On Non-Basic Rapoport-Zink Spaces

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 thir l-adic cohomology groups provide a realization of certain cases of local Laglands 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 some associated lower-dimensional (and in most favorable cases basic) Rapoport-Zink spaces. Such an equality was originally conjectured by Harris in [11](Conjecture 5.2, p.420).

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