pp-torsion coefficient systems for SL2(Qp){\rm SL}_2({\bf Q}_p) and GL2(Qp){\rm GL}_2({\bf Q}_p),

We show that the categories of smooth ${\rm SL}_2({\mathbb Q}_p)$-representations (resp. ${\rm GL}_2({\mathbb Q}_p)$-representations) of level $1$ on $p$-torsion modules are equivalent with certain explicitly described equivariant coefficient systems on the Bruhat-Tits tree; the coefficient system assigned to a representation $V$ assigns to an edge $\tau$ the invariants in $V$ under the pro-$p$-Iwahori subgroup corresponding to $\tau$. The proof relies on computations of the group cohomology of a compact open subgroup group $N_0$ of the unipotent radical of a Borel subgroup

Integral structures in automorphic line bundles on the pp-adic upper half plane

Given an automorphic line bundle ${\mathcal O}_X(k)$ of weight $k$ on the Drinfel'd upper half plane $X$ over a local field $K$, we construct a ${\rm GL}_2(K)$-equivariant integral lattice ${\mathcal O}_{\widehat{\mathfrak X}}(k)$ in ${\mathcal O}_X(k)\otimes_K\widehat{K}$, as a coherent sheaf on the formal model $\widehat{\mathfrak{X}}$ underlying $X\otimes_K\widehat{K}$. Here $\widehat{K}/K$ is ramified of degree $2$. This generalizes a construction of Teitelbaum from the case of even weight $k$ to arbitrary integer weight $k$. We compute $H^*(\widetilde{\mathfrak{X}},{\mathcal O}_{\widehat{\mathfrak X}}(k))$ and obtain applications to the de Rham cohomology $H_{dR}^1(\Gamma\backslash X,{\rm Sym}_K^k({\rm St}))$ with coefficients in the $k$-th symmetric power of the standard representation of ${\rm SL}_2(K)$ (where $k\ge0$) of projective curves $\Gamma\backslash X$ uniformized by $X$: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing $H_{dR}^1(\Gamma\backslash X,{\rm Sym}_K^k({\rm St}))$, we re-prove the Hodge decomposition of $H_{dR}^1(\Gamma\backslash X,{\rm Sym}_K^k({\rm St}))$ and show that the monodromy operator on $H_{dR}^1(\Gamma\backslash X,{\rm Sym}_K^k({\rm St}))$ respects integral de Rham structures and is induced by a "universal"{} monodromy operator defined on $\widehat{\mathfrak{X}}$, i.e. before passing to the $\Gamma$-quotient

Finiteness of de Rham cohomology in rigid analysis

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

Equivariant crystalline cohomology and base change

Given a perfect field $k$ of characteristic $p>0$, a smooth proper $k$-scheme $Y$, a crystal $E$ on $Y$ relative to $W(k)$ and a finite group $G$ acting on $Y$ and $E$, we show that, viewed as virtual $k[G]$-module, the reduction modulo $p$ of the crystalline cohomology of $E$ is the de Rham cohomology of $E$ modulo $p$. On the way we prove a base change theorem for the virtual $G$-representions associated with $G$-equivariant objects in the derived category of $W(k)$-modules

Frobenius and monodromy operators in rigid analysis, and Drinfel'd's symmetric space

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

Locally unitary principal series representations of GLd+1(F){\rm GL}_{d+1}(F)

For a local field $F$ we consider tamely ramified principal series representations $V$ of $G={\rm GL}_{d+1}(F)$ with coefficients in a finite extension $K$ of ${\mathbb Q}_p$. Let $I_0$ be a pro-$p$-Iwahori subgroup in $G$, let ${\mathcal H}(G,I_0)$ denote the corresponding pro-$p$-Iwahori Hecke algebra. If $V$ is locally unitary, i.e. if the ${\mathcal H}(G,I_0)$-module $V^{I_0}$ admits an integral structure, then such an integral structure can be chosen in a particularly well organized manner, in particular its modular reduction can be made completely explicit

Acyclic coefficient systems on buildings

For cohomological (resp. homological) coefficient systems ${\mathcal F}$ (resp. ${\mathcal V}$) on affine buildings $X$ with Coxeter data of type $\widetilde{A}_d$ we give for any $k\ge1$ a sufficient local criterion which implies $H^k(X,{\mathcal F})=0$ (resp. $H_k(X,{\mathcal V})=0)$. Using this criterion we prove a conjecture of de Shalit on the acyclicity of coefficient systems attached to hyperplane arrangements on the Bruhat-Tits building of the general linear group over a local field. We also generalize an acyclicity theorem of Schneider and Stuhler on coefficient systems attached to representations

Compactifications of Log Morphisms

We introduce the notion of a relative log scheme with boundary: a morphism of log schemes together with a (log schematically) dense open immersion of its source into a third log scheme. The sheaf of relative log differentials naturally extends to this compactification and there is a notion of smoothness for such data. We indicate how this weak sort of compactification may be used to develop useful de Rham and crystalline cohomology theories for semistable log schemes over the log point over a field which are not necessarily proper

De Rham cohomology of rigid spaces

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

Locally algebraic automorphisms of the PGL2(F){\rm PGL}_2(F)-tree and o{\mathfrak o}-torsion representations

For a local field $F$ and an Artinian local coefficient ring $\Lambda$ with the same positive residue characteristic $p$ we define, for any $e\in{\mathbb N}$, a category ${\mathfrak C}^{(e)}(\Lambda)$ of ${\rm GL}_2(F)$-equivariant coefficient systems on the Bruhat-Tits tree $X$ of ${\rm PGL}_2(F)$. There is an obvious functor from the category of ${\rm GL}_2(F)$-representations over $\Lambda$ to ${\mathfrak C}^{(e)}(\Lambda)$. If $F={\mathbb Q}_p$ then ${\mathfrak C}^{(1)}(\Lambda)$ is equivalent to the category of smooth ${\rm GL}_2({\mathbb Q}_p)$-representations over $\Lambda$ generated by their invariants under a pro-$p$-Iwahori subgroup. For general $F$ and $e$ we show that the subcategory of all objects in ${\mathfrak C}^{(e)}(\Lambda)$ with trivial central character is equivalent to a category of representations of a certain subgroup of ${\rm Aut}(X)$ consisting of "locally algebraic automorphisms of level $e$". For $e=1$ there is a functor from this category to that of modules over the (usual) pro-$p$-Iwahori Hecke algebra; it is a bijection between irreducible objects. Finally, we present a parallel of Colmez' functor $V\mapsto {\bf D}(V)$: to objects in ${\mathfrak C}^{(e)}(\Lambda)$ (for any $F$) we assign certain \'{e}tale $(\varphi,\Gamma)$-modules over an Iwasawa algebra ${\mathfrak o}[[\widehat{N}^{(1)}_{0,1}]]$ which contains the (usually considered) Iwasawa algebra ${\mathfrak o}[[{N}_{0}]]$. This assignment preserves finite generation