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RAPOPORT–ZINK SPACES OF HODGE TYPE

Part of: Algebraic groups (Co)homology theory

Published online by Cambridge University Press:  07 June 2018

WANSU KIM
WANSU KIM*
Affiliation:
Department of Mathematics, KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, South Korea; wansukim@kias.re.kr

Abstract

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When $p>2$, we construct a Hodge-type analogue of Rapoport–Zink spaces under the unramifiedness assumption, as formal schemes parametrizing ‘deformations’ (up to quasi-isogeny) of $p$-divisible groups with certain crystalline Tate tensors. We also define natural rigid analytic towers with expected extra structure, providing more examples of ‘local Shimura varieties’ conjectured by Rapoport and Viehmann.

MSC classification

Primary: 14L05: Formal groups, $p$-divisible groups
Secondary: 14F30: $p$-adic cohomology, crystalline cohomology
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e8
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2018

References

Artin, M., ‘Versal deformations and algebraic stacks’, Invent. Math. 27 (1974), 165–189. MR 0399094 (53 #2945).Google Scholar
Berthelot, P., ‘Cohomologie rigide et cohomologie rigide à support propre’. première partie, Prépublication IRMAR 96-03 (1996).Google Scholar
Berthelot, P., Breen, L. and Messing, W., Théorie de Dieudonné Cristalline. II, Lecture Notes in Mathematics, 930 (Springer, Berlin, 1982), MR 667344 (85k:14023).Google Scholar
Berthelot, P. and Ogus, A., Notes on Crystalline Cohomology, (Princeton University Press, Princeton, NJ, 1978), MR 0491705 (58 #10908).Google Scholar
Berthelot, P. and Ogus, A., ‘ F-isocrystals and de Rham cohomology. I’, Invent. Math. 72(2) (1983), 159–199. MR 700767 (85e:14025).Google Scholar
Bosch, S. and Lütkebohmert, W., ‘Formal and rigid geometry. II. Flattening techniques’, Math. Ann. 296(3) (1993), 403–429. MR 1225983 (94e:11070).Google Scholar
Brinon, O., ‘Représentations p-adiques cristallines et de de Rham dans le cas relatif’, Mém. Soc. Math. Fr. (N.S.) (112) (2008), vi+159 MR 2484979 (2010a:14034).Google Scholar
Bruhat, F. and Tits, J., ‘Groupes réductifs sur un corps local. I. Données radicielles valuées’, Inst. Hautes Études Sci. Publ. Math. (41) (1972), 5–251. MR 0327923.CrossRefGoogle Scholar
Bültel, O. and Pappas, G., ‘ $(G,\unicode[STIX]{x1D707})$ -displays and Rapoport-Zink spaces’, Preprint, 2017, arXiv:math/1702.00291, 44 pages.Google Scholar
Chase, S. U., ‘Direct products of modules’, Trans. Amer. Math. Soc. 97 (1960), 457–473. MR 0120260.Google Scholar
Chen, M., Kisin, M. and Viehmann, E., ‘Connected components of affine Deligne-Lusztig varieties in mixed characteristic’, Compos. Math. 151(9) (2015), 1697–1762. MR 3406443.Google Scholar
Colliot-Thélène, J.-L. and Sansuc, J.-J., ‘Fibrés quadratiques et composantes connexes réelles’, Math. Ann. 244 (1979), 105–134. MR 550842.Google Scholar
Conrad, B. and de Jong, A. J., ‘Approximation of versal deformations’, J. Algebra 255(2) (2002), 489–515. MR 1935511 (2004a:14003).Google Scholar
Conrad, B., Gabber, O. and Prasad, G., Pseudo-Reductive Groups, 2nd edn, New Mathematical Monographs, 26 (Cambridge University Press, Cambridge, 2015), MR 3362817.Google Scholar
de Jong, A. J., ‘Crystalline Dieudonné module theory via formal and rigid geometry’, Inst. Hautes Études Sci. Publ. Math. 82 (1995), 5–96. MR 1383213 (97f:14047).Google Scholar
de Jong, A. J., ‘Étale fundamental groups of non-Archimedean analytic spaces’, Compositio Math. 97(1–2) (1995), 89–118. Special issue in honour of Frans Oort. MR 1355119 (97c:32047).Google Scholar
de Jong, A. J. and Messing, W., ‘Crystalline Dieudonné theory over excellent schemes’, Bull. Soc. Math. France 127(2) (1999), 333–348. MR 1708635 (2001b:14075).Google Scholar
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., Corvallis, Ore., 1977), Part 2, Proceedings of Symposia in Pure Mathematics, XXXIII (American Mathematical Society, Providence, RI, 1979), 247–289. MR 546620 (81i:10032).Google Scholar
Faltings, G., ‘Integral crystalline cohomology over very ramified valuation rings’, J. Amer. Math. Soc. 12(1) (1999), 117–144. MR 1618483 (99e:14022).CrossRefGoogle Scholar
Fargues, L., ‘Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales’, Astérisque (291) (2004), 1–199. Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales.MR 2074714 (2005g:11110b).Google Scholar
Fontaine, J.-M., ‘Le corps des périodes p-adiques’, Astérisque (223) (1994), 59–111. With an appendix by Pierre Colmez, Périodes $p$ -adiques (Bures-sur-Yvette, 1988). MR 1293971 (95k:11086).Google Scholar
Grothendieck, A., ‘Eléments de géométrie algébrique, I–IV’, Inst. Hautes Études Sci. Publ. Math. (4, 8, 11, 17, 20, 24, 28, and 32) (1960–1967).Google Scholar
Grothendieck, A., Groupes de Barsotti-Tate et Cristaux de Dieudonné, Les Presses de l’Université de Montréal, Montreal, Que., 1974, Séminaire de Mathématiques Supérieures, 45 (Été, 1970), MR 0417192 (54 #5250).Google Scholar
Howard, B. and Pappas, G., ‘Rapoport-Zink spaces for spinor groups’, Compos. Math. 153(5) (2017), 1050–1118. MR 3705249.Google Scholar
Huber, R., Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Aspects of Mathematics, E30 (Friedr. Vieweg & Sohn, Braunschweig, 1996), MR 1734903 (2001c:14046).CrossRefGoogle Scholar
Illusie, L., ‘Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)’, Astérisque (127) (1985), 151–198. Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84). MR 801922.Google Scholar
Katz, N., ‘Serre-Tate local moduli’, inAlgebraic Surfaces (Orsay, 1976–78), Lecture Notes in Mathematics, 868 (Springer, Berlin, 1981), 138–202. MR 638600 (83k:14039b).Google Scholar
Kim, W., ‘The classification of p-divisible groups over 2-adic discrete valuation rings’, Math. Res. Lett. 19(01) (2012), 121–141. MR 2923180.CrossRefGoogle Scholar
Kim, W., ‘The relative Breuil-Kisin classification of p-divisible groups and finite flat group schemes’, Int. Math. Res. Not. 2015(17) (2015), 8152–8232. MR 3404012.Google Scholar
Kim, W., ‘Rapoport-Zink uniformisation of Hodge-type Shimura varieties’, Preprint.Google Scholar
Kisin, M., ‘Crystalline representations and F-crystals’, inAlgebraic Geometry and Number Theory, Progress in Mathematics, 253 (Birkhäuser Boston, Boston, MA, 2006), 459–496. MR 2263197 (2007j:11163).Google Scholar
Kisin, M., ‘Integral models for Shimura varieties of abelian type’, J. Amer. Math. Soc. 23(4) (2010), 967–1012. MR 2669706.Google Scholar
Knutson, D., Algebraic Spaces, Lecture Notes in Mathematics, 203 (Springer, Berlin, 1971), MR 0302647 (46 #1791).Google Scholar
Kottwitz, R. E., ‘Isocrystals with additional structure’, Compos. Math. 56(2) (1985), 201–220. MR 809866 (87i:14040).Google Scholar
Lau, E., ‘Dieudonné theory over semiperfect rings and perfectoid rings’, Preprint, 2016, arXiv:math/1603.07831.Google Scholar
Laumon, G. and Moret-Bailly, L., ‘Champs algébriques’, inErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 39 (Springer, Berlin, 2000), MR 1771927 (2001f:14006).Google Scholar
Matsumura, H., Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 8 (Cambridge University Press, Cambridge, 1986), Translated from the Japanese by M. Reid. MR 879273 (88h:13001).Google Scholar
Mazur, B. and Messing, W., Universal Extensions and One Dimensional Crystalline Cohomology, Lecture Notes in Mathematics, 370 (Springer, Berlin, 1974), MR 0374150 (51 #10350).Google Scholar
Messing, W., The Crystals Associated to Barsotti-Tate Groups: With Applications to Abelian Schemes, Lecture Notes in Mathematics, 264 (Springer, Berlin, 1972), MR 0347836 (50 #337).Google Scholar
Milne, J. S., ‘Introduction to Shimura varieties’, inHarmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings, 4 (American Mathematical Society, Providence, RI, 2005), 265–378. MR 2192012 (2006m:11087).Google Scholar
Moonen, B., ‘Models of Shimura varieties in mixed characteristics’, inGalois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Mathematical Society Lecture Note Series, 254 (Cambridge University Press, Cambridge, 1998), 267–350. MR 1696489 (2000e:11077).Google Scholar
Rapoport, M. and Richartz, M., ‘On the classification and specialization of F-isocrystals with additional structure’, Compos. Math. 103(2) (1996), 153–181. MR 1411570 (98c:14015).Google Scholar
Rapoport, M. and Viehmann, E., ‘Towards a theory of local Shimura varieties’, Münster J. Math. 7(1) (2014), 273–326. MR 3271247.Google Scholar
Rapoport, M. and Zink, T., Period Spaces for p-Divisible Groups, Annals of Mathematics Studies, 141 (Princeton University Press, Princeton, NJ, 1996), MR 1393439.Google Scholar
Schlessinger, M., ‘Functors of Artin rings’, Trans. Amer. Math. Soc. 130 (1968), 208–222. MR 0217093 (36 #184).Google Scholar
Scholze, P., ‘ p-adic Hodge theory for rigid-analytic varieties’, Forum Math. Pi 1 (2013), e1, 77. MR 3090230.CrossRefGoogle Scholar
Scholze, P., Perfectoid Spaces: a Survey, Current Developments in Mathematics, 2012 (Int. Press, Somerville, MA, 2013), 193–227. MR 3204346.Google Scholar
Scholze, P. and Weinstein, J., ‘Moduli of p-divisible groups’, Camb. J. Math. 1(2) (2013), 145–237. MR 3272049.Google Scholar
Valabrega, P., ‘A few theorems on completion of excellent rings’, Nagoya Math. J. 61 (1976), 127–133. MR 0407007 (53 #10790).Google Scholar
Vasiu, A., ‘Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic, part I’, Preprint, 2007, arXiv:0707.1668, 53 pages.Google Scholar
Vasiu, A., ‘A motivic conjecture of Milne’, J. Reine Angew. Math. 685 (2013), 181–247. MR 3181569.Google Scholar
Wedhorn, T., ‘On Tannakian duality over valuation rings’, J. Algebra 282(2) (2004), 575–609. MR 2101076 (2005j:18007).Google Scholar
Wintenberger, J.-P., ‘Un scindage de la filtration de Hodge pour certaines variétés algébriques sur les corps locaux’, Ann. of Math. (2) 119(3) (1984), 511–548. MR 744862 (86k:14015).Google Scholar
Zhu, X., ‘Affine Grassmannians and the geometric Satake in mixed characteristic’, Ann. of Math. (2) 185(2) (2017), 403–492. MR 3612002.Google Scholar