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Weight-monodromy conjecture for \(p\)-adically uniformized varieties. (English) Zbl 1154.14014
Summary: The aim of this paper is to prove the weight-monodromy conjecture (Deligne’s conjecture on the purity of monodromy filtration) for varieties \(p\)-adically uniformized by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply a positivity argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of \(p\)-adically uniformized varieties in terms of representation theoretic invariants. We also consider a \(p\)-adic analogue by using the weight spectral sequence of Mokrane.

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14F20 Étale and other Grothendieck topologies and (co)homologies
14D07 Variation of Hodge structures (algebro-geometric aspects)
14G35 Modular and Shimura varieties
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