On reduction of Hilbert-Blumenthal varieties

Yu, Chia-Fu

doi:10.5802/aif.2002

On reduction of Hilbert-Blumenthal varieties
[Sur la réduction des variétés de Hilbert-Blumenthal]

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)

Annales de l'Institut Fourier, Tome 53 (2003) no. 7, pp. 2105-2154.

Résumé
Abstract

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 $O_{𝐅}$ .

MR Zbl

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

Affiliations des auteurs :

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. https://aif.centre-mersenne.org/articles/10.5802/aif.2002/

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