Mod 𝑝 points on Shimura varieties of abelian type

Kisin, Mark

doi:10.1090/jams/867

Mod $p$ points on Shimura varieties of abelian type
HTML articles powered by AMS MathViewer

by Mark Kisin

J. Amer. Math. Soc. 30 (2017), 819-914

DOI: https://doi.org/10.1090/jams/867

Published electronically: January 11, 2017

PDF | Request permission

Abstract:

We show that the mod $p$ points on a Shimura variety of abelian type with hyperspecial level have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod $p$ point contains the reduction of a special point.

References

Don Blasius, A $p$-adic property of Hodge classes on abelian varieties, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 293–308. MR 1265557, DOI 10.1090/pspum/055.2/1265557
Mikhail Borovoi, Abelian Galois cohomology of reductive groups, Mem. Amer. Math. Soc. 132 (1998), no. 626, viii+50. MR 1401491, DOI 10.1090/memo/0626
Christophe Breuil, Groupes $p$-divisibles, groupes finis et modules filtrés, Ann. of Math. (2) 152 (2000), no. 2, 489–549 (French, with French summary). MR 1804530, DOI 10.2307/2661391
Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 207712
Miaofen Chen, Mark Kisin, and Eva Viehmann, Connected components of affine Deligne-Lusztig varieties in mixed characteristic, Compos. Math. 151 (2015), no. 9, 1697–1762. MR 3406443, DOI 10.1112/S0010437X15007253
Jean-Louis Colliot-Thélène and Venapally Suresh, Quelques questions d’approximation faible pour les tores algébriques, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 1, 273–288 (French, with English and French summaries). MR 2316239
Pierre Deligne, Variétés abéliennes ordinaires sur un corps fini, Invent. Math. 8 (1969), 238–243 (French). MR 254059, DOI 10.1007/BF01406076
Pierre Deligne, Travaux de Shimura, Séminaire Bourbaki, 23ème année (1970/1971), Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971, pp. Exp. No. 389, pp. 123–165 (French). MR 0498581
Pierre Deligne, 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., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 247–289 (French). MR 546620
A. J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Inst. Hautes Études Sci. Publ. Math. 82 (1995), 5–96 (1996). MR 1383213
Michael Demazure and Alexander Grothendieck, Schémas en groupes I,II,III, Lecture notes in Math., vol. 151–153, Springer, 1970.
Gerd Faltings, Integral crystalline cohomology over very ramified valuation rings, J. Amer. Math. Soc. 12 (1999), no. 1, 117–144. MR 1618483, DOI 10.1090/S0894-0347-99-00273-8
Gerd Faltings, Coverings of $p$-adic period domains, J. Reine Angew. Math. 643 (2010), 111–139. MR 2658191, DOI 10.1515/CRELLE.2010.046
Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353, DOI 10.1007/978-3-662-02632-8
Yasutaka Ihara, The congruence monodromy problems, J. Math. Soc. Japan 20 (1968), 107–121. MR 240105, DOI 10.2969/jmsj/02010107
Yasutaka Ihara, On congruence monodromy problems I,II,, thesis, Univ. of Tokyo, 1969.
Mark Kisin, Crystalline representations and $F$-crystals, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 459–496. MR 2263197, DOI 10.1007/978-0-8176-4532-8_{7}
Mark Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), no. 4, 967–1012. MR 2669706, DOI 10.1090/S0894-0347-10-00667-3
Robert E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), no. 4, 785–806. MR 683003
Robert E. Kottwitz, Shimura varieties and twisted orbital integrals, Math. Ann. 269 (1984), no. 3, 287–300. MR 761308, DOI 10.1007/BF01450697
Robert E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201–220. MR 809866
Robert E. Kottwitz, Shimura varieties and $\lambda$-adic representations, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 161–209. MR 1044820
Robert E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), no. 2, 373–444. MR 1124982, DOI 10.1090/S0894-0347-1992-1124982-1
Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. MR 1485921, DOI 10.1023/A:1000102604688
Mark Kisin and George J. Pappas, Integral models of Shimura varieties with parahoric level structure (2015), 81 pages pp., available at http://www.math.harvard.edu/~kisin/dvifiles/parahoric.pdf.
R. P. Langlands, Some contemporary problems with origins in the Jugendtraum, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R.I., 1976, pp. 401–418. MR 0437500
R. P. Langlands, Shimura varieties and the Selberg trace formula, Canadian J. Math. 29 (1977), no. 6, 1292–1299. MR 498400, DOI 10.4153/CJM-1977-129-2
R. P. Langlands, On the zeta functions of some simple Shimura varieties, Canadian J. Math. 31 (1979), no. 6, 1121–1216. MR 553157, DOI 10.4153/CJM-1979-102-1
R. P. Langlands, Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 205–246. MR 546619
R. P. Langlands and M. Rapoport, Shimuravarietäten und Gerben, J. Reine Angew. Math. 378 (1987), 113–220 (German). MR 895287
Laurent Moret-Bailly, Groupes de Picard et problèmes de Skolem. I, II, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 2, 161–179, 181–194 (French). MR 1005158
J. S. Milne, Points on Shimura varieties mod $p$, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 165–184. MR 546616
J. S. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 283–414. MR 1044823
James S. Milne, The points on a Shimura variety modulo a prime of good reduction, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 151–253. MR 1155229
Ben Moonen, Models of Shimura varieties in mixed characteristics, Galois representations in arithmetic algebraic geometry (Durham, 1996) London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 267–350. MR 1696489, DOI 10.1017/CBO9780511662010.008
Yasua Morita, Ihara’s conjectures and moduli space of abelian varieties, Thesis, Univ. Tokyo, 1970.
Matthias Pfau, The conjecture of Langlands and Rapoport for certain Shimura varieties of non-rational weight, J. Reine Angew. Math. 471 (1996), 165–199. MR 1374922, DOI 10.1515/crll.1996.471.165
Michael Rapoport, A guide to the reduction modulo $p$ of Shimura varieties, Astérisque 298 (2005), 271–318 (English, with English and French summaries). Automorphic forms. I. MR 2141705
Harry Reimann, The semi-simple zeta function of quaternionic Shimura varieties, Lecture Notes in Mathematics, vol. 1657, Springer-Verlag, Berlin, 1997. MR 1470457, DOI 10.1007/BFb0093995
M. Rapoport and M. Richartz, On the classification and specialization of $F$-isocrystals with additional structure, Compositio Math. 103 (1996), no. 2, 153–181. MR 1411570
M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439, DOI 10.1515/9781400882601
Harry Reimann and Thomas Zink, Der Dieudonnémodul einer polarisierten abelschen Mannigfaltigkeit vom CM-Typ, Ann. of Math. (2) 128 (1988), no. 3, 461–482 (German). MR 970608, DOI 10.2307/1971433
J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80 (French). MR 631309, DOI 10.1515/crll.1981.327.12
Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). MR 1324577, DOI 10.1007/BFb0108758
John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144. MR 206004, DOI 10.1007/BF01404549
Jean-Pierre Wintenberger, Torseur entre cohomologie étale $p$-adique et cohomologie cristalline; le cas abélien, Duke Math. J. 62 (1991), no. 3, 511–526 (French). MR 1104805, DOI 10.1215/S0012-7094-91-06221-6
Jean-Pierre Wintenberger, Propriétés du groupe tannakien des structures de Hodge $p$-adiques et torseur entre cohomologies cristalline et étale, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 5, 1289–1334 (French, with English and French summaries). MR 1600395
J.-P. Wintenberger, Existence de $F$-cristaux avec structures supplémentaires, Adv. Math. 190 (2005), no. 1, 196–224 (French, with English summary). MR 2104909, DOI 10.1016/j.aim.2003.12.006
Thomas Zink, Isogenieklassen von Punkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper, Math. Nachr. 112 (1983), 103–124 (German). MR 726854, DOI 10.1002/mana.19831120106
Thomas Zink, Windows for displays of $p$-divisible groups, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 491–518. MR 1827031