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On reduction of Hilbert-Blumenthal varieties. (English) Zbl 1117.14027
Summary: Let \(O_{{\mathbb{F}}}\) be the ring of integers of a totally real field \({\mathbb{F}}\) of degree \(g\). We study the reduction of the moduli space of separably polarized abelian \(O_{{\mathbb{F}}}\)-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 E. Z. Goren and F. Oort [J. Algebr. Geom. 9, 111–154 (2000; Zbl 0973.14010)] on the stratifications when \(p\) is unramified in \(O_{{\mathbb{F}}}\). We also prove the strong Grothendieck conjecture for the moduli space in some restricted cases, particularly when \(p\) is totally ramified in \(O_{{\mathbb{F}}}\).

MSC:
14G35 Modular and Shimura varieties
14L05 Formal groups, \(p\)-divisible groups
Citations:
Zbl 0973.14010
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