Hostname: page-component-7dd5485656-7jgsp Total loading time: 0 Render date: 2025-10-31T08:54:39.929Z Has data issue: false hasContentIssue false
  • English
  • Français

Rapoport–Zink spaces for spinor groups

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  10 April 2017

Benjamin Howard  and
Georgios Pappas
Benjamin Howard
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA email howardbe@bc.edu
Georgios Pappas
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA email pappas@math.msu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

After the work of Kisin, there is a good theory of canonical integral models of Shimura varieties of Hodge type at primes of good reduction. The first part of this paper develops a theory of Hodge type Rapoport–Zink formal schemes, which uniformize certain formal completions of such integral models. In the second part, the general theory is applied to the special case of Shimura varieties associated with groups of spinor similitudes, and the reduced scheme underlying the Rapoport–Zink space is determined explicitly.

Keywords

Shimura varietyspinor groupRapoport–Zink space

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties 14G35: Modular and Shimura varieties

Information

Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 5 , May 2017 , pp. 1050 - 1118
Copyright
© The Authors 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Asgari, M., Local L-functions for split spinor groups , Canad. J. Math. 54 (2002), 673–693; MR 1913914 (2003i:11062).CrossRefGoogle Scholar
Bass, H., Clifford algebras and spinor norms over a commutative ring , Amer. J. Math. 96 (1974), 156–206; MR 0360645 (50 #13092).CrossRefGoogle Scholar
Berthelot, P., Cohomologie rigide et cohomologie rigide à supports propres. Première partie. Preprint (1996), IRMAR 96-03, available at http://perso.univ-rennesl.fr/pierre.berthelot.Google Scholar
Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, vol. 930 (Springer, Berlin, 1982); MR 667344 (85k:14023).Google Scholar
Berthelot, P. and Messing, W., Théorie de Dieudonné cristalline. III. Théorèmes d’équivalence et de pleine fidélité , in The Grothendieck Festschrift, Vol. I, Progress in Mathematics, vol. 86 (Birkhäuser, Boston, 1990), 173–247; MR 1086886 (92h:14012).Google Scholar
Bhatt, B. and Scholze, P., Projectivity of the Witt vector affine Grassmannian. Preprint (2015), arXiv:1507.06490.Google Scholar
Chen, M., Kisin, M. and Viehmann, E., Connected components of affine Deligne–Lusztig varieties in mixed characteristic , Compositio Math. 151 (2015), 1697–1762; MR 3406443.Google Scholar
Chen, M. and Viehmann, E., Affine Deligne–Lusztig varieties and the action of  $J$ . Preprint (2015), arXiv:1507.02806.Google Scholar
Dat, J.-F., Orlik, S. and Rapoport, M., Period domains over finite and p-adic fields, Cambridge Tracts in Mathematics, vol. 183 (Cambridge University Press, Cambridge, 2010); MR 2676072 (2012a:22026).CrossRefGoogle Scholar
de Jong, A. J., Crystalline Dieudonné module theory via formal and rigid geometry , Publ. Math. Inst. Hautes Études Sci. 82 (1995), 5–96; MR 1383213 (97f:14047).Google Scholar
Deligne, P., Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques , in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, OR, 1977), Part 2, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 247–289; MR 546620 (81i:10032).Google Scholar
Drinfel’d, V. G., Coverings of p-adic symmetric domains , Funkcional. Anal. i Priložen. 10 (1976), 29–40; MR 0422290 (54 #10281).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I , Publ. Math. Inst. Hautes Études Sci. 20 (1964), 5–251; MR 0173675 (30 #3885).Google Scholar
Faltings, G., Integral crystalline cohomology over very ramified valuation rings , J. Amer. Math. Soc. 12 (1999), 117–144; MR 1618483 (99e:14022).Google Scholar
Gerstein, L., Basic quadratic forms, Graduate Studies in Mathematics, vol. 90 (American Mathematical Society, Providence, RI, 2008); MR 2396246 (2009e:11064).CrossRefGoogle Scholar
Görtz, U., Haines, T., Kottwitz, R. and Reuman, D., Dimensions of some affine Deligne–Lusztig varieties , Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), 467–511; MR 2265676 (2008e:14068).CrossRefGoogle Scholar
Görtz, U. and He, X., Basic loci of Coxeter type in Shimura varieties , Camb. J. Math. 3 (2015), 323–353; MR 3393024.Google Scholar
Hamacher, P., The almost product structure of Newton strata in the Deformation space of a Barsotti–Tate group with crystalline Tate tensors. Preprint (2016), arXiv:1601.03131.Google Scholar
Howard, B. and Pappas, G., On the supersingular locus of the GU(2,2) Shimura variety , Algebra Number Theory 8 (2014), 1659–1699; MR 3272278.Google Scholar
Kim, W., Rapoport–Zink spaces of Hodge type. Preprint (2013), arXiv:1308.5537.Google Scholar
Kim, W., Rapoport–Zink uniformization of Hodge-type Shimura varieties. Preprint (2014).Google Scholar
Kisin, M., Integral models for Shimura varieties of abelian type , J. Amer. Math. Soc. 23 (2010), 967–1012; MR 2669706.CrossRefGoogle Scholar
Kisin, M., Mod $p$ points on Shimura varieties of abelian type, J. Amer. Math. Soc., to appear. Preprint (2013).Google Scholar
Kisin, M. and Pappas, G., Integral models for Shimura varieties with parahoric level structure. Preprint (2015), arXiv:1512.01149.Google Scholar
Kitaoka, Y., Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, vol. 106 (Cambridge University Press, Cambridge, 1993); MR 1245266 (95c:11044).CrossRefGoogle Scholar
Kottwitz, R., Shimura varieties and twisted orbital integrals , Math. Ann. 269 (1984), 287–300; MR 761308 (87b:11047).CrossRefGoogle Scholar
Kottwitz, R., Isocrystals with additional structure , Compositio Math. 56 (1985), 201–220; MR 809866 (87i:14040).Google Scholar
Kottwitz, R., Points on some Shimura varieties over finite fields , J. Amer. Math. Soc. 5 (1992), 373–444; MR 1124982 (93a:11053).CrossRefGoogle Scholar
Kudla, S. S., Special cycles and derivatives of Eisenstein series , in Heegner points and Rankin L-series, Mathematical Sciences Research Institute Publications, vol. 49 (Cambridge University Press, Cambridge, 2004), 243–270; MR 2083214 (2005g:11108).Google Scholar
Kudla, S. S. and Rapoport, M., Arithmetic Hirzebruch–Zagier cycles , J. reine angew. Math. 515 (1999), 155–244; MR 1717613 (2002e:11076a).Google Scholar
Kudla, S. S. and Rapoport, M., Cycles on Siegel threefolds and derivatives of Eisenstein series , Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), 695–756; MR 1834500 (2002e:11076b).Google Scholar
Lang, S., Algebra, Graduate Texts in Mathematics, vol. 211, third edition (Springer, New York, 2002); MR 1878556 (2003e:00003).Google Scholar
Madapusi Pera, K., The Tate conjecture for K3 surfaces in odd characteristic , Invent. Math. 201 (2015), 625–668; MR 3370622.Google Scholar
Madapusi Pera, K., Integral canonical models for spin Shimura varieties , Compositio Math. 152 (2016), 769–824; MR 3484114.Google Scholar
Matsumura, H., Commutative algebra, Mathematics Lecture Note Series, vol. 56, second edition (Benjamin/Cummings Publishing Co., Reading, MA, 1980); MR 575344 (82i:13003).Google Scholar
Messing, W., The crystals associated to Barsotti–Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, vol. 264 (Springer, Berlin, New York, 1972); MR 0347836 (50 #337).Google Scholar
Moonen, B., Models of Shimura varieties in mixed characteristics , in Galois representations in arithmetic algebraic geometry (Durham, 1996), London Mathematical Society Lecture Note Series, vol. 254 (Cambridge University Press, Cambridge, 1998), 267–350; MR 1696489 (2000e:11077).Google Scholar
Nisnevich, Y. A., Étale cohomology and arithmetic of semisimple groups, ProQuest LLC, Ann Arbor, MI, 1982, PhD thesis, Harvard University; MR 2632405.Google Scholar
Ogus, A., F-isocrystals and de Rham cohomology. II. Convergent isocrystals , Duke Math. J. 51 (1984), 765–850; MR 771383 (86j:14012).CrossRefGoogle Scholar
Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139 (Academic Press, Boston, 1994), translated from the 1991 Russian original by R. Rowen; MR 1278263 (95b:11039).Google Scholar
Rapoport, M., A guide to the reduction modulo p of Shimura varieties , in Automorphic forms. I, Astérisque, vol. 298 (Société Mathématique de France, Paris, 2005), 271–318; MR 2141705 (2006c:11071).Google Scholar
Rapoport, M. and Richartz, M., On the classification and specialization of F-isocrystals with additional structure , Compositio Math. 103 (1996), 153–181; MR 1411570 (98c:14015).Google Scholar
Rapoport, M., Terstiege, U. and Wilson, S., The supersingular locus of the Shimura variety for GU(1, n - 1) over a ramified prime , Math. Z. 276 (2014), 1165–1188; MR 3175176.Google Scholar
Rapoport, M. and Viehmann, E., Towards a theory of local Shimura varieties , Münster J. Math. 7 (2014), 273–326; MR 3271247.Google Scholar
Rapoport, M. and Zink, Th., Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141 (Princeton University Press, Princeton, NJ, 1996); MR 1393439 (97f:14023).Google Scholar
Scholze, P. and Weinstein, J., Moduli of p-divisible groups , Camb. J. Math. 1 (2013), 145–237; MR 3272049.Google Scholar
Serre, J.-P., A course in arithmetic, Graduate Texts in Mathematics, vol. 7 (Springer, New York, 1973), translated from the French; MR 0344216 (49 #8956).CrossRefGoogle Scholar
Shimura, G., Arithmetic of quadratic forms, Springer Monographs in Mathematics (Springer, New York, 2010); MR 2665139 (2011m:11003).Google Scholar
Viehmann, E., Truncations of level 1 of elements in the loop group of a reductive group , Ann. of Math. (2) 179 (2014), 1009–1040; MR 3171757.Google Scholar
Vollaard, I., The supersingular locus of the Shimura variety for GU(1, s) , Canad. J. Math. 62 (2010), 668–720; MR 2666394.Google Scholar
Vollaard, I. and Wedhorn, T., The supersingular locus of the Shimura variety of GU(1, n - 1) II , Invent. Math. 184 (2011), 591–627; MR 2800696 (2012j:14035).CrossRefGoogle Scholar
Wortmann, D., The $\unicode[STIX]{x1D707}$ -ordinary locus for Shimura varieties of Hodge type. Preprint (2013), arXiv:1310.6444.Google Scholar
Zhang, C., Ekedahl–Oort strata for good reductions of Shimura varieties of Hodge type. Preprint (2013), arXiv:1312.4869.Google Scholar
Zhang, C., Stratifications and foliations for good reductions of Shimura varieties of Hodge type. Preprint (2015), arXiv:1512.08102.Google Scholar
Zhu, X., Affine Grassmannians and the geometric Satake in mixed characteristic , Ann. of Math. (2) 185 (2017), 403–492.Google Scholar
Zink, Th., Windows for displays of p-divisible groups, Moduli of abelian varieties (Texel Island, 1999), Progress in Mathematics, vol. 195 (Birkhäuser, Basel, 2001), 491–518; MR 1827031 (2002c:14073).Google Scholar
Zink, Th., The display of a formal p-divisible group , in Cohomologies p-adiques et applications arithmétiques, I, Astérisque, vol. 278 (Société Mathématique de France, Paris, 2002), 127–248; MR 1922825 (2004b:14083).Google Scholar