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Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture. (English) Zbl 1328.11070
The paper under review is part of a series of works dedicated to a proof of the André-Oort conjecture assuming the generalized Riemann hypothesis for CM fields. The proof is finished in another paper by B. Klingler and the second author [Ann. Math. (2) 180, No. 3, 867–925 (2014; Zbl 1377.11073)], some main ingredients of which are established in the present paper.
The André-Oort conjecture affirms that in any given Shimura variety $$S$$, the Zariski closure of a sequence of special subvarieties always equals a finite union of special subvarieties. Here special subvarieties can be characterized as geometric connected components of images of morphisms between Shimura varieties defined by morphisms between Shimura data, up to Hecke translation. In particular the conjecture predicts that the Zariski closure of a subset of special points, which are generalizations of CM points in Siegel modular varieties, remains a finite union of special subvarieties, which are in particular totally geodesic when viewed as a complex algebraic subvarieties of $$S$$ using the locally symmetric structure on $$S$$. It has aroused interests from several branches of arithmetic geometry, and witnessed progresses via various approaches, (cf. R. Noot [in: Séminaire Bourbaki. Volume 2004/2005. Exposés 938–951. Paris: Société Mathématique de France. 165–197, Exp. No. 942 (2006; Zbl 1175.14013)], T. Scanlon [in: Séminaire Bourbaki. Volume 2010/2011. Exposés 1027–1042. Avec table par noms d’auteurs de 1948/49 à 2009/10. Paris: Société Mathématique de France (SMF). 299–315 (2012; Zbl 1271.14030)]) etc.
As is explained in related works like [the second author, J. Théor. Nombres Bordx. 21, No. 2, 493–502 (2009; Zbl 1209.11055)], the approach adopted by Klingler, and the authors can be summarized as the “ergodic-Galois” alternative for a sequence of special subvarieties $$(S_n)$$ of $$S$$:
(a)
if the Galois orbits of the $$S_n$$’s are uniformly bounded, then the conjecture holds for $$(S_n)$$;
(b)
if the Galois orbits of the $$S_n$$’s are not bounded, then one may construct a new sequence $$(S_n')$$ of special subvarieties in $$S$$ such that $$S_n\subsetneq S_n'$$ for $$n$$ sufficiently large and that $$\bigcup_nS_n$$ and $$\bigcup_nS'_n$$ share the same Zariski closure.
The conjecture then follows by dimensional induction. Note that both (a) and (b) relies on the GRH.
The present paper treats (a) in detail, by establishing the following results:
(1)
a lower bound of degrees of Galois orbits of general special subvarieties;
(2)
equidistribution of a sequence of special subvarieties whose Mumford-Tate groups have a fixed $${\mathbb{Q}}$$-torus $${\mathbf{T}}$$ as their common connected centers;
(3)
charaterizing a sequence with bounded Galois orbits via the connected centers of their Mumford-Tate groups.
The authors introduce the notion of $${\mathbf{T}}$$-special subvarieties, motivated by the main result in [L. Clozel and the first author, Ann. Math. (2) 161, No. 3, 1571–1588 (2005; Zbl 1099.11031)]. Write $$({\mathbf{G}},X)$$ for the Shimura datum of the ambient Shimura variety $$S$$, a special subvariety $$S'$$ of $$S$$ is said to be $${\mathbf{T}}$$-special for a given $${\mathbb{Q}}$$-torus $${\mathbf{T}}\subset{\mathbf{G}}$$ if it is defined by some Shimura subdatum $$({\mathbf{G}}',X')\subset({\mathbf{G}},X)$$ with $${\mathbf{T}}$$ equal to the connected center of $${\mathbf{G}}'$$. Note that it suffices to treat the case where $${\mathbf{G}}$$ is semi-simple of adjoint type, with $${\mathbf{G}}'$$ being the generic Mumford-Tate group for $$S'$$. If $${\mathbf{T}}$$ is the trivial $${\mathbb{Q}}$$-torus (i.e., a single point), one is reduced to the notion of strongly special subvarieties studied in [Zbl 1099.11031]. When $${\mathbf{T}}$$ is non-trivial, one can show that the splitting field $$L_{\mathbf{T}}$$ of the connected center $${\mathbf{T}}$$ of $${\mathbf{G}}'$$ is a CM field, cf. [the second author, Duke Math. J. 132, No. 3, 393–407 (2006; Zbl 1097.11032)], which necessitates the GRH for CM fields. The $${\mathbb{Q}}$$-torus $${\mathbf{T}}$$ captures most quantitative information about Galois orbits of special subvarieties, and is closely related to the equidistribution of special subvarieties.
The result (1) is a lower bound of the degree of Galois orbits of special subvarieties computed against the automorphic line bundle on $$S$$, i.e., the line bundle defining the Baily-Borel compactification of $$S$$. Here the Galois orbits is taken with respect to the canonical model of $$S$$ over its reflex field $$E({\mathbf{G}},X)$$. The bound given in Section 2 of the paper (cf. Theorem 2.19 and Remark 2-20 formula (8)) actually only involves the counting of connected components in the Galois orbits, generalizing the case of special points treated in [Zbl 1097.11032]: $\deg_{\mathcal{L}}(\text{Gal}(\overline{\mathbb{Q}}/E)\cdot V)\geq c_N\prod_{p:K^{\mathrm{max}}_{{\mathbf{T}},p}\neq K_{{\mathbf{T}},p}}\max(1,B|K_{{\mathbf{T}},p}^{\mathrm{max}}/K_{{\mathbf{T}},p}|)\cdot(\log|\mathrm{disc}(L_{\mathbf{T}})|)^N$ where $$N$$ is any prescribed integer, $$c_N>0$$ is an absolute constant that only depends on $$N$$, and $$B>0$$ is an absolute constant independent of $$N$$ (lying in $$(0,1)$$). Here $$V$$ is a geometrically connected component of the Shimura subvariety in $$S$$ defined by $$({\mathbf{G}}',X')$$, $$K$$ is the compact open subgroup of $${\mathbf{G}}({\hat{\mathbb{Q}}})$$ defining the level structure for $$S=Sh_K({\mathbf{G}},X)$$, which is assumed to be a product $$K=\prod_pK_p$$ for $$K_p\subset{\mathbf{G}}({\mathbb{Q}}_p)$$, and $$K_{{\mathbf{T}},p}^{\mathrm{max}}$$ is the unique maximal compact open subgroup of $${\mathbf{T}}({\mathbb{Q}}_p)$$ in which the intersection $$K_{{\mathbf{T}},p}=K_p\cap{\mathbf{T}}({\mathbb{Q}}_p)$$ is of finite index.
Note that the term $$\log(|\mathrm{disc}(L_{\mathbf{T}})|)$$ comes from the estimation of the reciprocity map of the zero-dimensional Shimura variety associated to $$({\mathbf{C}},{\bar x})$$ with maximal compact open subgroup $$K_{\mathbf{C}} ^{\mathrm{max}}$$ as the level structure, where $${\mathbf{C}}={\mathbf{G}}'/{\mathbf{G}}'{}^{\mathrm{der}}$$ shares the same splitting field with $${\mathbf{T}}$$ as they are isogenous. This estimation relies on the GRH for the CM field $$L_{\mathbf{T}}$$. The term $$\prod_p{\mathrm{max}}\{1,B|K_{{\mathbf{T}} ,p}^{\mathrm{max}}/K_{{\mathbf{T}},p}|\}$$ only depends on the level structure $$K$$ and the $${\mathbb{Q}}$$-torus $${\mathbf{T}}$$.
The result (2), cf. Theorem 3.8 and Corollary 3.9, is reduced to the strongly special case studied in [Zbl 1099.11031], based on the useful fact that for a given $${\mathbb{Q}}$$-torus $${\mathbf{T}}$$ in $${\mathbf{G}}$$ arising as the connected center of some Shimura subdatum $$({\mathbf{G}}',X')$$, there exists a unique $${\mathbb{Q}}$$-subgroup $${\mathbf{H}}$$ of $$({\mathbf{G}},X)$$ having $${\mathbf{T}}$$ as the connected center, and finitely many $${\mathbf{T}}$$-special subdata of the form $$({\mathbf{H}},X_{\mathbf{H}})$$, such that any $${\mathbf{T}}$$-special subvariety lies in a maximal $${\mathbf{T}}$$-special subvariety defined by one of these subdata. Hence one is reduced to the case where $$({\mathbf{G}},X)$$ is $${\mathbf{T}}$$-special itself. Given a sequence of $${\mathbf{T}}$$-special subvarieties defined by $${\mathbf{T}}$$-special subdata $$({\mathbf{G}}_n,X_n)$$, one may reduce simultaneously by $${\mathbf{T}}$$, namely study the sequence of special subvarieties defined by $$({\mathbf{G}}_n/{\mathbf{T}},X'_n)$$ in $$({\mathbf{G}}/{\mathbf{T}},X')$$ with $$X'_n,X'$$ etc. deduced from $$X_n,X$$ respectively, which follows from the equidistribution of strongly special subvarieties. Note that this result itself is independent of the GRH.
Finally, assuming the GRH for CM fields, the authors prove that any sequence of special subvarieties $$S_n$$ in $$S$$ whose Galois orbits are of uniformly bounded degrees against the automorphic line bundle $${\mathcal{L}}$$ are $${\mathbf{T}} _i$$-special, with $${\mathbf{T}}_i$$ coming from a finite set of $${\mathbb{Q}}$$-tori in $${\mathbf{G}}$$, cf. Theorem 3.10 and Corollary 3.11. It suffices to consider special subvarieties with ${\mathrm{max}} \{1,B^{i({\mathbf{T}})}|K_{\mathbf{T}}^{\mathrm{max}}/K_{\mathbf{T}} |\}\log|\mathrm{disc}(L_{\mathbf{T}})|\leq M$ for some absolute constant $$M$$, where $${\mathbf{T}}$$ is the connected center of the Mumford-Tate group of for the special subvariety in question, and $$i({\mathbf{T}})$$ is the number of rational primes $$p$$ such that $$K_{{\mathbf{T}},p}\subsetneq K_{{\mathbf{T}},p}^{\mathrm{max}}$$. To establish the finiteness, the authors fix a rational representation $${\mathbf{G}}\rightarrow{\mathbf{GL}}_{n,{\mathbb{Q}} }$$ sending $$K$$ into $${\mathbf{GL}}_n({\hat{\mathbb{Z}}})$$, and prove that:
$$\bullet$$ when the discriminants $$|\mathrm{disc}(L_{\mathbf{T}})|$$ are uniformly bounded, the $${\mathbb{Q}}$$-tori $${\mathbf{T}}$$ involved lie in finitely many $${\mathbf{GL}}_n({\mathbb{Q}})$$-conjugacy classes of $${\mathbb{Q}}$$-tori in $${\mathbf{GL}}_{n,{\mathbb{Q}}}$$ (Lemma 3.13);
$$\bullet$$ when the products $${\mathrm{max}}\{1,B^{i({\mathbf{T}})}|K_{\mathbf{T}}^{\mathrm{max}}/K_{\mathbf{T}}|\}\log|\mathrm{disc}(L_{\mathbf{T}})|$$ are uniformly bounded, the $${\mathbb{Q}}$$-tori $${\mathbf{T}}$$ involved lie in finitely many $${\mathbf{GL}}_n(\mathbb{Z})$$-conjugacy classes of $${\mathbb{Q}}$$-tori in $${\mathbf{GL}}_{n,{\mathbb{Q}}}$$ (Proposition 3.14), and one reduces further to finitely many $$\Gamma$$-conjugacy classes, with $$\Gamma=K\cap{\mathbf{G}}({\mathbb{Q}})$$.
Hence the result (3) is proved.
To the end, we remark that the André-Oort conjecture has been recently proved unconditionally for the moduli space of principally polarized abelian varieties $$\mathcal A_g$$ by J. Tsimerman [“A proof of the Andre-Oort conjecture for $$\mathcal A_g$$”, Preprint, arXiv:1506.01466].
Reviewer: Xin Lu (Mainz)

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties
##### Keywords:
André-Oort conjecture; equidistribution; special varieties
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