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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