On reduction of Hilbert-Blumenthal varieties
Annales de l'Institut Fourier, Volume 53 (2003) no. 7, p. 2105-2154
Let O 𝐅 be the ring of integers of a totally real field 𝐅 of degree g. We study the reduction of the moduli space of separably polarized abelian O 𝐅 -varieties of dimension g modulo p for a fixed prime p. The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by a-types on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of Goren and Oort [J. Alg. Geom., 2000] on the stratifications when p is unramified in O 𝐅 . We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when p is totally ramified in O 𝐅 .
Soit O 𝐅 l’anneau des entiers d’un corps de nombres totalement réel 𝐅 de degré g. Nous étudions un nombre premier p fixé, la réduction modulo p de l’espace de modules classifiant les O 𝐅 -variétés abéliennes séparablement polarisées de dimension g. Nous construisons une stratification schématique par les types a du lieu de Rapoport et étudions sa relation avec la stratification par les pentes. En particulier, nous retrouvons les résultats principaux de Goren et Oort [J. Alg. Geom., 2000] sur les stratifications lorsque p n’est pas ramifié dans O 𝐅 . Nous démontrons également la conjecture de Grothendieck forte pour les espaces de modules dans certains cas, notamment lorsque p est totalement ramifié dans O 𝐅 .
DOI : https://doi.org/10.5802/aif.2002
Classification:  14G35,  14L05
Keywords: Hilbert-Blumenthal varieties, Dieudonné modules, stratifications, deformations
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     author = {Yu, Chia-Fu},
     title = {On reduction of Hilbert-Blumenthal varieties},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     pages = {2105-2154},
     doi = {10.5802/aif.2002},
     zbl = {02093468},
     mrnumber = {2044169},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_7_2105_0}
}
Yu, Chia-Fu. On reduction of Hilbert-Blumenthal varieties. Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 2105-2154. doi : 10.5802/aif.2002. http://www.numdam.org/item/AIF_2003__53_7_2105_0/

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