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On central leaves of Hodge-type Shimura varieties with parahoric level structure

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Abstract

Kisin and Pappas (Publ Math Inst Hautes Études Sci, 2018) constructed integral models of Hodge-type Shimura varieties with parahoric level structure at \(p>2\), such that the formal neighbourhood of a mod p point can be interpreted as a deformation space of p-divisible group with some Tate cycles (generalising Faltings’ construction). In this paper, we study the central leaf and the closed Newton stratum in the formal neighbourhoods of mod p points of Kisin–Pappas integral models with parahoric level structure; namely, we obtain the dimension of central leaves and the almost product structure of Newton strata. In the case of hyperspecial level structure (i.e., in the good reduction case), our main results were already obtained by Hamacher (Math Z 287(3–4):1255–1277, 2017), and we show that the result of this paper holds for ramified groups as well.

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Notes

  1. It is often convenient and natural to allow \(\mathfrak {X}\) to be an analytic space or a formal scheme. But it will be quite obvious how to adapt the subsequent discussion to these cases.

  2. By replacing \(\nu _{\dot{\tilde{w}}}\) with \(r\nu _{\dot{\tilde{w}}}\) for some r, the claim reduces to a familiar statement on cocharacters of split tori.

  3. We assume that \(p\not \mid |\pi _1(G^\mathrm {der})|\) to ensure that the local model \(\mathrm {M}^\mathrm {loc}_{\mathcal {G}^\circ ,\{\mu \}}\) is normal (so the “deformation ring” \(R_\mathcal {G}\) is normal). The rest of the assumptions are made because we need to have [24, Proposition 1.4.3] for the deformation theory, which is proved under the assumption that G splits after a tame extension and the adjoint group of G has no factor of type \(E_8\).

  4. As explained in Remark 4.2.8, this assumption can be arranged if \(R_\mathcal {G}\) came from some integral model of Shimura varieties constructed in [24].

  5. See also [10, §3.2] for the statement and argument which works for more general connected reductive groups over \(\mathbb {Z}_p\) instead of \(\mathrm {GL}(\Lambda )\).

  6. As explained in Remark 4.2.8, this assumption can be arranged if \(R_\mathcal {G}\) came from some integral model of Shimura varieties constructed in [24].

  7. Here, one can give an alternative construction of \(\jmath _\xi \) without using Proposition 5.2.3, as follows. Since \(X_\xi \) has constant Newton polygon, it is known that there exists a unique isomorphism of F-isocrystals \(M_\xi [\tfrac{1}{p}] \xrightarrow {\sim }W(R)\otimes _{\breve{\mathbb {Z}}_p}\varvec{\mathrm {M}}[\tfrac{1}{p}]\) that reduces to the identity map on \(\varvec{\mathrm {M}}[\tfrac{1}{p}]\) via \(W(R)\twoheadrightarrow W(\overline{\mathbb {F}}_p) = \breve{\mathbb {Z}}_p\). (The existence of such isomorphism is proved in [18, Theorem 2.7.4], and the uniqueness follows from [36, Lemma 3.9]). Now, by the Dieudonné theory over R (cf. [1, Corollaire 3.4.3]), the above isomorphism of F-isocrystals gives rise to a unique quasi-isogeny \(X_\xi \dashrightarrow \mathbb {X}_R\) of p-divisible groups over \(SpecR\), which should recover \(\jmath _\xi \) by uniqueness.

References

  1. Berthelot, P.: Théorie de Dieudonné sur un anneau de valuation parfait. Ann. Sci. École Norm. Sup. (4) 13(2), 225–268 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée. Inst. Hautes Études Sci. Publ. Math. 60, 197–376 (1984)

    Article  MATH  Google Scholar 

  3. Caraiani, A., Scholze, P.: On the generic part of the cohomology of compact unitary Shimura varieties. Ann. Math. 186(3), 649–766 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chai, C.-L.: Hecke orbits on Siegel modular varieties, Geometric Methods in Algebra and Number Theory, Progress in Mathematics, vol. 235. Birkhäuser, Boston, pp. 71–107 (2005)

  5. de Jong, A.J.: Crystalline Dieudonné module theory via formal and rigid geometry. Inst. Hautes Études Sci. Publ. Math. 82, 5–96 (1995)

    Article  MATH  Google Scholar 

  6. Deligne, P.: Hodge cycles on abelian varieties, Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, vol. 900. Springer, Berlin (1982)

  7. Faltings, G.: Integral crystalline cohomology over very ramified valuation rings. J. Am. Math. Soc. 12(1), 117–144 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fargues, L., Fontaine, J.-M.: Vector bundles and \(p\)-adic Galois representations. In: Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt. 1, vol. 2, Amer. Math. Soc., Providence pp. 77–113 (2012)

  9. Haines, T.J., Rapoport, M.: Shimura varieties with \(\Gamma_1(p)\)-level via Hecke algebra isomorphisms: the Drinfeld case. Ann. Sci. Éc. Norm. Supér. (4) 45(5), 719–785 (2013)

    Article  MATH  Google Scholar 

  10. Hamacher, P.: The almost product structure of Newton strata in the deformation space of a Barsotti-Tate group with crystalline Tate tensors. Math. Z. 287(3–4), 1255–1277 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hamacher, P.: On the Newton stratification in the good reduction of Shimura varieties (preprint) (2016). arXiv:1605.05540

  12. Hamacher, P., Kim, W.: \(l\)-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients (preprint) (2017). arXiv:1711.07123

  13. Harris, M., Taylor, R.: The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001). With an appendix by Vladimir G. Berkovich

  14. Hartl, U., Viehmann, E.: Foliations in deformation spaces of local \(G\)-shtukas. Adv. Math. 229(1), 54–78 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. He, X.: Geometric and homological properties of affine Deligne-Lusztig varieties. Ann. Math. 179(1), 367–404 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. He, X.: Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties. Ann. Sci. Éc. Norm. Supér. (4) 49(5), 1125–1141 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Howard, B., Pappas, G.: Rapoport-Zink spaces for spinor groups. Compos. Math. 153(5), 1050–1118 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Katz, N.M.: Slope filtration of \(F\)-crystals. Journées de Géométrie Algébrique de Rennes (Rennes, 1978), vol. I. Astérisque, vol. 63, Soc. Math. France, Paris, pp. 113–163 (1979)

  19. Katz, N.: Serre–Tate Local Moduli, Algebraic Surfaces (Orsay, 1976–78), Lecture Notes in Math., vol. 868. Springer, Berlin, pp. 138–202 (1981)

  20. Kim, W.: Rapoport–Zink spaces of Hodge type. Forum Math. Sigma 6, e8 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kim, W.: Rapoport–Zink uniformisation of Hodge-type Shimura varieties (preprint) (2015)

  22. Kisin, M.: Integral models for Shimura varieties of abelian type. J. Am. Math. Soc. 23(4), 967–1012 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kisin, M.: Mod\(p\) points on Shimura varieties of abelian type. J. Amer. Math. Soc 30(3), 819–914 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kisin, M., Pappas, G.: Integral models of Shimura varieties with parahoric level structure. Publ. Math. Inst. Hautes Études Sci. (2018). https://doi.org/10.1007/s10240-018-0100-0

  25. Kottwitz, R.E.: Isocrystals with additional structure. Compos. Math. 56(2), 201–220 (1985)

    MathSciNet  MATH  Google Scholar 

  26. Kottwitz, R.E.: Isocrystals with additional structure. II. Compos. Math. 109(3), 255–339 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lau, E.: Relations between Dieudonné displays and crystalline Dieudonné theory. Algebra Number Theory 8(9), 2201–2262 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mantovan, E.: On certain unitary group Shimura varieties, Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales, Astérisque, vol. 291, Soc. Math. France, Paris, pp. 201–331 (2004)

  29. Mantovan, E.: On the cohomology of certain PEL-type Shimura varieties. Duke Math. J. 129(3), 573–610 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Moonen, B.: Models of Shimura varieties in mixed characteristics, Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., vol. 254. Cambridge University Press, Cambridge, pp. 267–350 (1998)

  31. Neupert, S.: Foliations and the cohomology of moduli spaces of bounded global \(G\)-shtukas. (preprint) arXiv:1610.95935

  32. Oort, F.: Foliations in moduli spaces of abelian varieties. J. Am. Math. Soc. 17(2), 267–296 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Oort, F.: Foliations in moduli spaces of abelian varieties and dimension of leaves, Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin., vol. II, Progr. Math., vol. 270. Birkhäuser, Boston, pp. 465–501 (2009)

  34. Pappas, G., Zhu, X.: Local models of Shimura varieties and a conjecture of Kottwitz. Invent. Math. 194(1), 147–254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rapoport, M.: A guide to the reduction modulo \(p\) of Shimura varieties. Automorphic forms. I. Astérisque, vol. 298, Soc. Math. France, Paris, pp. 271–318 (2005)

  36. Rapoport, M., Richartz, M.: On the classification and specialization of \(F\)-isocrystals with additional structure. Compos. Math. 103(2), 153–181 (1996)

    MathSciNet  MATH  Google Scholar 

  37. Rapoport, M., Viehmann, E.: Towards a theory of local Shimura varieties. Münster J. Math. 7(1), 273–326 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Rapoport, M., Zink, T.: Period spaces for \(p\)-divisible groups, Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996)

    MATH  Google Scholar 

  39. Scholze, P., Weinstein, J.: Moduli of \(p\)-divisible groups. Camb. J. Math. 1(2), 145–237 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Viehmann, E., Wu, H.: Central leaves in loop groups. Math. Res. Letters (2018) (to appear)

  41. Zhou, R.: Mod-\(p\) isogeny classes on Shimura varieties with parahoric level structure (preprint) (2017). arXiv:1707.09685

  42. Zink, T.: A Dieudonné theory for \(p\)-Divisible Groups, Class Field Theory—Its Centenary and Prospect (Tokyo, 1998), Adv. Stud. Pure Math., vol. 30. Math. Soc. Japan, Tokyo, pp. 139–160 (2001)

  43. Zink, T.: On the slope filtration. Duke Math. J. 109(1), 79–95 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author thanks Paul Hamacher for many helpful discussions, and Julien Hauseux for his help with group theory used in the proof of Proposition 3.1.4. The author would also like to thank the anonymous referee whose comments were greatly helpful. This work was supported by the EPSRC (Engineering and Physical Sciences Research Council) in the form of EP/L025302/1.

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  1. KIAS, Department of Mathematics, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, South Korea

    Wansu Kim

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Kim, W. On central leaves of Hodge-type Shimura varieties with parahoric level structure. Math. Z. 291, 329–363 (2019). https://doi.org/10.1007/s00209-018-2086-1

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