Special subvarieties of mixed Shimura varieties

Special subvarieties of mixed Shimura varieties
[Sous-variétés spéciales des variétés de Shimura mixtes]

Chen, Ke

Directeur de thèse : Ullmo, Emmanuel

Membres du jury : Clozel, Laurent ; Ullmo, Emmanuel ; Wildeshaus, Jörg ; Yafaev, Andrei

Thèses d'Orsay, no. 773 (2009) , 122 p.

Résumé
Abstract

Cette thèse est dédiée à l’étude de la conjecture d’André-Oort pour les variétés de Shimura mixtes. On montre que dans une variété de Shimura mixte $M$ définie par une donnée de Shimura mixte $(𝐏, Y)$ , soient $𝐂$ un $ℚ$ -tore dans $𝐏$ et $Z$ une sous-variété fermée quelconque dans $M$ , alors l’ensemble des sous-variétés $𝐂$ -spéciales maximales contenues dans $Z$ est fini. La démonstration suit la stratégie de L. Clozel, E. Ullmo, et A. Yafaev dans le cas pure, qui dépend de la théorie de Ratner sur des propriétés ergodiques des flots unipotents sur des espaces homogènes. D’ailleurs, une minoration sur le degré de l’orbite sous Galois d’une sous-variété pure est montrée dans le cas mixte, adaptée du cas pure établi par E. Ullmo et A. Yafaev. Enfin, une version relative de la conjecture de Manin-Mumford est démontrée en caractéristique nul: soit $A$ un $S$ -schéma abélien en caractéristique nul, alors l’adhérence de Zariski d’une suite de sous-schémas de torsion dans $A$ égale une réunion finie de sous-schémas de torsion.

This thesis studies the Andre-Oort conjecture for mixed Shimura varieties. The main result is: let $M$ be a mixed Shimura variety defined by a mixed Shimura datum $(𝐏, Y)$ , $𝐂$ a fixed $ℚ$ -torus of $𝐏$ , and $Z$ an arbitrary closed subvariety in $M$ , then the set of maximal $𝐂$ -special subvarieties of $M$ contained in $Z$ is finite. The proof follows the strategy applied by L. Clozel, E. Ullmo, and A. Yafaev in the pure case, which relies on Ratner’s theory on ergodic properties of unipotent flows on homogeneous spaces. Besides, a minoration on the degree of the Galois orbit of a special subvariety is proved in the mixed case, adapted from the pure case established by E. Ullmo and A. Yafaev. Finally, a relative version of the Manin-Mumford conjecture is proved in characteristic zero: let $A$ be an abelian $S$ -scheme of characteristic zero, then the Zariski closure of a sequence of torsion subschemes in $A$ remains a finite union of torsion subschemes.

Classification : 14G35(11G18), 14K05
Keywords: Diophantine approximation, mixed Shimura varieties, special subvarieties, André-Oort conjecture, Manin-Mumford conjecture, equidistribution, Ratner’s theory.
Mot clés : approximation diophantienne, variétés de Shimura mixtes, sous-variétés spéciales, conjecture d’André-Oort, conjecture de Manin-Mumford, équidistribution, theorie de Ratner.

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     title = {Special subvarieties of mixed {Shimura} varieties},
     series = {Th\`eses d'Orsay},
     publisher = {Universite Paris-Sud Facult\'e des Sciences d'Orsay},
     number = {773},
     year = {2009},
     language = {en},
     url = {http://archive.numdam.org/item/BJHTUP11_2009__0773__A1_0/}
}

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Chen, Ke. Special subvarieties of mixed Shimura varieties. Thèses d'Orsay, no. 773 (2009), 122 p. http://numdam.org/item/BJHTUP11_2009__0773__A1_0/

Sommaire

1 Preliminaries	p. 16
1.1 Shimura data and Shimura varieties	p. 16
1.1.10 Fibers over a pure section	p. 23
1.1.11 Special sections	p. 24
1.2 Canonical models and reciprocity maps	p. 31
1.3 Hecke correspondences and special subvarieties	p. 34
2 Introduction to the André-Oort conjecture	p. 41
2.1 Statement of the conjecture	p. 41
2.2 Reductions and Reformulations	p. 44
2.3 An approach under GRH	p. 46
2.3.1 Some terminologies	p. 47
2.3.2 The approach under GRH	p. 48
2.4 Main results	p. 48
2.4.1 General setting	p. 49
2.4.2 The (weakly) homogeneous case	p. 49
2.4.3 The estimation of the degree of Galois orbits	p. 51
2.4.4 A generalization of the Manin-Mumford conjecture	p. 54
3 Equidistribution of special subvarieties: the homogeneous case	p. 56
3.1 Introduction	p. 56
3.1.1 The strategy of L. Clozel and E. Ullmo	p. 56
3.1.2 Our strategy in the mixed case	p. 57
3.2 Preliminaries on groups and ergodic theory	p. 60
3.3 Equidistribution of lattice subspaces	p. 64
3.4 $S$ -spaces and special $S$ -subspaces	p. 67
3.5 The Dani-Margulis argument	p. 71
3.6 The general case of a homogeneous sequence of special subvarieties	p. 76
4 The degree of the Galois orbit of a special subvariety	p. 80
4.1 Outline of the estimation of E. Ullmo and A. Yafaev	p. 81
4.2 On the factor $I_{2}^{w} (M')$	p. 87
4.3 The case of mixed special subvarieties	p. 92
5 Further perspectives	p. 94
5.1 Motivation	p. 94
5.2 Prerequisites on abelian schemes	p. 96
5.3 Generalized Manin-Mumford conjecture: the uniform case	p. 98
5.4 Generalized Manin-Mumford conjecture: the non-uniform case	p. 105
5.4.4 Products and fibrations: further remarks	p. 107

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