We introduce a category of 'rigid spaces with overconvergent structure sheaf' which we call dagger spaces -this is the correct category in which de Rham cohomology in rigid analysis should be studied. We compare it with the (usual) category of rigid spaces, give Serre and Poincaré duality theorems and explain the relation with Berthelot's rigid cohomology.

We define Frobenius and monodromy operators on the de Rham cohomology of K-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction Y , over a complete discrete valuation ring K of mixed characteristic. For this we introduce log rigid cohomology and generalize the so called Hyodo-Kato isomorphism to versions for non-proper Y , for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of Y . We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel'd's symmetric space X and its quotients. Our results are used in a critical way in the recent proof of the monodromy-weight conjecture for quotients of X given by de Shalit [7].

We define de Rham cohomology groups for rigid spaces over non-archimedean fields of characteristic zero, based on the notion of dagger space introduced in [12]. We establish some functorial properties and a finiteness result, and discuss the relation to the rigid cohomology as defined by P. Berthelot [2].

For a large class of smooth dagger spaces-rigid spaces with overconvergent structure sheaf-we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of P. Berthelot's rigid cohomology also in the nonsmooth case. We need a careful study of de Rham cohomology in situations of semistable reduction.

Let o be the ring of integers in a finite extension K of Q p , let k be its residue field. Let G be a split reductive group over Q p , let T be a maximal split torus in G. Let H(G, I 0 ) be the prop-Iwahori Hecke o-algebra. Given a semiinfinite reduced chamber gallery (alcove walk) C We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of G over K. For d = 1 we recover Colmez' functor (when restricted to o-torsion GL 2 (Q p )-representations generated by their pro-p-Iwahori invariants). Contents IntroductionIn his remarkable opus [5] on the p-adic local Langlands correspondence for GL 2 (Q p ), Colmez established a bijection between certain representations of GL 2 (Q p ) and certain two-dimensional representations of the absolute Galois group Gal Qp of the field Q p of p-adic numbers. These representations have coefficients either in a finite extension of F p , or in a finite extension of Q p . In either case, the theory of (ϕ, Γ)-modules as developed by Fontaine [7] provides the required intermediate objects in order to pass from one side to the other. Prior to Colmez' work the characteristic p correspondence had been suggested by Breuil as an explicit "by hand" matching between the objects on either side; it was then astonishing to see this correspondence being realized even by a functorial relationship between GL 2 (Q p )-representations and (ϕ, Γ)-modules. A certain functor D from o-torsion representations of GL 2 (Q p ) to (ϕ, Γ)-modules over o constitutes one half of this relationship. Here o is the ring of integers in a finite extension K of Q p . Although Colmez does not phrase it in these terms, his functor D may be viewed as factoring through a functor from certain coefficient systems on the Bruhat Tits tree X of PGL 2 (Q p ) to (ϕ, Γ)-modules. The purpose of the present paper is to suggest an extension of this latter functor to certain coefficient systems on the Bruhat Tits building X of a general split reductive group G over Q p . Such coefficient systems can in particular be attached to (ofinite-length) modules over the pro-p-Iwahori Hecke o-algebra H(G, I 0 ), formed with respect to a pro-p-Iwahori subgroup I 0 in G. The entire construction depends on a certain choice, and for each such choice we end up (Theorem 7.5) with an exact functor from such H(G, I 0 )-modules to (ϕ r , Γ)-modules, with r ∈ N depending on that choice.Let v 0 denote the vertex of X fixed by GL 2 (Z p ). In GL 2 (Q p ) consider the element ϕ = p 0 0 1 and the subgroups N 0 = 1 Z p 0 1 and Γ = Z × p 0 0 1 . The orbit of v 0 under the submonoid ⌊N 0 , ϕ, Γ⌋ of G generated by N 0 , ϕ and Γ defines a halftree X + inside X: its edges are those whose both vertices belong to that orbit. Adding the unique edge with only one vertex (namely v 0 ) in that orbit we obtain the half tree. Under a suitable finiteness conditions these are compact Now suppose we are given a G-equivariant coefficient system V on X. By what we said, in order to pass from V to anétale (...