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Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points on Shimura varieties?

The question of course does not always make sense for ${\bf{Q}}$-points: a theorem of Shimura shows that a quaternionic Shimura curve has no ${\bf{R}}$-points, and a theorem of Mazur shows that the modular curve $Y_0(N)$ has no ${\bf{Q}}$-points for $N$ sufficiently large. The question does however seem to make sense for certain abelian extensions of CM fields. (For instance, in the setting of a quaternionic Shimura curve $M$ defined over a totally real field $F$, if a totally imaginary quadratic extension $K/F$ embeds into the underlying quaternion algebra, then there is a supply of CM points on $M$ defined over certain abelian extensions of $K$). In particular, I should like to know more about the following questions:

(i) Over which number fields $k$ does a given Shimura variety $S(G, X)$ have a $k$-rational point?

(ii) For which such number fields $k$ will $S(G, X)(k)$ be Zariski dense?

(iii) To what extent are such $k$-rational points accounted for by CM points (or similar constructions)?

Sorry if these questions are imprecise or wrongly formulated, I would be happy to at least have an indication of where to look in the literature if someone has already thought about this.

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You might find Milne's article at jmilne.org/math/xnotes/svh.pdf useful. He shows that for most classical groups $G$ one can give a motivic interpretation of the points of $Sh(G,X)$. –  Keerthi Madapusi Pera Mar 25 '11 at 0:13
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1 Answer

up vote 7 down vote accepted

If you haven't, you should first think about these questions just for modular curves, which are the simplest Shimura varieties. Then there are only finitely many $N$ for which the modular curve of level $N$ has genus $< 2$. Once the genus is at least $2$, there are only finitely many points over any fixed number field (by Mordell's conjecture/Faltings theorem).

For higher dimensional Shimura varieties, once the level gets large enough the variety will become of general type, and Lang's conjecture will (presumably) apply, so as to give restrictions on the rational points. In particular, the answer to (ii) will presumably be never if the level is large enough. The $\overline{\mathbb Q}$-points are Zariski dense by the Nullstellensatz, and most of them won't be CM points (unless the Shimura variety is associated to a torus, so that every point is CM!), so eventually (i.e. if you make $k$ large enough) you will get points that are not CM.

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Many thanks for these excellent comments! I hadn't considered the issue with genus for modular curves, which is a good observation to make here. –  jvo Mar 25 '11 at 9:04
Regarding (iii), the André-Oort conjecture also seems relevant: the Zariski-closure of the CM points is conjectured (and often known, at least under GRH) to be itself a union of irreducible special sub-varieties. So as Matt Emerton said, most $\bar{\mathbb Q}$-points are definitely not special. –  Olivier Mar 25 '11 at 12:55
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