On reduction of Hilbert-Blumenthal varieties
[Sur la réduction des variétés de Hilbert-Blumenthal]
Annales de l'Institut Fourier, Tome 53 (2003) no. 7, pp. 2105-2154.

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 .

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 .

DOI : 10.5802/aif.2002
Classification : 14G35, 14L05
Keywords: Hilbert-Blumenthal varieties, Dieudonné modules, stratifications, deformations
Mot clés : variétés de Hilbert-Blumenthal, modules de Dieudonné, stratifications, déformations
Yu, Chia-Fu 1

1 National Tsing-Hua University, National Center for Theoretical Sciences, 3rd General Bldg, 101 Sec. Kuang-Fu road, Tsinchu 30043 (Taiwan)
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Yu, Chia-Fu. On reduction of Hilbert-Blumenthal varieties. Annales de l'Institut Fourier, Tome 53 (2003) no. 7, pp. 2105-2154. doi : 10.5802/aif.2002. http://www.numdam.org/articles/10.5802/aif.2002/

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