• You are most welcome! The type of bounds you are seeking appear to be difficult to obtain for higher-dimensional CM abelian varieties. But I believe they are certainly interesting and should even have applications to "Unlikely Intersections". – Ariyan Javanpeykar Dec 10 '13 at 18:02
• Thanks! These are the kind of applications I was thinking of... – user42721 Dec 10 '13 at 20:05