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Deligne–Lusztig varieties and basic EKOR strata

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  29 June 2020

Haining Wang
Haining Wang*
Affiliation:
Department of Mathematics, McGill University, 805 Sherbrooke St W, Montreal, QCH3A 0B9, Canada
*
e-mail: [email protected]
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Abstract

Using the axioms of He and Rapoport for the stratifications of Shimura varieties, we explain a result of Görtz, He, and Nie that the EKOR strata contained in the basic loci can be described as a disjoint union of Deligne–Lusztig varieties. In the special case of Siegel modular varieties, we compare their descriptions to that of Görtz and Yu for the supersingular Kottwitz-Rapoport strata and to the descriptions of Harashita and Hoeve for the supersingular Ekedahl–Oort strata.

Keywords

Deligne-Lusztig varietiesShimura varieties

MSC classification

Primary: 14G35: Modular and Shimura varieties
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 349 - 367
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Copyright
© Canadian Mathematical Society 2020

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References

Bhatt, B. and Scholze, P., Projectivity of the Witt vector affine Grassmannian . Invent. Math. 209(2017), no. 2, 329–423. https://doi.org/10.1007/s00222-016-0710-4 CrossRefGoogle Scholar
Deligne, P. and Lusztig, G., Representations of reductive groups over finite fields . Ann. of Math. (2) 103(1976), no. 1, 103–161. https://doi.org/10.2307/1971021 CrossRefGoogle Scholar
Genestier, A. and Ngô, B. C., Alcôves et $p$ -rang des variétés abéliennes . Ann. Inst. Fourier (Grenoble) vol. 52(2002), no. 6, 1665–1680.Google Scholar
Görtz, U., On the flatness of models of certain Shimura varieties of PEL-type . Math. Ann. 321(2001), no. 3, 689–727. https://doi.org/10.1007/s002080100250 CrossRefGoogle Scholar
Görtz, U., On the flatness of local models for the symplectic group . Adv. Math. 351(2003), no. 1, 89–115. https://doi.org/10.1016/S0001-8708(02)00062-2 CrossRefGoogle Scholar
Görtz, U. and He, X.-H., Basic loci of Coxeter type in Shimura varieties . Camb. J. Math. 3(2015), no. 3, 323–353.CrossRefGoogle Scholar
Görtz, U. and He, X.-H., Erratum to: Basic loci in Shimura varieties of Coxeter type. Camb. J. Math. 6(2018), no. 1, 89–92. https://doi.org/10.4310/CJM.2018.v6.n1.e4 CrossRefGoogle Scholar
Görtz, U., He, X.-H., and Nie, S.-A., Fully Hodge-Newton decomposable Shimura varieties . Peking Math. J. 2(2019), 99–154. https://doi.org/10.1007/s42543-019-00013-2 CrossRefGoogle Scholar
Görtz, U. and Hoeve, M., Ekedahl-Oort strata and Kottwitz-Rapoport strata . J. Algebra 351(2012), 160–174. https://doi.org/10.1016/j.jalgebra.2011.10.039 CrossRefGoogle Scholar
Görtz, U. and Yu, C.-F., Supersingular Kottwitz-Rapoport strata and Deligne–Lusztig varieties . J. Inst. Math. Jussieu 9(2010), no. 2, 357–390. https://doi.org/10.1017/S1474748009000218 CrossRefGoogle Scholar
Görtz, U. and Yu, C.-F., The supersingular locus in Siegel modular varieties with Iwahori level structure . Math. Ann. 353(2012), no. 2, 465–498. https://doi.org/10.1007/s00208-011-0689-5 CrossRefGoogle Scholar
Hamacher, P. and Kim, W., l-Adic etale cohomology of Shimura varieties of Hodge type with non-trivial coefficient . Math. Ann. 375(2019), no. 3–4, 973–1044. https://doi.org/10.1007/s00208-019-01815-6 CrossRefGoogle Scholar
Harashita, S., Ekedahl-Oort strata contained in the supersingular locus and Deligne-Lusztig varieties . J. Algebr. Geom. 19(2010), no. 3, 419–438. https://doi.org/10.1090/S1056-3911-09-00519-0 CrossRefGoogle Scholar
He, X.-H., Geometric and homological properties of affine Deligne-Lusztig varieties . Ann. of Math. (2) 179(2014), no. 1, 367–404. https://doi.org/10.4007/annals.2014.179.1.6 CrossRefGoogle Scholar
He, X.-H., Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties . Ann. Sci. Éc. Norm. Supér. (4) 49(2016), no. 5, 1125–1141. https://doi.org/10.24033/asens.2305 CrossRefGoogle Scholar
He, H., Li, C., and Zhu, Y.-H., Fine Deligne–Lusztig varieties and arithmetic fundamental lemmas . Preprint, 2019. arXiv:1901.02870.CrossRefGoogle Scholar
He, X.-H. and Rapoport, M., Stratifications in the reduction of Shimura varieties . Manuscripta Math. 152(2017), no. 3–4, 317–343. https://doi.org/10.1017/fms.2019.45 CrossRefGoogle Scholar
Helm, D., Tian, Y.-C., and Xiao, L., Tate cycles on some unitary Shimura varieties mod p . Algebra Number Theory 11(2017), no. 10, 2213–2288. https://doi.org/10.2140/ant.2017.11.2213 CrossRefGoogle Scholar
Hoeve, M., Ekedahl-Oort strata in the supersingular locus . J. Lond. Math. Soc. (2) 81(2010), no. 1, 129–141. https://doi.org/10.1112/jlms/jdp061 CrossRefGoogle Scholar
Kisin, M., Mod p points on Shimura varieties of abelian type . J. Amer. Math. Soc. 30(2017), no. 3, 819–914. https://doi.org/10.1090/jams/867 CrossRefGoogle Scholar
Langlands, R. and Rapoport, M., Shimuravarietäten und Gerben . J. Reine Angew. Math. 378 (1987), 113–220.Google Scholar
Moonen, B., Group schemes with additional structures and Weyl group cosets . In: Moduli of abelian varieties (Texel Island, 1999), Progr. Math., 195, Birkhäuser, Basel, 2001, pp. 255–298.CrossRefGoogle Scholar
Rapoport, M., A guide to the reduction modulo $p$ of Shimura varieties . Automorphic forms. I. Astérisque 298(2005), 271–318.Google Scholar
Rapoport, M. and Richartz, M., On the classification and specialization of $F$ -isocrystals with additional structure . Compositio Math. 103(1996), no. 2, 153–181.Google Scholar
Rapoport, M., Terstiege, U., and Zhang, W., On the arithmetic fundamental lemma in the minuscule case . Compos. Math. 149(2013), no. 10, 1631–1666. https://doi.org/10.1112/S0010437X13007239 CrossRefGoogle Scholar
Rapoport, M. and Viehmann, E., Towards a theory of local Shimura varieties . Münster J. Math. 7(2014), no. 1, 273–326.Google Scholar
Rapoport, M. and Zink, T., Period spaces for $p$ -divisible groups. Ann. Math. Stud., 141, Princeton University Press, Princeton, NJ, 1996. https://doi.org/1015/9781400882601 Google Scholar
Shen, X., Yu, C.-F., and Zhang, C., EKOR strata for Shimura varieties with parahoric level structure. Preprint, 2019.Google Scholar
Viehmann, E., Truncations of level 1 of elements in the loop group of a reductive group . Ann. of Math. (2) 179(2014), no. 3, 1009–1040. https://doi.org/10.4007/annals.2014.179.3.3 CrossRefGoogle Scholar
Vollaard, I., The supersingular locus of the Shimura variety for $\omega\in^{c}W$ . Canad. J. Math. 62(2010), no. 3, 668–720. https://doi.org/10.4153/CJM-2010-031-2 CrossRefGoogle Scholar
Vollaard, I. and Wedhorn, T., The supersingular locus of the Shimura variety of $\omega\in^{c}W.$ II . Invent. Math. 184(2011), no. 3, 591–627. https://doi.org/10.1007/s00222-010-0299-y CrossRefGoogle Scholar
Xiao, L. and Zhu, X.-W., Cycles on Shimura varieties via geometric Satake . Preprint, 2017.Google Scholar
Zhou, R., Mod-p isogeny classes on Shimura varieties with parahoric level structure . Preprint, 2019.CrossRefGoogle Scholar
Zhu, X.-W., Affine Grassmannians and the geometric Satake in mixed characteristic . Ann. of Math. (2) 185(2017), no. 2, 403–492. https://doi.org/10.4007/annals.2017.185.2.2 CrossRefGoogle Scholar