On the de Rham and crystalline cohomology of quotients of Ω_{K}^{(d+1)}

Let k be a perfect field of characteristic p > 0 and let K be a finite extension of K_{0} = W (k)[p^{−1}]. For a proper (strictly) semi-stable (formal) scheme X over a O_{K} we have the Hyodo-Kato isomorphism ρ between the de Rham cohomology H_{dR}^{∗}(X_{K}) of the generic fibre X_{K} and the Hyodo-Kato cohomology H_{HK}^{∗}(Y) of the special fibre Y of X, endowing (a K_{0}-lattice of) H_{dR}^{∗}(X_{K}) with a Frobenius operator ϕ and a monodromy operator N. On the other hand, the components of Y give rise to a canonical admissible open covering of X_{K}, viewed as a rigid analytic space, and thus to a canonical Cech-filtration (F_{C}^{r})_{r≥0} on H_{dR}^{∗}(X_{K}). I want to talk about a rigid analytic construction of ρ; it can be used to relate (F_{C}^{r})_{r≥0} with N. Then I specialize to the case where [K : ℚ_{p}] is finite and X_{K} is the quotient of Drinfel’d’s padic symmetric space Ω_{K}^{(d+1)} by a cocompact discrete torsionfree subgroup of PGL_{d+1}(K). Here the analytic construction of ρ allows us to identify (F_{C}^{r})_{r≥0} with the slope and the weight filtration for ϕ on H_{dR}^{d}(X_{K}). Together with the now proven C_{st}-conjecture we get a new proof for the fact that (F_{C}^{r})_{r≥0} is opposite to the Hodge filtration, as was shown earlier by Spiess and Iovita.