On reduction of Hilbert-Blumenthal varieties

Yu, Chia-Fu

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 $O_{𝐅}$.

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