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Deformation spaces of one-dimensional formal modules and their cohomology. (English) Zbl 1140.22017

From the abstract: Let \(\mathcal M _m\) be the formal scheme which represents the functor of deformations of a one-dimensional formal module over \(\bar {\mathbb F}_p\) equipped with a level-\(m\)-structure. By work of Boyer (in equal characteristic) and Harris and Taylor, the \(\ell \)-adic étale cohomology of the generic fibre \(M_m\) of \(\mathcal M _m\) realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper the author shows without the use of global moduli spaces that the Jacquet-Langlands correspondence is realized by the Euler-Poincaré characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the spaces \(M_m\).

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
20G25 Linear algebraic groups over local fields and their integers
14G35 Modular and Shimura varieties
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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