Correspondances de Hecke, action de Galois et la conjecture d'André-Oort  [ Hecke correspondences, Galois action and the André-Oort conjecture ]
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque no. 307  (2006), Talk no. 942, p. 165-197

Let M be a Shimura variety, Z an irreducible closed subset of M and SZ() a Zariski dense set of special points. The André-Oort conjecture states that Z is a subvariety of Hodge type. For instance, if M is a moduli space of abelian varieties, then S is a set of points corresponding to abelian varieties of CM-type and Z should parametrize a family of abelian varieties characterized by certain Hodge classes. Edixhoven and Yafaev have proved special cases of this conjecture using the actions of the Hecke algebra and of the Galois group.

Soient M une variété de Shimura, ZM fermée et irréductible et SZ() un ensemble Zariski dense de points spéciaux. Selon la conjecture d’André-Oort, Z est une sous-variété de type Hodge. Par exemple, si M est un espace de modules de variétés abéliennes, S est un ensemble de points correspondant à des variétés de type CM et Z doit paramétrer des variétés abéliennes munies de certaines classes de Hodge. En utilisant les actions de l’algèbre de Hecke et du groupe de Galois, Edixhoven et Yafaev montrent certains cas de la conjecture.

Classification:  11G18,  14G35
Keywords: Shimura variety, modular variety, subvariety, Hecke correspondence
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     author = {Noot, Rutger},
     title = {Correspondances de Hecke, action de Galois et~la~conjecture d'Andr\'e-Oort},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:942},
     pages = {165-197},
     zbl = {1175.14013},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2004-2005__47__165_0}
}
Noot, Rutger. Correspondances de Hecke, action de Galois et la conjecture d'André-Oort, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 942, pp. 165-197. http://www.numdam.org/item/SB_2004-2005__47__165_0/

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