Sheaves of bounded p-adic logarithmic differential forms

Abstract

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...

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

“…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.…”

“…In particular this means that the inclusions F r M −λ j +j → F r M −λ j−1 +j are quasiisomorphisms. Moreover, as explained in [25] (and recalled in [13]) it follows from 1.1 that there are differential operators…”

“…f s M as defined here is gr λ 0 −r M +s M resp. f λ 0 −r M +s M as defined in [13]. We have f r M +1 M = 0 and f r M −λ 0 M = M .…”

Integral $p$-adic étale cohomology of Drinfeld symmetric spaces

“…(Grosse-Klönne, [13,Th. 4.5], [15,Prop. 4.5]) For i > 0, j ≥ 0, we have H i ét (X, Ω j X ) = 0 and d = 0 on H 0 ét (X, Ω j X ).…”

Homological vanishing theorems for locally analytic representations

“…Generalizing the integral structures of Iovita and Spieß to arbitrary coefficients, Große-Klönne proved the conjecture for arbitrary d when M is the restriction of the regular representation of GL d+1 (L) or its dual (cf. [16]). Finally, in [17], Große-Klönne used global methods to prove part (ii) of Conjecture 4.1 under the assumption that Γ is of arithmetic type in the sense of [17], §4.…”