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It is known that if $X$ is a curve over a number field $F$ equipped with a flat regular model over $O_F$ the integer ring, one can define, using a suitable ample line bundle with an Hermitian metric, the notion of height for points in $X$. In particular, there are only finitely many algebraic points with height lower than a given constant.

What happens when one studies the heights of CM points on the compactified modular curve, with respect to the ample line bundle defining modular forms of large weight (plus suitable Hermitian metric)? Is it possible to study the distribution of CM cycles through height functions? Does one have similar results as in the previous example, say bounding the number of CM points by heights?

This is also motivated by the Bogomolov conjecture. Roughly speaking, for $A$ an abelian variety over a number field, and $Z\subset A$ an irreducible closed subvariety that is not a torsion subvariety (abelian subvariety translated by a torsion point), then the "number" (or some other numerical invariants) of torsion subvarieties in $Z$ is bounded in terms of suitable constants in heights, degrees, that only depends on $Z$. It might be of interest to work out an analogue for modular curves and perhaps other moduli spaces like Shimura varieties.

Excuse me for the vague statements above and thanks in advance for comments and references.

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For your titular question, Beats me. Personally I'm not aware of anyone who's studied the distribution of CM points with respect to height in the way you describe.

What I have seen papers that study the distribution of CM points with respect to things other than height and papers that look at the height of CM points (as well as papers that say quite a lot about the Faltings height of a CM abelian variety).

Here are a few different papers of note on those topics:

Equidistribution of CM points:

http://www.math.ucla.edu/~wdduke/preprints/modud.pdf - Gives an amazing asymptotic result on the trace of a CM $j$-invariant (i.e. a result on $X(1)$).

http://www.math.columbia.edu/~szhang/papers/ZhangIMRN.pdf - Gives an analogue on more general modular curves and quaternionic Shimura Curves as well as a connection to the Andre-Oort conjecture

Also good for an overview is the "Equidistribution in Number Theory" volume edited by Granville and Rudnick http://www.springer.com/mathematics/numbers/book/978-1-4020-5403-7

Heights of CM points/cycles:

http://www.math.columbia.edu/~szhang/papers/HCMI.pdf - Heights of CM Points I--The Gross-Zagier formula

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.4204 - This attempts to identify affine elliptic modular curves by the presence of CM points of large enough height(logarithmic height on $\overline{\mathbf{Q}}$).

I'd like to emphasize that this does not pretend to be a complete reference list and I don't pretend to know much about the theory of heights. Rather this is a smattering of exciting ideas in the area.

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thanks a lot. The paper of Breuer is helpful. and of course in general one should turn to otherr tools like, i guess, Gross-Zagier formulas and the likes. –  genshin Nov 5 '10 at 7:31
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