On Non-Archimedean Uniformization

Mustafin, G A

doi:10.1070/rm1978v033n01abeh002251

Cited by 23 publications

(37 citation statements)

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“…Passing to a subgroup with finite index in Γ we may and will assume that Γ is torsionfree and that the quotient Q Γ = Γ\Q has strictly semistable reduction. By [25] it even algebraizes to a projective A-scheme, and the covering map Q → Q Γ isétale. Let X Γ = Γ\X be the generic fibre, let Q Γ = Γ\Q be the special fibre of Q Γ .…”

Section: Hodge Decomposition On Quotients Ofmentioning

confidence: 99%

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

Grosse-Klönne¹

2005

J. Algebraic Geom.

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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].

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Section: Hodge Decomposition On Quotients Ofmentioning

confidence: 99%

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

Grosse-Klönne¹

2005

J. Algebraic Geom.

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“…By [M,Theorem 2.2], if Y is convex, then p is semistable (i.e., S Y is nonsingular) and S Y has smooth irreducible components and normal crossings. Following [F], we refer to S Y as the Deligne scheme.…”

Section: Definition 111mentioning

confidence: 99%

Geometry of Chow quotients of Grassmannians

Keel

Tevelev

2006

Duke Math. J.

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“…In the second part of this paper (sections 5 and 6) we develop general criteria for conjectures of Schneider raised in [16]. Let Γ ⊂ SL d+1 (K) be a cocompact discrete (torsionfree) subgroup; thus the quotient X Γ = Γ\X of X is a projective K-scheme [14]. Let M be a K[Γ]-module with dim K (M ) < ∞.…”

Section: Introductionmentioning

confidence: 99%

Sheaves of bounded p-adic logarithmic differential forms

Grosse-Klönne

2007

Annales Scientifiques de l’École Normale Supérieure

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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...

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On Non-Archimedean Uniformization

Cited by 23 publications

References 1 publication

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

Geometry of Chow quotients of Grassmannians

Sheaves of bounded p-adic logarithmic differential forms

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