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The André-Oort conjecture. (English) Zbl 1377.11073
This paper proves the André-Oort conjecture for arbitrary Shimura varieties, under the assumption of the generalized Riemann hypothesis for CM fields or a technical hypothesis on the set of specials points in question. Recall that a Shimura variety $$\text{Sh}(\mathbf{G},X)$$ is an inverse system of smooth quasi-projective projective varieties over $$\mathbb{C}$$ attached to a connected reductive group $$\mathbf{G}$$ over $$\mathbb{Q}$$ and a $$\mathbf{G}(\mathbb{R})$$-conjugacy class of homomorphisms $$X \subset \text{Hom}(\text{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_m, \mathbf{G}_{\mathbb{R}})$$ satisfying certain axioms (a Shimura datum), indexed by sufficiently small compact open subgroups $$K \subset \mathbf{G}(\mathbb{A}_f)$$. The formation of $$\text{Sh}(\mathbf{G},X)$$ is functorial with respect to an obvious notion of morphism of Shimura data. The group $$\mathbf{G}(\mathbb{A}_f)$$ also acts on the right on the tower $$\text{Sh}(\mathbf{G},X)$$ by algebraic correspondences, called Hecke correspondences.
In the context of Shimura varieties, one may define a subvariety $$V \subset \text{Sh}_K(\mathbf{G},X_{\mathbf{G}})$$ to be special if there exists a Shimura datum $$(\mathbf{H},X_{\mathbf{H}})$$, a morphism $$(\mathbf{H},X_{\mathbf{H}}) \to (\mathbf{G},X_{\mathbf{G}})$$, and an element $$g \in \mathbf{G}(\mathbb{A}_f)$$ such that $$V$$ is an irreducible component of the image of the composite $\text{Sh}(\mathbf{H},X_{\mathbf{H}}) \to \text{Sh}(\mathbf{G},X_{\mathbf{G}}) \underset{\longrightarrow}{\cdot g} \text{Sh}(\mathbf{G},X_{\mathbf{G}}{)} \text{Sh}_K(\mathbf{G},X_{\mathbf{G}}).$ (As is pointed out in the paper’s introduction, the notion of special can also be understood more generally, in abstract Hodge-theoretic terms.) A special point is a special subvariety of dimension zero.
If $$V \subset \text{Sh}_K(\mathbf{G},X)$$ is special, then one can show that the special points in $$\text{Sh}_K(\mathbf{G},X)(\mathbb{C})$$ contained in $$V$$ form a dense subset of $$V$$ for the strong (and hence for the Zariski) topology. The André-Oort conjecture asserts the converse: that every irreducible component of the Zariski-closure of a collection $$\Sigma \subset \text{Sh}_K(\mathbf{G},X)(\mathbb{C})$$ of special points is a special subvariety.
The main result of the paper under review is to prove the André-Oort conjecture under the assumption of the generalized Riemann hypothesis for CM fields, or under a technical hypothesis on the Mumford-Tate groups of the points in $$\Sigma$$. In fact, the authors prove a natural generalization where $$\Sigma$$ may be any collection of special subvarieties, not just points (subject to analogous assumptions).
The proof is rooted in the strategy of B. Edixhoven and the second author [Ann. Math. (2) 157, No. 2, 621–645 (2003; Zbl 1053.14023)], who proved the conjecture for curves in Shimura varieties containing infinite sets of special points satisfying the technical hypothesis alluded to above. Two main difficulties arise here. One is the question of irreducibility of transforms of subvarieties under Hecke correspondences. The other is the issue of higher dimensional special subvarieties, which the authors deal with via an inductive technique based on previous work of E. Ullmo and the second author [Ann. Math. (2) 180, No. 3, 823–865 (2014; Zbl 1328.11070)].

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties
##### Keywords:
André-Oort conjecture; Shimura varieties
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##### References:
 [1] E. Cattani, P. Deligne, and A. Kaplan, ”On the locus of Hodge classes,” J. Amer. Math. Soc., vol. 8, iss. 2, pp. 483-506, 1995. · Zbl 0851.14004 [2] C. Voisin, ”Hodge loci and absolute Hodge classes,” Compos. Math., vol. 143, iss. 4, pp. 945-958, 2007. · Zbl 1124.14015 [3] P. Deligne, ”Travaux de Shimura,” in Séminaire Bourbaki, 23ème Année (1970/71), Exp. No. 389, New York: Springer-Verlag, 1971, vol. 244, pp. 123-165. [4] P. Deligne, ”Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques,” in Automorphic Forms, Representations and $$L$$-Functions, Part 2, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 247-289. [5] Milne, J. S., Introduction to Shimura varietiesProvidence, RI: Amer. Math. Soc., 2005. [6] E. Ullmo and A. Yafaev, ”Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture,” Ann. of Math., vol. 180, pp. 823-865, 2014. · Zbl 1328.11070 [7] N. Ratazzi and E. Ullmo, ”Galois + équidistribution = Manin-Mumford,” in Arithmetic Geometry, Providence, RI: Amer. Math. Soc., 2009, vol. 8, pp. 419-430. [8] B. Edixhoven, ”On the André-Oort conjecture for Hilbert modular surfaces,” in Moduli of Abelian Varieties, Basel: Birkhäuser, 2001, vol. 195, pp. 133-155. · Zbl 1029.14007 [9] L. Clozel and E. Ullmo, ”Équidistribution de sous-variétés spéciales,” Ann. of Math., vol. 161, iss. 3, pp. 1571-1588, 2005. · Zbl 1099.11031 [10] E. Ullmo, ”Equidistribution de sous-variétés spéciales. II,” J. Reine Angew. Math., vol. 606, pp. 193-216, 2007. · Zbl 1137.11043 [11] B. Edixhoven and A. Yafaev, ”Subvarieties of Shimura varieties,” Ann. of Math., vol. 157, iss. 2, pp. 621-645, 2003. · Zbl 1053.14023 [12] A. Yafaev, ”A conjecture of Yves André’s,” Duke Math. J., vol. 132, iss. 3, pp. 393-407, 2006. · Zbl 1097.11032 [13] B. Edixhoven, ”Special points on products of modular curves,” Duke Math. J., vol. 126, iss. 2, pp. 325-348, 2005. · Zbl 1072.14027 [14] A. Yafaev, ”On a result of Moonen on the moduli space of principally polarised abelian varieties,” Compos. Math., vol. 141, iss. 5, pp. 1103-1108, 2005. · Zbl 1093.14034 [15] M. Ratner, ”On Raghunathan’s measure conjecture,” Ann. of Math., vol. 134, iss. 3, pp. 545-607, 1991. · Zbl 0763.28012 [16] S. G. Dani and G. A. Margulis, ”Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces,” Proc. Indian Acad. Sci. Math. Sci., vol. 101, iss. 1, pp. 1-17, 1991. · Zbl 0731.22008 [17] S. Zhang, ”Equidistribution of CM-points on quaternion Shimura varieties,” Int. Math. Res. Not., vol. 59, pp. 3657-3689, 2005. · Zbl 1096.14016 [18] E. Ullmo, ”Manin-Mumford, André-Oort, the equidistribution point of view,” in Equidistribution in Number Theory, an Introduction, New York: Springer-Verlag, 2007, vol. 237, pp. 103-138. · Zbl 1181.11041 [19] B. Edixhoven, ”Special points on the product of two modular curves,” Compositio Math., vol. 114, iss. 3, pp. 315-328, 1998. · Zbl 0928.14019 [20] A. Borel, Introduction aux Groupes Arithmétiques, Paris: Hermann, 1969, vol. XV. · Zbl 0186.33202 [21] R. Pink, Arithmetical compactification of mixed Shimura varietiesBonn: Universität Bonn, Mathematisches Institut, 1990. · Zbl 0748.14007 [22] Schémas en groupes. III: Structure des schémas en groupes réductifs, Demazure, M. and Grothendieck, A., Eds., New York: Springer-Verlag, 1970, vol. 153. · Zbl 0212.52810 [23] B. Weisfeiler, ”Strong approximation for Zariski-dense subgroups of semisimple algebraic groups,” Ann. of Math., vol. 120, iss. 2, pp. 271-315, 1984. · Zbl 0568.14025 [24] M. V. Nori, ”On subgroups of $${ GL}_n({\mathbf F_p)}$$,” Invent. Math., vol. 88, iss. 2, pp. 257-275, 1987. · Zbl 0632.20030 [25] R. Lazarsfeld, Positivity in Algebraic Geometry. I. Classical Setting: Line Bundles and Linear Series, New York: Springer-Verlag, 2004, vol. 48. · Zbl 1093.14501 [26] W. L. Baily Jr. and A. Borel, ”Compactification of arithmetic quotients of bounded symmetric domains,” Ann. of Math., vol. 84, pp. 442-528, 1966. · Zbl 0154.08602 [27] I. Satake, ”A note on holomorphic imbeddings and compactification of symmetric domains,” Amer. J. Math., vol. 90, pp. 231-247, 1968. · Zbl 0187.02903 [28] E. B. Dynkin, ”Semisimple subalgebras of semisimple Lie algebras,” Mat. Sbornik N.S., vol. 30(72), pp. 349-462 (3 plates), 1952. · Zbl 0048.01701 [29] S. Ihara, ”Holomorphic imbeddings of symmetric domains,” J. Math. Soc. Japan, vol. 19, pp. 261-302, 1967. · Zbl 0159.11102 [30] I. Satake, ”On some properties of holomorphic imbeddings of symmetric domains,” Amer. J. Math., vol. 91, pp. 289-305, 1969. · Zbl 0182.11301 [31] J. Milne, Kazhdan’s theorem on arithmetic varieties. [32] B. Moonen, ”Linearity properties of Shimura varieties. I,” J. Algebraic Geom., vol. 7, iss. 3, pp. 539-567, 1998. · Zbl 0956.14016 [33] R. Pink, ”Strong approximation for Zariski dense subgroups over arbitrary global fields,” Comment. Math. Helv., vol. 75, iss. 4, pp. 608-643, 2000. · Zbl 0981.20036 [34] V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Boston: Academic Press, 1994. · Zbl 0841.20046 [35] F. Bruhat and J. Tits, ”Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée,” Inst. Hautes Études Sci. Publ. Math., vol. 60, pp. 197-376, 1984. · Zbl 0597.14041 [36] F. Bruhat and J. Tits, ”Groupes réductifs sur un corps local,” Inst. Hautes Études Sci. Publ. Math., vol. 41, pp. 5-251, 1972. · Zbl 0254.14017 [37] P. Cartier, ”Representations of $$p$$-adic groups: a survey,” in Automorphic Forms, Representations and $$L$$-Functions, Part 1, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 111-155. [38] G. J. Heckman and E. M. Opdam, ”Harmonic analysis for affine Hecke algebras,” in Current Developments in Mathematics, 1996, Boston: Internat. Press, 1997, pp. 37-60. · Zbl 0932.22006 [39] J. Tits, ”Reductive groups over local fields,” in Automorphic Forms, Representations and $$L$$-Functions, Part 1, Providence, RI: Amer. Math. Soc., 1979, vol. XXXIII, pp. 29-69. [40] A. Borel, ”Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup,” Invent. Math., vol. 35, pp. 233-259, 1976. · Zbl 0334.22012 [41] W. Fulton, Intersection Theory, Second ed., New York: Springer-Verlag, 1998, vol. 2. · Zbl 0885.14002
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