In this talk, I will give a proof of the weight-monodromy conjecture (Deligne's conjecture on the purity of monodromy filtration) for varieties with p-adic uniformization 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 an argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink.
As an application, by combining this result with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theoretic invariants.
Tetsushi Ito, MPI / Univ. of Tokyo
p.s. Although I already had talks on this topic in the same title at several places, I'll try to include some 'latest' results... if possible.