On non-basic Rapoport-Zink spaces
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 5, pp. 671-716.

In this paper we study certain moduli spaces of Barsotti-Tate groups constructed by Rapoport and Zink as local analogues of Shimura varieties. More precisely, given an isogeny class of Barsotti-Tate groups with unramified additional structures, we investigate how the associated (non-basic) moduli spaces compare to the (basic) moduli spaces associated with its isoclinic constituents. This aspect of the geometry of the Rapoport-Zink spaces is closely related to Kottwitz’s prediction that their l-adic cohomology groups provide a realization of certain cases of local Langlands correspondences and in particular to the question of whether they contain any supercuspidal representations. Our results are compatible with this prediction and identify many cases when no supercuspidal representations appear. In those cases, we prove that the l-adic cohomology of the non-basic spaces is equal (in the appropriate sense) to the parabolic induction of the l-adic cohomology of some associated lower-dimensional (and in the most favorable cases basic) Rapoport-Zink spaces. Such an equality was originally conjectured by Harris in [11] (Conjecture 5.2, p. 420).

Dans cet article, on considère certains espaces de Rapoport-Zink non-ramifiés, associés à des groupes p-divisibles non-basiques et on étudie leur géométrie vis-à-vis de celle des espaces de Rapoport-Zink basiques correspondants. L’origine de ce problème se situe, d’une part, dans la conjecture de Kottwitz concernant la réalisation des correspondances de Langlands locales dans la cohomologie étale l-adique des espaces de Rapoport-Zink et, d’autre part, plus simplement dans la question d’identifier pour lesquels de ces espaces la partie supercuspidale de la cohomologie n’est pas vide. Nos résultats sont compatibles avec cette conjecture et, dans certains cas particuliers, ils répondent à la dernière question. En particulier, dans ces cas, on établit une formule reliant la cohomologie de ces espaces à l’induction parabolique de celle de certains espaces de Rapoport-Zink de dimension inférieure (et dans les cas plus favorables basiques). Cette formule a été précédemment conjecturée par Harris dans [11] (Conjecture 5.2, p. 420).

DOI: 10.24033/asens.2079
Classification: 14G35, 14L05, 11Fxx
Keywords: $p$-divisible groups, Rapoport-Zink spaces, Shimura varieties, Langlands correspondences
Mot clés : groupes $p$-divisibles, espaces de Rapoport-Zink, variétés de Shimura, correspondances de Langlands
@article{ASENS_2008_4_41_5_671_0,
     author = {Mantovan, Elena},
     title = {On non-basic {Rapoport-Zink} spaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {671--716},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {5},
     year = {2008},
     doi = {10.24033/asens.2079},
     zbl = {1236.11101},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2079/}
}
TY  - JOUR
AU  - Mantovan, Elena
TI  - On non-basic Rapoport-Zink spaces
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2008
SP  - 671
EP  - 716
VL  - 41
IS  - 5
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/asens.2079/
DO  - 10.24033/asens.2079
LA  - en
ID  - ASENS_2008_4_41_5_671_0
ER  - 
%0 Journal Article
%A Mantovan, Elena
%T On non-basic Rapoport-Zink spaces
%J Annales scientifiques de l'École Normale Supérieure
%D 2008
%P 671-716
%V 41
%N 5
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/asens.2079/
%R 10.24033/asens.2079
%G en
%F ASENS_2008_4_41_5_671_0
Mantovan, Elena. On non-basic Rapoport-Zink spaces. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 5, pp. 671-716. doi : 10.24033/asens.2079. http://archive.numdam.org/articles/10.24033/asens.2079/

[1] V. G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math. I.H.É.S. 78 (1993), 5-161. | Numdam | MR | Zbl

[2] V. G. Berkovich, Vanishing cycles for formal schemes, Invent. Math. 115 (1994), 539-571. | MR | Zbl

[3] V. G. Berkovich, Vanishing cycles for formal schemes. II, Invent. Math. 125 (1996), 367-390. | MR | Zbl

[4] P. Boyer, Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. Math. 138 (1999), 573-629. | MR | Zbl

[5] P. Colmez & J.-M. Fontaine, Construction des représentations p-adiques semi-stables, Invent. Math. 140 (2000), 1-43. | Zbl

[6] M. Demazure, Lectures on p-divisible groups, Lecture Notes in Math. 302, Springer, 1972. | MR | Zbl

[7] V. G. DrinfelʼD, Coverings of p-adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), 29-40. | MR | Zbl

[8] L. Fargues, Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales, Astérisque 291 (2004), 1-199. | MR | Zbl

[9] J.-M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, in Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. III, Astérisque 65, Soc. Math. France, 1979, 3-80. | Numdam | MR | Zbl

[10] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Séminaire de Mathématiques Supérieures, No. 45 (Été 1970), Presses de l'Univ. Montréal, 1974. | MR | Zbl

[11] M. Harris, Local Langlands correspondences and vanishing cycles on Shimura varieties, in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, 2001, 407-427. | MR | Zbl

[12] M. Harris & R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies 151, Princeton University Press, 2001. | Zbl

[13] L. Illusie, Déformations de groupes de Barsotti-Tate (d'après A. Grothendieck), Astérisque 127 (1985), 151-198. | Numdam | MR | Zbl

[14] A. J. D. Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. I.H.É.S. 82 (1995), 5-96. | Numdam | MR | Zbl

[15] N. M. Katz, Slope filtration of F-crystals, in Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. I, Astérisque 63, Soc. Math. France, 1979, 113-163. | Numdam | MR | Zbl

[16] N. M. Katz & B. Mazur, Arithmetic moduli of elliptic curves, Annals of Math. Studies 108, Princeton University Press, 1985. | Zbl

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

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

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

[20] R. E. Kottwitz, On the Hodge-Newton decomposition for split groups, Int. Math. Res. Not. 26 (2003), 1433-1447. | MR | Zbl

[21] J. I. Manin, Theory of commutative formal groups over fields of finite characteristic, Uspehi Mat. Nauk 18 (1963), 3-90; Russ. Math. Surveys 18 (1963), 1-80. | MR | Zbl

[22] E. Mantovan, On certain unitary group Shimura varieties, Astérisque 291 (2004), 201-331. | MR | Zbl

[23] E. Mantovan, On the cohomology of certain PEL-type Shimura varieties, Duke Math. J. 129 (2005), 573-610. | MR | Zbl

[24] E. Mantovan & E. Viehmann, On the Hodge-Newton filtration for p-divisible 𝒪-modules, preprint, arXiv:0710.4194, to appear in Math. Z. | Zbl

[25] W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Math. 264, Springer, 1972. | MR | Zbl

[26] B. Moonen, Serre-Tate theory for moduli spaces of PEL type, Ann. Sci. École Norm. Sup. 37 (2004), 223-269. | Numdam | MR | Zbl

[27] D. Mumford, Lectures on curves on an algebraic surface, Annals of Math. Studies, No. 59, Princeton University Press, 1966. | MR | Zbl

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

[29] F. Oort & T. Zink, Families of p-divisible groups with constant Newton polygon, Doc. Math. 7 (2002), 183-201. | Zbl

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

[31] M. Rapoport & T. Zink, Period spaces for p-divisible groups, Annals of Mathematics Studies 141, Princeton University Press, 1996. | Zbl

[32] T. Wedhorn, Ordinariness in good reductions of Shimura varieties of PEL-type, Ann. Sci. École Norm. Sup. 32 (1999), 575-618. | Numdam | MR | Zbl

[33] T. Zink, On the slope filtration, Duke Math. J. 109 (2001), 79-95. | MR | Zbl

Cited by Sources: