Logarithmic differential forms on p-adic symmetric spaces

Iovită, Adrian; Spiess, Michael

doi:10.1215/s0012-7094-01-11023-5

Cited by 19 publications

(26 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Hdg ) j≥0 and (F r Γ ) r≥0 are opposite to each other whenever F admits an integral lattice has been shown earlier by Iovita and Spiess [18] by completely different methods, and yet another proof is due to Alon and de Shalit.…”

Section: The Hodge Filtration (Fmentioning

confidence: 83%

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

Grosse-Klönne¹

2005

J. Algebraic Geom.

View full text Add to dashboard Cite

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].

show abstract

Section: The Hodge Filtration (Fmentioning

confidence: 83%

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

Grosse-Klönne¹

2005

J. Algebraic Geom.

View full text Add to dashboard Cite

show abstract

“…Iovita and Spiess [15] first proved all these conjectures for the trivial G-representation M = K (and there is another proof by Alon and de Shalit), later we proved it furthermore for the standard representation M = K d+1 of G and its dual [13]. In [11] we showed that if Φ is as in (a)(v) then the filtration F • Γ is just the slope (resp.…”

Section: Introductionmentioning

confidence: 63%

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

Grosse-Klönne

2011

J. Algebraic Geom.

View full text Add to dashboard Cite

Let K be a finite extension of Q p and let X be Drinfel d's symmetric space of dimension d over K. Let Γ ⊂ SL d+1 (K) be a cocompact discrete (torsionfree) subgroup and let X Γ = Γ\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. download from IP 130.74.92.202.License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf 152 E. GROSSE-KLÖNNE (a) Let X be the natural strictly semistable G-equivariant formal O Kscheme underlying X, let X Γ = Γ\X. Let M be a K[Γ]-module with dim K (M ) < ∞. The associated Γ-equivariant constant sheaf M on the rigid space X, respectively the formal scheme X (in its Zariski topology), descends to a locally constant sheaf M Γ on X Γ (or even a local system, in the ordinary topological sense, on the Berkovich analytic space associated with X Γ ), respectively on the formal scheme X Γ . (Thus M Γ does not mean the subspace of Γ-invariants, or its associated constant sheaf, of the abstractAs these complexes have interesting global cohomology only in middle degree d, our central object of study is the de Rham cohomology groupendowed with additional structure elements as follows.(i) There is a covering spectral sequence

show abstract

“…We may pass to the base field extension K →K. Then we have inclusions of sheaf complexes Remarks: (1) The decomposition (51) was proven for the trivial representation M = K for the first time by Iovita and Spiess [10]. Our present proof appears to provide a geometric underpinning of the one given in [10].…”

Section: Remarksmentioning

confidence: 70%

“…every element of H 0 (X, Ω s X ) ⊗ K, is in fact logarithmic, in particular it is closed. Thus H 0 (X, Ω s X ) ⊗ K must be the space of bounded logarithmic differential s-forms on X studied in [10] (if char(K) = 0).…”

Section: Coherent Cohomology Via Logarithmic Differential Formsmentioning

confidence: 99%

Sheaves of bounded p-adic logarithmic differential forms

Grosse-Klönne

2007

Annales Scientifiques de l’École Normale Supérieure

View full text Add to dashboard Cite

Let K be a local field, X the Drinfel'd symmetric space X of dimension d over K and X the natural formal O K -scheme underlying X; thus G = GL d+1 (K) acts on X and X. Given a K-rational G-representation M we construct a G-equivariant subsheaf M 0 OK of O K -lattices in the constant sheaf M on X. We study the cohomology of sheaves of logarithmic differential forms on X (or X) with coefficients in M 0 OK . 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. for their interest in this work. I am grateful to the referee for his very careful reading.1 finding and analysing equivariant subsheaves in M ⊗ Ω i X , e.g. those of exact, closed or logarithmic differential forms. All this motivates the main objetive of the first part of this paper, the study of equivariant integral structures in the vector bundles M ⊗ Ω i X and in their sub sheaves of logarithmic differential forms. The central question concerning the de Rham cohomology with coefficients in M of a projective quotient X Γ of X is that for the position of its Hodge filtration (e.g. due to p-adic Hodge theory its knowledge in case d = 1 is another crucial point in [2]); the second part of this paper is devoted to this question.We discuss the content in more detail. Let now more generally K be a non-Archimedean local field with ring of integers O K and residue field k. Let M be a rational representation of G = GL d+1 (K), i.e. a finite dimensional K-vector space M together with a morphism of K-group varieties GL d+1 → GL(M ). It is well known that for any compact open subgroup H of G there exists an H-stable free O K -module lattice in M ; we fix one such choice M 0 for H = GL d+1 (O K ). We choose a totally ramified extensionK of K of degree d + 1 and twist the action of G on M ⊗ KK by a suitableK-valued character of G. We show that the choice of M 0 determines for any other maximal open compact subgroup H ⊂ G a distinguished H-stable OK -lattice in M ⊗ KK and the collection of these lattices can be assembled into a G-equivariant coefficient system on the Bruhat-Tits building BT of PGL d+1 . In fact this is only a reinterpretation of our Proposition 3.1. We do not mention BT at all, we rather work with the G-equivariant semistable formal O K -scheme X underlying Drinfel'd's symmetric space X over K of dimension d + 1, as constructed in [14]. It is well known that the intersections of the irreducible components of X⊗ k are in natural bijection with the simplices of BT . Thus what we do is to construct from M 0 a constructible G-equivariant subsheaf M 0 OK of the constant sheaf with value M ⊗ KK on X such that M 0 OK (U ) for quasicompact open U ⊂ X is an OK -lattice in M ⊗ KK . We then consider the coherent O X ⊗ O K OK -module sheaf M 0 OK ⊗ O K O X and compute explicitly its reduction (M 0 OK ⊗ O K O X ) ⊗ OK k. See Theorem 3.3 for our res...

show abstract

Logarithmic differential forms on p-adic symmetric spaces

Cited by 19 publications

References 10 publications

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

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

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

Sheaves of bounded p-adic logarithmic differential forms

Contact Info

Product

Resources

About