Kottwitz-Rapoport and $p$-rank strata in the reduction of Shimura varieties of PEL type

Hartwig, Philipp

doi:10.5802/aif.2951

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]

Hartwig, Philipp ¹

¹ Universität Duisburg-Essen Fakultät für Mathematik 45117 Essen (Germany)

Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1031-1103.

Résumé
Abstract

Nous étudions la réduction de certains modèles entiers des variétés de Shimura de type PEL à structure de niveau Iwahori. Sur ces espaces on a la stratification de Kottwitz-Rapoport et la stratification de $p$ -rang. Nous montrons que le $p$ -rang est constant sur un strate de Kottwitz-Rapoport, généralisant un résultat de Ngô et Genestier. Nous montrons une formule abstraite, uniforme pour le $p$ -rang sur un strate de Kottwitz-Rapoport. Dans les cas symplectique et unitaire nous trouvons des formules explicites pour sa valeur. Nous appliquons ces formules à la question de la densité du lieu ordinaire et à la question de la dimension du lieu de $p$ -rang 0.

We study the reduction of certain integral models of Shimura varieties of PEL type with Iwahori level structure. On these spaces we have the Kottwitz-Rapoport and the $p$ -rank stratification. We show that the $p$ -rank is constant on a KR stratum, generalizing a result of Ngô and Genestier. We prove an abstract, uniform formula for the $p$ -rank on a KR stratum. In the symplectic and in the unitary case we derive explicit formulas for its value. We apply these formulas to the question of the density of the ordinary locus and to the question of the dimension of the $p$ -rank 0 locus.

DOI : 10.5802/aif.2951

Classification : 14G35, 14K10
Keywords: Abelian varieties, $p$-rank stratification, Kottwitz-Rapoport stratification, Iwahori decomposition, ordinary locus, Hilbert-Blumenthal modular varieties, affine Deligne-Lusztig varieties
Mots-clés : Variétés abéliennes, stratification de $p$-rang, stratification de Kottwitz-Rapoport, décomposition d’Iwahori, lieu ordinaire, espaces de modules de Hilbert-Blumenthal, variétés de Deligne-Lusztig affines

Affiliations des auteurs :

Hartwig, Philipp ¹

¹ Universität Duisburg-Essen Fakultät für Mathematik 45117 Essen (Germany)

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Hartwig, Philipp. Kottwitz-Rapoport and $p$-rank strata in the reduction of Shimura varieties of PEL type. Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1031-1103. doi : 10.5802/aif.2951. https://aif.centre-mersenne.org/articles/10.5802/aif.2951/

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