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Integral models for Shimura varieties of abelian type
Author:
Mark Kisin
Journal:
J. Amer. Math. Soc. 23 (2010), 967-1012
MSC (2010):
Primary 11G18; Secondary 14G35
Posted:
April 21, 2010
MathSciNet review:
2669706
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Abstract: We construct smooth integral models of Shimura varieties of abelian type at primes where the level structure is hyperspecial.
- [Bl]
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
(95j:14022)
- [BO]
P.
Berthelot and A.
Ogus, 𝐹-isocrystals and de Rham cohomology. I, Invent.
Math. 72 (1983), no. 2, 159–199. MR 700767
(85e:14025), http://dx.doi.org/10.1007/BF01389319
- [Br]
C. Breuil, Schémas en groupes et corps des normes (unpublished), 1998, 13 pages.
- [BT]
F.
Bruhat and J.
Tits, Groupes réductifs sur un corps local. II.
Schémas en groupes. Existence d’une donnée radicielle
valuée, Inst. Hautes Études Sci. Publ. Math.
60 (1984), 197–376 (French). MR 756316
(86c:20042)
- [CS]
J.-L.
Colliot-Thélène and J.-J.
Sansuc, Fibrés quadratiques et composantes connexes
réelles, Math. Ann. 244 (1979), no. 2,
105–134 (French). MR 550842
(81c:14010), http://dx.doi.org/10.1007/BF01420486
- [Ch]
Claude
Chevalley, Deux théorèmes
d’arithmétique, J. Math. Soc. Japan 3
(1951), 36–44 (French). MR 0044570
(13,440a)
- [CSu]
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
(2008b:11047)
- [De 1]
Pierre
Deligne, Travaux de Shimura, Séminaire Bourbaki,
23ème année (1970/71), Exp. No. 389, Springer, Berlin, 1971,
pp. 123–165. Lecture Notes in Math., Vol. 244 (French). MR 0498581
(58 #16675)
- [De 2]
Pierre
Deligne, James
S. Milne, Arthur
Ogus, and Kuang-yen
Shih, Hodge cycles, motives, and Shimura varieties, Lecture
Notes in Mathematics, vol. 900, Springer-Verlag, Berlin, 1982. MR 654325
(84m:14046)
- [De 3]
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), Part 2, Proc.
Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979,
pp. 247–289 (French). MR 546620
(81i:10032)
- [De 4]
Pierre
Deligne, Équations différentielles à points
singuliers réguliers, Lecture Notes in Mathematics, Vol. 163,
Springer-Verlag, Berlin, 1970 (French). MR 0417174
(54 #5232)
- [DG]
Schémas en groupes. I: Propriétés
générales des schémas en groupes,
Séminaire de Géométrie Algébrique du Bois Marie
1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture
Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin, 1970 (French). MR 0274458
(43 #223a)
Schémas en groupes. II: Groupes de type multiplicatif, et
structure des schémas en groupes généraux,
Séminaire de Géométrie Algébrique du Bois Marie
1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture
Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin, 1970 (French). MR 0274459
(43 #223b)
Schémas en groupes. III: Structure des schémas en
groupes réductifs, Séminaire de Géométrie
Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M.
Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 153,
Springer-Verlag, Berlin, 1970 (French). MR 0274460
(43 #223c)
- [Fa]
Gerd
Faltings, Integral crystalline cohomology over
very ramified valuation rings, J. Amer. Math.
Soc. 12 (1999), no. 1, 117–144. MR 1618483
(99e:14022), http://dx.doi.org/10.1090/S0894-0347-99-00273-8
- [FC]
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
(92d:14036)
- [Fo]
Jean-Marc
Fontaine, Représentations 𝑝-adiques des corps
locaux. I, The Grothendieck Festschrift, Vol. II, Progr. Math.,
vol. 87, Birkhäuser Boston, Boston, MA, 1990,
pp. 249–309 (French). MR 1106901
(92i:11125)
- [GRR]
Groupes de monodromie en géométrie algébrique.
I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin,
1972 (French). Séminaire de Géométrie
Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé
par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. MR 0354656
(50 #7134)
- [Ki 1]
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
(2007j:11163), http://dx.doi.org/10.1007/978-0-8176-4532-8_7
- [Ki 2]
Mark
Kisin, Potentially semi-stable deformation
rings, J. Amer. Math. Soc.
21 (2008), no. 2,
513–546. MR 2373358
(2009c:11194), http://dx.doi.org/10.1090/S0894-0347-07-00576-0
- [Ki 3]
Mark
Kisin, Modularity of 2-adic Barsotti-Tate representations,
Invent. Math. 178 (2009), no. 3, 587–634. MR 2551765
(2010k:11089), http://dx.doi.org/10.1007/s00222-009-0207-5
- [Ki 4]
Mark
Kisin, Integral canonical models of Shimura varieties, J.
Théor. Nombres Bordeaux 21 (2009), no. 2,
301–312 (English, with English and French summaries). MR 2541427
(2010i:11087)
- [Ko]
Robert
E. Kottwitz, Points on some Shimura varieties over
finite fields, J. Amer. Math. Soc.
5 (1992), no. 2,
373–444. MR 1124982
(93a:11053), http://dx.doi.org/10.1090/S0894-0347-1992-1124982-1
- [La]
R.
P. Langlands, Some contemporary problems with origins in the
Jugendtraum, Mathematical developments arising from Hilbert problems
(Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb,
Ill., 1974), Amer. Math. Soc., Providence, R. I., 1976,
pp. 401–418. MR 0437500
(55 #10426)
- [Me]
William
Messing, The crystals associated to Barsotti-Tate groups: with
applications to abelian schemes, Lecture Notes in Mathematics, Vol.
264, Springer-Verlag, Berlin, 1972. MR 0347836
(50 #337)
- [MFK]
D.
Mumford, J.
Fogarty, and F.
Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der
Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related
Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
(95m:14012)
- [Mi 1]
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
(94g:11041)
- [Mi 2]
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
(91a:11027)
- [Mi 3]
J. Milne, On the conjecture of Langlands and Rapoport, available at www.arxiv.org (2007).
- [Mi 4]
J.
S. Milne, Shimura varieties and motives, Motives (Seattle, WA,
1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc.,
Providence, RI, 1994, pp. 447–523. MR 1265562
(95c:11076)
- [Mo]
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
(2000e:11077), http://dx.doi.org/10.1017/CBO9780511662010.008
- [PY]
Gopal
Prasad and Jiu-Kang
Yu, On quasi-reductive group
schemes, J. Algebraic Geom.
15 (2006), no. 3,
507–549. With an appendix by Brian Conrad. MR 2219847
(2007c:14047), http://dx.doi.org/10.1090/S1056-3911-06-00422-X
- [RZ]
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 (97f:14023)
- [Sa]
Neantro
Saavedra Rivano, Catégories Tannakiennes, Lecture Notes
in Mathematics, Vol. 265, Springer-Verlag, Berlin, 1972 (French). MR 0338002
(49 #2769)
- [Sp]
T.
A. Springer, Reductive groups, Automorphic forms,
representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon
State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math.,
XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR 546587
(80h:20062)
- [Ti]
J.
Tits, Reductive groups over local fields, Automorphic forms,
representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon
State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math.,
XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588
(80h:20064)
- [Va 1]
Adrian
Vasiu, Integral canonical models of Shimura varieties of preabelian
type, Asian J. Math. 3 (1999), no. 2,
401–518. MR 1796512
(2002b:11087)
- [Va 2]
A. Vasiu, Integral Canonical Models of Shimura Varieties of Preabelian Type. (Fully corrected version.), available at www.arxiv.org (2003), 135 pages.
- [Va 3]
A. Vasiu, A motivic conjecture of Milne, available at www.arxiv.org (2003), 46 pages.
- [Va 4]
A. Vasiu, Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic I, available at www.arxiv.org (2007), 48 pages.
- [Va 5]
A. Vasiu, Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic II, available at www.arxiv.org (2007), 33 pages.
- [Wa]
William
C. Waterhouse, Introduction to affine group schemes, Graduate
Texts in Mathematics, vol. 66, Springer-Verlag, New York, 1979. MR 547117
(82e:14003)
- [Zi 1]
Thomas
Zink, The display of a formal 𝑝-divisible group,
Astérisque 278 (2002), 127–248. Cohomologies
𝑝-adiques et applications arithmétiques, I. MR 1922825
(2004b:14083)
- [Zi 2]
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
(2002c:14073)
- [Bl]
- D. Blasius, A
-adic property of Hodge classes on abelian varieties, Motives (Seattle 1991), 293-308, Proc. Sympos. Pure Math. 55, Part 2, Amer. Math. Soc., pp. 293-308, 1994. MR 1265557 (95j:14022)
- [BO]
- P. Berthelot, A. Ogus,
-isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983), 159-199. MR 700767 (85e:14025)
- [Br]
- C. Breuil, Schémas en groupes et corps des normes (unpublished), 1998, 13 pages.
- [BT]
- F. Bruhat, J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d'une donnée radicielle valuée, Pub. Math. IHES 60 (1984), 197-376. MR 756316 (86c:20042)
- [CS]
- J.-L. Colliot-Thélène, J.-J. Sansuc, Fibrés quadratiques et composantes connexes réelles, Math. Ann. 244 (1979), 105-134. MR 550842 (81c:14010)
- [Ch]
- C. Chevalley, Deux théorèmes d'arithmétique, J. Math. Soc. Japan 3 (1951), 36-44. MR 0044570 (13:440a)
- [CSu]
- J.-L. Colliot-Thélène, V. Suresh, Quelques questions d'approximation faible pour les tores algébriques, Ann. Inst. Fourier 57 (2007), 273-288. MR 2316239 (2008b:11047)
- [De 1]
- P. Deligne, Travaux de Shimura, Exp. 389, Séminaire Bourbaki (1970/1971), Lecture Notes in Math. 244, Springer, pp. 123-165, 1971. MR 0498581 (58:16675)
- [De 2]
- P. Deligne, Hodge cycles on abelian varieties, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, pp. 9-100, Springer-Verlag, 1982. MR 0654325 (84m:14046)
- [De 3]
- P. Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and
-functions (Corvallis 1977), Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., pp. 247-289, 1979. MR 546620 (81i:10032)
- [De 4]
- P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer, 1970. MR 0417174 (54:5232)
- [DG]
- M. Demazure, A. Grothendieck, Schémas en groupes I,II,III, Lecture Notes in Math. 151-153, Springer, 1962-1970. MR 0274458 (43:223a); MR 0274459 (43:223b); MR 0274460 (43:223c)
- [Fa]
- G. Faltings, Integral crystalline cohomology over very ramified valuation rings, J. Amer. Math. Soc. 12 (1999), 117-144. MR 1618483 (99e:14022)
- [FC]
- G. Faltings, C.-L. Chai, Degeneration of Abelian varieties, Ergeb. der Math. 22, Springer, 1990. MR 1083353 (92d:14036)
- [Fo]
- J.-M. Fontaine, Représentations
-adiques des corps locaux, Grothendieck Festschrift II, Prog. Math. 87, Birkhäuser, pp. 249-309, 1991. MR 1106901 (92i:11125)
- [GRR]
- A. Grothendieck, M. Raynaud, D. Rim, Groupes de monodromie en géométrie algébrique (SGA 7 I), Lecture Notes in Math. 288, Springer, 1972. MR 0354656 (50:7134)
- [Ki 1]
- M. Kisin, Crystalline representations and
-crystals, Algebraic geometry and number theory, Progr. Math 253, Birkhäuser Boston, pp. 459-496, 2006. MR 2263197 (2007j:11163)
- [Ki 2]
- M. Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), 513-546. MR 2373358 (2009c:11194)
- [Ki 3]
- M. Kisin, Modularity of
-adic Barsotti-Tate representations, Invent. Math. 178 (2009), 587-634. MR 2551765
- [Ki 4]
- M. Kisin, Integral canonical models of Shimura varieties (Journées Arithmétiques 2007), J. Th. Nombres Bordeaux 21 (2009), 301-312. MR 2541427
- [Ko]
- R. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373-444. MR 1124982 (93a:11053)
- [La]
- R. Langlands, Some contemporary problems with origins in the Jugendtraum, Mathematical developments arising from Hilbert problems (De Kalb 1974), Proc. Sympos. Pure Math. XXVIII, Amer. Math. Soc., pp. 401-418, 1976. MR 0437500 (55:10426)
- [Me]
- W. Messing, The crystals associated to Barsotti-Tate groups, with applications to Abelian schemes, Lecture Notes in Math. 264, Springer, 1972. MR 0347836 (50:337)
- [MFK]
- D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Ergeb. der Math. 34, Springer, 1994. MR 1304906 (95m:14012)
- [Mi 1]
- J. Milne, The points on a Shimura variety modulo a prime of good reduction, The zeta functions of Picard modular surfaces, 1992, pp. 151-253. MR 1155229 (94g:11041)
- [Mi 2]
- J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, Automorphic forms, Shimura varieties. and
-functions I (Ann Arbor 1988), Perspectives in Math. 10, pp. 284-414, 1990. MR 1044823 (91a:11027)
- [Mi 3]
- J. Milne, On the conjecture of Langlands and Rapoport, available at www.arxiv.org (2007).
- [Mi 4]
- J. Milne, Shimura varieties and motives, Motives (Seattle 1991), Proc. Sympos. Pure Math. 55, Part 2, Amer. Math. Soc., pp. 447-523, 1994. MR 1265562 (95c:11076)
- [Mo]
- B. Moonen, Models of Shimura varieties in mixed characteristics, Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, pp. 267-350, 1998. MR 1696489 (2000e:11077)
- [PY]
- G. Prasad, J.-K. Yu, On quasi-reductive group schemes. With an appendix by Brian Conrad, J. Algebraic Geom. 15 (2006), 507-549. MR 2219847 (2007c:14047)
- [RZ]
- M. Rapoport, T. Zink, Period space for
-divisible groups, Ann. Math. Studies 141, Princeton University Press, 1996. MR 1393439 (97f:14023)
- [Sa]
- N. Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Math. 265, Springer, 1972. MR 0338002 (49:2769)
- [Sp]
- T. Springer, Reductive groups, Automorphic forms, representations and
-functions (Corvallis, 1977), Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., pp. 1-27, 1979. MR 546587 (80h:20062)
- [Ti]
- J. Tits, Reductive groups over local fields, Automorphic forms, representations and
-functions (Corvallis, 1977), Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., pp. 29-69, 1979. MR 546588 (80h:20064)
- [Va 1]
- A. Vasiu, Integral Canonical Models of Shimura Varieties of Preabelian Type, Asian J. Math. 3 (1999), 401-518. MR 1796512 (2002b:11087)
- [Va 2]
- A. Vasiu, Integral Canonical Models of Shimura Varieties of Preabelian Type. (Fully corrected version.), available at www.arxiv.org (2003), 135 pages.
- [Va 3]
- A. Vasiu, A motivic conjecture of Milne, available at www.arxiv.org (2003), 46 pages.
- [Va 4]
- A. Vasiu, Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic I, available at www.arxiv.org (2007), 48 pages.
- [Va 5]
- A. Vasiu, Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic II, available at www.arxiv.org (2007), 33 pages.
- [Wa]
- W. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, Springer, 1997. MR 547117 (82e:14003)
- [Zi 1]
- T. Zink, The display of a formal
-divisible group, Cohomologies -adiques et application arithmétiques I, Astérisque 278, Soc. Math. France, pp. 127-248, 2002. MR 1922825 (2004b:14083)
- [Zi 2]
- T. Zink, Windows for displays of
-divisible groups, Moduli of abelian varieties (Text 1999), Prog. in Math. 195, Birkhäuser, pp. 491-518, 2001. MR 1827031 (2002c:14073)
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Additional Information
Mark Kisin
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Address at time of publication:
Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
Email:
kisin@math.harvard.edu
DOI:
http://dx.doi.org/10.1090/S0894-0347-10-00667-3
PII:
S 0894-0347(10)00667-3
Received by editor(s):
November 4, 2008
Received by editor(s) in revised form:
December 16, 2009
Posted:
April 21, 2010
Additional Notes:
The author was partially supported by NSF grant DMS-0017749000
Article copyright:
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