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.

@phdthesis{BJHTUP11_2009__0773__A1_0,
     author = {Chen, Ke},
     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://www.numdam.org/item/BJHTUP11_2009__0773__A1_0/}
}

TY  - BOOK
AU  - Chen, Ke
TI  - Special subvarieties of mixed Shimura varieties
T3  - Thèses d'Orsay
PY  - 2009
IS  - 773
PB  - Universite Paris-Sud Faculté des Sciences d'Orsay
UR  - http://www.numdam.org/item/BJHTUP11_2009__0773__A1_0/
LA  - en
ID  - BJHTUP11_2009__0773__A1_0
ER  -

%0 Book
%A Chen, Ke
%T Special subvarieties of mixed Shimura varieties
%S Thèses d'Orsay
%D 2009
%N 773
%I Universite Paris-Sud Faculté des Sciences d'Orsay
%U http://www.numdam.org/item/BJHTUP11_2009__0773__A1_0/
%G en
%F BJHTUP11_2009__0773__A1_0

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

Bibliographie
Cité par

[André-1] Y. André, G -functions and geometry, Aspects of Mathematics Vol. 13. Vieweg 1989 | MR | Zbl

[André-2] Y. André, Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part, Compositio Mathematica, 1992, Vol. 82 (1), pp. 1-24 | Zbl | MR | Numdam

[André-3] Y. André, Finitude des couples d'invariants modulaires singuliers sur une courbe algébrique plane non-modulaire, Journal für die reine und angewandte Mathematik, 1998, Vol. 505, pp. 203-208 | MR | Zbl | DOI

[André-4] Y. André, Shimura varieties, subvarieties, and CM points Six lectures at the Franco-Taiwan Arithmetic Festival, 2001. c.f. http://www.math.umd.edu/ yu/notes.shtml

[B-HC] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Bulletin of the American mathematics society, 1961, Vol. 67 (6), pp. 579-583 | MR | Zbl | DOI

[Bo-1] M.V. Borovoi, The Langlands conjecture on the conjugation of Shimura varieties, Functional Analysis and Applications, 1982, Vol. 16, pp. 292-294 | Zbl | DOI

[Bo-2] M.V. Borovoi Conjugation of Shimura varieties, Proceedings of the International Congress of Mathematicians, Berkeley 1986, Part.I, pp. 783-790 | MR | Zbl

[Ch-1] K. Chen, On the equidistribution of special subvarieties in a mixed Shimura variety, preprint

[Ch-2] K. Chen, Equidistribution and the Manin-Mumford conjecture, draft

[CU-1] L. Clozel and E. Ullmo, Correspondances modulaires et mesures invariants, Journal für die reine und angewandte Mathematik, 2003, Vol. 558, pp 47-83 | MR | Zbl

[CU-2] L. Clozel and E. Ullmo, Équidistribution des points de Hecke, in Contribution to automorphic forms, geometry, and number theory, pp. 193-254, John Hopkins University Press, 2004 | MR | Zbl

[CU-3] L. Clozel and E. Ullmo, Équidistribution de sous-variétés spéciales, Annals of Mathematics, 161 (3), pp 1571-1588, 2005 | MR | Zbl

[CU-4] L. Clozel and E. Ullmo, Rigidité de l'action des opérateurs de Hecke sur les espaces de réseaux, Mathematischen Annalen, 2003, Vol. 326, No. 2, pp. 209-236 | MR | Zbl | DOI

[CU-5] L. Clozel and E. Ullmo, Équidistribution adélique des tores et équidistribution des points CM, Documenta Mathematica, 2006, Volume in honor of J. Coates, pp. 233-260 | MR | Zbl

[DM] S. Dani and G. Margulis, Asymptotic behaviors of trajectories ofunipo- tent flows on homogeneous spaces, Proceedings of Indian academy of sciences (mathematical sciences), 1991, Vol. 101 (1), pp. 1-17 | MR | Zbl | DOI

[D-1] P. Deligne, Travaux de Shimura Séminaire bourbaki 1971, Exposé 389, Lecture Notes in Mathematics, Vol. 255, pp. 123-165 Springer Verlag 1972 | MR | Zbl | Numdam

[D-2] P. Deligne, Variétés de Shimura : interprétation modulaire, et techniques de construction de modèles canoniques in Automorphic forms, representations, and L-functions edited by A. Borel and W. Casselman, Proceedings of Symposia in Pure Mathematics, Vol. 33, Part 2, pp. 247-289. American Mathematics Society 1979 | MR | Zbl | DOI

[Duke-1] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Inventiones Mathematica, 1988, Vol. 92, no. 2, pp. 385-401 | MR | Zbl

[Duke-2] W. Duke, Modular functions and the uniform distributions of CM points, Mathematische Annalen, 2006, Vol. 334, pp. 241-252 | MR | Zbl | DOI

[E-1] B. Edixhoven, Special points on the product of two modular curves, Compositio Mathematica, 1998, Vol. 114 (3), pp. 315-328 | MR | Zbl

[E-2] B. Edixhoven, On the André-Oort conjecture for Hilbert modular surfaces, in Moduli of abelian varieties (Texel Island 1999) edited by G.van der Geer, C. Faber, F. Oort, Progress in Mathematics Vol. 199, pp. 133-155, Birkhäuser 2001 | MR | Zbl

[E-3] B. Edixhoven, Special points on products of modular curves Duke Mathematics Journal, 1998, Vol. 126 (2), pp. 621-645 | MR | Zbl

[EY] B. Edixhoven and A. Yafaev, Subvarieties of Shimura varieties, Annals of Mathematics, 2003, Vol. 157 (2), pp. 621-645 | MR | Zbl

[EGA IV3] J. Dieudonné and A. Grothendieck, Éléments de géométrie algébrique : IV. études locales des schémas et des morphismes des schémas, troisième partie, Publications mathématiques de l'I.H.E.S, 1966, tome 28, pp. 5-255 | Zbl | Numdam | MR | DOI

[E-M-S] A. Eskin, S. Mozes, N. Shah, Non-divergence of translates of certain algebraic measures, Geometric analysis and functional analysis, 1990, Vol. 7, pp. 48-80 | MR | Zbl | DOI

[Hind] M. Hindry, Autour d'une conjecture de Serge Lang, Inventiones Mathematica, 1988, vol. 94, pp. 575-603 | MR | Zbl | DOI

[G] A. Grothendieck, Un théorème sur les homomorphismes de schémas abéliens, Inventiones Mathematica, 1966, vol. 2, pp. 59-78 | MR | Zbl | DOI

[GIT] J. Fogarty and D. Mumford, Geometric invariant theory, Springer 1982 | MR | Zbl

[JLZ] D. Jiang, J. Li, and S. Zhang, Periods and distributions of cycles on Hilbert modular varieties, Pure and applied mathematics quaterly, 2006, Vol. 2, No. 1, pp. 219-277 | MR | Zbl | DOI

[Ku] M. Kuga, Fiber varieties over a symmetric space whose fibers are abelian varieties, Vol. I and Vol. II, The University of Chicago, 1963-1964 | MR | Zbl

[KY] B. Klingler and A. Yafaev, On the André-Oort conjecture preprint 2008 | Zbl

[LNM900] P. Deligne, J. Milne, A. Ogus, and K. Shih, Hodge cycles, motives, and Shimura varieties, Lecture notes in mathematics Vol. 900, 1982, Springer Verlag | MR | Zbl

[Milne-0] J. Milne, Canonical models of Shimura varieties and automorphic vector bundles, Automorphic forms, Shimura varieties, and L-functions Vol. 1, edited by L. Clozel and J. Milne, Academeic Press 1990 | MR | Zbl

[Milne-1] J. Milne, Introduction to Shimura varieties, Harmonic analysis, Trace formula, and Shimura varieties, Clay Institute and American Mathematical Society | Zbl

[Milne-2] J. Milne, Algebraic groups and arithmetic groups, course notes, cf. http://www.jmilne.org/math

[Mo] B. Moonen, Models of Shimura varieties in mixed characteristics in Galois representations in arithmetic algebraic geometry (Durham 1996) edited by AJ. Scholl and R. Taylor, London Mathematics Society Lecture Notes Series Vol. 254, pp 267-350. Cambridge University Press, 1998 | MR | Zbl

[MS] S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic theory and dynamic systems, 1995, Vol. 15, pp. 149-159 | MR | Zbl | DOI

[Mum] D. Mumford, Abelian varieties, Tata Institute. | MR | Zbl

[Mu] J-P. Murre, Representation of unramified junctors, applications (according to unpublished results of A. Grothendieck), Séminaire Bourbaki, Exposé 294, 1964-1965 | Zbl | Numdam | MR

[Nori] M. Nori, On subgroups of $G L_{n} (𝐅_{p})$ , Inventiones Mathematica, 1987, Vol. 88 (2), pp. 257-275 | MR | Zbl

[Noot] R. Noot, Correspondances de Hecke, actions de Galois, et la conjecture d'André-Oort, Séminaire Bourbaki, Exposé 942, 2004-2005 | MR | Zbl | Numdam

[Oort-1] F. Oort, Some questions in algebraic geometry, 1995. c.f. http://www.math.uu.nl/people/oort

[Oort-2] F. Oort, Canonical liftings and dense sets of CM-points, in Arithmetic geometry (Cortona 1994) , Symposia Mathematica, Vol. 37, pp. 228-234, Cambridge University Press 1997 | MR | Zbl

[Pink-0] R. Pink, Arithmetical compactifications of mixed shimura varieties, Ph.D Thesis. | MR | Zbl

[Pink-1] R. Pink, A combination of the conjecture of Mordell-Lang and André-Oort in Geometric methods in algebra and number theory, Progress in Mathematics, Vol. 235, pp. 251-282. Birkhäuser, 2005 | MR | Zbl

[PR] V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Pure and applied mathematics, Academic Press, 1990 | Zbl | MR

[Ragh] M.S. Raughunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematic und ihrer Grenzgebiete, Band 68, Springer Verlag 1972 | MR | Zbl

[Rat-1] M. Ratner, On Raghunathan's measure conjecture, Annals of mathematics, 1991, Vol. 194, pp. 545-607 | MR | Zbl | DOI

[Rat-2] M. Ratner, Raghunathan's topological conjecture and distributions of unipotentflows, Duke mathematics journal, 1991, Vol. 63, pp. 235-280 | MR | Zbl | DOI

[R-1] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Inventiones Mathematica, 1983, Vol. 71 (1), pp. 207-233 | MR | Zbl | DOI

[R-2] M. Raynaud, Sous-variétés dune variété abélienne et points de torsion, in Arithmetic and Geometry, Vol. I, edited by M. Artin and J. Tate, Progress in Mathematics, Vol. 35, pp. 327-352, Birkhäuser 1983 | MR | Zbl | DOI

[RT] D. Rockmore and K. Tan, A note on the order of finite subgroups of ${𝐆𝐋}_{n} (ℤ)$ , Archiv der Mathematik, 1995, vol. 64 (4), pp. 283-288 | MR | Zbl | DOI

[RU] N. Ratazzi and E. Ullmo, Galois + Équidistribution = Manin-Mumford, notes of lectures at Göttingen Summer School of Arithmetic Geometry, 2006, cf. http://www.math.u-psud.fr/~ullmo | Zbl

[S] N. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Mathematischen Annalen, Vol. 289 (1991), pp. 315-334 | MR | Zbl | DOI

[Sh] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures I, Princeton University Press, 1971 | MR | Zbl

[SGA 4] M. Artin, A. Grothendieck, J-L. Verdier, Séminaire de géométrie algébrique du Bois-Marie : Théorie des topos et cohomologie étale des schémas, Tome 3, Lecture notes in mathematics Vol. 305,1970, Springer Verlag

[Spr] T.A. Springer, Linear algebraic groups, Second edition, Progress in Mathematics, Vol. 9, Birkhäuser | MR | Zbl

[SUZ] L. Szpiro, E. Ullmo, and S. Zhang, Equidistribution des petits points, Inventiones Mathematica, 1997, Vol. 127, pp. 337-347 | MR | Zbl | DOI

[U-1] E. Ullmo, Théorie ergodique et géométrie arithmétique in Proceedings of the International Congress of Mathematicians (Beijing 2002) Vol. II, pp. 197-206, Higher Education Press 2002. | MR | Zbl

[U-2] E. Ullmo, Points rationnels des variétés de Shimura, Intematianl Mathematical Research Notices, 2004, Vol. 76, pp. 4109-4125 | MR | Zbl | DOI

[U-3] E. Ullmo, Équidistribution de sous-variétés spéciales II, Journal für die reine und angewandte Mathematik, 2007, Vol. 606, pp. 193-216 | MR | Zbl

[U-3] E. Ullmo, Manin-Mumford, André-Oort : the equidistribution point of view, notes of a course at the Summer Scool "Equidistribution in Number theory" | Zbl

[U-4] E. Ullmo, Autour de la conjecture d'André-Oort, notes of a course at Paris-Sud University, 2007.

[UY-1] E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties : towards the André-Oort conjecture, preprint 2006, with an appendix by P. Gille and L. Morel-Bailly entitled Actions algébriques des groupes arithmétiques, cf. http://www.math.u-psud.fr/~ullmo | MR | Zbl

[UY-2] E. Ullmo and A. Yafaev, The André-Oort conjecture for products of modular curves, notes of lectures at Gottingen Summer School of Arithmetic Geometry, 2006, cf. http://www.math.u-psud.fr/~ullmo | Zbl

[UY-3] E. Ullmo and A. Yafaev, Points rationnels des variétés de Shimura : un principe du tout ou rien, preprint 2007, cf. http://www.math.u-psud.fr/~ullmo | MR | Zbl

[Y-1] A. Yafaev, Special points on products of two Shimura curves, Manuscripta Mathematica, 2001, Vol. 104 (2), pp163-171 | MR | Zbl | DOI

[Y-2] A. Yafaev, On a result of Moonen on the moduli space of principally polarized abelian varieties, Compositio Mathematica, 2005, Vol. 141 (5), pp. 1103-1108 | MR | Zbl | DOI

[Y-3] A. Yafaev, A conjecture of Yves André, Duke Mathematics Journal, 2006, Vol. 132 (3), pp. 393-407 | MR | Zbl | DOI

[Zh] S. Zhang, Equidistribution of small points on abelian varieties, Annals of Mathematics, 1998, Vol. 147 (1), 159-165 | MR | Zbl

[Zim] R. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, Vol. 81, Birkhäuser 1984 | MR | Zbl