This thesis studies the Andre-Oort conjecture for mixed Shimura varieties. The main result is: let be a mixed Shimura variety defined by a mixed Shimura datum , a fixed -torus of , and an arbitrary closed subvariety in , then the set of maximal -special subvarieties of contained in 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 be an abelian -scheme of characteristic zero, then the Zariski closure of a sequence of torsion subschemes in remains a finite union of torsion subschemes.
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 définie par une donnée de Shimura mixte , soient un -tore dans et une sous-variété fermée quelconque dans , alors l’ensemble des sous-variétés -spéciales maximales contenues dans 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 un -schéma abélien en caractéristique nul, alors l’adhérence de Zariski d’une suite de sous-schémas de torsion dans égale une réunion finie de sous-schémas de torsion.
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://archive.numdam.org/item/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/
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