Integral models of Shimura varieties with parahoric level structure

Kisin, M.; Pappas, G.

doi:10.1007/s10240-018-0100-0

Kisin, M. ; Pappas, G.

Publications Mathématiques de l'IHÉS, Tome 128 (2018), pp. 121-218.

Résumé

For a prime $p > 2$ , we construct integral models over $p$ for Shimura varieties with parahoric level structure, attached to Shimura data $(G, X)$ of abelian type, such that $G$ splits over a tamely ramified extension of $𝐐_{p}$ . The local structure of these integral models is related to certain “local models”, which are defined group theoretically. Under some additional assumptions, we show that these integral models satisfy a conjecture of Kottwitz which gives an explicit description for the trace of Frobenius action on their sheaf of nearby cycles.

Reçu le : 2015-11-27
Accepté le : 2018-03-28
Première publication : 2018-04-30
Publié le : 2018-11-09

DOI : 10.1007/s10240-018-0100-0

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     author = {Kisin, M. and Pappas, G.},
     title = {Integral models of {Shimura} varieties with parahoric level structure},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {121--218},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {128},
     year = {2018},
     doi = {10.1007/s10240-018-0100-0},
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Kisin, M.; Pappas, G. Integral models of Shimura varieties with parahoric level structure. Publications Mathématiques de l'IHÉS, Tome 128 (2018), pp. 121-218. doi : 10.1007/s10240-018-0100-0. http://www.numdam.org/articles/10.1007/s10240-018-0100-0/

Bibliographie
Cité par

[1.] Anantharaman, S. Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 5-79

[2.] Bayer-Fluckiger, E.; Parimala, R. Galois cohomology of the classical groups over fields of cohomological dimension $\leq 2$ , Invent. Math., Volume 122 (1995), pp. 195-229 | DOI | MR | Zbl

[3.] Bayer-Fluckiger, E.; Parimala, R. Classical groups and the Hasse principle, Ann. Math. (2), Volume 147 (1998), pp. 651-693 | DOI | MR | Zbl

[4.] Berthelot, P.; Ogus, A. $F$ -isocrystals and de Rham cohomology. I, Invent. Math., Volume 72 (1983), pp. 159-199 | DOI | MR | Zbl

[5.] Blasius, D. A $p$ -adic property of Hodge classes on abelian varieties, Motives (1994), pp. 293-308 | DOI

[6.] Bosch, S.; Lütkebohmert, W.; Raynaud, M. Néron Models, Springer, Berlin, 1990 | DOI | Zbl

[7.] Bourbaki, N. Lie Groups and Lie Algebras, Springer, Berlin, 2005 (Chaps. 7–9, translated from the 1975 and 1982 French originals by Andrew Pressley) | Zbl

[8.] Breuil, C. Groupes $p$ -divisibles, groupes finis et modules filtrés, Ann. Math. (2), Volume 152 (2000), pp. 489-549 | DOI | MR | Zbl

[9.] Broshi, M. $G$ -torsors over a Dedekind scheme, J. Pure Appl. Algebra, Volume 217 (2013), pp. 11-19 | DOI | MR | Zbl

[10.] Bruhat, F.; Tits, J. Groupes réductifs sur un corps local, Publ. Math. IHÉS, Volume 41 (1972), pp. 5-251 | DOI | Zbl

[11.] Bruhat, F.; Tits, J. Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Publ. Math. IHÉS, Volume 60 (1984), pp. 197-376 | DOI | Zbl

[12.] Bruhat, F.; Tits, J. Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. Fr., Volume 112 (1984), pp. 259-301 | DOI | Zbl

[13.] Bruhat, F.; Tits, J. Schémas en groupes et immeubles des groupes classiques sur un corps local. II. Groupes unitaires, Bull. Soc. Math. Fr., Volume 115 (1987), pp. 141-195 | DOI | Zbl

[14.] Colliot-Thélène, J.-L. Un théorème de finitude pour le groupe de Chow des zéro-cycles d’un groupe algébrique linéaire sur un corps $p$ -adique, Invent. Math., Volume 159 (2005), pp. 589-606 | DOI | MR | Zbl

[15.] Colliot-Thélène, J.-L. Résolutions flasques des groupes linéaires connexes, J. Reine Angew. Math., Volume 618 (2008), pp. 77-133 | MR | Zbl

[16.] Colliot-Thélène, J.-L.; Sansuc, J.-J. Fibrés quadratiques et composantes connexes réelles, Math. Ann., Volume 244 (1979), pp. 105-134 | DOI | MR | Zbl

[17.] B. Conrad, Reductive group schemes, Notes for the SGA3 Summer School, Luminy 2011, http://math.stanford.edu/~conrad/papers/luminysga3.pdf.

[18.] de Jong, A. J. The moduli spaces of principally polarized abelian varieties with $Γ_{0} (p)$ -level structure, J. Algebraic Geom., Volume 2 (1993), pp. 667-688 | MR | Zbl

[19.] de Jong, A. J. Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. Inst. Hautes Études Sci., Volume 82 (1995), pp. 5-96 | DOI | Zbl

[20.] Deligne, P. Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic Forms, Representations and $L$ -functions (Proc. Sympos. Pure Math., Oregon State Univ.), Part 2 (1977), pp. 247-289

[21.] Deligne, P.; Pappas, G. Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compos. Math., Volume 90 (1994), pp. 59-79 | Zbl

[22.] Edixhoven, B. Néron models and tame ramification, Compos. Math., Volume 81 (1992), pp. 291-306 | Zbl

[23.] Faltings, G.; Chai, C.-L. Degeneration of Abelian Varieties, Springer, Berlin, 1990 (with an Appendix by David Mumford) | DOI | Zbl

[24.] Gabber, O.; Orgogozo, F. Sur la $p$ -dimension des corps, Invent. Math., Volume 174 (2008), pp. 47-80 | DOI | MR | Zbl

[25.] Gaitsgory, D. Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math., Volume 144 (2001), pp. 253-280 | DOI | MR | Zbl

[26.] Gan, W. T.; Yu, J.-K. Schémas en groupes et immeubles des groupes exceptionnels sur un corps local. I. Le groupe $G_{2}$ , Bull. Soc. Math. Fr., Volume 131 (2003), pp. 307-358 | DOI

[27.] Gan, W. T.; Yu, J.-K. Schémas en groupes et immeubles des groupes exceptionnels sur un corps local. II. Les groupes $F_{4}$ et $E_{6}$ , Bull. Soc. Math. Fr., Volume 133 (2005), pp. 159-197 | DOI

[28.] Gille, P. Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique $\leq 2$ , Compos. Math., Volume 125 (2001), pp. 283-325 | DOI | Zbl

[29.] Gille, P. Torseurs sur la droite affine, Transform. Groups, Volume 7 (2002), pp. 231-245 | DOI | MR | Zbl

[30.] Gille, P. Serre’s conjecture II: a survey, Quadratic Forms, Linear Algebraic Groups, and Cohomology, Springer, New York, 2010, pp. 41-56 | DOI

[31.] Görtz, U. On the flatness of local models for the symplectic group, Adv. Math., Volume 176 (2003), pp. 89-115 | DOI | MR | Zbl

[32.] Görtz, U.; Haines, T. The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties, J. Reine Angew. Math., Volume 609 (2007), pp. 161-213 | MR | Zbl

[33.] Grothendieck, A. Le groupe de Brauer. I, II, III, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 46-188

[34.] Haines, T. Introduction to Shimura varieties with bad reduction of parahoric type, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., Am. Math. Soc., Providence, 2005, pp. 583-642

[35.] Haines, T. The base change fundamental lemma for central elements in parahoric Hecke algebras, Duke Math. J., Volume 149 (2009), pp. 569-643 (Corrigendum, on the author’s web page) | DOI | MR | Zbl

[36.] Haines, T. The stable Bernstein center and test functions for Shimura varieties, Automorphic Forms and Galois Representations, vol. 2, London Math. Soc., London, 2014, pp. 118-186 | DOI

[37.] Haines, T.; Ngô, B. C. Nearby cycles for local models of some Shimura varieties, Compos. Math., Volume 133 (2002), pp. 117-150 | DOI | MR | Zbl

[38.] T. Haines and M. Rapoport, On parahoric subgroups, 2008. Appendix to [57].

[39.] Illusie, L. Autour du théorème de monodromie locale, Astérisque, Volume 223 (1994), pp. 9-57 Périodes $p$ -adiques (Bures-sur-Yvette, 1988) | Zbl

[40.] L. Illusie, Y. Laszlo and F. Orgogozo, Travaux de Gabber sur l’uniformisation locale et la cohomologie etale des schemas quasi-excellents. Seminaire a l’Ecole polytechnique 2006–2008, | arXiv

[41.] Jantzen, J. C. Representations of Algebraic Groups, Am. Math. Soc., Providence, 2003 | Zbl

[42.] Katz, N. Travaux de Dwork, Séminaire Bourbaki, 24ème année (1971/1972), 1973, pp. 167-200 (Exp. No. 409) | DOI

[43.] Kisin, M. Integral models for Shimura varieties of abelian type, J. Am. Math. Soc., Volume 23 (2010), pp. 967-1012 | DOI | MR | Zbl

[44.] Kisin, M. $\mod p$ points on Shimura varieties of abelian type, J. Am. Math. Soc., Volume 30 (2017), pp. 819-914 | DOI | Zbl

[45.] Kottwitz, R. Shimura varieties and twisted orbital integrals, Math. Ann., Volume 269 (1984), pp. 287-300 | DOI | MR | Zbl

[46.] Kottwitz, R. Points on some Shimura varieties over finite fields, J. Am. Math. Soc., Volume 5 (1992), pp. 373-444 | DOI | MR | Zbl

[47.] Kuzumaki, T. The cohomological dimension of the quotient field of the two-dimensional complete local domain, Compos. Math., Volume 79 (1991), pp. 157-167 | MR | Zbl

[48.] Landvogt, E. Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math., Volume 518 (2000), pp. 213-241 | MR | Zbl

[49.] Langlands, R. P. Some contemporary problems with origins in the Jugendtraum, Mathematical Developments Arising from Hilbert Problems (1976), pp. 401-418 | DOI

[50.] Lau, E. Relations between Dieudonné displays and crystalline Dieudonné theory, Algebra Number Theory, Volume 8 (2014), pp. 2201-2262 | DOI | MR | Zbl

[51.] Lusztig, G. Singularities, character formulas, and a $q$ -analog of weight multiplicities, Analysis and Topology on Singular Spaces, II, III (1983), pp. 208-229

[52.] K. Madapusi Pera, Toroidal compactifications of integral models of Shimura varieties of Hodge type. Preprint, | arXiv

[53.] Milne, J. S. Canonical models of (mixed) Shimura varieties and automorphic vector bundles, Automorphic Forms, Shimura Varieties, and $L$ -functions, vol. I (1990), pp. 283-414

[54.] Milne, J. S. The points on a Shimura variety modulo a prime of good reduction, The Zeta Functions of Picard Modular Surfaces, Univ. Montréal, Montreal, 1992, pp. 151-253

[55.] Moret-Bailly, L. Groupes de Picard et problèmes de Skolem. I, II, Ann. Sci. Éc. Norm. Supér. (4), Volume 22 (1989), pp. 161-179 (181–194) | DOI | Zbl

[56.] Pappas, G. On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebraic Geom., Volume 9 (2000), pp. 577-605 | MR | Zbl

[57.] Pappas, G.; Rapoport, M. Twisted loop groups and their affine flag varieties, Adv. Math., Volume 219 (2008), pp. 118-198 (with an Appendix by T. Haines and M. Rapoport) | DOI | MR | Zbl

[58.] Pappas, G.; Rapoport, M.; Smithling, B. Local models of Shimura varieties. I. Geometry and combinatorics, Handbook of Moduli, vol. III, International Press, Somerville, 2013, pp. 135-217

[59.] Pappas, G.; Zhu, X. Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math., Volume 194 (2013), pp. 147-254 | DOI | MR | Zbl

[60.] Prasad, G.; Yu, J-K. On finite group actions on reductive groups and buildings, Invent. Math., Volume 147 (2002), pp. 545-560 | DOI | MR | Zbl

[61.] Prasad, G.; Yu, J-K. On quasi-reductive group schemes, J. Algebraic Geom., Volume 15 (2006), pp. 507-549 (with an Appendix by Brian Conrad) | DOI | MR | Zbl

[62.] Rapoport, M. On the bad reduction of Shimura varieties, Automorphic Forms, Shimura Varieties, and $L$ -functions, vol. II (1990), pp. 253-321

[63.] Rapoport, M. A guide to the reduction modulo $p$ of Shimura varieties, Astérisque, Volume 298 (2005), pp. 271-318 (Automorphic forms. I) | MR | Zbl

[64.] Rapoport, M.; Zink, Th. Period spaces for $p$ -divisible groups, Princeton University Press, Princeton, 1996 | Zbl

[65.] Raynaud, M.; Gruson, L. Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | DOI | MR | Zbl

[66.] Satake, I. Symplectic representations of algebraic groups satisfying a certain analyticity condition, Acta Math., Volume 117 (1967), pp. 215-279 | DOI | MR | Zbl

[67.] Serre, J.-P. Galois Cohomology, Springer, Berlin, 1994 | DOI | Zbl

[68.] Seshadri, C. S. Triviality of vector bundles over the affine space $K^{2}$ , Proc. Natl. Acad. Sci. USA, Volume 44 (1958), pp. 456-458 | DOI | Zbl

[69.] Tits, J. Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math., Volume 247 (1971), pp. 196-220 | MR | Zbl

[70.] Tits, J. Reductive groups over local fields, Automorphic Forms, Representations and $L$ -functions (Proc. Sympos. Pure Math.) (1979), pp. 29-69 | DOI

[71.] Wintenberger, J.-P. Existence de $F$ -cristaux avec structures supplémentaires, Adv. Math., Volume 190 (2005), pp. 196-224 | DOI | MR | Zbl

[72.] J.-K. Yu, Smooth models associated to concave functions in Bruhat-Tits theory, 2002. Preprint.

[73.] Zink, Th. A Dieudonné theory for $p$ -divisible groups, Class Field Theory—Its Centenary and Prospect (2001), pp. 139-160 | DOI

[74.] Zink, Th. Windows for displays of $p$ -divisible groups, Moduli of Abelian Varieties (2001), pp. 491-518 | DOI

[75.] Zink, Th. The display of a formal $p$ -divisible group, Astérisque, Volume 278 (2002), pp. 127-248 (Cohomologies $p$ -adiques et applications arithmétiques, I.) | MR | Zbl

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