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This, in view of the analogies between CM points on Shimura curves and torsion points on elliptic curves, is a sequel to an earlier question I had asked: The torsion point count in higher dimension .

Mazur proved that the number of $\mathbb{Q}$-rational torsion points of an elliptic curve is bounded by $16$. Recall in comparison that the modular curve $\mathcal{A}_1 = X_0(1) = \mathbb{A}_j^1$ has exactly $13$ CM points rational over $\mathbb{Q}$; those correspond to the $\bar{\mathbb{Q}}$-isomorphism classes of CM elliptic curves defined over $\mathbb{Q}$. (In addition to the full integer rings of the nine quadratic imaginary fields of class number one, there are also the orders of conductor $2$ in $\mathbb{Q}(\sqrt{-1}), \mathbb{Q}(\sqrt{-3}), \mathbb{Q}(\sqrt{-7})$ and the order of conductor $3$ in $\mathbb{Q}(\sqrt{-3})$.)

It was observed by Will Sawin in the other question that under standard conjectures generalizing Serre's open image theorem, the number of torsion points of degree $\leq d$ over $\mathbb{Q}$ in an abelian variety should be proportional to $d^{\alpha}$ for some exponent $\alpha$ depending on the Mumford-Tate group. For elliptic curves, the two possible exponents are $3/2$ and $2$.

Consider now the same question for the special points $x \in X$ of a Shimura variety having $[\mathbb{Q}(x):\mathbb{Q}] \leq d$. (The special points are the zero-dimensional subvarieties of Hodge type, or the points whose Mumford-Tate group is a torus. ) What should be expected of the asymptotic count of these points as $d \to \infty$? Are they again proportional to a power of $d$? Any good estimates in interesting special cases such as the modular curves $X_0(N)$?

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For the $j$-line, a special point corresponds to a CM-elliptic curve, so to an order in an imaginary quadratic field. As you are looking at the degree of the point, this corresponds to the degree of the Hilbert class field of the order, which can be estimated using the Brauer-Siegel formula in terms of the discriminant. So you get a growth between powers of $d$ (whether you get an asymptotic is not so clear). Level structures should be easy to handle and not change much. I expect this should work too in higher dimensions but the technical details will be much harder. Special points corresponding to simple abelian varieties are also CM and hopefully the non-simple ones can be incorporated in an error term.

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  • $\begingroup$ Thank you. It would be good to know if the available results on average class numbers of quadratic fields imply a precise asymptotic $\sim c d^{\alpha}$ for the CM point count on the $j$-line, and what the exponent $\alpha$ and proportionality constant $c$ would be. $\endgroup$ – Vesselin Dimitrov Aug 15 '14 at 13:32

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