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Kottwitz-Rapoport and \(p\)-rank strata in the reduction of Shimura varieties of PEL type. (Strates de Kottwitz-Rapoport et de \(p\)-rang dans la réduction des variétés de Shimura de type PEL.) (English. French summary) Zbl 1345.14032
This paper investigates the geometry of the special fibre of suitable integral models of Shimura varieties of PEL type with Iwahori level structure at the chosen prime. It is proved that the \(p\)-rank is constant on every Kottwitz-Rapoport stratum. This generalizes a result of B. C. Ngô and A. Genestier [Ann. Inst. Fourier 52, No. 6, 1665–1680 (2002; Zbl 1046.14023)].
An abstract uniform formula for the \(p\)-rank on a given KR stratum in combinatorial terms is given, and is made explicit in the symplectic and unitary cases. From this, results about the density of the ordinary locus and about the dimension of the \(p\)-rank \(0\) stratum are derived.
Furthermore, the Hilbert-Blumenthal case corresponding to a totally real extension \(F/\mathbb Q\) of degree \(g\) is discussed in detail. This generalizes the results of H. Stamm in the case \(g=2\) [Forum Math. 9, No. 4, 405–455 (1997; Zbl 0916.14022)].

MSC:
14G35 Modular and Shimura varieties
14K10 Algebraic moduli of abelian varieties, classification
11G18 Arithmetic aspects of modular and Shimura varieties
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