arXiv subjects

Elmar Grosse-Klönne

Publications and source records attributed to Elmar Grosse-Klönne.

At least 19 recordsLinked to original sources

arXiv2017

From pro-$p$ Iwahori-Hecke modules to $(φ,Γ)$-modules, II

Let ${\mathfrak o}$ be the ring of integers in a finite extension field of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let ${\mathcal H}(G,I_0)$ be its pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. In \cite{dfun} we introduced a general principle how to assign to a certain additionally chosen datum $(C^{(\bullet)},ϕ,τ)$ an exact functor $M\mapsto{\bf D}(Θ_*{\mathcal V}_M)$ from finite length ${\mathcal H}(G,I_0)$-modules to $(φ^r,Γ)$-modules. In the present paper we concretely work out such data $(C^{(\bullet)},ϕ,τ)$ for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular ${\mathcal H}(G,I_0)\otimes_{\mathfrak o}k$-modules with the set of $(φ^r,Γ)$-modules satisfying a certain symmetry condition.

math.NT↗

arXiv2015

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

For a local field $F$ and an Artinian local coefficient ring $Λ$ with the same positive residue characteristic $p$ we define, for any $e\in{\mathbb N}$, a category ${\mathfrak C}^{(e)}(Λ)$ 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 $Λ$ to ${\mathfrak C}^{(e)}(Λ)$. If $F={\mathbb Q}_p$ then ${\mathfrak C}^{(1)}(Λ)$ is equivalent to the category of smooth ${\rm GL}_2({\mathbb Q}_p)$-representations over $Λ$ generated by their invariants under a pro-$p$-Iwahori subgroup. For general $F$ and $e$ we show that the subcategory of all objects in ${\mathfrak C}^{(e)}(Λ)$ 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 ${\mathfrak C}^{(e)}(Λ)$ (for any $F$) we assign certain étale $(φ,Γ)$-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.

math.NT↗

arXiv2015

From pro-$p$ Iwahori-Hecke modules to $(φ,Γ)$-modules I

Let ${\mathfrak o}$ be the ring of integers in a finite extension $K$ of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let $T$ be a maximal split torus in $G$. Let ${\mathcal H}(G,I_0)$ be the pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. Given a semiinfinite reduced chamber gallery (alcove walk) $C^{({\bullet})}$ in the $T$-stable apartment, a period $ϕ\in N(T)$ of $C^{({\bullet})}$ of length $r$ and a homomorphism $τ:{\mathbb Z}_p^{\times}\to T$ compatible with $ϕ$, we construct a functor from the category ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ of finite length ${\mathcal H}(G,I_0)$-modules to étale $(φ^r,Γ)$-modules over Fontaine's ring ${\mathcal O}_{\mathcal E}$. If $G={\rm GL}_{d+1}({\mathbb Q}_p)$ there are essentially two choices of ($C^{({\bullet})}$, $ϕ$, $τ$) with $r=1$, both leading to a functor from ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ to étale $(φ,Γ)$-modules and hence to ${\rm Gal}_{{\mathbb Q}_p}$-representations. Both induce a bijection between the set of absolutely simple supersingular ${\mathcal H}(G,I_0)\otimes_{\mathfrak o} k$-modules of dimension $d+1$ and the set of irreducible representations of ${\rm Gal}_{{\mathbb Q}_p}$ over $k$ of dimension $d+1$. 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 ${\mathfrak o}$-torsion ${\rm GL}_{2}({\mathbb Q}_p)$-representations generated by their pro-$p$-Iwahori invariants)

math.NT↗

arXiv2014

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.

math.AG↗

arXiv2014

Rigid analytic spaces with overconvergent structure sheaf

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.

math.AG↗

arXiv2014

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.

math.AG↗

arXiv2014

On Families of Pure Slope $L$-Functions

Let $R$ be the ring of integers in a finite extension $K$ of $\mathbb{Q}_p$, let $k$ be its residue field and let $χ:π_1(X)\to R^{\times}=GL_{1}(R)$ be a "geometric" rank one representation of the arithmetic fundamental group of a smooth affine $k$-scheme $X$. We show that the locally $K$-analytic characters $κ:R^{\times}\to\mathbb{C}_p^{\times}$ are the $\mathbb{C}_p$-valued points of a $K$-rigid space ${\cal W}$ and that $$L(κ\circχ,T)=\prod_{\overline{x}\in X}\frac{1}{1-(κ\circχ)(Frob_{\overline{x}})T^{°(\overline{x})}},$$viewed as a two variable function in $T$ and $κ$, is meromorphic on $\mathbb{A}_{\mathbb{C}_p}^1\times{\cal W}$. On the way we prove, based on a construction of Wan, a slope decomposition for ordinary overconvergent (finite rank) $σ$-modules, in the Grothendieck group of nuclear $σ$-modules.

math.NT↗

arXiv2014

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.

math.AG↗

arXiv2014

Integral structures in automorphic line bundles on the $p$-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 $H_{dR}^1(Γ\backslash X,{\rm Sym}_K^k({\rm St}))$ with coefficients in the $k$-th symmetric power of the standard representation of ${\rm SL}_2(K)$ (where $k\ge0$) of projective curves $Γ\backslash X$ uniformized by $X$: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing $H_{dR}^1(Γ\backslash X,{\rm Sym}_K^k({\rm St}))$, we re-prove the Hodge decomposition of $H_{dR}^1(Γ\backslash X,{\rm Sym}_K^k({\rm St}))$ and show that the monodromy operator on $H_{dR}^1(Γ\backslash X,{\rm Sym}_K^k({\rm St}))$ respects integral de Rham structures and is induced by a "universal"{} monodromy operator defined on $\widehat{\mathfrak{X}}$, i.e. before passing to the $Γ$-quotient.

math.NT↗

arXiv2014

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.

math.AG↗

arXiv2014

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.

math.RT↗

arXiv2014

Integral structures in the $p$-adic holomorphic discrete series

For a local non-Archimedean field $K$ we construct ${\rm GL}_{d+1}(K)$-equivariant coherent sheaves ${\mathcal V}_{{\mathcal O}_K}$ on the formal ${\mathcal O}_K$-scheme ${\mathfrak X}$ underlying the symmetric space $X$ over $K$ of dimension $d$. These ${\mathcal V}_{{\mathcal O}_K}$ are ${\mathcal O}_K$-lattices in (the sheaf version of) the holomorphic discrete series representations (in $K$-vector spaces) of ${\rm GL}_{d+1}(K)$ as defined by P. Schneider \cite{schn}. We prove that the cohomology $H^t({\mathfrak X},{\mathcal V}_{{\mathcal O}_K})$ vanishes for $t>0$, for ${\mathcal V}_{{\mathcal O}_K}$ in a certain subclass. The proof is related to the other main topic of this paper: over a finite field $k$, the study of the cohomology of vector bundles on the natural normal crossings compactification $Y$ of the Deligne-Lusztig variety $Y^0$ for ${\rm GL}_{d+1}/k$ (so $Y^0$ is the open subscheme of ${\mathbb P}_k^d$ obtained by deleting all its $k$-rational hyperplanes).

math.RT↗

arXiv2014

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.

math.RT↗

arXiv2014

The Cech filtration and monodromy in log crystalline cohomology

For a strictly semistable log scheme $Y$ over a perfect field $k$ of characteristic $p$ we investigate the canonical Čech spectral sequence $(C)_T$ abutting to the Hyodo-Kato (log crystalline) cohomology $H_{crys}^*(Y/T)_{\mathbb{Q}}$ of $Y$ and beginning with the log convergent cohomology of its various component intersections $Y^i$. We compare the filtration on $H_{crys}^*(Y/T)_{\mathbb{Q}}$ arising from $(C)_T$ with the monodromy operator $N$ on $H_{crys}^*(Y/T)_{\mathbb{Q}}$. We also express $N$ through residue maps and study relations with singular cohomology. If $Y$ lifts to a proper strictly semistable (formal) scheme $X$ over a finite totally ramified extension of $W(k)$, with generic fibre $X_K$, we obtain results on how the simplicial structure of $X_K^{an}$ (as analytic space) is reflected in $H_{dR}^*(X_K)=H_{dR}^*(X_K^{an})$.

math.AG↗

arXiv2014

On the crystalline cohomology of Deligne-Lusztig varieties

Let $X\to Y^0$ be an abelian prime-to-$p$ Galois covering of smooth schemes over a perfect field $k$ of characteristic $p>0$. Let $Y$ be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on $Y$. We describe a logarithmic $F$-crystal on $Y$ whose rational crystalline cohomology is the rigid cohomology of $X$, in particular provides a natural $W[F]$-lattice inside the latter; here $W$ is the Witt vector ring of $k$. If a finite group $G$ acts compatibly on $X$, $Y^0$ and $Y$ then our construction is $G$-equivariant. As an example we apply it to Deligne-Lusztig varieties. For a finite field $k$, if ${\mathbb G}$ is a connected reductive algebraic group defined over $k$ and ${\mathbb L}$ a $k$-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant $W[F]$-lattice in the cohomology ($\ell$-adic or rigid) of the corresponding Deligne-Lusztig variety and an expression of its reduction modulo $p$ in terms of equivariant Hodge cohomology groups.

math.RT↗

arXiv2014

Sheaves of bounded $p$-adic logarithmic differential forms

Let $K$ be a local field, $X$ the Drinfel'd symmetric space $X$ of dimension $d$ over $K$ and ${\mathfrak X}$ the natural formal ${\mathcal O}_K$-scheme underlying $X$; thus $G={\rm GL}\sb {d+1}(K)$ acts on $X$ and ${\mathfrak X}$. Given a $K$-rational $G$-representation $M$ we construct a $G$-equivariant subsheaf ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$ of ${\mathcal O}_K$-lattices in the constant sheaf $M$ on ${\mathfrak X}$. We study the cohomology of sheaves of logarithmic differential forms on $X$ (or ${\mathfrak X}$) with coefficients in ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$. In the second part we give general criteria for two conjectures of P. Schneider on $p$-adic Hodge decompositions of the cohomology of $p$-adic local systems on projective varieties uniformized by $X$. Applying the results of the first part we prove the conjectures in certain cases.

math.AG↗

arXiv2014

On the $p$-adic cohomology of some $p$-adically uniformized varieties

Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's symmetric space of dimension $d$ over $K$. Let $Γ\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let ${X}_Γ=Γ\backslash {X}$, a smooth projective ${K}$-variety. In this paper we investigate the de Rham and log crystalline (log convergent) cohomology of local systems on $X_Γ$ arising from $K[Γ]$-modules. (I) We prove the monodromy weight conjecture in this context. To do so we work out, for a general strictly semistable proper scheme of pure relative dimension $d$ over a cdvr of mixed characteristic, a rigid analytic description of the $d$-fold iterate of the monodromy operator acting on de Rham cohomology. (II) In cases of arithmetical interest we prove the (weak) admissibility of this cohomology (as a filtered $(ϕ,N)$-module) and the degeneration of the relevant Hodge spectral sequence.

math.AG↗

arXiv2014

$p$-torsion coefficient systems for ${\rm SL}_2({\bf Q}_p)$ and ${\rm GL}_2({\bf Q}_p)$,

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 $τ$ the invariants in $V$ under the pro-$p$-Iwahori subgroup corresponding to $τ$. 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.

math.RT↗