On reduction of Hilbert-Blumenthal varieties

[Sur la réduction des variétés de Hilbert-Blumenthal]

Yu, Chia-Fu

Annales de l'Institut Fourier, Tome 53 (2003), p. 2105-2154 / Harvested from Numdam

Résumé

Soit $O_{{\bf F}}$ l’anneau des entiers d’un corps de nombres totalement réel ${\bf F}$ de degré $g$. Nous étudions un nombre premier $p$ fixé, la réduction modulo $p$ de l’espace de modules classifiant les $O_{{\bf F}}$-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 $O_{{\bf F}}$. Nous démontrons également la conjecture de Grothendieck forte pour les espaces de modules dans certains cas, notamment lorsque $p$ est totalement ramifié dans $O_{{\bf F}}$.

Abstract

Let $O_{{\bf F}}$ be the ring of integers of a totally real field ${\bf F}$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_{{\bf F}}$-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 $O_{{\bf F}}$. We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when $p$ is totally ramified in $O_{{\bf F}}$.

Détail

Publié le : 2003-01-01

MR 2044169 | Zbl 02093468

DOI : https://doi.org/10.5802/aif.2002
Classification:  14G35,  14L05
Mots clés: variétés de Hilbert-Blumenthal, modules de Dieudonné, stratifications, déformations

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