On non-basic Rapoport-Zink spaces

Elena Mantovan

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 5, page 671-716
  • ISSN: 0012-9593

Abstract

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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).

How to cite

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Mantovan, Elena. "On non-basic Rapoport-Zink spaces." Annales scientifiques de l'École Normale Supérieure 41.5 (2008): 671-716. <http://eudml.org/doc/272181>.

@article{Mantovan2008,
abstract = {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).},
author = {Mantovan, Elena},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {$p$-divisible groups; Rapoport-Zink spaces; Shimura varieties; Langlands correspondences},
language = {eng},
number = {5},
pages = {671-716},
publisher = {Société mathématique de France},
title = {On non-basic Rapoport-Zink spaces},
url = {http://eudml.org/doc/272181},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Mantovan, Elena
TI - On non-basic Rapoport-Zink spaces
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 5
SP - 671
EP - 716
AB - 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).
LA - eng
KW - $p$-divisible groups; Rapoport-Zink spaces; Shimura varieties; Langlands correspondences
UR - http://eudml.org/doc/272181
ER -

References

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