We study Deligne’s conjecture on the monodromy weight filtration on the nearby cycles in the mixed characteristic case, and reduce it to the nondegeneracy of certain pairings in the semistable case. We also prove a related conjecture of Rapoport and Zink which uses only the image of the Cech restriction morphism, if Deligne’s conjecture holds for a general hyperplane section. In general we show...

As an example of relative p-adic Hodge theory, we sketch the construction of the universal admissible filtration of an isocrystal (φ-module) over the completion of the maximal unramified extension of Qp, together with the associated universal crystalline local system. Mathematics Subject Classification (2000). Primary 14G22; Secondary 11G25.