A guide to the reduction modulo $p$ of Shimura varieties

A guide to the reduction modulo

p

of Shimura varieties

Rapoport, Michael

Formes automorphes (I) - Actes du semestre du centre Émile Borel, printemps 2000, Astérisque, no. 298 (2005), pp. 271-318.

MR Zbl | 7 citations dans Numdam

@incollection{AST_2005__298__271_0,
     author = {Rapoport, Michael},
     title = {A guide to the reduction modulo $p$ of {Shimura} varieties},
     booktitle = {Formes automorphes (I) - Actes du semestre du centre \'Emile Borel, printemps 2000},
     editor = {Tilouine Jacques and Carayol Henri and Harris Michael and Vign\'eras Marie-France},
     series = {Ast\'erisque},
     pages = {271--318},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {298},
     year = {2005},
     mrnumber = {2141705},
     zbl = {1084.11029},
     language = {en},
     url = {http://www.numdam.org/item/AST_2005__298__271_0/}
}

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Rapoport, Michael. A guide to the reduction modulo $p$ of Shimura varieties, dans Formes automorphes (I) - Actes du semestre du centre Émile Borel, printemps 2000, Astérisque, no. 298 (2005), pp. 271-318. http://www.numdam.org/item/AST_2005__298__271_0/

Bibliographie
Cité par

[BD] A. Beilinson & V. Drinfeld - Quantization of Hitchhrs integrable system and Hecke eigensheaves, available at http://www.math.uchicago.edu/~benzvi/.

[BT1] F. Bruhat & J. Tits - Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41 (1972), p. 5-251. | DOI | EuDML | Numdam | MR | Zbl

[BT2] F. Bruhat & J. Tits, Groupes réductifs sur un corps local. II Schémas en groupes. Existence d'une donnée radicielle valuée, Publ. Math. Inst. Hautes Études Sci. 60 (1984), p. 197-376. | EuDML | Numdam | MR | Zbl

[C1] C.-L. Chai - Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli, Invent. Math. 121 (1995), p. 439-479. | DOI | EuDML | MR | Zbl

[C2] C.-L. Chai, Newton polygons as lattice points, Amer. J. Math. 122 (2000), p. 967-990. | MR | Zbl

[CN1] C.-L. Chai & P. Norman - Bad reduction of the Siegel moduli scheme of genus two with $Γ_{0} (p)$ -level structure, Amer. J. Math. 112 (1990), p. 1003-1071. | DOI | MR | Zbl

[CN2] C.-L. Chai & P. Norman, Singularities of the $Γ_{0} (p)$ -level structure, J. Algebraic Geom., 1 (1992), p. 251-278. | MR | Zbl

[DL] P. Deligne & G. Lusztig - Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), p. 103-161. | DOI | MR | Zbl

[DP] P. Deligne & G. Pappas - Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compositio Math. 90 (1994), p. 59-79. | EuDML | Numdam | MR | Zbl

[DR] P. Deligne & M. Rapoport - Les schémas de modules de courbes elliptiques, in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lect. Notes in Math., vol. 349, Springer, Berlin, 1973, p. 143-316. | DOI | MR | Zbl

[DM] F. Digne & J. Michel - Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. | MR | Zbl

[D] V. G. Drinfeld - Coverings of $p$ -adic symmetric domains, Funktsional. Anal. i Prilozhen. 10 (1976), p. 29-40, Russian. | MR | Zbl

[F1] G. Faltings - Explicit resolution of local singularities of moduli-spaces. J. reine angew. Math. 483 (1997), p. 183-196. | EuDML | MR | Zbl

[F2] G. Faltings, Toroidal resolutions for some matrix singularities, in Moduli of abelian varieties (Texel Island, 1999), Progress in Math., vol. 195, Birkhäuser, Basel, 2001, p. 157-184. | DOI | MR | Zbl

[Ge] A. Genestier - Un modèle semi-stable de la variété de Siegel de genre $3$ avec structures de niveau de type $Γ_{0} (p)$ , Compositio Math. 123 (2000), p. 303-328. | DOI | MR | Zbl

[G1] U. Görtz - On the flatness of models of certain Shimura varieties of $P E L$ -type, Math. Ann. 321 (2001), p. 689-727. | DOI | MR | Zbl

[G2] U. Görtz, On the flatness of local models for the symplectic group, Adv. in Math. 176 (2003), p. 89-115. | DOI | MR | Zbl

[G3] U. Görtz, Topological flatness of local models in the ramified case, math.AG/0205050, 2002. | Zbl

[H1] Th. Haines - On matrix coefficients of the Satake isomorphism: complements to the paper of M. Rapoport, Manuscripta Math. 101 (2000), p. 167-174. | DOI | MR | Zbl

[H2] Th. Haines, Test functions for Shimura varieties: the Drinfeld case, Duke Math. J. 106 (2001), p. 19-40. | DOI | MR | Zbl

[H3] Th. Haines, The combinatorics of Bernstein functions, Trans. Amer. Math. Soc. 353 (2001), p. 1251-1278. | DOI | MR | Zbl

[HN1] Th. Haines & B. C. Ngo - Alcoves associated to special fibers of local models, Amer. J. Math. 124 (2002), p. 1125-1152. | DOI | MR | Zbl

[HN2] Th. Haines & B. C. Ngo, Nearby cycles for local models of some Shimura varieties, Compositio Math. 133 (2002), p. 117-150. | DOI | MR | Zbl

[HR] Th. Haines & M. Rapoport - paper in preparation.

[IM] N. Iwahori & H. Matsumoto - On some Bruhat decomposition and the structure of the Hecke rings of $p$ -adic Chevalley groups, Publ. Math. Inst. Hautes Etudes Sci. 25 (1965), p. 5-48. | DOI | EuDML | Numdam | MR | Zbl

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

[JO] A. De Jong & F. Oort - Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), p. 209-241. | DOI | MR | Zbl

[Ka] C. Kaiser - Ein getwistetes fundamentales Lemma für die $G S p_{4}$ , Bonner Mathematische Schriften, vol. 303, Universität Bonn, Mathematisches Institut, Bonn, 1997. | MR | Zbl

[K1] R. Kottwitz - Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), p. 287-300. | DOI | EuDML | MR | Zbl

[K2] R. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), p. 201-220. | EuDML | Numdam | MR | Zbl

[K3] R. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), p. 373-444. | DOI | MR | Zbl

[K4] R. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), p. 255-339. | DOI | MR | Zbl

[KR1] R. Kottwitz & M. Rapoport - Minuscule alcoves for $G L_{n}$ and $G S p_{2 n}$ , Manuscripta Math. 102 (2000), p. 403-428. | DOI | MR | Zbl

[KR2] R. Kottwitz & M. Rapoport, On the existence of $F$ -crystals, Comment. Math. Helv. 78 (2003), p. 153-184. | DOI | MR | Zbl

[L] L. Lafforgue - Pavages de polyèdres, recollement de cellules de Schubert et compactification d'espaces de configurations, in preparation.

[LR] R. Langlands & M. Rapoport - Shimuravarietäten und Gerben, J. reine angew. Math. 378 (1987), p. 113-220. | EuDML | MR | Zbl

[LO] K.-Z. Li & F. Oort - Moduli of super singular abelian varieties, Lect. Notes in Math., vol. 1680, Springer-Verlag, Berlin, 1998. | MR | Zbl

[M1] J. Milne - The conjecture of Langlands and Rapoport for Siegel modular varieties, Bull. Amer. Math. Soc. (N.S.) 24 (1991), p. 335-341. | DOI | MR | Zbl

[M2] J. Milne, The points on a Shimura variety modulo a prime of good reduction, in The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, p. 151-253. | MR | Zbl

[M3] J. Milne, Motives over finite fields, in Motives (Seattle, WA, 1991), Part 1, Proc. Sympos. Pure Math., vol. 55, American Mathematical Society, Providence, RI, 1994, p. 401-459. | MR | Zbl

[NG] B. C. Ngo & A. Genestier - Alcoves et $p$ -rang des variétés abéliennes, Ann. Inst. Fourier (Grenoble) 52 (2002), p. 1665-1680. | DOI | EuDML | Numdam | MR | Zbl

[O] F. Oort - Newton polygon strata in the moduli space of abelian varieties, in Moduli of abelian varieties (Texel Island, 1999). Progress in Math., vol. 195, Birkhäuser, Basel, 2001, p. 417-440. | DOI | MR | Zbl

[P] G. Pappas - On the arithmetic moduli schemes of $P E L$ Shimura varieties, J. Algebraic Geom. 9 (2000), p. 577-605. | MR | Zbl

[PR1] G. Pappas & M. Rapoport - Local models in the ramified case I. The $E L$ -case, J. Algebraic Geom. 12 (2003), p. 107-145. | DOI | MR | Zbl

[PR2] G. Pappas & M. Rapoport, Local models in the ramified case II. Splitting models, math.AG/0205021, 2002. | Zbl

[Pf] M. Pfau - The conjecture of Langlands and Rapoport for certain Shimura varieties of non-rational weight, J. reine angew. Math. 471 (1996), p. 165-199. | EuDML | MR | Zbl

[R1] M. Rapoport - On the local zeta function of quaternionic Shimura varieties with bad reduction, Math. Ann. 279 (1988), p. 673-697. | DOI | EuDML | MR | Zbl

[R2] M. Rapoport, On the bad reduction of Shimura varieties, in Automorphic forms, Shimura varieties, and $L$ -functions, Vol. II (Ann Arbor, MI, 1988), Perspect. Math., vol. 11, Academic Press, Boston, MA, 1990, p. 253-321. | MR | Zbl

[R3] M. Rapoport, A positivity property of the Satake isomorphism, Manuscripta Math. 101 (2000), p. 153-166. | DOI | MR | Zbl

[RR] M. Rapoport & M. Richartz - On the classification and specialization of $F$ -isocrystals with additional structure, Compositio Math. 103 (1996), p. 153-181. | EuDML | Numdam | MR | Zbl

[RZ1] M. Rapoport & Th. Zink - Über die lokale Zetafunktion von Shimuravarietäten. Monodrorniefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), p. 21-101. | DOI | EuDML | MR | Zbl

[RZ2] M. Rapoport & Th. Zink, Period spaces for $p$ -divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. | MR | Zbl

[Re1] H. Reimann - The semi-simple zeta function of quaternionic Shimura varieties, Lect. Notes in Math., vol. 1657, Springer-Verlag, Berlin, 1997. | MR | Zbl

[Re2] H. Reimann, On the zeta function of quaternionic Shimura varieties, Math. Ann. 317 (2000), p. 41-55. | DOI | MR | Zbl

[Re3] H. Reimann, Reduction of Shimura varieties at parahoric levels, Manuscripta Math. 107 (2002), p. 355-390. | DOI | MR | Zbl

[Ri] M. Richartz - Klassifikation von selbstdualen Dieudonnégittern in einem dreidimensionalen polarisierten supersingulären Isokristall, Diss., Universität Bonn, 1998. | MR | Zbl

[S] H. Stamm - On the reduction of the Hilbert-Blumenthal-moduli scheme with $Γ_{0} (p)$ -level structure, Forum Math. 9 (1997), p. 405-455. | EuDML | MR | Zbl

[T] J. Tits - Reductive groups over local fields, Proc. Sympos. Pure Math., vol. 33, American Mathematical Society, Providence, RI, 1979, p. 29-69. | DOI | MR | Zbl

[W] T. Wedhorn - Ordinariness in good reductions of Shimura varieties of $P E L$ -type, Ann. scient. Éc. Norm. Sup. 4e série 32 (1999), p. 575-618. | DOI | EuDML | Numdam | MR | Zbl

[BW] O. Bültel & T. Wedhorn - Congruence relations for Shimura varieties associated to some unitary groups, arXiv: math. AG/0202026, 2002. | MR | Zbl

[FR] J.-M. Fontaine & M. Rapoport - Existence de filtrations admissibles sur des isocristaux, Bull. Soc. math. France, to appear; arXiv: math.NT/0204293. | EuDML | Numdam | MR | Zbl

[K5] R. Kottwitz - On the Hodge-Newton decomposition for split groups, Internat. Math. Res. Notices 26 (2003), p. 1433-1447. | DOI | MR | Zbl

[Le] C. Leigh - A converse to Mazur's inequality for split classical groups, arXiv: math. NT/0211327, 2002.

[R4] M. Rapoport - On the Newton stratification, in Séminaire Bourbaki 2001/2002, Astérisque, vol. 290, Société Mathématique de France, 2003, Exp. n° 903, p. 207-224. | EuDML | Numdam | MR | Zbl

[Reu1] D. Reuman - Determining whether certain affine Deligne-Lusztig sets are empty, arXiv: math.RT/0211434, 2002. | MR

[Reu2] D. Reuman, Formulas for the dimensions of some affine Deligne-Lusztig varieties, arXiv: math.RT/0303146, 2003. | MR | Zbl

[Wi] J.-P. Wintenberger - Existence de $F$ -cristaux avec structures supplémentaires, arXiv: math.NT/0303026, 2003. | MR | Zbl