On a Conjecture of Rapoport and Zink

Urs Hartl ^{∗}

January 23, 2013

Abstract

In their book, Rapoport and Zink constructed rigid analytic period spaces F^{wa} for Fontaine’s filtered isocrystals, and period morphisms from PEL moduli spaces of pdivisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism F^{a} → F^{wa} of rigid analytic spaces and of a universal local system of ℚ_{p}-vector spaces on F^{a}. Such a local system would give rise to a tower of étale covering spaces ĔK̃ of F^{a}, equipped with a Hecke-action, and an action of the automorphism group J(ℚ_{p}) of the isocrystal with extra structure.

For Hodge-Tate weights n−1 and n we construct in this article an intrinsic Berkovich open subspace F^{0} of F^{wa} and the universal local system on F^{0}. We show that only in exceptional cases F^{0} equals all of F^{wa} and when the Shimura group is GL_{n} we determine all these cases. We conjecture that the rigid-analytic space associated with F^{0} is the maximal possible F^{a}, and that F^{0} is connected. We give evidence for these conjectures. For those period spaces possessing PEL period morphisms, we show that F^{0} equals the image of the period morphism. Then our local system is the rational Tate module of the universal p-divisible group and carries a J(ℚ_{p})-linearization. We construct the tower ĔK̃ of étale covering spaces, and we show that it is canonically isomorphic in a Hecke and J(ℚ_{p})-equivariant way to the tower constructed by Rapoport and Zink using the universal p-divisible group. Mathematics Subject Classification (2000): 11S20, (11G18, 14L05, 14M15)

Contents

1 Introduction 2

2 The Conjecture of Rapoport and Zink 4

3 Fontaine’s Rings 10

4 φ-Modules 12

5 Constructing φ-Modules from Filtered Isocrystals 15

6 The Minuscule Case 19

7 Relation with Period Morphisms 26 8 PEL period morphisms 33

∗The author acknowledges support of the Deutsche Forschungsgemeinschaft in form of DFG-grant HA3006/2-1 and SFB 878.

9 When does weakly admissible imply admissible ? 41

10 Conjectures 48

A Berkovich’s Analytic Spaces 50

References 52

1 Introduction

The conjecture of Rapoport and Zink, we discuss in this article, is concerned with padic period spaces, the existence of local systems of ℚ_{p}-vector spaces on them and their resulting étale covering spaces. Rapoport and Zink fix a reductive group G over ℚ_{p}, an element b ∈ G(K_{0}) for K_{0} := W (𝔽_{p}^{alg})[\frac{1}{p}], and a conjugacy class \{μ\} of cocharacters of G. They consider the projective variety F̆ over a finite extension Ĕ/K_{0} parametrizing the cocharacters belonging to this conjugacy class. They show that the set of weakly admissible cocharacters is a rigid analytic subspace (F̆_{b}^{wa})^{rig} of F̆, which is called a p-adic period space; see [RZ96, Proposition 1.36]. On [RZ96, p. 29] they conjecture the existence of an étale morphism (F̆_{b}^{a})^{rig} → (F̆_{b}^{wa})^{rig} of rigid analytic spaces which is bijective on rigid analytic points, and the existence of a tensor functor \underline{V} from ℚ_{p}-rational representations ρ : G → GL(V) to local systems of ℚ_{p}-vector spaces on (F̆_{b}^{a})^{rig}, such that at every point μ of (F̆_{b}^{a})^{rig} the fiber of the local system \underline{V}(ρ) is the crystalline Galois representation, which by the Colmez-Fontaine Theorem [CF00] corresponds to the filtration on the isocrystal (V ⊗_{ℚ_{p}} K_{0} , ρ(b) · φ) given by ρ ∘ μ. Here φ is the p-Frobenius on K_{0}.

The interest in such a tensor functor comes from the fact that it defines a tower (ĔK̃)_{K̃⊂G̃(ℚ_{p})} of étale covering spaces of (F̆_{b}^{a})^{rig}, on which an inner form G̃(ℚ_{p}) of G acts through Hecke-correspondences, and the group J(ℚ_{p}) = {g ∈ G(K_{0}) : g^{−1}bφ(g) = b} acts horizontally; see Remark 2.7 for details. The first instance of such a tower was constructed by Drinfeld [Dri76]. There G is the group of units in the central division algebra over ℚ_{p} with Hasse invariant −\frac{1}{n}, the group J(ℚ_{p}) equals GL_{n}(ℚ_{p}), and the period space is the Drinfeld upper halfspace Ω which is the complement of all ℚ_{p}-rational hyperplanes in ℙ^{n−1}; see [RZ96, 1.44–1.46]. It has been shown that the ℓ-adic cohomology of Drinfeld’s tower realizes local Langlands and Jaquet-Langlands correspondences [Car90, Har97, HT01]. Inspired by this case, Rapoport [Rap95, Hope 4.2] asked whether such towers also exist over general p-adic period spaces. For those period spaces possessing PEL period morphisms, Rapoport and Zink [RZ96, 5.32–5.39] constructed a tower and conjectured that it consists of étale covering spaces of the image of the period morphism [RZ96, 1.37]. We prove this conjecture in Theorem 8.4 by determining the image of the period morphism, by constructing the tensor functor and the associated tower (ĔK̃)_{K̃⊂G̃(ℚ_{p})} of étale covering spaces over this image, and by showing that the latter tower coincides with the tower of [RZ96, 5.32– 5.39]. We show in Example 6.7 and Theorem 9.3 that the distinction between (F̆_{b}^{wa})^{rig} and (F̆_{b}^{a})^{rig} is crucial for this, because the image of the period morphism is smaller than (F̆_{b}^{wa})^{rig} except in a few low-dimensional cases. Therefore one should in general not expect that the tensor functor and the tower exists over all of (F̆_{b}^{wa})^{rig}. Let us give more details.

De Jong [dJ95] pointed out that to study local systems and étale covering spaces, it is best to work in the category of Berkovich’s Ĕ-analytic spaces rather than rigid analytic spaces. For convenience of the reader we review Berkovich’s definition and the relation with rigid analytic spaces in Appendix A. We show that in fact (F̆_{b}^{wa})^{rig} is the rigid analytic space associated with an open Ĕ-analytic subspace F̆_{b}^{wa} of F̆ (Proposition 10.1). In Section 2 we recall the definition of (F̆_{b}^{wa})^{rig} and the precise formulation of the conjecture. We slightly strengthen the conjecture and reformulate it in terms of an Ĕ-analytic space F̆_{b}^{a} in Conjecture 2.3. We also explain its interpretation in terms of the étale fundamental group, and we construct the tower (ĔK̃)_{K̃⊂G̃(ℚ_{p})} of étale covering spaces of F̆_{b}^{a} in Section 2.

We further contribute to the conjecture in the situation where G has a faithful representation ρ such that all Hodge-Tate weights of ρ ∘ μ are n − 1 and n for some integer n. Namely, we construct an intrinsic open Ĕ-analytic subspace F̆_{b}^{0} of F̆_{b}^{wa} (Theorem 6.6) and the universal local system on F̆_{b}^{0} (Theorem 6.8). We conjecture that F̆_{b}^{0} is connected (Conjecture 6.5) and equals the largest possible F̆_{b}^{a} as above (Conjecture 6.10). We provide some evidence for these conjectures in Section 10. The construction of F̆_{b}^{0} proceeds as follows. With any analytic point μ of F̆ (these are the ones of which Ĕ-analytic spaces consist; see Definition A.1) we associate in Section 5 a φ-module M_{μ} over the ring B̃_{rig}^{†} and we let F̆_{b}^{0} be the set of those μ for which M_{μ} is isoclinic of slope zero. This explicit condition is approachable by combinatorial properties of the Newton slopes of b as we exemplify by our computations in Example 6.7 and Theorem 9.3. The ring B̃_{rig}^{†}, which is a kind of maximal unramified extension of the Robba ring, is defined in Section 3, and the notion of φ-modules and their elementary properties are recalled in Section 4. Our construction is inspired by Berger’s [Ber08] construction which associates with any rigid analytic point μ ∈ F̆ (the ones whose residue field H(μ) is finite over Ĕ) a (φ, Γ)-module over the Robba ring. Due to the restriction to rigid analytic points, Berger’s approach works even without the above assumption on the Hodge-Tate weights. However, the techniques we use in Section 6, where we prove that F̆_{b}^{0} is an open Ĕ-analytic subspace of F̆, require in an essential way to work with Ĕ-analytic spaces. These also have points whose residue field is not finite over Ĕ. Moreover, they are topological spaces in the classical sense and we show that the complement F̆ ∖ F̆_{b}^{0} is the image of a compact set under a continuous map (Theorem 6.6). It is worth noting that this crude map is in fact only continuous and not a morphism of Ĕ-analytic or rigid analytic spaces.

For Hodge-Tate weights −1 and 0 Rapoport and Zink also study a period morphism π̆ : Ğ^{an} → F̆_{b}^{wa} starting at the Ĕ-analytic space Ğ^{an} associated with a Rapoport-Zink space Ğ, that is, a PEL moduli space of p-divisible groups in a fixed isogeny class. The problem to determine the image of such period morphisms was already mentioned by Grothendieck [Gro71]. By the Colmez-Fontaine Theorem [CF00] it contains every classical rigid analytic point. When G = GL(V), de Jong [dJ95, §§6 and 7] observed that the rational Tate module of the universal p-divisible group over Ğ^{an} descends to a local system V of ℚ_{p}-vector spaces on the image π̆(Ğ^{an}) of π̆; compare Theorem 7.3 where we also show that π̆(Ğ^{an}) = F̆_{b}^{0} and that Ğ^{an} is identified with the space of ℤ_{p}-lattices inside V. This uses a recent result of Faltings [Fal10], or alternatively a result of Scholze and Weinstein [SW13]. In the general PEL-situation we prove in Theorem 8.4 that π̆(Ğ^{an}) = F̆_{b}^{0} and that the rational Tate-module of the universal p-divisible group over Ğ^{an} induces a tensor functor \underline{V} from ℚ_{p}-rational representations of G to local systems of ℚ_{p}-vector spaces on F̆_{b}^{0} as in the conjecture of Rapoport and Zink. We conjecture that F̆_{b}^{0} is the largest open Ĕ-analytic subset on which such a tensor functor exists (Conjectures 7.6 and 8.6) and we give some evidence for this in Section 10. We further show in Theorems 7.3 and 8.4 that the tensor functor \underline{V} carries a canonical J(ℚ_{p})-linearization which gives rise to a horizontal J(ℚ_{p})-action on the associated tower (ĔK̃)_{K̃⊂G̃(ℚ_{p})} of étale covering spaces. And we prove

that (ĔK̃)_{K̃⊂G̃(ℚ_{p})} is canonically and J(ℚ_{p}) × G̃(ℚ_{p})-equivariantly isomorphic to the tower constructed by Rapoport and Zink [RZ96, 5.32–5.39] using the universal p-divisible group on Ğ^{an}.

Note that on the associated rigid analytic spaces, the inclusion π̆(Ğ^{an}) = F̆_{b}^{0} ⊂ F̆_{b}^{wa} induces an étale morphism π̆(Ğ^{an})^{rig} → (F̆_{b}^{wa})^{rig} which is bijective on classical rigid analytic points. It was noticed in [RZ96] and [dJ95] as a peculiarity that this étale morphism might nevertheless fail to be an isomorphism. We give a natural explanation of this phenomenon. Indeed, we show that in general π̆(Ğ^{an}) = F̆_{b}^{0} is strictly contained in F̆_{b}^{wa} (Example 6.7) and hence π̆(Ğ^{an})^{rig} → (F̆_{b}^{wa})^{rig} in general is not an isomorphism (Proposition 2.5). For G = GL_{n} we even compile a complete list of those b ∈ GL_{n}(K_{0}) for which π̆(Ğ^{an}) = F̆_{b}^{wa} in Theorem 9.3. The fact that the inclusion π̆(Ğ^{an}) ⊂ F̆_{b}^{wa} may be strict while it induces a bijection on rigid analytic points by the Colmez-Fontaine Theorem [CF00] again shows that one must take into considerations also the points of F̆_{b}^{wa} whose residue field is not finite over Ĕ. In other words one should work with Berkovich’s Ĕ-analytic spaces rather than rigid analytic spaces.

To end this introduction let us mention that some of the results presented here were announced in [Har08] and already summarized in [DOR10, §11.4]. The ideas in this article are inspired by our analogous theory in equal characteristic [Har11], or [Har09, §6.2], where we were able to prove the analog of the Rapoport-Zink Conjecture and the surjectivity of the period morphism onto F̆_{b}^{0} for arbitrary Hodge-Tate weights. In the beginning we had hoped to be able to extend our approach here also to Hodge-Tate weights other than n − 1 and n, but we did not succeed. Not even the construction of the φ-module M_{μ} over B̃_{rig}^{†}, let alone the openness result could be established along this line. The reason lies in the fact that for analytic points μ ∈ F̆ whose residue field H(μ) is not finite over Ĕ the ring B̃_{rig}^{†} is not a H(μ)-algebra (like Fontaine’s field B_{dR} is not a ℂ_{p}-algebra). So Berger’s construction could not be adapted. We also mention that Kedlaya [Ked10] has announced a strategy to overcome these difficulties and to extend our results to arbitrary Hodge-Tate weights; see also Remarks 10.2 and 2.4(a).

Acknowledgments. The author would like to thank the anonymous referee for his comments and explanations. They helped to improve the exposition. He also would like to thank F. Andreatta, L. Berger, V. Berkovich, O. Brinon, J.-M. Fontaine, A. Genestier, U. Görtz, M. Kisin, V. Lafforgue, J. Pottharst, M. Rapoport, E. Viehmann, and T. Zink for many helpful discussions and for their interest in this work.

2 The Conjecture of Rapoport and Zink

In this section we recall the construction of Rapoport’s and Zink’s p-adic period spaces. We strengthen their conjecture mentioned above and interpret it in terms of the étale fundamental group. We also explain its relation with towers of étale covering spaces. Let us first recall the definition of filtered isocrystals. We denote by 𝔽_{p}^{alg} an algebraic closure of the finite field with p elements and by K_{0} := W (𝔽_{p}^{alg})[\frac{1}{p}] the fraction field of the ring of Witt vectors over 𝔽_{p}^{alg}. Let φ = W (Frob_{p}) be the Frobenius lift on K_{0}.

Definition 2.1. An F -isocrystal over 𝔽_{p}^{alg} is a finite dimensional K_{0}-vector space D equipped with a φ-linear automorphism φ_{D}. If L is a field extension of K_{0} and F il^{•}D_{L} is an exhaustive separated decreasing filtration of D_{L} := D ⊗_{K_{0}} L by L-subspaces we say that \underline{D} = (D, φ_{D}, F il^{•}D_{L}) is a filtered isocrystal over L. The integers i for which F il^{−i}D_{L} ≠ F il^{−i+1}D_{L} are called the Hodge-Tate weights of \underline{D}. We let t_{N}(\underline{D}) be the p-adic valuation of det φ_{D} (with respect to any basis of D) and we let

t_{H}(\underline{D}) = ∑_{i∈ℤ} i · dim_{L} gr_{F il^{•}}^{i}(D_{L}) .

The filtered isocrystal \underline{D} is called weakly admissible^{1} if

t_{H}(\underline{D}) = t_{N}(\underline{D})  and  t_{H}(\underline{D}′) ≤ t_{N}(\underline{D}′)  (2.1)

for any subobject \underline{D}′ = (D′, φ_{D}|_{D′}, F il^{•}D′_{L}) of \underline{D}, where D′ is any φ_{D}-stable K_{0}-subspace of D which is equipped with the induced filtration F il^{i}D′_{L} = D′_{L} ∩F il^{i}D_{L} on D′_{L} := D′ ⊗_{K_{0}} L.

To construct period spaces let G be a reductive linear algebraic group over ℚ_{p}. Fix a conjugacy class {μ} of cocharacters

μ : 𝔾_{m} → G

defined over subfields of ℂ_{p} (the p-adic completion of an algebraic closure of K_{0}). Let E be the field of definition of the conjugacy class. It is a finite extension of ℚ_{p}. Two cocharacters in this conjugacy class are called equivalent if they induce the same weight filtration on the category \underline{Rep}_{ℚ_{p}}G of finite dimensional ℚ_{p}-rational representations of G. There is a projective variety F over E whose ℂ_{p}-valued points are in bijection with the equivalence classes of cocharacters (from the fixed conjugacy class {μ}). Namely let V in \underline{Rep}_{ℚ_{p}}G be any faithful representation of G. With a cocharacter μ defined over a field L one associates the filtration F il_{μ}^{i}V_{L} := ⨁_{j≥i} V_{L,j} of V_{L} := V ⊗_{ℚ_{p}} L given by the weight spaces

V_{L,j} := { v ∈ V_{L} : μ(z) · v = z^{j}v for all z ∈ 𝔾_{m}(L) } .

This defines a closed embedding of F into a partial flag variety of V

F ⊂⟶ Flag(V) ⊗_{ℚ_{p}} E ,  (2.2)

where the points of Flag(V) with value in a ℚ_{p}-algebra R are the filtrations F^{i} of V ⊗_{ℚ_{p}} R by R-submodules which are direct summands such that rk_{R} gr_{F^{•}}^{i} is the multiplicity of the weight i of the conjugacy class {μ} on V.

Now let b ∈ G(K_{0}). For any V in \underline{Rep}_{ℚ_{p}}G one obtains an F -isocrystal (V ⊗_{ℚ_{p}} K_{0} , b·φ). Its automorphism group contains the group from [RZ96, Proposition 1.12]

J(ℚ_{p}) = { g ∈ G(K_{0}) : g^{−1}bφ(g) = b } .  (2.3)

1This used to be the terminology until Colmez–Fontaine [CF00] showed that for L/K_{0} finite weakly admissible implies admissible. Since we consider also infinite extensions L/K_{0} for which the Colmez–Fontaine Theorem fails we stick to the old terminology.

A pair (b, μ) with an element b ∈ G(K_{0}) and a cocharacter μ : 𝔾_{m} → G defined over L ⊃ K_{0} is called weakly admissible if for some faithful representation ρ : G ↪ GL(V) in \underline{Rep}_{ℚ_{p}}G the filtered isocrystal

\underline{D}_{b,μ}(V) := (V ⊗_{ℚ_{p}} K_{0} , ρ(b)·φ , F il_{μ}^{•}V_{L})  (2.4)

is weakly admissible. In fact this then holds for any V in \underline{Rep}_{ℚ_{p}}G; see [RZ96, 1.18]. If (b, μ) is weakly admissible and L/K_{0} is finite then the filtered isocrystal \underline{D}_{b,μ}(V) is admissible, that is it arises from a crystalline Galois-representation Gal(L^{alg}/L) → GL(U) via Fontaine’s covariant functor

\underline{D}_{b,μ}(V) ≅ D_{cris}(U) := (U ⊗_{ℚ_{p}} B_{cris})^{Gal(L^{alg}/L)}

which by the Colmez-Fontaine theorem [CF00] is an equivalence of categories from crystalline representations of Gal(L^{alg}/L) to weakly admissible filtered isocrystals over L. We write V_{cris}(\underline{D}_{b,μ}(V)) := U for Fontaine’s covariant inverse functor. The assignment

\underline{Rep}_{ℚ_{p}}G ⟶ \underline{Rep}_{ℚ_{p}}(Gal(L^{alg}/L)) , V ↦ V_{cris}(\underline{D}_{b,μ}(V))  (2.5)

defines a tensor functor from \underline{Rep}_{ℚ_{p}}G to the category of continuous Gal(L^{alg}/L)-represen-tations in finite dimensional ℚ_{p}-vector spaces.

Let Ĕ = EK_{0} be the completion of the maximal unramified extension of E. In what follows we consider cocharacters μ defined over complete extensions L of Ĕ. Let F̆^{rig} be the rigid analytic space over Ĕ associated with the variety F̆ = F ⊗_{E} Ĕ. Rapoport and Zink define the p-adic period space associated with (G, b, {μ}) as

(F̆_{b}^{wa})^{rig} := { μ ∈ F̆^{rig} : (b, μ) is weakly admissible } .

It is acted on by the group J(ℚ_{p}) from (2.3) via μ ↦ γμγ^{−1} for γ ∈ J(ℚ_{p}). Rapoport and Zink show that (F̆_{b}^{wa})^{rig} is an admissible open rigid analytic subspace of F̆^{rig}; see [RZ96, Proposition 1.36]. For the proof they reduce to the case where b is decent with some positive integer s, that is

b · φ(b) · . . . · φ^{s−1}(b) = sν(p) .  (2.6)

where ν ∈ Hom_{K_{0}}(𝔾_{m}, G) ⊗_{ℤ} ℚ is the slope quasi-cocharacter [RZ96, 1.7] and s satisfies sν ∈ Hom_{K_{0}}(𝔾_{m}, G). If b is decent with integer s then (F̆_{b}^{wa})^{rig} has a natural structure of rigid analytic subspace of (F ⊗_{E} E_{s})^{rig} over the field E_{s} = E · W (𝔽_{p^{s}})[\frac{1}{p}] from which it arises by base change to Ĕ. In [RZ96, p. 29] Rapoport and Zink make a conjecture on the existence of a universal local system of ℚ_{p}-vector spaces on (F̆_{b}^{wa})^{rig} which we refine in 2.3 below.

Stimulated by [RZ96], the notion of local systems of ℚ_{p}-vector spaces on rigid analytic spaces was studied by de Jong [dJ95, §4] who pointed out that this is best done working with Berkovich’s Ĕ-analytic spaces rather than rigid analytic spaces. So let F̆^{an} be the Ĕ-analytic space associated with the variety F̆. See Appendix A for the notion of Ĕanalytic space and the relation with rigid analytic spaces. In fact the argument of [RZ96, Proposition 1.36] shows that there exists an open Ĕ-analytic subspace F̆_{b}^{wa} of F̆^{an} whose associated rigid analytic space is the period space (F̆_{b}^{wa})^{rig}; we explain the proof of this fact in Proposition 10.1.

Local systems of ℚ_{p}-vector spaces on an Ĕ-analytic space Y are related to the category \underline{Rep}_{ℚ_{p}}(π_{1}^{ét}(Y, ȳ)) of finite dimensional continuous ℚ_{p}-linear representations of the étale fundamental group π_{1}^{ét}(Y, ȳ) of Y as follows. This group was defined by de Jong [dJ95].

Proposition 2.2. ([dJ95, Theorem 4.2].) For any geometric base point ȳ of Y there is a natural ℚ_{p}-linear tensor functor

ωȳ : ℚ_{p}-\underline{Loc}_{Y} → \underline{Rep}_{ℚ_{p}}(π_{1}^{ét}(Y, ȳ))

which assigns to a local system V the π_{1}^{ét}(Y, ȳ)-representation Vȳ. It is an equivalence if Y is connected.

In particular, for any point y ∈ Y (L) and geometric base point ȳ lying above y, any local system V of ℚ_{p}-vector spaces on Y gives rise to a Galois representation

Gal(L^{alg}/L) = π_{1}^{ét}(y, ȳ) → π_{1}^{ét}(Y, ȳ) → GL(Vȳ) .  (2.7)

We denote this Galois representation by V_{y}. Now we propose the following refinement of the conjecture of Rapoport and Zink [RZ96, p. 29]; see also [DOR10, Conjecture 11.4.4].

Conjecture 2.3. There exists a unique largest arcwise connected dense open Ĕ-analytic subspace F̆_{b}^{a} ⊂ F̆_{b}^{wa} invariant under J(ℚ_{p}) with F̆_{b}^{a}(L) = F̆_{b}^{wa}(L) for all finite extensions L/Ĕ, and a tensor functor \underline{V} from \underline{Rep}_{ℚ_{p}}G to the category ℚ_{p}-\underline{Loc}F̆_{b}^{a} of local systems of

ℚ_{p}-vector spaces on F̆_{b}^{a} with the following property:

For any point μ ∈ F̆_{b}^{a}(L) with L/Ĕ finite, the tensor functor \underline{Rep}_{ℚ_{p}}G → \underline{Rep}_{ℚ_{p}}(Gal(L^{alg}/L)) , V ↦ V_{cris}(\underline{D}_{b,μ}(V)) from (2.5) is isomorphic to the tensor functor V ↦ \underline{V}(V)_{μ} from (2.7) which associates with a representation V ∈ \underline{Rep}_{ℚ_{p}}G the fiber at μ of the corresponding local system \underline{V}(V).

(2.8)

Remark 2.4. (a) There seem to be reasons to expect that the conjecture is false if μ is not minuscule; see [Sch12, SW13].

(b) Note that any Ĕ-analytic space is locally arcwise connected by [Beh90, Theorem 3.2.1]. Moreover, every open Ĕ-analytic subspace of F̆^{an} is paracompact by Lemma A.3.

(c) Forgetting the action of Gal(L^{alg}/L), Condition (2.8) implies that

for any point μ ∈ F̆_{b}^{a}(L) with L/Ĕ finite, the fiber functors \underline{Rep}_{ℚ_{p}}G → (ℚ_{p}-vector spaces) , V ↦ V_{cris}(\underline{D}_{b,μ}(V)) and V ↦ \underline{V}(V)_{μ} are isomorphic. (2.9)

But note that (2.9) for G = GL_{n} simply says that dim_{ℚ_{p}} \underline{V}(V) = dim_{ℚ_{p}} V and also in general (2.9) is weaker than Condition (2.8).

(d) In a preprint version of this article (Remark 7.2 in arXiv:math.NT/0605254v1) we stated (and [DOR10, Remark 11.4.5] quoted this) that the existence of a unique largest dense open Ĕ-analytic subspace F̆_{b}^{a} ⊂ F̆_{b}^{wa} satisfying (2.8) is automatic. We actually do not know whether this is correct. The problem is that on the same open subspace U ⊂ F̆_{b}^{wa} there might be two different tensor functors satisfying (2.8), because the groups Gal(H(x)^{alg}/H(x)) are not necessarily dense in π_{1}^{ét}(U, ȳ) when x ranges over all points of U, and even less when in addition H(x)/Ĕ is finite. Indeed, these groups lie in the kernel of π_{1}^{ét}(U, ȳ) → π_{1}^{top}(U, ȳ) which however has dense image; see [dJ95, Theorem 2.10(iv)]. Therefore the tensor functors on two different open subspaces do not automatically glue to a tensor functor on the union of the open subspaces.

To derive the original conjecture of Rapoport and Zink from Conjecture 2.3 we in addition note the following

Proposition 2.5. If Conjecture 2.3 holds, then the open immersion F̆_{b}^{a} ⊂ F̆_{b}^{wa} induces an étale morphism of rigid analytic spaces (F̆_{b}^{a})^{rig} → (F̆_{b}^{wa})^{rig} which is bijective on rigid analytic points. It is an isomorphism if and only if F̆_{b}^{a} = F̆_{b}^{wa}.

Proof. The functor ( )^{rig} takes étale morphisms to étale morphisms. The rest is a consequence of Theorem A.2 since F̆_{b}^{a} and F̆_{b}^{wa} are paracompact by Lemma A.3.

The nature of such a morphism (F̆_{b}^{a})^{rig} → (F̆_{b}^{wa})^{rig} may seem somewhat obscure since it is a non-quasi-compact refinement of the Grothendieck topology on the same underlying set of points F̆_{b}^{a}(Ĕ^{alg}) = F̆_{b}^{wa}(Ĕ^{alg}). It is less mysterious on the level of Ĕ-analytic spaces because there one can see the points in the complement F̆_{b}^{wa} ∖ F̆_{b}^{a}; compare Example 6.7 below.

After recalling some of Fontaine’s rings and some facts on φ-modules in the next two sections we will construct in Sections 5 and 6 for Hodge-Tate weights n − 1 and n an open Ĕ-analytic subspace F̆_{b}^{0} of F̆_{b}^{wa} which is a candidate for the space searched for in Conjecture 2.3.

Let us end this section by explaining the significance of Conjecture 2.3 in terms of the étale fundamental group and in terms of étale covering spaces. Let μ ∈ F̆_{b}^{a}(L) be a point with values in a finite extension L/Ĕ and let μ̄ be a geometric base point of F̆_{b}^{a} lying above μ. Then Rapoport and Zink [RZ96, 1.19] consider a cohomology class

cl(b, μ) = κ(b) − μ^{#} ∈ H^{1}(ℚ_{p}, G)

where κ(b) ∈ π_{1}(G)_{Gal(\overline{ℚ}_{p}/ℚ_{p})} is the Kottwitz point [Kot97, §7] and μ^{#} is the image of μ under the natural map X_{∗}(T) → π_{1}(G) → π_{1}(G)_{Gal(\overline{ℚ}_{p}/ℚ_{p})}. By the weak admissibility of the pair (b, μ), the difference κ(b) − μ^{#} lies in (π_{1}(G)_{Gal(\overline{ℚ}_{p}/ℚ_{p})})_{tors} which Kottwitz [Kot97, 3.2] identifies with H^{1}(ℚ_{p}, G).

Let ω_{0} be the forgetful fiber functor \underline{Rep}_{ℚ_{p}}G → (ℚ_{p}-vector spaces). The cohomology class cl(b, μ) defines a G-torsor over ℚ_{p} which by [DM82, Theorem 3.2] is of the form Isom^{⊗}(ω_{0}, ω̃) for a fiber functor ω̃ : \underline{Rep}_{ℚ_{p}}G → (ℚ_{p}-vector spaces). Consider the image of

cl(b, μ) in H^{1}(ℚ_{p}, G^{ad}) and let G̃ be the associated inner form of G. This yields a noncanonical isomorphism of ℚ_{p}-group schemes G̃ ≅ Aut^{⊗}(ω̃) by [DM82, Proposition 2.8] which we fix in the sequel.

Corollary 2.6. Let Y ⊂ F̆_{b}^{wa} be an open connected Ĕ-analytic subspace, and let μ̄ be a geometric base point of Y. Then the set of isomorphism classes of tensor functors \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}_{Y} satisfying (2.9) is in bijection with the set of isomorphism classes of continuous group homomorphisms

π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p})

where two homomorphisms are isomorphic if they differ by composition with an inner automorphism of the ℚ_{p}-group scheme G̃.

Proof. Let \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}_{Y} be a tensor functor and consider the fiber functor from \underline{Rep}_{ℚ_{p}}G to (ℚ_{p}-vector spaces) obtained as the composition ωμ̄ ∘ \underline{V} with the forgetful fiber functor ωμ̄ on \underline{Rep}_{ℚ_{p}}(π_{1}^{ét}(Y, μ̄)) from Proposition 2.2. By [dJ95, Theorem 2.9] the fiber functors ωμ̄ and ω_{}_{μ̄′} at any two geometric base points μ̄ and μ̄′ are isomorphic. So without changing the isomorphism class of ωμ̄ ∘ \underline{V} we may assume that μ̄ maps to a point μ ∈ Y (L) with values in a finite extension L/Ĕ. Then by a theorem of Wintenberger [Win97, Corollary to Proposition 4.5.3] (see also [CF00, Proposition on page 4]) Condition (2.9) at μ is equivalent to the assertion that the cohomology class of the G-torsor Isom^{⊗}(ω_{0}, ωμ̄ ∘ \underline{V}) equals cl(b, μ) ∈ H^{1}(ℚ_{p}, G). This yields a non-canonical isomorphism of fiber functors β_{0} : ω̃ ^{∼} ⟶ ωμ̄ ∘ \underline{V} by [DM82, Theorem 3.2], and a non-canonical isomorphism of ℚ_{p}- group schemes G̃ ≅ Aut^{⊗}(ω̃) ≅ Aut^{⊗}(ωμ̄ ∘ \underline{V}) which is uniquely determined only up to an inner automorphism of G̃. Any element γ ∈ π_{1}^{ét}(Y, μ̄) yields a tensor automorphism of ωμ̄ ∘ \underline{V}. This defines a group homomorphism f : π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p}) which is unique up to composition with an inner automorphism of the group scheme G̃. Since for all ρ ∈ \underline{Rep}_{ℚ_{p}}G

the induced homomorphism π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p}) → GL(ωμ̄ ∘ \underline{V}(ρ))(ℚ_{p}) is continuous, also f is continuous.

Conversely let f : π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p}) be a continuous group homomorphism. Then we define a tensor functor \underline{Rep}_{ℚ_{p}}G → \underline{Rep}_{ℚ_{p}}(π_{1}^{ét}(Y, μ̄)) by sending a representation ρ of G to the representation

ρ′ : π_{1}^{ét}(Y, μ̄) ⟶ GL(ω̃(ρ))(ℚ_{p}) , γ ↦ ω̃(ρ)(f(γ)) .

Here ω̃(ρ)(f(γ)) is the automorphism by which f(γ) ∈ G̃(ℚ_{p}) ≅ Aut^{⊗}(ω̃)(ℚ_{p}) acts on the ℚ_{p}-vector space ω̃(ρ). Note that ρ′ is continuous because G̃(ℚ_{p}) → GL(ω̃(ρ))(ℚ_{p}) is continuous. Let \underline{V}(ρ) := ω−1 ; μ̄(ρ′) be the local system of ℚ_{p}-vector spaces on Y induced from ρ′ via Proposition 2.2. This defines a tensor functor \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}_{Y}. The composition ωμ̄ ∘ \underline{V} with the forgetful fiber functor ωμ̄ equals ω̃. Since the cohomology class of Isom^{⊗}(ω_{0}, ω̃) equals cl(b, μ), Wintenberger’s result mentioned above shows that the tensor functor \underline{V} satisfies condition (2.9) at μ. Moreover, Wintenberger’s theorem implies that the isomorphism class of the fiber functor (2.5) is constant on Y. Also for any two geometric base points μ̄ and μ̄′ of Y the fiber functors ωμ̄ and ω_{}_{μ̄′} are isomorphic by [dJ95, Theorem 2.9] since Y is connected. Therefore the isomorphism class of the fiber functor ωμ̄ ∘ \underline{V} is likewise constant on Y and condition (2.9) holds at any point μ ∈ F̆_{b}^{a}(L) with L/Ĕ finite. Clearly the assignments \underline{V} ↦ f and f ↦ \underline{V} are inverse to each other.

Remark 2.7. (a) In the situation of Corollary 2.6 any tensor functor \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}_{Y} induces a tower of étale covering spaces of Y with Hecke action. More precisely, let K̃ ⊂ G̃(ℚ_{p}) be a compact open subgroup and let ĔK̃ be the space representing K̃-level structures on \underline{V}, that is residue classes modulo K̃ of tensor isomorphisms

(β : ω̃ ^{∼} ⟶ ωμ̄ ∘ \underline{V}) mod K̃ ∈ Isom^{⊗}(ω̃, ωμ̄ ∘ \underline{V})/K̃

such that the class βK̃ is invariant under the étale fundamental group of ĔK̃. Then ĔK̃ is the étale covering space of Y corresponding to the discrete π_{1}^{ét}(Y, μ̄)-set Isom^{⊗}(ω̃, ωμ̄ ∘ \underline{V})/K̃. Any choice of a fixed tensor isomorphism β_{0} ∈ Isom^{⊗}(ω̃, ωμ̄ ∘ \underline{V}) associates with \underline{V} a representation π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p}) as in the proof of Corollary 2.6, and induces an identification of the π_{1}^{ét}(Y, μ̄)-sets Isom^{⊗}(ω̃, ωμ̄ ∘ \underline{V})/K̃ and G̃(ℚ_{p})/K̃.

On the tower (ĔK̃)_{K̃⊂G̃(ℚ_{p})} the group G̃(ℚ_{p}) acts via Hecke correspondences: Let g ∈

G̃(ℚ_{p}) and let K̃, K̃′ ⊂ G̃(ℚ_{p}) be compact open subgroups. Then the Hecke correspondence π(g)_{K̃,K̃′} is given by the diagram

\xymatrix{ & Ĕ_{K̃′∩g^{−1}K̃g} \ar[dl] \ar[dr] &  &  & β mod (K̃′ ∩ g^{−1}K̃g) \ar@{|->}[dl] \ar[dr] \\ ĔK̃′ &  & ĔK̃ \ar@{-->}[ll] & β mod K̃′ &  & β g^{−1} mod K̃}  (2.10)

The whole construction does not depend on the choice of the base point μ̄ by [dJ95, Theorem 2.9] and hence also applies if Y is not connected.

(b) Assume moreover, that a group Γ acts on Y and let \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}_{Y} be a tensor functor which carries a Γ-linearization, that is, for every γ ∈ Γ an isomorphism φ_{γ} : γ^{∗}\underline{V} ^{∼} ⟶ \underline{V} of tensor functors (where γ^{∗}\underline{V} is the pullback of \underline{V} under the morphism γ : Y → Y), satisfying the cocycle condition φ_{γ} ∘ γ^{∗}φ_{δ} = φ_{δγ} for all γ, δ ∈ Γ.

Then the tower of étale covering spaces ĔK̃ inherits an action of Γ over Y as follows. Recall that ĔK̃ corresponds to the discrete π_{1}^{ét}(Y, μ̄)-set Isom^{⊗}(ω̃, ωμ̄ ∘ \underline{V})/K̃. By functoriality of the fundamental group, γ^{∗}ĔK̃ := ĔK̃ ×_{Y,γ} Y corresponds to the discrete π_{1}^{ét}(Y, γ^{−1}μ̄)-set Isom^{⊗}(ω̃, ω_{γ^{−1}μ̄}∘γ^{∗}\underline{V})/K̃ which as a set equals Isom^{⊗}(ω̃, ωμ̄∘\underline{V})/K̃. By [dJ95, Theorem 2.9] there exists an isomorphism of fiber functors ψ_{γ} : ω_{γ^{−1}μ̄} ^{∼} ⟶ ωμ̄ providing a change of base point isomorphism

Isom^{⊗}(ω̃, ω_{γ^{−1}μ̄} ∘ γ^{∗}\underline{V})/K̃ ^{∼} ⟶ Isom^{⊗}(ω̃, ωμ̄ ∘ γ^{∗}\underline{V})/K̃ , β′K̃ ⟼ (ψ_{γ} ∘ γ^{∗}\underline{V}) ∘ β′K̃ .

This shows that the covering space γ^{∗}ĔK̃ of Y corresponds to the discrete π_{1}^{ét}(Y, μ̄)-set Isom^{⊗}(ω̃, ωμ̄ ∘ γ^{∗}\underline{V})/K̃. Then the isomorphism

Isom^{⊗}(ω̃, ωμ̄ ∘ \underline{V})/K̃ ^{∼} ⟶ Isom^{⊗}(ω̃, ωμ̄ ∘ γ^{∗}\underline{V})/K̃ , βK̃ ⟼ ωμ̄(φ_{γ}^{−1}) ∘ βK̃  (2.11)

of π_{1}^{ét}(Y, μ̄)-sets corresponds to an isomorphism ĔK̃ ^{∼} ⟶ γ^{∗}ĔK̃ of étale covering spaces of Y. We compose the latter with the projection onto ĔK̃ to obtain an isomorphism γ := γĔ_{K} of

ĔK̃ which makes the following diagram commutative

[figure: ĔK̃; γĔ_{K}; ĔK̃; Y; γ; Y .]

These isomorphisms for varying K̃ are compatible with the projection morphisms of the tower. Furthermore, from the explicit descriptions in (2.10) and (2.11) one sees that the action of Γ on the tower (ĔK̃)_{K̃⊂G̃(ℚ_{p})} commutes with the action of G̃(ℚ_{p}) through Hecke correspondences.

3 Fontaine’s Rings

We recall from Colmez [Col08] some of the rings used in p-adic Hodge theory [Fon79, Fon82, Fon94, FW79a, FW79b]. Let O_{L} be a complete valuation ring of rank one which is an extension of ℤ_{p} and let L be its fraction field. Let v_{p} be the valuation on O_{L} which we assume to be normalized so that v_{p}(p) = 1. In p-adic Hodge theory it is usually assumed that O_{L} is discretely valued with perfect residue field. However, we make neither of these assumptions here, because in Section 5 we need this more general situation. Let C be the completion of an algebraic closure L^{alg} of L and let O_{C} be the valuation ring of C. For n ∈ ℕ let ε^{(n)} be a primitive p^{n}-th root of unity chosen in such a way that (ε^{(n+1)})^{p} = ε^{(n)} for all n. We define

Ẽ^{+} := Ẽ^{+}(C) := { x = (x^{(n)})_{n∈ℕ_{0}} : x^{(n)} ∈ O_{C} , (x^{(n+1)})^{p} = x^{(n)} for all n }.

Fix the elements ε := (1, ε^{(1)}, ε^{(2)}, …) and \overline{π} := ε − 1 of Ẽ^{+}. With the multiplication xy := (x^{(n)}y^{(n)})_{n∈ℕ_{0}}, the addition x + y := (lim_{m→∞}(x^{(m+n)} + y^{(m+n)})^{p^{m}})_{n∈ℕ_{0}}, and the valuation v_{E}(x) := v_{p}(x^{(0)}), Ẽ^{+}(C) becomes a complete valuation ring of rank one with algebraically closed fraction field, called Ẽ := Ẽ(C), of characteristic p. Next we define

Ã^{+} := Ã^{+}(C) := W (Ẽ^{+}(C)) and

Ã := Ã(C) := W (Ẽ(C)) the rings of Witt vectors,

B̃^{+} := B̃^{+}(C) := Ã^{+}(C)[\frac{1}{p}] and

B̃ := B̃(C) := Ã(C)[\frac{1}{p}] the fraction field of Ã(C).

By Witt vector functoriality there is the Frobenius lift φ := W (Frob_{p}) on these four rings. For x ∈ Ẽ(C) we let [x] ∈ Ã(C) denote the Teichmüller lift. Set π := [ε] − 1. If x = ∑_{i=0}^{∞} p^{i}[x_{i}] ∈ Ã(C) then we set w_{k}(x) := min{ v_{E}(x_{i}) : i ≤ k }. For r > 0 let

Ã^{(0,r]} := Ã^{(0,r]}(C) := \{ x ∈ Ã(C) : lim_{k→+∞} w_{k}(x) + \frac{k}{r} = +∞ \},

B̃^{(0,r]} := B̃^{(0,r]}(C) := Ã^{(0,r]}(C)[\frac{1}{p}] ,

B̃^{†} := B̃^{†}(C) := ⋃_{r→0} B̃^{(0,r]}(C) .

One has Ã^{+}(C) ⊂ Ã^{(0,r]}(C) ⊂ Ã^{(0,s]}(C) for all r ≥ s > 0. On B̃^{(0,r]}(C) there is a valuation defined for x = ∑_{i≫−∞}^{∞} p^{i}[x_{i}] as

v^{(0,r]}(x) := min\{ w_{k}(x) + \frac{k}{r} : k ∈ ℤ \} = min\{ v_{E}(x_{i}) + \frac{i}{r} : i ∈ ℤ \}.

Now one defines B̃^{]0,r]}(C) as the Fréchet completion of B̃^{(0,r]}(C) with respect to the family of semi-valuations v^{[s,r]}(x) := min{ v^{(0,s]}(x), v^{(0,r]}(x) } for 0 < s ≤ r. This means in concrete terms that a sequence of elements x_{n} ∈ B̃^{(0,r]}(C) converges in B̃^{]0,r]}(C) if and only if lim_{n→∞} v^{[s,r]}(x_{n+1} − x_{n}) = +∞ for all 0 < s ≤ r. Also if r ≥ s we let B̃^{[s,r]}(C) be the completion of B̃^{(0,r]}(C) with respect to v^{[s,r]}. We view B̃^{]0,r]}(C) as a subring of B̃^{I}(C) for any closed subinterval I ⊂ (0, r]. Let

B̃_{rig}^{†} := B̃_{rig}^{†}(C) := ⋃_{r→0} B̃^{]0,r]}(C) .

4 φ-MODULES 12

The morphism φ extends to bicontinuous isomorphisms of rings B̃^{]0,pr]}(C) → B̃^{]0,r]}(C) and B̃^{[ps,pr]}(C) → B̃^{[s,r]}(C) defining an automorphism of B̃_{rig}^{†}(C). The rings B̃^{]0,r]}(C) and

B̃_{rig}^{†}(C) were studied in detail by Berger [Ber02, §2] who called the first B̃_{rig}^{†,(p−1)/pr} instead. Note that the ring B̃_{rig}^{†}(C) is denoted Γ_{an,con}^{alg} by Kedlaya [Ked05, Definition 2.4.8, Remark 2.4.13] and is an integral domain.

There is a homomorphism θ : B̃^{(0,1]}(C) → C sending ∑_{i≫−∞}^{∞} p^{i}[x_{i}] to ∑_{i≫−∞}^{∞} p^{i}x_{i}^{(0)} which extends by continuity to B̃^{]0,1]}(C) and even to B̃^{[1,1]}(C). The series

t := log[ε] = ∑_{n=1}^{∞} \frac{(−1)^{n−1}}{n} π^{n} ∈ B̃^{]0,1]}(C)

satisfies φ(t) = pt and t B̃^{[1,1]}(C) = ker(θ : B̃^{[1,1]}(C) → C). Let

B_{cris}^{+} := B_{cris}^{+}(C) be the p-adic completion of B̃^{+}(C)[\frac{w^{n}}{n!} : w ∈ ker θ, n ∈ ℕ] and

B_{cris} := B_{cris}(C) := B_{cris}^{+}(C)[\frac{1}{t}].

The morphism φ extends to endomorphisms of B_{cris}^{+}(C) and B_{cris}(C). Let

B̃_{rig}^{+} := B̃_{rig}^{+}(C) := ⋂_{n∈ℕ_{0}} φ^{n}B_{cris}^{+}(C) and

B̃_{rig} := B̃_{rig}(C) := B̃_{rig}^{+}[\frac{1}{t}] .

One easily sees that B̃_{rig}^{+}(C) ⊂ B̃^{]0,r]}(C) for any r > 0. More precisely, B̃^{]0,1]}(C) equals the p-adic completion of B̃_{rig}^{+}(C)[\frac{p}{[π̄]}] and hence is a flat B̃_{rig}^{+}(C)-algebra.

4 φ-Modules

Let L and C be as in the previous section. To shorten notation we will drop the denotation (C) from the rings introduced there. We recall some definitions and facts from Kedlaya [Ked05]. Let a be a positive integer.

Definition 4.1. A φ^{a}-module over B̃_{rig}^{†} is a finite free B̃_{rig}^{†}-module M with a bijective φ^{a}-semilinear map φ_{M} : M → M. The rank of M as a B̃_{rig}^{†}-module is denoted rk M. A morphism of φ^{a}-modules is a morphism of the underlying B̃_{rig}^{†}-modules which commutes with the φ_{M}’s. We denote the set of morphism between two φ^{a}-modules M and M′ by Hom_{φ^{a}}(M, M′).

Example 4.2. Let c, d ∈ ℤ with d > 0 and (c, d) = 1. Define the φ^{a}-module M(c, d) over B̃_{rig}^{†} as M(c, d) = ⨁_{i=1}^{d} B̃_{rig}^{†}e_{i} equipped with

φ_{M}(e_{1}) = e_{2} , …, φ_{M}(e_{d−1}) = e_{d} , φ_{M}(e_{d}) = p^{c}e_{1} .  (4.1)

By abuse of notation we also write M(nc, nd) := M(c, d)^{⊕n} for n ∈ ℕ_{>0}.

The category of φ^{a}-modules over B̃_{rig}^{†} is a rigid additive tensor category with unit object M(0, 1). For a positive integer b there is a restriction of Frobenius functor [b]_{∗} from φ^{a}- modules over B̃_{rig}^{†} to φ^{ab}-modules over B̃_{rig}^{†} sending (M, φ_{M}) to (M, φ_{M}^{b}). Kedlaya proved the following structure theorem.

Theorem 4.3. ([Ked05, Theorem 4.5.7]) Any φ^{a}-module M over B̃_{rig}^{†} is isomorphic to a direct sum of φ^{a}-modules M(c_{i}, d_{i}) for uniquely determined pairs (c_{i}, d_{i}) up to permutation with gcd(c_{i}, d_{i}) = 1. It satisfies ∧^{d}M ≅ M(c, 1) where c = ∑_{i} c_{i} and d = rk M = ∑_{i} d_{i}.

Definition 4.4. If det M := ∧^{rk M} M ≅ M(c, 1) we define the degree and the weight of M as deg M := c and wt M := \frac{deg M}{rk M}.

Proposition 4.5. ([Ked05, Lemma 3.4.9]) Every φ^{a}-submodule M′ ⊂ M(c, d)^{⊕n} over B̃_{rig}^{†} satisfies wt M′ ≥ wt M(c, d)^{⊕n} = \frac{c}{d}.

Let us record some facts about the M(c, d).

Lemma 4.6. ([Ked05, Lemmas 4.1.2 and 3.2.4]) The φ^{a}-modules M(c, d) over B̃_{rig}^{†} satisfy

(a) M(c, d) ⊗ M(c′, d′) ≅ M(cd′ + c′d, dd′),

(b) M(c, d)^{∨} ≅ M(−c, d) and

(c) [d]_{∗}M(c, d) ≅ M(c, 1)^{⊕d}.

Definition 4.7. For a φ^{a}-module M over B̃_{rig}^{†} we define the set of φ^{a}-invariants as

H_{φ^{a}}^{0}(M) := { x ∈ M : φ_{M}(x) = x } .

It is a vector space over H_{φ^{a}}^{0}(M(0, 1)) = ℚ_{p^{a}} := W (𝔽_{p^{a}})[p^{−1}]; use [Ked05, Proposition 3.3.4]. We have Hom_{φ^{a}}(M, M′) = H_{φ^{a}}^{0}(M^{∨} ⊗ M′) for φ^{a}-modules M and M′ over B̃_{rig}^{†}.

Proposition 4.8. If a ≥ c > 0 then the φ^{a}-module M(−c, 1) over B̃_{rig}^{†} satisfies

H_{φ^{a}}^{0}(M(−c, 1)) = { ∑_{ν∈ℤ} p^{cν} ∑_{j=0}^{c−1} p^{j}φ^{−aν}([x_{j}]) : x_{0}, …, x_{c−1} ∈ Ẽ , v_{E}(x_{j}) > 0 } .

Proof. One easily verifies that the series on the right converges (even in B̃^{]0,r]} for all r > 0) and is a φ^{a}-invariant of M(−c, 1).

Conversely let x ∈ B̃_{rig}^{†} satisfy x = p^{−c}φ^{a}(x). Then x ∈ B̃_{rig}^{+} by [Ber04, Proposition I.4.1] and hence possesses an expansion x = ∑_{j=−∞}^{∞} p^{j}[x_{j}] with uniquely determined x_{j} ∈ Ẽ satisfying v_{E}(x_{j}) > 0 by [Fou05, Lemme 3.9.17 and Corollaire 3.9.9]. The uniqueness implies φ^{a}([x_{j}]) = [x_{j−c}] and the proposition follows.

Let us make a remark on radii of convergence. Let M^{]0,r]} be a free B̃^{]0,r]}-module and for 0 < s ≤ r let M^{]0,s]} := M^{]0,r]} ⊗_{B̃^{]0,r]},ι} B̃^{]0,s]} be obtained by base change via the natural inclusion ι : B̃^{]0,r]} ↪ B̃^{]0,s]}. Let further

φ_{M}^{]0,rp^{−a}]} : M^{]0,r]} ⊗_{B̃^{]0,r]},φ^{a}} B̃^{]0,rp^{−a}]} ^{∼} ⟶ M^{]0,rp^{−a}]}

be an isomorphism of B̃^{]0,rp^{−a}]}-modules. We say that a φ^{a}-module (M, φ_{M}) over B̃_{rig}^{†} is

represented by the pair (M^{]0,r]}, φ_{M}^{]0,rp^{−a}]}) if (M, φ_{M}) = (M^{]0,r]} ⊗B̃^{]0,r]} B̃_{rig}^{†}, φ_{M}^{]0,rp^{−a}]} ⊗ id).

Proposition 4.9. If M is represented by (M^{]0,r]}, φ_{M}^{]0,rp^{−a}]}) then

H_{φ^{a}}^{0}(M) = { x ∈ M^{]0,r]} : φ_{M}^{]0,rp^{−a}]}(x ⊗_{φ^{a}} 1) = x ⊗_{ι} 1 } .

Proof. The inclusion “⊃” follows from the inclusion M^{]0,r]} ⊂ M.

To prove the opposite inclusion “⊂” let x ∈ H_{φ^{a}}^{0}(M). Since B̃_{rig}^{†} = ⋃_{s>0} B̃^{]0,s]} there exists a 0 < s ≤ r with x ∈ M^{]0,s]}. Choose a B̃^{]0,r]}-basis of M^{]0,r]} and write x with respect to this basis as a vector v = (v_{1}, …, v_{n})^{T} ∈ (B̃^{]0,s]})^{⊕n}. Let A ∈ GL_{n}(B̃^{]0,rp^{−a}]}) be the matrix by which the isomorphism φ_{M}^{]0,rp^{−a}]} acts on this basis. The equation x = φ_{M}(x) translates into A^{−1}v = (φ^{a}(v_{1}), …, φ^{a}(v_{n}))^{T}. Thus if s ≤ rp^{−a} we find φ^{a}(v_{i}) ∈ B̃^{]0,s]}, whence v_{i} ∈ B̃^{]0,sp^{a}]}. Continuing in this way we see that in fact v_{i} ∈ B̃^{]0,r]}, that is x ∈ M^{]0,r]}.

Proposition 4.10. Let c, d > 0 and let the φ-module M = M(−c, d) over B̃_{rig}^{†}(C) be represented by M^{]0,1]} = ⨁_{i=1}^{d} B̃^{]0,1]}e_{i} and φ^{]0,p^{−1}]} as in (4.1). Then θ : B̃^{]0,1]}(C) → C induces the following exact sequence of ℚ_{p}-vector spaces

[figure: 0; H_{φ}^{0}(M(d − c, d)); T_{−c,d}; H_{φ}^{0}(M(−c, d)); θ_{M}; ⨁_{i=1}^{d} C e_{i}; 0 ,; ∑_{i=1}^{d} x_{i}e_{i}; ∑_{i=1}^{d} p^{i−1}t x_{i}e_{i} , ∑_{i=1}^{d} y_{i}e_{i}; ∑_{i=1}^{d} θ(y_{i})e_{i} .]

In particular θ_{M} is an isomorphism for 0 < c < d.

Proof. The elements of H_{φ}^{0}(M(−c, d)) are of the form ∑_{i=1}^{d} φ^{i−1}(y)e_{i} for y = p^{−c}φ^{d}(y) and y ∈ B̃_{rig}^{+} ⊂ B̃_{max}^{+} ⊂ B̃_{rig}^{†} by [Ber04, Proposition I.4.1]. The map θ_{M} sends this element to ∑_{i=1}^{d} θ(φ^{i−1}(y))e_{i}. As {φ^{i−1} : 1 ≤ i ≤ d} = Gal(ℚ_{p^{d}}/ℚ_{p}), the sequence is exact on the right by [Col02, Lemme 8.16]. Since φ(t) = pt and B̃_{rig}^{†} is an integral domain the sequence is well defined and exact on the left. Finally, the exactness in the middle follows from t · B̃_{max}^{+} = { y ∈ B̃_{max}^{+} : θ(φ^{n}(y)) = 0 for all n ∈ N_{0} }; see [Col02, Proposition 8.10]. If 0 < c < d then H_{φ}^{0}(M(d − c, d)) = (0) by [Ked05, Proposition 4.1.3] and θ_{M} is an isomorphism.

Corollary 4.11. If A : M → M(c, d) is a homomorphism of φ-modules over B̃_{rig}^{†} represented by A^{]0,1]} : M^{]0,1]} → (B̃^{]0,1]})^{⊕d} with θ_{M(c,d)} ∘ A^{]0,1]}(M^{]0,1]}) = (0) then A factors as A = T ∘ B for a morphism B : M → M(c + d, d) where T = T_{c,d} : M(c + d, d) → M(c, d) is induced by multiplication with t as in Proposition 4.10.

Proof. By Theorem 4.3 and Lemma 4.6 we write M ≅ ∑_{i=1}^{r} M(c_{i}, d_{i}) and

Hom_{φ}(M, M(c, d)) ≅ ⨁_{i=1}^{r} H_{φ}^{0}(M(cd_{i} − c_{i}d, dd_{i})) .

Then the assertion follows from Proposition 4.10.

Proposition 4.12. The φ^{a}-modules M(c, d) over B̃_{rig}^{†} satisfy

(a) dim_{ℚ_{p^{a}}} H_{φ^{a}}^{0}(M(c, d)) = \{ \begin{matrix}0 & if c > 0 \\ 1 & if c = 0, d = 1 \\ ∞ & if c < 0 \end{matrix}

(b) dim_{ℚ_{p^{a}}} Hom_{φ^{a}}(M(c, d), M(c′, d′)) = \{ \begin{matrix}0 & if c/d < c′/d′ , \\ d^{2} & if c = c′, d = d′ , \\ ∞ & if c/d > c′/d′ . \end{matrix}

5 CONSTRUCTING φ-MODULES FROM FILTERED ISOCRYSTALS 15

(c) End_{φ^{a}}(M(c, d)) is a central division algebra over ℚ_{p^{a}} of dimension d^{2} and Hasse invariant \frac{c}{d} if (c, d) = 1.

Proof. (a) The assertions for c > 0 and c = 0, d = 1 were proved by Kedlaya [Ked05, Propositions 4.1.3 and 3.3.4]. The case c < 0 follows from Proposition 4.10 since dim_{ℚ_{p^{a}}} C = ∞. Note that Proposition 4.10 was proved only for a = 1 but the surjectivity of θ_{M}, which only relies on [Col02, Lemme 8.16], holds also for a > 1.

(b) follows from (a) and

Hom_{φ^{a}}(M(c, d) , M(c′, d′)) = H_{φ^{a}}^{0}(M(−c, d) ⊗ M(c′, d′)) ≅ H_{φ^{a}}^{0}(M(c′d − cd′, dd′)) .

(c) was proved in [Ked05, Remark 4.1.5]. The Hasse invariant can be computed as in [HP04, Proposition 8.6].

5 Constructing φ-Modules from Filtered Isocrystals

We keep the notation from Section 3. In the situation where the minimal and maximal Hodge-Tate weights differ by at most 1 we will associate with any analytic point μ ∈ F̆^{an} a φ-module M_{μ} over B̃_{rig}^{†}(C) where C is the completion of an algebraic closure of H(μ). In case H(μ)/Ĕ finite this construction parallels Berger’s construction [Ber08, §II] that works for arbitrary Hodge-Tate weights and even produces an “H(μ)-rational” version of M_{μ}. We begin with the following

Proposition 5.1. Let N be a φ-module over B̃_{rig}^{†} represented by a free B̃^{]0,1]}-module N^{]0,1]}

and an isomorphism φ_{N}^{]0,p^{−1}]} : N^{]0,1]} ⊗_{B̃^{]0,1]},φ} B̃^{]0,p^{−1}]} ^{∼} ⟶ N^{]0,p^{−1}]}. Consider the morphism θ_{N} = id_{N} ⊗θ : N^{]0,1]} → N^{]0,1]} ⊗_{B̃^{]0,1]},θ} C =: W_{N} and let W_{M} be a C-subspace of W_{N}. Then there exists a uniquely determined φ-submodule M ⊂ N over B̃_{rig}^{†} with tN ⊂ M which is represented by a B̃^{]0,1]}-submodule tN^{]0,1]} ⊂ M^{]0,1]} ⊂ N^{]0,1]} such that θ_{N}(M^{]0,1]}) = W_{M}.

Proof. If I is a closed subinterval of (0, 1] set N^{I} := N^{]0,1]} ⊗B̃^{]0,1]} B̃^{I}. Also if I = [s, r] we let pI := [ps, pr]. The isomorphism φ_{N}^{]0,p^{−1}]} induces an isomorphism φ_{N}^{I} : N^{pI} ⊗_{B̃^{pI},φ} B̃^{I} ^{∼} ⟶ N^{I}. Let I_{n} := [p^{−n−1}, p^{−n}] for n ∈ ℕ_{0}. We define M′^{I_{0}} as the preimage of W_{M} in N^{I_{0}} under the canonical morphism N^{I_{0}} → N^{I_{0}} ⊗_{B̃^{I_{0}},θ} C = W_{N} and we let M^{I_{0}} be the intersection of

M′^{I_{0}} with φ_{N}^{[p^{−1},p^{−1}]}(M′^{I_{0}} ⊗_{B̃^{I_{0}},φ} B̃^{[p^{−1},p^{−1}]}) inside N^{[p^{−1},p^{−1}]}, that is M^{I_{0}} equals

ker( M′^{I_{0}} ↪ N^{[p^{−1},p^{−1}]} ⟶ N^{[p^{−1},p^{−1}]}/φ_{N}^{[p^{−1},p^{−1}]}(M′^{I_{0}} ⊗_{B̃^{I_{0}},φ} B̃^{[p^{−1},p^{−1}]}) ) .

Since B̃^{I_{0}} is a principal ideal domain by [Ked05, Proposition 2.6.8] we see that M^{I_{0}} is a free B̃^{I_{0}}-submodule of N^{I_{0}} of full rank. For n ≥ 1 we define M^{I_{n}} as the image of M^{I_{0}} ⊗_{B̃^{I_{0}},φ^{n}} B̃^{I_{n}} under the isomorphism

φ_{N}^{I_{n}} ∘ … ∘ φ_{N}^{I_{1}} : N^{I_{0}} ⊗_{B̃^{I_{0}},φ^{n}} B̃^{I_{n}} ^{∼} ⟶ N^{I_{n}}.

In the terminology of Kedlaya [Ked05, §2.8] the collection M^{I_{n}} for n ∈ ℕ_{0} defines a vector bundle over B̃^{]0,1]} which by [Ked05, Theorem 2.8.4] corresponds to a free B̃^{]0,1]}-submodule M^{]0,1]} of N^{]0,1]}. By construction it satisfies tN^{]0,1]} ⊂ M^{]0,1]} and φ_{N}^{]0,p^{−1}]} restricts to an isomorphism on M^{]0,1]}. This makes M := M^{]0,1]}⊗B̃^{]0,1]} B̃_{rig}^{†} into a φ-module with the desired properties. Clearly M^{]0,1]} is uniquely determined by the subspace W_{M} ⊂ W_{N} and by the requirements that tN^{]0,1]} ⊂ M^{]0,1]} ⊂ N^{]0,1]} and that φ_{N}^{I_{n}}(M^{I_{n−1}} ⊗_{B̃^{I_{n−1}},φ} B̃^{I_{n}}) = M^{I_{n}}.

Now assume that L is a (not necessarily finite) extension of K_{0} and let n be an integer. Let (D, φ_{D}) be an F -isocrystal over 𝔽_{p}^{alg} and let F il^{n}D_{L} be an L-subspace of D_{L} = D ⊗_{K_{0}} L. Let F il^{n−1}D_{L} = D_{L} and F il^{n+1}D_{L} = (0). Then \underline{D} = (D, φ_{D}, F il^{•}D_{L}) is a filtered isocrystal over L with Hodge-Tate weights −n and −n + 1.

Definition 5.2. We set D^{]0,1]} := D ⊗_{K_{0}} B̃^{]0,1]} and D(\underline{D}) := (D, φ_{D}) := (D, φ_{D}) ⊗_{K_{0}} B̃_{rig}^{†}. Multiplication with t^{n} defines natural isomorphisms id_{D} ⊗t^{n} : t^{−n}D^{]0,1]} ^{∼} ⟶ D ⊗_{K_{0}} B̃^{]0,1]} and t^{−n}D(\underline{D}) ^{∼} ⟶ (D, p^{−n}φ_{D}) ⊗_{K_{0}} B̃_{rig}^{†}. We let M(\underline{D}) be the φ-submodule of t^{−n}D over B̃_{rig}^{†} represented by the B̃^{]0,1]}-submodule M^{]0,1]} of t^{−n}D^{]0,1]} with θ_{t^{−n}D}(M^{]0,1]}) = (F il^{n}D_{L}) ⊗_{L} C under the map θ_{t^{−n}D} : t^{−n}D^{]0,1]} → t^{−n}D^{]0,1]} ⊗_{B̃^{]0,1]},θ} C = D_{C}, whose existence was established in Proposition 5.1.

The following lemma is immediate from the definition.

Lemma 5.3. For all integers m consider the Tate object

1l(m) := (D = K_{0} , φ_{D} = p^{m} , K_{0} = F il^{m} ⊃ F il^{m+1} = (0)).

Then the canonical isomorphism id_{D} ⊗t^{m} : D(\underline{D} ⊗ 1l(m)) ^{∼} ⟶ t^{m}D(\underline{D}) induces an isomorphism M(\underline{D} ⊗ 1l(m)) ≅ M(\underline{D}).

The construction is compatible with dualizing in the following sense.

Proposition 5.4. There is a canonical isomorphism M(\underline{D}^{∨}) ≅ M(\underline{D})^{∨}, where the φmodule M(\underline{D})^{∨} := Hom_{φ}(M(\underline{D}), M(0, 1)) is the inner hom of φ-modules into the unit object M(0, 1).

Proof. If \underline{D} is a filtered isocrystal with F il^{n−1}D_{L} = D_{L} ⊃ F il^{n}D_{L} ⊃ F il^{n+1}D_{L} = (0) then

\underline{D}^{∨} has F il^{−n}DL^{∨} = DL^{∨} ⊃ F il^{−n+1}DL^{∨} ⊃ F il^{−n+2}DL^{∨} = (0) because F il^{i}DL^{∨} := \{λ ∈ DL^{∨} :

λ(F il^{1−i}D_{L}) = (0)} for all i. Thus the canonical pairing D × D^{∨} → K_{0} induces a pairing ψ as follows

[figure: M(\underline{D})^{]0,1]}; ×; M(\underline{D}^{∨})^{]0,1]}; t^{−n}D(\underline{D})^{]0,1]}; ≅; ×; t^{n−1}D(\underline{D}^{∨})^{]0,1]}; ≅; ψ; M(−1, 1)^{]0,1]}; ≅; (D, p^{−n}φ_{D}) ⊗_{K_{0}} B̃^{]0,1]}; θ_{t^{−n}D(\underline{D})}; ×; (D^{∨}, p^{n−1}φ_{D^{∨}}) ⊗_{K_{0}} B̃^{]0,1]}; θ_{t^{n−1}D(\underline{D}^{∨})}; (K_{0}, p^{−1}φ_{K_{0}}) ⊗_{K_{0}} B̃^{]0,1]}; θ_{M(−1,1)}; D_{C}; ×; DC^{∨}; C]

Since θ_{t^{−n}D(\underline{D})}(M(\underline{D})^{]0,1]}) = F il^{n}D_{C} and θ_{t^{n−1}D(\underline{D}^{∨})}(M(\underline{D}^{∨})^{]0,1]}) = F il^{−n+1}DC^{∨} we obtain

θ_{M(−1,1)} ∘ ψ(M(\underline{D})^{]0,1]} ⊗B̃^{]0,1]} M(\underline{D}^{∨})^{]0,1]}) = (0). By Corollary 4.11 the pairing M(\underline{D}) ⊗B̃_{rig}^{†} M(\underline{D}^{∨}) → M(−1, 1) factors through M(0, 1). This implies M(\underline{D}^{∨}) ↪ M(\underline{D})^{∨}. Since both φ-modules have rank equal to dim D and by Theorem 5.5 below their degree satisfies

deg M(\underline{D}^{∨}) = t_{N}(\underline{D}^{∨}) − t_{H}(\underline{D}^{∨}) = −t_{N}(\underline{D}) + t_{H}(\underline{D}) = − deg M(\underline{D}) = deg M(\underline{D})^{∨} ,

[Ked05, Lemma 3.4.2] yields M(\underline{D}^{∨}) ^{∼} ⟶ M(\underline{D})^{∨}.

Unfortunately we do not know how to construct M(\underline{D}) for filtered isocrystals \underline{D} with Hodge-Tate weights other than −n and −n + 1. Therefore we cannot make \underline{D} ↦ M(\underline{D}) into a tensor functor and we do not follow Berger’s argument [Ber08, Théorème IV.2.1] to prove the next theorem.

Theorem 5.5. deg M(\underline{D}) = t_{N}(\underline{D}) − t_{H}(\underline{D}).

Proof. By the Dieudonné-Manin classification [Man63] there exists an F -isocrystal (D′, φ_{D′}) over 𝔽_{p}^{alg} of rank one with φ_{D′} = p^{t_{N} (\underline{D})} · φ which is isomorphic to det(D, φ_{D}). So by construction deg D(\underline{D}) = t_{N}(\underline{D}). Moreover,

t_{H}(\underline{D}) = (n − 1) · dim_{L}(D_{L}/F il^{n}D_{L}) + n · dim_{L} F il^{n}D_{L}
= n · dim_{L} D_{L} − dim_{C}(D_{L}/F il^{n}D_{L}) ⊗_{L} C
= n · dim_{K_{0}} D − dim_{C}(t^{−n}D^{]0,1]}/M^{]0,1]}) ⊗_{B̃^{]0,1]},θ} C .

The following lemma implies that

deg t^{i}D(\underline{D}) − deg t^{i−1}D(\underline{D}) = dim_{K_{0}} D for all i and
deg M(\underline{D}) − deg t^{−n}D(\underline{D}) = dim_{C}(t^{−n}D^{]0,1]}/M^{]0,1]}) ⊗_{B̃^{]0,1]},θ} C

and this proves the theorem.

Lemma 5.6. Let M_{1} and M_{2} be φ-modules over B̃_{rig}^{†} represented by B̃^{]0,1]}-modules M_{i}^{]0,1]}.

Assume that M_{1}^{]0,1]} ⊃ M_{2}^{]0,1]} ⊃ tM_{1}^{]0,1]}. Then

deg M_{2} − deg M_{1} = dim_{C}(M_{1}^{]0,1]}/M_{2}^{]0,1]}) ⊗_{B̃^{]0,1]},θ} C .

Proof. Clearly the equality holds for M_{2} = tM_{1} ≅ M_{1} ⊗M(1, 1) since deg tM_{1} −deg M_{1} = rk M_{1}. We claim that it suffices to prove the inequality

deg M_{2} − deg M_{1} ≥ dim_{C} W  (5.1)

where we abbreviate W := (M_{1}^{]0,1]}/M_{2}^{]0,1]}) ⊗_{B̃^{]0,1]},θ} C. Indeed we apply the inequality to the two inclusions M_{1} ⊃ M_{2} ⊃ tM_{1} and M_{2} ⊃ tM_{1} ⊃ tM_{2} and conclude using the exact sequence of B̃^{]0,1]}-modules

0 ⟶ M_{2}^{]0,1]}/tM_{1}^{]0,1]} ⟶ M_{1}^{]0,1]}/tM_{1}^{]0,1]} ⟶ M_{1}^{]0,1]}/M_{2}^{]0,1]} ⟶ 0 .

To prove the inequality (5.1) we argue by induction on dim_{C} W. Let dim_{C} W = 1. Since det M_{1} ⊃ det M_{2} we know that deg M_{2} ≥ deg M_{1} from Proposition 4.12. If we had deg M_{2} = deg M_{1} then M_{2} = M_{1} by [Ked05, Lemma 3.4.2]. So deg M_{2} − deg M_{1} ≥ 1 = dim_{C} W as desired.

Let now dim_{C} W > 1 and choose a C-subspace W′ of dimension 1 of W. By Proposition 5.1 there is a unique B̃_{rig}^{†}-submodule M_{2} ⊂ M_{3} ⊂ M_{1} corresponding to W′. By induction deg M_{3} − deg M_{1} ≥ dim_{C}(W/W′) and deg M_{2} − deg M_{3} ≥ dim_{C} W′ proving the lemma.

Remark 5.7. As J. Pottharst pointed out to us, one could nevertheless follow [Ber08, Théorème IV.2.1] to prove Theorem 5.5 by working in the exact tensor category \underline{MP}_{C}^{φ} of triples (D, φ_{D}, L) with (D, φ_{D}) an F -isocrystal over 𝔽_{p}^{alg} and L a B_{dR}^{+}-lattice inside D ⊗_{K_{0}} B_{dR}, where B_{dR}^{+} = lim_{⟵} B̃^{+}/(ker θ)^{i}, B_{dR} = B_{dR}^{+}[t^{−1}], and θ : B_{dR}^{+}/tB_{dR}^{+} ^{∼} ⟶ C. Note that to \underline{MP}_{C}^{φ} also the functions t_{N} and t_{H} extend via t_{N}(D, φ_{D}, L) := t_{N}(D, φ_{D}) and t_{H}(D, φ_{D}, L) := m if ∧^{dim D}L = t^{−m} ∧^{dim D} D ⊗_{K_{0}} B_{dR}^{+}. From \underline{MP}_{C}^{φ} there are exact tensor functors F_{1} to the category of filtered isocrystals over C by defining

F il^{i}D_{C} = (t^{i}L ∩ D ⊗_{K_{0}} B_{dR}^{+}) / (t^{i}L ∩ t · D ⊗_{K_{0}} B_{dR}^{+}) = id_{D} ⊗θ(t^{i}L ∩ D ⊗_{K_{0}} B_{dR}^{+}) ,

and F_{2} to the category of φ-modules over B̃_{rig}^{†} similar to our construction in Proposition 5.1. More precisely F_{2} assigns to (D, φ_{D}, L) the φ-module

M(D, φ_{D}, L) := M(D, φ_{D}, L)^{]0,1]} ⊗B̃^{]0,1]} B̃_{rig}^{†}  where
M(D, φ_{D}, L)^{]0,1]} := { m ∈ D ⊗_{K_{0}} B̃^{]0,1]}[t^{−1}] : φ_{D}^{i}(m ⊗_{φ^{i}} 1) ∈ L for all i ≤ 0 } .

Restricted to the subcategories where the Hodge-Tate weights are −n and −n + 1, respectively where t^{−n+1}D ⊗_{K_{0}} B_{dR}^{+} ⊂ L ⊂ t^{−n}D ⊗_{K_{0}} B_{dR}^{+}, the functor F_{1} is an equivalence. The inverse functor F_{1}^{−1} assigns to the filtered isocrystal \underline{D} the B_{dR}^{+}-lattice L such that t^{n}L is the preimage of F il^{n}D_{C} under the map id_{D} ⊗θ : D ⊗_{K_{0}} B_{dR}^{+} → D_{C}. Our functor \underline{D} ↦ M(\underline{D}) from Definition 5.2 is then the composition F_{2} ∘ F_{1}^{−1}.

Remark 5.8. The construction of M(\underline{D}) is functorial with respect to automorphisms of C. Therefore M(\underline{D}) gives rise to a Galois representation

Gal(H(x)^{alg}/H(x)) ⟶ GL(H_{φ}^{0}(M(\underline{D}))) .

Of course this representation is of most interest when M(\underline{D}) ≅ M(0, 1)^{⊕ dim D} since then H_{φ}^{0}(M(\underline{D})) ≅ ℚ_{p}^{⊕ dim D}.

If L is a discretely valued extension of K_{0} one can check that M(\underline{D}) equals the φ-module over B̃_{rig}^{†} constructed by Berger [Ber08, §II]. One of Berger’s main theorems is the following criterion.

Theorem 5.9. ([Ber08, §IV.2]) Let L be a discretely valued extension of K_{0}. Then \underline{D} is weakly admissible if and only if M(\underline{D}) ≅ M(0, 1)^{⊕ dim D}. In this case the crystalline Galois representation associated with \underline{D} is the representation H_{φ}^{0}(M(\underline{D})) from Remark 5.8.

The theorem indicates what the local system on F̆_{b}^{a} could be. We will discuss this in detail in the next sections. Let us first make explicit what happens if M(\underline{D}) ≅ M̸(0, 1)^{⊕ dim D}.

Proposition 5.10. Assume that t_{N}(\underline{D}) = t_{H}(\underline{D}) and that the Hodge-Tate weights of \underline{D} are −n and −n+1. Then M(\underline{D}) ≅ M̸(0, 1)^{⊕ dim D} if and only if for some (any) integer e ≥ (dim D) − 1 there exists a non-zero x ∈ H_{φ^{e}}^{0}(M(1, 1) ⊗ [e]_{∗}(t^{−n}D)) with θ_{t^{−n}D}(φ_{t^{−n}D}^{m}(x)) ∈ (F il^{n}D_{L}) ⊗_{L} C for all m = 0, …, e − 1.

Note that t^{−n}D can be represented by (D ⊗_{K_{0}} B̃^{]0,r]}, p^{−n}b·φ) for arbitrarily large r > 0. Hence by Proposition 4.9 the element x actually belongs to D ⊗_{K_{0}} B̃^{]0,r]} and the expression θ_{t^{−n}D}(φ_{t^{−n}D}^{m}(x)) makes sense.

Proof. Let M(\underline{D}) ≅ M̸(0, 1)^{⊕ dim D} and let e ≥ (dim D)−1 be any integer. By Theorems 5.5 and 4.3 there is a φ-module M(c, d) over B̃_{rig}^{†} with c < 0 and d < dim D which is a summand of M(\underline{D}). Thus by Proposition 4.12 there exists a non-zero morphism of φmodules f : M(−1, e) → M(c, d) ⊂ M(\underline{D}) ⊂ t^{−n}D. Let e_{1}, …, e_{e} be a basis of M(−1, e) satisfying (4.1) and let x be the image of e_{1} in t^{−n}D under f. Then φ_{t^{−n}D}^{e}(x) = p^{−1}x, that is x ∈ H_{φ^{e}}^{0}(M(1, 1) ⊗ [e]_{∗}(t^{−n}D)). Moreover, f(e_{m+1}) = φ_{t^{−n}D}^{m}(x) in t^{−n}D for 0 ≤ m < e. Now the fact that the morphism f factors through M(\underline{D}) amounts by Definition 5.2 to θ_{t^{−n}D}(φ_{t^{−n}D}^{m}(x)) ∈ (F il^{n}D_{L}) ⊗_{L} C.

Conversely assume that for some integer e ≥ (dim D)−1 there exists a non-zero element x in H_{φ^{e}}^{0}(M(1, 1)⊗[e]_{∗}(t^{−n}D)) with θ(φ_{t^{−n}D}^{m}(x)) ∈ (F il^{n}D_{L})⊗_{L}C for all 0 ≤ m < e. Define the non-trivial morphism of φ-modules f : M(−1, e) → t^{−n}D by f(e_{m+1}) := φ_{t^{−n}D}^{m}(x) for 0 ≤ m < e. Since θ_{t^{−n}D}(φ_{t^{−n}D}^{m}(x)) ∈ (F il^{n}D_{L}) ⊗_{L} C the morphism f factors through M(\underline{D}) by Definition 5.2. By Proposition 4.12 this implies M(\underline{D}) ≅ M̸(0, 1)^{⊕ dim D}.

6 The Minuscule Case

We retain the notation from Section 2. In particular G is a reductive group, b ∈ G(K_{0}), and {μ} is a conjugacy class of cocharacters of G. Recall from [RZ96, 1.7] that there is a slope quasi-cocharacter ν ∈ Hom_{K_{0}}(𝔾_{m}, G) ⊗_{ℤ} ℚ associated with b. We make the following assumption on the triple (G, b, {μ}):

ν^{#} = μ^{#} in (π_{1}(G)_{Gal(\overline{ℚ}_{p}/ℚ_{p})}) ⊗_{ℤ} ℚ, and
there is an n ∈ ℤ and a faithful ℚ_{p}-rational representation V of G,  (6.1)

such that all the weights of {μ} on V are n − 1 and n, and the isocrystal \underline{D}_{b,μ}(V) from (2.4) has Newton slopes in the interval [n − 1, n].

The significance of ν^{#} = μ^{#} is explained by the following

Lemma 6.1. Let ν be the slope quasi-cocharacter in Hom_{K_{0}}(𝔾_{m}, G) ⊗_{ℤ} ℚ associated with b; see [RZ96, 1.7], and let ρ : G → GL(V) be in \underline{Rep}_{ℚ_{p}}G. Then under the canonical identification Hom(𝔾_{m}, 𝔾_{m}) = ℤ we have

det_{V} ∘ρ ∘ ν = t_{N}(\underline{D}_{b,μ}(V))  and  det_{V} ∘ρ ∘ μ = t_{H}(\underline{D}_{b,μ}(V)) .

In particular, if the images ν^{#} and μ^{#} of ν and μ in (π_{1}(G)_{Gal(\overline{ℚ}_{p}/ℚ_{p})}) ⊗_{ℤ} ℚ coincide then t_{N}(\underline{D}_{b,μ}(V)) = t_{H}(\underline{D}_{b,μ}(V)) for all V ∈ \underline{Rep}_{ℚ_{p}}G.

Proof. The statement about t_{N} follows from the construction of ν in [Kot85, §4.2]. The statement about t_{H} is immediate from the definition of \underline{D}_{b,μ}(V). If ν^{#} = μ^{#} holds in (π_{1}(G)_{Gal(\overline{ℚ}_{p}/ℚ_{p})}) ⊗_{ℤ} ℚ then (ρ ∘ ν)^{#} = (ρ ∘ μ)^{#} in (π_{1}(GL(V))_{Gal(\overline{ℚ}_{p}/ℚ_{p})}) ⊗_{ℤ} ℚ ≅ ℚ. Under the last isomorphism we have (ρ ∘ ν)^{#} = det ∘ρ ∘ ν and likewise for μ. This proves the lemma.

If (G, {μ}) satisfies (6.1) with n = 1 then μ is a minuscule coweight by Serre [Ser79, §3]. Furthermore, assume that G does not possess a proper normal subgroup through which μ factors. Then all weights of μ in V form one orbit under the Weyl group, the group G has no simply connected factor group, and all irreducible components of G^{der} are of type A_{n}, B_{n}, C_{n} or D_{n}. Moreover, in case V is an irreducible representation, dim V is even, except possibly when all irreducible factors of G^{der} are of type A.

Assume now that (G, b, {μ}) satisfies condition (6.1) and let ρ : G ↪ GL(V) =: G′ be the faithful representation from (6.1). This defines a closed embedding F ↪ F′ := Flag(V) into the flag variety from (2.2). Here Flag(V) is actually the Grassmannian over K_{0} of c-dimensional subspaces of V_{K_{0}} = V ⊗_{ℚ_{p}} K_{0}, where c is the multiplicity of the weight n of μ on V. By [RZ96, 1.18] we obtain a closed embedding of Ĕ-analytic period spaces F̆_{b}^{wa} ↪ F̆′ _{ρ(b)}^{wa}, μ ↦ ρ ∘ μ.

Let μ ∈ F̆^{an} be an analytic point. Let L = H(μ) and let C be the completion of an algebraic closure of L. Let \underline{D}_{μ} := \underline{D}_{b,μ}(V) := (V_{K_{0}}, b·φ, F il_{μ}^{•}D_{L}) be the filtered isocrystal from (2.4). In particular F il_{μ}^{n−1}D_{L} = D_{L} and F il_{μ}^{n+1}D_{L} = (0). Now we let M_{μ} := M_{b,μ}(V) := M(\underline{D}_{b,μ}(V)) be the φ-module over B̃_{rig}^{†}(C) from Definition 5.2. Note that deg M_{μ} = 0 under our assumption (6.1) by Theorem 5.5 and Lemma 6.1. We define

F̆_{b}^{0} := F̆_{b}^{0}(V) := { μ ∈ F̆^{an} analytic points : M_{b,μ}(V) ≅ M(0, 1) ⊗_{ℚ_{p}} V }. (6.2)

If b′ = g b φ(g^{−1}) for some g ∈ G(K_{0}) one easily checks that μ ↦ gμg^{−1} =: μ′ maps F̆_{b}^{0} isomorphically onto F̆_{b′}^{0}, because g induces isomorphisms \underline{D}_{b′,μ′}(V) ≅ \underline{D}_{b,μ}(V) and M_{b′,μ′}(V) ≅ M_{b,μ}(V). In particular F̆_{b}^{0} is invariant under the group J(ℚ_{p}) from (2.3).

Proposition 6.2. The set F̆_{b}^{0} is independent of the representation V of G with (6.1).

Proof. Let ρ : G ↪ GL(V) and ρ̃ : G ↪ GL(Ṽ) be two representations of G satisfying (6.1) for integers n respectively ñ. Let μ ∈ F̆^{an} and assume M_{b,μ}(V) ≅ M(0, 1) ⊗_{ℚ_{p}} V. We must show that also M_{b,μ}(Ṽ) ≅ M(0, 1) ⊗_{ℚ_{p}} Ṽ. Since V is a faithful representation, Ṽ is a direct summand of ⨁_{i} V^{⊗n_{i}} ⊗_{ℚ_{p}} V^{∨}^{⊗m_{i}} for suitable positive integers n_{i}, m_{i}. As in Remark 5.7 consider the object (D, φ_{D}, L) := (V_{K_{0}} , ρ(b)φ , t^{−n}( id_{V_{K_{0}}} ⊗θ)^{−1}F il^{n}D_{C}) ∈ \underline{MP}_{C}^{φ} associated with \underline{D}_{b,μ}(V) under the inverse F_{1}^{−1} of the functor F_{1} for Hodge-Tate weights −n and −n + 1. Set

(D̃, φD̃, L̃) := (Ṽ_{K_{0}} , ρ̃(b)φ , Ṽ ⊗_{ℚ_{p}} B_{dR} ∩ ⨁_{i} L^{⊗n_{i}} ⊗_{B_{dR}^{+}} L^{∨}^{⊗m_{i}})

Since F_{1} is a tensor functor it assigns to (D̃, φD̃, L̃) the filtered isocrystal \underline{D}_{b,μ}(Ṽ) = (D̃, φD̃, F il_{μ}^{•}D̃_{C}) over C. The latter has Hodge-Tate weights −ñ and −ñ + 1, and hence t^{−ñ+1}D̃ ⊗_{K_{0}} B_{dR}^{+} ⊂ L̃ ⊂ t^{−ñ}D̃ ⊗_{K_{0}} B_{dR}^{+} and the inverse F_{1}^{−1} of F_{1} for Hodge-Tate weights −ñ and −ñ + 1 assigns to \underline{D}_{b,μ}(Ṽ) the object (D̃, φD̃, L̃) of \underline{MP}_{C}^{φ}. Therefore under the tensor functor F_{2} we have

M_{b,μ}(Ṽ) = F_{2}(D̃, φD̃, L̃) = Ṽ ⊗_{ℚ_{p}} B̃_{rig}^{†}[t^{−1}] ∩ ⨁_{i} M_{b,μ}(V)^{⊗n_{i}} ⊗B̃_{rig}^{†} M_{b,μ}(V)^{∨}^{⊗m_{i}}

as φ-modules over B̃_{rig}^{†}. By Theorem 4.3 we can write M_{b,μ}(Ṽ) ≅ ⨁_{j} M(c̃_{j}, d̃_{j}). From M_{b,μ}(V) ≅ M(0, 1) ⊗_{ℚ_{p}} V it follows that also ⨁_{i} M_{b,μ}(V)^{⊗n_{i}} ⊗B̃_{rig}^{†} M_{b,μ}(V)^{∨}^{⊗m_{i}} is isomor-

phic to a direct sum of M(0, 1). By Proposition 4.12 also its direct summand M_{b,μ}(Ṽ) is isomorphic to M(0, 1) ⊗_{ℚ_{p}} Ṽ as desired.

Proposition 6.3. The set F̆_{b}^{0} is invariant under the group J(ℚ_{p}) from (2.3) and contained in F̆_{b}^{wa} with F̆_{b}^{0}(L) = F̆_{b}^{wa}(L) for any finite extension L/Ĕ.

Proof. The invariance under J(ℚ_{p}) was remarked before Proposition 6.2.

Let μ ∈ F̆_{b}^{0} be an analytic point and set L = H(μ). Consider a faithful representation V as in (6.1) and let \underline{D}_{μ} = \underline{D}_{b,μ}(V) = (D, φ_{D}, F il_{μ}^{•}D_{L}). Let D′ ⊂ D be a φ_{D}-stable K_{0}-subspace and let F il_{μ}^{i}D′_{L} := D′_{L} ∩ F il_{μ}^{i}D_{L}. We have to show that the inequality t_{H}(\underline{D}′) ≤ t_{N}(\underline{D}′) holds for the subobject \underline{D}′ = (D′, φ_{D}|_{D′}, F il_{μ}^{•}D′_{L}) ⊂ \underline{D}_{μ} with equality if \underline{D}′ = \underline{D}_{μ}. Consider the φ-submodule M′ := M(\underline{D}′) ⊂ M(\underline{D}_{μ}) = M_{μ}. Then

rk M′ · wt M′ = deg M′ = t_{N}(\underline{D}′) − t_{H}(\underline{D}′)

by Theorem 5.5. Since μ ∈ F̆_{b}^{0} we have M_{μ} ≅ M(0, 1) ⊗_{ℚ_{p}} V and hence t_{H}(\underline{D}_{μ}) = t_{N}(\underline{D}_{μ}). Moreover, wt M′ ≥ wt M_{μ} = 0 by Proposition 4.5 proving t_{N}(\underline{D}′) ≥ t_{H}(\underline{D}′). This shows that F̆_{b}^{0} ⊂ F̆_{b}^{wa}. The equality on L-valued points follows from Theorem 5.9.

We expect that the equality on L-valued points holds for many more L.

Conjecture 6.4. If L/Ĕ is a complete valued extension such that the value group of L is finitely generated then F̆_{b}^{0}(L) = F̆_{b}^{wa}(L).

We will see in a moment that F̆_{b}^{0} ⊂ F̆_{b}^{wa} is open. We also make the

Conjecture 6.5. F̆_{b}^{0} is arcwise connected.

We give evidence for these two conjectures in Section 10. In particular we show in Proposition 10.1 that Conjecture 6.5 follows from Conjecture 6.4. It is known in some cases that Conjecture 6.5 holds; see Remarks 7.4(a) and 8.5(b).

Theorem 6.6. The set F̆_{b}^{0} is a paracompact open Ĕ-analytic subspace of F̆^{an}. If b is decent with the integer s; cf. (2.6); then F̆_{b}^{0} has a natural structure of paracompact open E_{s}-analytic subspace of (F ⊗_{E} E_{s})^{an} from which it arises by base change to Ĕ.

Proof. (a) Since F̆^{an} has a finite covering by affine spaces, all its open Ĕ-analytic subspaces are paracompact by Lemma A.3. To prove that F̆_{b}^{0} ⊂ F̆^{an} is open consider a faithful representation V of dimension h as in (6.1) and let G′ = GL(V). This defines a closed embedding F ↪ F′ := Flag(V) ⊗_{ℚ_{p}} E into the flag variety from (2.2). Here Flag(V) is a Grassmannian isomorphic to G′/S′ where S′ = Stab_{G′}(V_{0}) is the stabilizer of an appropriate subspace V_{0} of V. By definition F̆_{b}^{0} = F̆^{an} ∩ F̆′_{b}^{0}. So it suffices to prove the theorem for G′ instead of G. Since G′ is connected we may assume by [Kot85] that b is decent, say with integer s. We let F′_{s}^{an} := (F′ ⊗_{E} E_{s})^{an} and define the subset F′_{b}^{0} ⊂ F′_{s}^{an} by the same condition as in (6.2). We will show that it is open. For μ ∈ F′_{s}^{an} we set \underline{D}_{μ} = \underline{D}_{b,μ}(V). By Lemma 6.1 our assumption (6.1) implies t_{N}(\underline{D}_{μ}) = t_{H}(\underline{D}_{μ}). We choose an integer e ≥ (dim V) − 1 which is a multiple of s. Since \underline{D}_{μ} has Hodge-Tate weights −n and −n + 1, the set F′_{s}^{an} ∖ F′_{b}^{0} is by Proposition 5.10 equal to

{ μ ∈ F′_{s}^{an} analytic points : there exists an algebraically closed complete

extension C of H(μ) and an element x ∈ H_{φ^{e}}^{0}(M(1, 1) ⊗ [e]_{∗}(t^{−n}D)), x ≠ 0

with θ_{t^{−n}D}(φ_{t^{−n}D}^{m}(x)) ∈ (F il_{μ}^{n}V_{H(μ)}) ⊗_{H(μ)} C for all m = 0, …, e − 1 } .

Here D := D(\underline{D}_{μ}) = (D, φ_{D}) ⊗_{K_{0}} B̃_{rig}^{†}(C) is the φ-module over B̃_{rig}^{†}(C) associated with \underline{D}_{μ} in Definition 5.2 and [e]_{∗}(t^{−1}D) = (D, p^{−e}φ_{D}^{e}) ⊗_{K_{0}} B̃_{rig}^{†}(C).

(b) We identify V ⊗_{ℚ_{p}} 𝔸_{E_{s}}^{1} with affine h-space 𝔸_{E_{s}}^{h} over E_{s}. For ζ ∈ E_{s} consider the E_{s}-analytic polydisc with radii (|ζ|, …, |ζ|)

𝔻(ζ)^{eh} = M(E_{s}⟨\frac{w_{im}}{ζ} : i = 1, …, h; m = 0, …, e − 1⟩) ⊂ (𝔸_{E_{s}}^{h})^{e}.

We will construct in (c) below a constant ζ ∈ E_{s} and a compact subset Z of 𝔻(ζ)^{eh} with the following property: If C is an algebraically closed complete extension of E_{s} and x ∈ H_{φ^{e}}^{0}(M(1, 1) ⊗ [e]_{∗}(t^{−n}D)) with x ≠ 0 then for some integer N

(w_{m})_{m=0}^{e−1} := ((w_{1m}, …, w_{hm})^{T} )_{m=0}^{e−1} := (p^{N}θ_{t^{−n}D}(φ_{t^{−n}D}^{m}(x)))_{m=0}^{e−1}

is a C valued point of Z and Z consists precisely of those points.

Now let G′_{s}^{an} be the E_{s}-analytic space associated with the group scheme G′ ⊗_{ℚ_{p}} E_{s} and consider the morphism of E_{s}-analytic spaces

β : G′_{s}^{an} ×_{E_{s}} 𝔻(ζ)^{eh} ⟶ (𝔸_{E_{s}}^{h})^{e} ≅ (V ⊗_{ℚ_{p}} 𝔸_{E_{s}}^{1})^{e}
( g , (w_{m})_{m=0}^{e−1} ) ⟼ (g^{−1}w_{m})_{m=0}^{e−1} .

Let Y be the closed subset of G′_{s}^{an} ×_{E_{s}} 𝔻(ζ)^{eh} defined by the condition that (w_{m})_{m=0}^{e−1} belongs to Z and that β(Y) ⊂ (V_{0} ⊗_{ℚ_{p}} 𝔸_{E_{s}}^{1})^{e}. Furthermore consider the projection map

pr_{1} : G′_{s}^{an} ×_{E_{s}} 𝔻(ζ)^{eh} ⟶ G′_{s}^{an}

onto the first factor and the canonical map γ : G′_{s}^{an} → F′_{s}^{an} coming from the isomorphism F′_{s} ≅ G′_{s}/ Stab_{G′_{s}}(V_{0}). Then μ ∈ F′_{s}^{an} does not belong to F′_{b}^{0} if and only if μ ∈ γ ∘ pr_{1}(Y). Since 𝔻(ζ)^{eh} is quasi-compact the projection pr_{1} is a proper map of topological Hausdorff spaces by [Beh90, Proposition 3.3.2]. Thus in particular it is closed and pr_{1}(Y) is closed. Note that F′_{s}^{an} carries the quotient topology under γ since γ is a smooth morphism of schemes, hence open by [Beh93, Proposition 3.5.8 and Corollary 3.7.4]. Since by construction pr_{1}(Y) = γ^{−1}(γ ∘ pr_{1}(Y)) we conclude that F′_{b}^{0} = F′_{s}^{an} ∖ γ ∘ pr_{1}(Y) is open in F′_{s}^{an} as desired.

(c) It remains to construct the compact set Z. Since b is decent, the φ^{e}-module M(1, 1) ⊗ [e]_{∗}(t^{−n}D) is isomorphic to ⨁_{i=1}^{h} M(−c_{i}, 1) for suitable integers c_{i}. We assume that the identification of V ⊗_{ℚ_{p}} 𝔸_{E_{s}}^{1} with 𝔸_{E_{s}}^{h} in (b) was chosen compatibly with this direct sum decomposition. Let

c_{1}, …, c_{k} > 0 = c_{k+1} = … = c_{ℓ} > c_{ℓ+1}, …, c_{h} .

Since all Newton slopes of D, respectively t^{−n}D, lie in [n − 1, n], respectively [−1, 0], we have e − 1 ≥ c_{i} ≥ −1 for all i. Then by Propositions 4.8 and 4.12

H_{φ^{e}}^{0}(M(1, 1) ⊗ [e]_{∗}(t^{−n}D)) ≅ ⨁_{i=1}^{k}{ ∑_{ν∈ℤ} p^{c_{i}ν} ∑_{j=0}^{c_{i}−1} p^{j}φ^{−eν}([u_{ij}]) : u_{ij} ∈ Ẽ, v_{E}(u_{ij}) > 0 }

⊕ ⨁_{i=k+1}^{ℓ} W (𝔽_{p^{e}})[\frac{1}{p}]

⊕ ⨁_{i=ℓ+1}^{h}(0) .

For 1 ≤ i ≤ k, 0 ≤ j ≤ c_{i} − 1 and all integers λ ≥ 0 consider the compact E_{s}-analytic spaces which we just view as compact sets

U_{ij}^{(0)} := M(E_{s}⟨^{\frac{u_{ij}^{(0)}}{p}} ⟩) = \{ |u_{ij}^{(0)}| ≤ |p| \}  and
U_{ij}^{(λ)} := M(E_{s}⟨u_{ij}^{(λ)}⟩) = { |u_{ij}^{(λ)}| ≤ 1 }  for λ > 0 .

Then the sets

U_{ij} := { (u_{ij}^{(λ)})_{λ∈ℕ_{0}} ∈ ∏_{λ∈ℕ_{0}} U_{ij}^{(λ)} : (u_{ij}^{(λ+1)})^{p} = u_{ij}^{(λ)} for all λ ≥ 0 } and

U := ∏_{i=1}^{k} ∏_{j=0}^{c_{i}−1} U_{ij} × ∏_{i=k+1}^{ℓ} W (𝔽_{p^{e}}) × ∏_{i=ℓ+1}^{h}{0}

are compact by Tychonoff’s theorem. For an arbitrary algebraically closed complete extension C of E_{s} consider a C-valued point u of U given by

(((u_{ij}^{(λ)})_{λ∈ℕ_{0}})_{i=1,...,k; j=0,...,c_{i}−1} , (a_{i})_{i=k+1,...,ℓ} , (0)_{i=ℓ+1,...,h})

with u_{ij}^{(λ)} ∈ C and a_{i} ∈ W (𝔽_{p^{e}}). We assign to u the C-valued point y of 𝔸_{E_{s}}^{eh} with (w_{m})_{m=0}^{e−1} = (θ_{t^{−n}D}(φ_{t^{−n}D}^{m}(x)))_{m=0}^{e−1} where x is the element of H_{φ^{e}}^{0}(M(1, 1)⊗[e]_{∗}(t^{−n}D)) associated with the u_{ij}^{(λ)} and a_{i}. In concrete terms this means

w_{m} := b · φ(b) ⋯ φ^{m−1}(b) · ( \begin{matrix}( ∑_{j=0}^{c_{i}−1} p^{j}(∑_{ν>0} p^{c_{i}ν}u_{ij}^{(eν−m)} + ∑_{ν≤0} p^{c_{i}ν}(u_{ij}^{(0)})^{p^{m−νe}}))_{i=1}^{k} \\ (φ^{m}(a_{i}))_{i=k+1}^{ℓ} \\ (0)_{i=ℓ+1}^{h} \end{matrix} )

for m = 0, …, e−1. This defines a map α : U → 𝔸_{E_{s}}^{eh}, u ↦ y of topological Hausdorff spaces. We claim that α is continuous. By definition of the topology on 𝔸_{E_{s}}^{eh} (see Appendix A) the map α is continuous if and only if for any polynomial f ∈ E_{s}[w_{im} : 1 ≤ i ≤ h, 0 ≤ m ≤ e−1] and any open interval I ⊂ ℝ the preimage under α of the open set

W = { y ∈ 𝔸_{E_{s}}^{eh} : |f|_{y} ∈ I }

is open in U. Since |u_{ij}^{(0)}|_{u} ≤ |p| there is a constant ζ ∈ E_{s} such that |w_{im}|_{α(u)} ≤ |ζ| for all i and m and all u ∈ U. Write f = ∑\underline{n} b\underline{n} \underline{w}\underline{n} where b\underline{n} ∈ E_{s} for every multi index \underline{n} ∈ ℕ_{0}^{eh}. If we set w_{im} = w′_{im} + w′′_{im} we obtain from the Taylor expansion of f in powers of \underline{w}′′ a bound δ such that for all y ∈ 𝔸_{E_{s}}^{eh} the condition |w′′_{im}|_{y} ≤ δ for all i, m implies

|f(\underline{w})|_{y} ∈ I and |\underline{w}|_{y} ≤ |ζ|  ⟺  |f(\underline{w}′)|_{y} ∈ I and |\underline{w}′|_{y} ≤ |ζ| .

Now we fix a positive integer r and set for all m = 0, …, e − 1

(w′′_{im})_{i=1}^{h} := b · φ(b) ⋯ φ^{m−1}(b) · ( \begin{matrix}(∑_{j=0}^{c_{i}−1} p^{j} ∑_{ν>r} p^{c_{i}ν}u_{ij}^{(eν−m)})_{i=1}^{k} \\ (0)_{i=k+1}^{ℓ} \\ (0)_{i=ℓ+1}^{h} \end{matrix} )  and

(w′_{im})_{i=1}^{h} := b · φ(b) ⋯ φ^{m−1}(b) · ( \begin{matrix}(∑_{j=0}^{c_{i}−1} p^{j} ∑_{ν≤r} p^{c_{i}ν}(u_{ij}^{(er)})^{p^{m+(r−ν)e}})_{i=1}^{k} \\ (φ^{m}(a_{i}))_{i=k+1}^{ℓ} \\ (0)_{i=ℓ+1}^{h} \end{matrix} ) .

Since |u_{ij}^{(λ)}|_{u} < 1 for all λ, i, j and for all points u ∈ U we can find a large enough integer r such that |w′′_{im}|_{α(u)} ≤ δ for all i, m and for all u ∈ U. This shows that

α^{−1}(W) = W′ := { u ∈ U : |f(w′_{im})|_{α(u)} ∈ I } .

To prove that W′ is open in U observe that the map

α′_{m} : U′ := ∏_{i=1}^{k} ∏_{j=0}^{c_{i}−1} U_{ij}^{(er)} × ∏_{i=k+1}^{ℓ} 𝔸_{E_{s}}^{1} ⟶ 𝔸_{E_{s}}^{h}

( (u_{ij}^{(er)})_{i,j} , (φ^{m}(a_{i}))_{i} ) ⟼ (w′_{im})_{i=1}^{h}

is a morphism of E_{s}-analytic spaces, and in particular continuous. Furthermore, the projection maps α_{ij} : U_{ij} → U_{ij}^{(er)} and the inclusion

α_{i} : W (𝔽_{p^{s}}) ⟶ 𝔸_{E_{s}}^{e} = M(E_{s}[T_{0}, …, T_{e−1}]) , a ⟼ (T_{m} ↦ φ^{m}(a))

for i = k + 1, …, ℓ are continuous. Since

α′ := (α′_{0} × … × α′_{e−1}) ∘ ( ∏_{i=1}^{k} ∏_{j=0}^{c_{i}−1} α_{ij} × ∏_{i=k+1}^{ℓ} α_{i}) : U ⟶ 𝔸_{E_{s}}^{eh}

maps u to α′(u) = (w′_{m})_{m=0}^{e−1} the set W′ = (α′)^{−1}(W) is open in U and this proves that α is continuous.

Now multiplying (w_{im})_{i,m} with p amounts to replacing u_{ij}^{(λ)} by u_{i,j−1}^{(λ)} for j = 1, …, c_{i}−1,

and u_{i,0}^{(λ)} by (u_{i,c_{i}−1}^{(λ)})^{p^{e}}, and a_{i} by pa_{i}. Thus we may take

Z := α(U ∖ { u ∈ U : |u_{ij}^{(0)}|_{u} < |p|^{p^{e}} , a_{i} ∈ pW (𝔽_{p^{e}}) for all i and j }) .

Then Z is the continuous image of a compact set and satisfies the property required in (b). This proves the theorem.

We give an example showing that the inclusion F̆_{b}^{0} ⊂ F̆_{b}^{wa} may be strict. Similar examples were independently obtained by A. Genestier and V. Lafforgue. In Section 9 we will determine all b ∈ GL_{n}(K_{0}) for which F̆_{b}^{0} = F̆_{b}^{wa}.

Example 6.7. Let G = GL_{5} and consider the conjugacy class {μ} of cocharacters containing μ : 𝔾_{m} → G , z ↦ diag(z, z, 1, 1, 1). We have E = ℚ_{p} and F ≅ Grass(2, 5) the Grassmannian. Let

b = ( \begin{matrix}0 & 0 & 0 & 0 & p^{−2} \\ p & 0 & 0 & 0 & 0 \\ 0 & p & 0 & 0 & 0 \\ 0 & 0 & p & 0 & 0 \\ 0 & 0 & 0 & p & 0 \end{matrix} ) ∈ G(K_{0}) .  (6.3)

The element b is decent with integer s = 5 and (G, b, {μ}) satisfies (6.1) for n = 1. Since the isocrystal (K_{0}^{⊕5}, b·φ) is simple every cocharacter in {μ} is weakly admissible, that is F̆_{b}^{wa} = F̆^{an}.

Let μ ∈ F̆^{an} and let C be the completion of an algebraic closure of H(μ). The φ-module M_{μ} = M(K_{0}^{⊕5}, b·φ, F il_{μ}^{•}H(μ)^{⊕5}) constructed from b and μ satisfies deg M_{μ} = 0 and

t^{−1}D = D ⊗ M(−1, 1) = M(−3, 5) ⊃ M_{μ} ⊃ D ≅ M(2, 5)

where D = D(K_{0}^{⊕5}, b · φ) = (K_{0}^{⊕5}, b · φ) ⊗_{K_{0}} B̃_{rig}^{†}(C) and t^{−1}D = D(K_{0}^{⊕5}, p^{−1}b · φ). Since by Proposition 4.12 the weight of every summand of M_{μ} lies between −3/5 and 2/5, either M_{μ} ≅ M(0, 1)^{⊕5} or M_{μ} ≅ M(−1, 2) ⊕ M(1, 3). (The first entries must sum to 0 = deg M_{μ} and the second entries must sum to 5 = rk M_{μ}.) Now the bad situation M_{μ} ≅ M(−1, 2) ⊕ M(1, 3) occurs if and only if

(0) ≠ Hom_{φ}(M(−1, 2) , M_{μ}) =
= { f ∈ Hom_{φ}(M(−1, 2) , t^{−1}D): θ_{t^{−1}D}(im f) ⊂ F il_{μ}^{1}H(μ)^{⊕5} } .

We compute Hom_{φ}(M(−1, 2) , M(−3, 5)) =

= \{ A = (a_{i,j})_{i=1...5, j=1,2} ∈ B̃_{rig}^{†}(C)^{5×2} : p^{−1}b · φ(A) = A · ( \begin{matrix}0 & p^{−1} \\ 1 & 0 \end{matrix} )\}

with b as in (6.3). This implies p^{−3}φ(a_{5,1}) = a_{1,2}, p^{−3}φ(a_{5,2}) = p^{−1}a_{1,1} and φ(a_{i−1,1}) = a_{i,2}, φ(a_{i−1,2}) = p^{−1}a_{i,1} for 2 ≤ i ≤ 5. So a_{1,2} must satisfy p^{−1} · φ^{10}(a_{1,2}) = a_{1,2}, that is, x := a_{1,2} ∈ H_{φ^{10}}^{0}(M(−1, 1)). By Proposition 4.8 it follows that x = ∑_{ν∈ℤ} p^{ν}φ^{−10ν}([u]) for some u ∈ Ẽ(C) with v_{E}(u) > 0. We conclude that Hom_{φ}(M(−1, 2) , M(−3, 5)) =

= \{ A = ( \begin{matrix}φ^{5}(x) & x \\ φ^{11}(x) & φ^{6}(x) \\ φ^{17}(x) & φ^{12}(x) \\ φ^{23}(x) & φ^{18}(x) \\ φ^{29}(x) & φ^{24}(x) \end{matrix} ) \begin{matrix}: x = ∑_{ν∈ℤ} p^{ν}φ^{−10ν}([u]) ∈ H_{φ^{10}}^{0}(M(−1, 1)) , \\ u ∈ Ẽ(C), v_{E}(u) > 0 \end{matrix} \} .

Thus the points μ ∈ F̆^{an} for which the columns of θ_{t^{−1}D}(A) are contained in F il_{μ}^{1}H(μ)^{⊕5} do not belong to F̆_{b}^{0}. We may therefore take any nonzero u ∈ Ẽ(C) with v_{E}(u) > 0, define the matrix A correspondingly, and let μ ∈ F̆^{an}(C) be a point for which F il_{μ}^{1}H(μ)^{⊕5} contains the columns of θ_{t^{−1}D}(A). All these points lie in the complement of F̆_{b}^{0} in F̆_{b}^{wa}.

Note that rk θ_{t^{−1}D}(A) = 2 and μ is uniquely determined by A and hence by u. Indeed, assume the contrary and consider ∧^{2}A ∈ Hom_{φ}(∧^{2}M(−1, 2), ∧^{2}M(−3, 5)) which is nonzero because A is an inclusion M(−1, 2) ↪ M(−3, 5). Here ∧^{2}M(−1, 2) = M(−1, 1) and ∧^{2}M(−3, 5) ≅ M(−6, 5)^{⊕2} by [Ked05, Proposition 4.6.3]. Then θ_{∧^{2}t^{−1}D}(∧^{2}A) = ∧^{2}θ_{t^{−1}D}(A) = 0 implies that ∧^{2}A = T ∘ B for 0 ≠ B ∈ Hom_{φ}(M(−1, 1) , M(−1, 5)^{⊕2}) by Corollary 4.11. But since −1 < −\frac{1}{5}, the later space is zero by Proposition 4.12 which gives the contradiction to our assumption.

In particular the map α : { u ∈ Ẽ : 0 < |u| < 1 } → F̆^{an}, u ↦ μ parametrizes the complement F̆^{an} ∖ F̆_{b}^{0}. The nature of the map α is somewhat mysterious. It is continuous by similar arguments as in part (c) of the proof of Theorem 6.6 but its image does not meet the dense subset of points x ∈ F̆^{an} with H(x) finite over K_{0}.

We conclude this section by stating the following theorem which we will prove at the end of Section 7.

Theorem 6.8. For any representation V of G as in (6.1) there is a local system V of ℚ_{p}- vector spaces on F̆_{b}^{0} such that the fiber of V at any point μ ∈ F̆_{b}^{0}(L) for a finite extension L/Ĕ induces via (2.7) the crystalline Galois representation V_{cris}(\underline{D}_{b,μ}(V)) associated with the (weakly) admissible filtered isocrystal \underline{D}_{b,μ}(V) = (V_{K_{0}}, ρ(b)·φ, F il_{μ}^{•}V_{L}).

Remark 6.9. Note that this theorem is weaker than Conjecture 2.3 in two respects. Firstly we do not know in general whether V ↦ V is a tensor functor \underline{Rep}_{ℚ_{p}} G → ℚ_{p}-\underline{Loc}F̆_{b}^{0} as required in Conjecture 2.3. However this is true if G = GL_{n} (see Theorem 7.3) or if G is the group associated to an EL or PEL situation (see Theorem 8.4).

Secondly we do neither know whether F̆_{b}^{0} is connected nor whether it is the largest open subspace of F̆_{b}^{wa} with this property. However, we conjecture that this is true.

Conjecture 6.10. F̆_{b}^{0} is the largest open Ĕ-analytic subspace of F̆_{b}^{wa} carrying a local system V that has the properties stated in the theorem.

We give some evidence for this in Section 10.

7 Relation with Period Morphisms

For G = GL(V), Rapoport and Zink study a period morphism π̆ : Ğ^{an} → F̆_{b}^{wa}. We keep the notation of Section 6. In particular let b̃ ∈ GL(V)(K_{0}), let {μ̃} be a conjugacy class of cocharacters 𝔾_{m} → GL(V). Then Ĕ = K_{0}. We assume that V satisfies (6.1) for an integer ñ. We set b := p^{1−ñ}·b̃ ∈ G(K_{0}) and define μ : 𝔾_{m} → G by μ(z) = z^{1−ñ}·μ̃(z). Then \underline{D}_{b,μ}(V) = \underline{D}_{b̃,μ̃}(V) ⊗ 1l(1 − ñ) and M_{b,μ}(V) = M_{b̃,μ̃}(V) by Lemma 5.3. This implies that the assignment μ̃ ↦ μ is an isomorphism F̆ 0 ; b̃ ^{∼} ⟶ F̆_{b}^{0}.

Now (G, b, {μ}) and V satisfy (6.1) for n = 1. Therefore (V_{K_{0}}, b · φ) is the covariant Dieudonné-module (D, φ_{D}) of a p-divisible group X = X_{b}(V) over 𝔽_{p}^{alg}; see Messing [Mes72]. This means D := 𝔻(X) is the tensor product with K_{0} of the Lie algebra of the universal vector extension of some (any) lift of X to a p-divisible group over W (𝔽_{p}^{alg}). On X there is the Verschiebung morphism V er_{X} : Frob_{p}^{∗} X → X. By the crystalline nature of 𝔻(X) it induces the φ-semilinear isomorphism φ_{D} := 𝔻(V er_{X}) ∘ φ : D → D. Moreover, (6.1) and Lemma 6.1 imply

dim X = dim_{K_{0}} D − t_{N}(\underline{D}_{b,μ}(V)) = dim_{K_{0}} D − t_{H}(\underline{D}_{b,μ}(V)) = dim V − dim_{L} F il^{1}D_{L} =: d

and F̆^{an} is the Grassmannian of (dim V − d)-dimensional subspaces of V.

The period morphism is constructed as follows. We denote by Nilp_{W (𝔽_{p}^{alg})} the category

of W (𝔽_{p}^{alg})-schemes on which p is locally nilpotent. For S ∈ Nilp_{W (𝔽_{p}^{alg})} denote by S̄ the closed subscheme defined by the ideal pO_{S}. We set XS̄ := X ×_{𝔽_{p}^{alg}} S̄.

Proposition 7.1 ([RZ96, Theorem 2.16]). The contravariant functor Nilp_{W (𝔽_{p}^{alg})} ⟶ Sets

S ⟼ { Isomorphism classes of (X, η : XS̄ → XS̄) where

X is a p-divisible group over S and

η is a quasi-isogeny over S̄ to XS̄ = X ×_{S} S̄ }

is representable by a formal scheme Ğ locally formally of finite type over Spf W (𝔽_{p}^{alg}).

The group J(ℚ_{p}) from (2.3) equals the quasi-isogeny group of X. It acts on Ğ by γ: (X, η) ↦ (X, η ∘ γ^{−1}) for γ ∈ J(ℚ_{p}).

Rapoport and Zink also study formal moduli schemes corresponding to additional data on (X, η) of type EL and PEL; see Section 8. Their period morphisms are derived from the following prototype.

Let (X, η : X\overline{Ğ} → X\overline{Ğ}) be the universal p-divisible group with quasi-isogeny over Ğ. Let

𝔻(X)Ğ be the Lie algebra of the universal vector extension of X over Ğ. It sits in the exact sequence

0 ⟶ (Lie X^{∨})^{∨} ⟶ 𝔻(X)Ğ ⟶ Lie X ⟶ 0 .

Let Ğ^{an} be the K_{0}-analytic space associated with Ğ; see Theorem A.2. The quasi-isogeny η induces an isomorphism

𝔻(η) : D ⊗_{K_{0}} OĞ^{an} = 𝔻(X)_{K_{0}} ⊗_{K_{0}} OĞ^{an} ^{∼} ⟶ 𝔻(X)Ğ^{an}

see [RZ96, Proposition 5.15]. The kernel of the morphism

V_{K_{0}} ⊗_{K_{0}} OĞ^{an} ^{∼} ⟶ 𝔻(X)Ğ^{an} ⟶ (Lie X)Ğ^{an}

defines the Ğ^{an}-valued point 𝔻(η)^{−1}(Lie X^{∨})Ğ^{an}^{∨} of the Grassmannian F̆^{an}. This is the desired period morphism π̆ : Ğ^{an} → F̆^{an}. It is equivariant with respect to the J(ℚ_{p})- action on F̆^{an} under which γ ∈ J(ℚ_{p}) ⊂ G(K_{0}) sends the universal subspace F il^{1} on F̆^{an} to γ(F il^{1}). Rapoport and Zink show that π̆ factors through F̆_{b}^{wa}. They noticed that π̆ : Ğ^{an} → F̆_{b}^{wa} is in general not quasi-compact [RZ96, 5.53]. We will give an explanation for this fact in Remark 8.5(d) below.

Proposition 7.2. The period morphism factors through F̆_{b}^{0} and is surjective on the level of rigid analytic points.

Proof. We fix the tautological representation V of G, set D = V_{K_{0}}, and consider F̆^{an} as the Grassmannian of (dim V − d)-dimensional subspaces of V. Let x ∈ Ğ^{an} be an analytic point and let μ = π̆(x) ∈ F̆^{an}. Let L = H(x) and let C be the completion of an algebraic closure of L. Let X_{x} be the fiber of the universal p-divisible group X at x and consider the Tate module T_{p}X_{x} of X_{x}. An element λ ∈ T_{p}X_{x} corresponds to a morphism of p-divisible groups λ : ℚ_{p}/ℤ_{p} → X_{O_{C}} over O_{C}. By functoriality of the universal vector extension this yields the following diagram of C-vector spaces

[figure: 0; 𝔻(ℚ_{p}/ℤ_{p})_{C}; 𝔻(ℚ_{p}/ℤ_{p})_{C}; 𝔻(λ)_{C}; 0; 0; (F il_{μ}^{1}D_{L}) ⊗_{L} C; 𝔻(X)_{O_{L}} ⊗_{O_{L}} C; (Lie X_{x})_{C}; 0 .]  (7.1)

Note that Lie ℚ_{p}/ℤ_{p} = (0) and that F il_{μ}^{1}D_{L} is the L-subspace of D_{L} associated with X_{x} via the period morphism. We also obtain a morphism of crystals 𝔻(λ) : 𝔻(ℚ_{p}/ℤ_{p}) → 𝔻(X_{O_{C}}) which we evaluate on the pd-thickening B_{cris}^{+}(C) of O_{C}. Since 𝔻(ℚ_{p}/ℤ_{p})_{B_{cris}^{+}(C)} = B_{cris}^{+}(C) (because the universal vector extension of ℚ_{p}/ℤ_{p} over B_{cris}^{+}(C) is obtained from the sequence 0 → ℤ_{p} → ℚ_{p} → ℚ_{p}/ℤ_{p} → 0 by pushout via ℤ_{p} → B_{cris}^{+}(C)) we obtain a morphism

[figure: T_{p}X_{x} ⊗_{ℤ_{p}} B_{cris}^{+}(C); 𝔻(X_{x})_{B_{cris}^{+}(C)}; D ⊗_{K_{0}} B_{cris}^{+}(C); ∼; λ ⊗ a; 𝔻(λ)(a) ,]

which is compatible with the Frobenius on both sides. Here the isomorphism on the right arises from the quasi-isogeny η_{x} by the same reasoning as above. By Faltings’s [Fal99, Theorem 7] the morphism on the left is injective. Since the elements of T_{p}X_{x} are φ-invariant inside T_{p}X_{x} ⊗_{ℤ_{p}} B_{cris}^{+}(C) and B̃_{rig}^{+}(C) equals ⋂_{n∈ℕ_{0}} φ^{n}B_{cris}^{+}(C) we get a monomorphism

\xymatrix{T_{p}X_{x} ⊗_{ℤ_{p}} B̃_{rig}^{+}(C) \ar@{^{(}->}[r] & D ⊗_{K_{0}} B̃_{rig}^{+}(C) .}

It gives rise to a monomorphism

[figure: T_{p}X_{x}; T_{p}X_{x} ⊗_{ℤ_{p}} B̃^{]0,1]}(C); D ⊗_{K_{0}} B̃^{]0,1]}(C); λ; λ ⊗ 1; 𝔻(λ)(1)]

since B̃^{]0,1]}(C) is a flat B̃_{rig}^{+}(C)-algebra. Consider the morphism θ : D⊗_{K_{0}} B̃^{]0,1]} → D⊗_{K_{0}} C. Diagram (7.1) shows that θ(T_{p}X_{x}) ⊂ (F il_{μ}^{1}D_{L}) ⊗_{L} C and so by Proposition 5.1, the φmodule T_{p}X_{x} ⊗_{ℤ_{p}} B̃_{rig}^{†}(C) = M(0, 1) ⊗_{ℤ_{p}} T_{p}X_{x} is in fact contained in the φ-module M_{μ} := M(D, φ_{D}, F il_{μ}^{1}D_{L}). But then M_{μ} ≅ M(0, 1) ⊗_{ℚ_{p}} V by [Ked05, Lemma 3.4.2] because both φ-modules have the same rank and deg M_{μ} = 0. This proves that π̆ factors through F̆_{b}^{0}.

It remains to show that Ğ^{an} → F̆_{b}^{0} ↪ F̆_{b}^{wa} is surjective on the level of rigid analytic points. This follows from the Colmez-Fontaine Theorem [CF00] and Kisin’s proof [Kis06, Theorem 0.3] of Fontaine’s conjecture that every crystalline representation with Hodge-Tate weights 0 and −1 arises from a p-divisible group. See also Breuil [Bre00, Theorem 1.4].

Using a result of Faltings [Fal10], or alternatively a result of Scholze and Weinstein [SW13], we show in Theorem 7.3 below that F̆_{b}^{0} actually equals the image of the period morphism π̆. In order to formulate part (c) of the theorem we recall from Remark 2.7(a) that with each tensor functor \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}F̆_{b}^{0} one associates a tower of étale covering spaces

{ĔK̃}_{K̃⊂G̃(ℚ_{p})} of F̆_{b}^{0}. In the situation we treat in this section, G is the algebraic group GL(V) over ℚ_{p} and in the notation introduced before Corollary 2.6 we use Hilbert 90 (see [Mil80, Proposition III.4.9]) to identify ω̃ = ω_{0}, Ṽ = ω̃(V) = ω_{0}(V) = V, and G̃ = G = GL(Ṽ). The situation will be more general in the next section. Similarly to the covering spaces ĔK̃, Rapoport and Zink [RZ96, 5.32–5.39] constructed a tower of coverings Ğan ; K̃ of Ğ^{an}: For any compact open subgroup K̃ ⊂ G̃(ℚ_{p}) let Ğan ; K̃ be the Ĕ-analytic space representing K̃-level structures on the universal p-divisible group XĞ^{an}, that is classes modulo K̃ of trivializations α : Ṽ ^{∼} ⟶ (V_{p}XĞ^{an})μ̄ of the rational Tate module (at some geometric base point μ̄ of F̆_{b}^{0}), such that α mod K̃ is invariant under the étale fundamental group of Ğan ; K̃. The Ğan ; K̃ do not depend on the choice of μ̄. After fixing a ℤ_{p}-lattice Λ̃ in Ṽ and a preimage x̄ ∈ Ğ^{an} of μ̄ we can identify Ğ_{GL(Λ̃)}^{an} with Ğ^{an} by considering on Ğ^{an} the unique residue class of trivializations

(α : Λ̃ ^{∼} ⟶ (T_{p}XĞ^{an})x̄) mod GL(Λ̃), where α is any isomorphism. For K̃ ⊂ GL(Λ̃) the space Ğan ; K̃ is then a finite étale covering space of Ğ^{an}. For any K̃ and any g ∈ G̃(ℚ_{p}) there are F̆_{b}^{0}-isomorphisms iK̃(g) : Ğan ; K̃ ^{∼} ⟶ Ğ_{g^{−1}K̃g}^{an} given by (α mod K̃) ↦ (α g mod g^{−1}K̃g). This

proves that the Ğan ; K̃ are Ĕ-analytic spaces and Rapoport and Zink [RZ96, 5.39] show that they are independent of the choice of Λ̃ and x̄. On the tower (Ğan ; K̃)_{K̃⊂G̃(ℚ_{p})} the group

G̃(ℚ_{p}) acts through Hecke correspondences; see [RZ96, 5.34]. Also the action of J(ℚ_{p}) on Ğ^{an} lifts to an action on each Ğan ; K̃ by γ: (X, η, αK̃) ↦ (X, η ∘γ^{−1}, αK̃) for γ ∈ J(ℚ_{p}), which commutes with the Hecke-action. Rapoport and Zink conjectured that the Ğan ; K̃ are étale covering spaces of the image π̆(Ğ^{an}); compare [RZ96, 1.37]. We prove this in part (c) of the following

Theorem 7.3. Let G be the algebraic group GL(V) over ℚ_{p} and set G̃ = G.

(a) The image π̆(Ğ^{an}) of the period morphism π̆ for GL(V) equals F̆_{b}^{0}.

(b) The rational Tate module V_{p}XĞ^{an} of the universal p-divisible group XĞ^{an} over Ğ^{an} descends to a local system V of ℚ_{p}-vector spaces on F̆_{b}^{0}. This induces a tensor functor \underline{V} from \underline{Rep}_{ℚ_{p}}G to the category ℚ_{p}-\underline{Loc}F̆_{b}^{0} of local systems of ℚ_{p}-vector spaces on F̆_{b}^{0} satisfying (2.8) of Conjecture 2.3 and \underline{V}(V) ≅ V.

(c) The tower of Ĕ-analytic spaces (Ğan ; K̃)_{K̃⊂G̃(ℚ_{p})} is canonically isomorphic over F̆_{b}^{0} in a

Hecke equivariant way to the tower of étale covering spaces (ĔK̃)_{K̃⊂G̃(ℚ_{p})} of F̆_{b}^{0} asso-

ciated with the tensor functor \underline{V}. In particular, Ğ^{an} is (non-canonically) isomorphic to the space of ℤ_{p}-lattices inside the local system V.

(d) The tensor functor \underline{V} carries a canonical J(ℚ_{p})-linearization which by Remark 2.7(b) induces an action of J(ℚ_{p}) on the tower (ĔK̃)_{K̃⊂G̃(ℚ_{p})}. The isomorphism from (c) is equivariant for the J(ℚ_{p})-action on both towers.

A previously known example for this theorem is the Lubin-Tate situation where F̆_{b}^{0} = F̆_{b}^{wa} = F̆^{an} = ℙ_{K_{0}}^{h−1}. In this case the theorem was proved by de Jong [dJ95, Proposition 7.2] .

Proof of Theorem 7.3. (a) The inclusion π̆(Ğ^{an}) ⊂ F̆_{b}^{0} was proved in Proposition 7.2. So let now μ ∈ F̆_{b}^{0} and let C be the completion of an algebraic closure of L := H(μ). Consider the twisted dual

\underline{D̃} := ((D, φ_{D}, F il_{μ}^{•}D_{L}) ⊗ 1l(−1))^{∨} = Hom((D, φ_{D}, F il_{μ}^{•}D_{L}), 1l(1)).

It satisfies F il^{0}\underline{D̃}_{C} = \underline{D̃}_{C} and F il^{2}\underline{D̃}_{C} = (0). We abbreviate D̃ := D(\underline{D̃}) and M̃ := M(\underline{D̃}). Then M̃ ≅ M(0, 1) ⊗_{ℚ_{p}} V^{∨} by Lemma 5.3 and Proposition 5.4. Consider the morphisms

α : (B̃_{rig}^{†}) ⊗_{ℚ_{p}} V^{∨} = M(0, 1) ⊗_{ℚ_{p}} V^{∨} ≅ M̃ ↪ t^{−1}D̃ and β : D̃ ↪ M̃ ≅ (B̃_{rig}^{†}) ⊗_{ℚ_{p}} V^{∨}

and note that t^{−1}D̃ ≅ D(\underline{D̃} ⊗ 1l(−1)). Let h = dim V^{∨} and fix a basis of V^{∨}. This induces a basis of D̃ = V^{∨} ⊗_{ℚ_{p}} B̃_{rig}^{†}. With respect to this basis, α and β are represented by matrices A, B ∈ M_{h}(B̃_{rig}^{†}) satisfying AB = t Id_{h}. Then A, B ∈ M_{h}(B̃_{rig}^{+}) ⊂ M_{h}(B_{cris}^{+}) by [Ber04, Proposition I.4.1]. So A defines an isomorphism α_{cris} : B_{cris}^{⊕h} ^{∼} ⟶ (\underline{D̃} ⊗ 1l(−1)) ⊗_{K_{0}} B_{cris} compatible with the Frobenius on both sides. Moreover, it maps (B_{cris}^{+})^{⊕h} onto the preimage of F il_{μ}^{1}\underline{D̃}_{C} under the map id ⊗θ : (\underline{D̃} ⊗ 1l(−1)) ⊗_{K_{0}} B_{cris}^{+} → \underline{D̃} ⊗_{K_{0}} C. We denote this preimage by F il^{1}(\underline{D̃} ⊗_{K_{0}} B_{cris}). Tensoring α_{cris} with 1l(1) ⊗_{K_{0}} B_{cris} and precomposing with B_{cris}^{⊕h} ^{∼} ⟶ 1l(1) ⊗_{K_{0}} B_{cris}^{⊕h}, x ↦ t^{−1}x yields an isomorphism B_{cris}^{⊕h} ^{∼} ⟶ \underline{D̃} ⊗_{K_{0}} B_{cris} compatible with Frobenius which maps F il^{1}B_{cris}^{⊕h} := (t · B_{cris}^{+})^{⊕h} onto F il^{1}(\underline{D̃} ⊗_{K_{0}} B_{cris}). This means that \underline{D̃} is admissible in the sense of Faltings [Fal10, Definition 1].

By [Fal10, Theorems 9 and 14], the filtered isocrystal \underline{D̃} comes from a p-divisible group X over O_{L} and a quasi-isogeny η : X_{O_{L}/(p)} → X_{O_{L}/(p)}. Note that Faltings uses contravariant Dieudonné modules and that his assertions for the covariant Dieudonné functor 𝔻 mean F il_{μ}^{1}D_{L} = 𝔻(η^{−1})(Lie X^{∨})L^{∨}. Therefore μ lies in the image of π̆^{an} as desired.

(b) This is proved by de Jong [dJ95]. Indeed by [dJ95, Proposition 6.2] the rational Tate module V_{p}XĞ^{an} is a local system of ℚ_{p}-vector spaces on Ğ^{an}. In order that it defines a local system V on F̆_{b}^{0} it suffices by [dJ95, Definition 4.1] to show that

(i) π̆ : Ğ^{an} → F̆_{b}^{0} is a covering for the étale topology.

(ii) There is a descent datum ψ̃ : pr_{1}^{∗}V_{p}XĞ^{an} ^{∼} ⟶ pr_{2}^{∗}V_{p}XĞ^{an} over Ğ^{an} ×F̆_{b}^{0} Ğ^{an} where

pr_{i} : Ğ^{an} ×F̆_{b}^{0} Ğ^{an} → Ğ^{an} is the projection onto the i-th factor, such that ψ̃ satisfies

the cocycle condition on Ğ^{an} ×F̆_{b}^{0} Ğ^{an} ×F̆_{b}^{0} Ğ^{an}.

Statement (i) was proved by Fargues [Far04, Lemma 2.3.24] and also follows from Faltings’s result [Fal10, Theorem 14] which says that F̆_{b}^{0} has an open covering F̆_{b}^{0} = ⋃_{i} U_{i} such that the morphism ∐_{i} U_{i} → F̆_{b}^{0} factors through π̆ : Ğ^{an} → F̆_{b}^{0}.

For (ii) consider the universal filtered isocrystal \underline{D} = (V_{K_{0}}, bφ, F il^{•}) over F̆_{b}^{0} and its canonical descent datum id : pr_{1}^{∗}π̆^{∗}\underline{D} = pr_{2}^{∗}π̆^{∗}\underline{D} over Ğ^{an} ×F̆_{b}^{0} Ğ^{an}. De Jong considers

pr_{i}^{∗}XĞ^{an} as an object of the stack BT_{ℚ,Ğ^{an}×F̆^{0}_{b} Ğ^{an}}^{rig} of p-divisible groups up to isogeny over

Ğ^{an} ×F̆_{b}^{0} Ğ^{an}. The functor from this stack to filtered isocrystals sends pr_{i}^{∗}XĞ^{an} to pr_{i}^{∗}π̆^{∗}\underline{D}. Since this functor is fully faithful by [dJ95, Proposition 6.6] one obtains a descent datum ψ : pr_{1}^{∗}XĞ^{an} ^{∼} ⟶ pr_{2}^{∗}XĞ^{an} in BT_{ℚ,Ğ^{an}×F̆^{0}_{b} Ğ^{an}}^{rig}. One now applies the Tate-module functor V_{p}

and takes ψ̃ = V_{p}ψ : pr_{1}^{∗}V_{p}XĞ^{an} ^{∼} ⟶ pr_{2}^{∗}V_{p}XĞ^{an}. This yields the local system V on F̆_{b}^{0}.

The local system V induces the desired tensor functor \underline{V} because V is a tensor generator of \underline{Rep}_{ℚ_{p}}GL(V). Let us give some more details. We do not know whether F̆_{b}^{0} is connected, although we expect this to hold; see Conjecture 6.5. Therefore we consider each connected component Y of it separately and fix a geometric base point μ̄ of Y. Then V induces a representation

π_{1}^{ét}(Y, μ̄) ⟶ GL(Vμ̄)(ℚ_{p})

by Proposition 2.2. We use the notation introduced before Corollary 2.6. Since dim V = dim Vμ̄ and H^{1}(ℚ_{p}, GL(V)) = (0) by Hilbert 90, we can take ω̃ = ω_{0} and G̃ = GL(V) ≅ GL(Vμ̄). This defines a continuous group homomorphism π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p}) and via Corollary 2.6 a tensor functor \underline{V} : \underline{Rep}_{ℚ_{p}}GL(V) → ℚ_{p}-\underline{Loc}_{Y} satisfying (2.9). Tracing through the proof of Corollary 2.6 we see that its composition with the forgetful fiber functor ωμ̄ satisfies ωμ̄ ∘\underline{V} ≅ ω̃ = ω_{0}. Hence ωμ̄ ∘\underline{V}(V) ≅ V ≅ Vμ̄ and \underline{V}(V) ≅ V. Moreover, for every μ ∈ F̆_{b}^{0}(L) with L/K_{0} finite the Gal(L^{alg}/L)-representations satisfy

\underline{V}(V)_{μ} ≅ V_{p}X_{O_{L}} ≅ V_{cris}(𝔻(X_{O_{L}}), (Lie XO_{L}^{∨})^{∨}) ≅ V_{cris}(\underline{D}_{b,μ}(V)) .

This holds on all connected components Y and therefore gives the tensor functor \underline{V} satisfying (2.8) of Conjecture 2.3 on all of F̆_{b}^{0} as claimed.

(c) We fix a geometric base point μ̄ of F̆_{b}^{0} and consider the canonical family of morphisms fK̃ : Ğan ; K̃ → ĔK̃ which sends αK̃ for α : Ṽ ^{∼} ⟶ (V_{p}XĞ^{an})μ̄ to the K̃-residue class βK̃ of tensor isomorphisms β : ω̃ ^{∼} ⟶ ωμ̄ ∘ \underline{V} induced by β_{V} := α : ω̃(V) = Ṽ ^{∼} ⟶ (V_{p}XĞ^{an})μ̄ = ωμ̄ ∘ \underline{V}(V) via the fact that V is a tensor generator of \underline{Rep}_{ℚ_{p}}G. Again fK̃ is independent of the choice of μ̄ by [dJ95, Theorem 2.9]. By construction the family (fK̃)K̃ is equivariant for the Hecke action of G̃(ℚ_{p}) on both towers. For any algebraically closed complete extension C of Ĕ the morphism fK̃ is bijective on C-valued points because the fibers of Ğan ; K̃(C) and ĔK̃(C) over a fixed C-valued point of F̆_{b}^{0} are both isomorphic to the coset G̃(ℚ_{p})/K̃ by [RZ96, Proposition 5.37] and Remark 2.7(a). Hence fK̃ is quasi-finite by [Beh93, Proposition 3.1.4]. Since Ğan ; K̃ and ĔK̃ are étale over F̆_{b}^{0} the morphisms fK̃ are étale by [Beh93, Corollary 3.3.9] and hence isomorphisms by Proposition A.4.

The statement about Ğ^{an} comes from the identification Ğ^{an} ≅ Ğ_{GL(Λ̃)}^{an} described before

the theorem and the fact that Ĕ_{GL(Λ̃)} is the space of ℤ_{p}-lattices inside V.

(d) Recall that the action γ : Ğ → Ğ of γ ∈ J(ℚ_{p}) is defined using the universal property of Ğ by requiring that γ^{∗}(XĞ, η) ≅ (XĞ, η ∘ γ^{−1}), i.e., there is an isomorphism Φ_{γ}: γ^{∗}XĞ ^{∼} ⟶ XĞ with (Φ_{γ})\overline{Ğ} ∘ γ^{∗}η = η ∘ γ^{−1}. By rigidity of quasi-isogenies the isomorophism Φ_{γ} is uniquely determined and a straight forward calculation shows that Φ_{γ} ∘γ^{∗}Φ_{δ} = Φ_{δγ}. By definition of the local system V as V_{p}XĞ^{an} this yields a canonical J(ℚ_{p})-linearization V_{p}Φ_{γ} : γ^{∗}\underline{V} ^{∼} ⟶ \underline{V} on \underline{V}. Now the equivariance of the isomorphism fK̃: Ğan ; K̃ ^{∼} ⟶ ĔK̃ , (X, η, αK̃) ↦ βK̃ from (c) follows from the explicit description of the J(ℚ_{p})-actions given in (2.11) and before Theorem 7.3.

Remark 7.4. (a) If the Newton polygon and the Hodge polygon of b have no point in common except for the two endpoints, then Chen [Che11, Lemme 5.1.1.1 and Théorème 5.1.1.1] proved that the image of the period morphism F̆_{b}^{0} is (arcwise) connected.

(b) In the case when J is an inner form of G, that is, when the isocrystal (V_{K_{0}}, b · φ) is isoclinic, Kottwitz has formulated a conjecture on how the compactly supported ℓ-adic étale cohomology of the towers in Theorem 7.3(c) decomposes as a representation of G̃(ℚ_{p}) × J(ℚ_{p}) × W_{E}, where W_{E} is the Weil group of E; see [Rap95, Conjecture 5.1].

(c) Let \underline{D} := (V_{K_{0}}, b·φ, F il_{μ}^{•}) with μ ∈ F̆^{an} be a filtered isocrystal satisfying (6.1) for n = 1. One can show that Faltings’s notion of admissibility [Fal10, Definition 1] for \underline{D} is equivalent to our condition M(\underline{D}) ≅ M(0, 1) ⊗_{ℚ_{p}} V from (6.2). In fact, our proof of Theorem 7.3(a) shows that our condition implies Faltings’s admissibility for \underline{D̃} := Hom(\underline{D}, 1l(1)). Note also that \underline{D̃} is admissible if and only if \underline{D} is by [Fal10, Remarks after Definition 1], and that M(\underline{D̃}) ≅ M(\underline{D})^{∨} by Lemma 5.3 and Proposition 5.4.

To prove the converse we use the notation of the proof of Theorem 7.3(a) and assume that \underline{D̃} is admissible in the sense of Faltings via an isomorphism B_{cris}^{⊕h} ^{∼} ⟶ \underline{D̃} ⊗_{K_{0}} B_{cris} compatible with Frobenius and the filtrations F il^{1}. Tensoring this isomorphism with 1l(−1) ⊗_{K_{0}} B_{cris} and precomposing with B_{cris}^{⊕h} ^{∼} ⟶ 1l(−1) ⊗_{K_{0}} B_{cris}^{⊕h}, x ↦ tx yields an isomorphism α_{cris} : B_{cris}^{⊕h} ^{∼} ⟶ (\underline{D̃} ⊗ 1l(−1)) ⊗_{K_{0}} B_{cris} compatible with Frobenius, which maps (B_{cris}^{+})^{⊕h} onto F il^{1}(\underline{D̃} ⊗_{K_{0}} B_{cris}) ⊂ (\underline{D̃} ⊗1l(−1))⊗_{K_{0}} B_{cris}^{+}. Since B̃_{rig}^{+} = ⋂_{n∈ℕ_{0}} φ^{n}B_{cris}^{+} the isomorphism α_{cris} comes from a monomorphism (B̃_{rig}^{+})^{⊕h} ↪ (\underline{D̃} ⊗ 1l(−1)) ⊗_{K_{0}} B̃_{rig}^{+}, which by flatness of B̃_{rig}^{†} over B̃_{rig}^{+} induces a monomorphism

M(0, 1)^{⊕h} ↪ (\underline{D̃} ⊗ 1l(−1)) ⊗_{K_{0}} B̃_{rig}^{†} ≅ t^{−1}D(\underline{D̃}).

By construction of M(\underline{D̃}) in 5.2 and the fact that α_{cris}(B_{cris}^{+})^{⊕h} = F il^{1}(\underline{D̃} ⊗_{K_{0}} B_{cris}), this monomorphism maps M(0, 1)^{⊕h} isomorphically onto M(\underline{D̃}). This proves that

M(\underline{D}) ≅ M(\underline{D̃})^{∨} ≅ (M(0, 1)^{⊕h})^{∨} ≅ M(0, 1)^{⊕h}.

Therefore Faltings’s notion of admissibility is equivalent to ours. Using this equivalence, the openness of F̆_{b}^{0} in Theorem 6.6 also follows from [Fal10, Corollary 10].

(d) There is an alternative proof by Scholze and Weinstein [SW13], not relying on Faltings’s [Fal10] results, which shows that F̆_{b}^{0} equals the image of the period morphism π̆. By using this alternative proof and our Theorem 6.6, one could make the whole treatment in this article independent of [Fal10].

Corollary 7.5. Let G = GL(V), let b̃ ∈ G(K_{0}), and let {μ̃} be a conjugacy class of cocharacters 𝔾_{m} → G such that V satisfies (6.1) for some integer ñ. Then there is a tensor functor \underline{Ṽ} from \underline{Rep}_{ℚ_{p}}G to the category ℚ_{p}-\underline{Loc}F̆ 0 ; b̃ of local systems of ℚ_{p}-vector spaces on

F̆ 0 ; b̃ satisfying (2.8) of Conjecture 2.3.

Proof. Like at the beginning of this section we set b := p^{1−ñ} ·b̃ ∈ G(K_{0}) and define the cocharacter μ: 𝔾_{m} → G by μ(z) = z^{1−ñ}·μ̃(z) to obtain identifications \underline{D}_{b,μ}(V) = \underline{D}_{b̃,μ̃}(V)⊗ 1l(1 − ñ) and F̆ 0 ; b̃ ^{∼} ⟶ F̆_{b}^{0} , μ̃ ↦ μ. Then Theorem 7.3(b) yields a tensor functor \underline{V}, which for every L-valued point μ̃ ∈ F̆ 0 ; b̃ for L/K_{0} finite satisfies

\underline{V}(V)_{μ} ≅ V_{cris}(\underline{D}_{b,μ}(V)) = V_{cris}(\underline{D}_{b̃,μ̃}(V)) ⊗ ℚ_{p}(ñ − 1)

as Gal(L^{alg}/L)-representations because V_{cris}(1l(1)) = ℚ_{p}(−1). Thus it remains to twist \underline{V} with the cyclotomic character. In concrete terms this means the following. On each connected component Y of F̆ 0 ; b̃ with some geometric base point μ̄ the tensor functor \underline{V} corresponds to a representation ρ′ : π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p}) ≅ GL_{h}(ℚ_{p}). We now consider the representation ρ̃′ : π_{1}^{ét}(Y, μ̄) → GL_{h}(ℚ_{p}) defined by ρ̃′(γ) := χ_{cyc}(γ)^{1−ñ} · ρ′(γ) where χ_{cyc} : π_{1}^{ét}(Y, μ̄) → Gal(Ĕ^{alg}/Ĕ) ⟶^{χ_{cyc}} ℤp^{×} is the cyclotomic character of the base field Ĕ. Let \underline{Ṽ} be the tensor functor from \underline{Rep}_{ℚ_{p}}G to ℚ_{p}-\underline{Loc}F̆ 0 ; b̃ associated with ρ̃′ by Corollary 2.6.

It is independent of the base point μ̄ and hence exists on all of F̆ 0 ; b̃. Then there is an isomorphism of Gal(L^{alg}/L)-representations \underline{Ṽ}(V)μ̃ ≅ V_{cris}(\underline{D}_{b̃,μ̃}(V)) for all μ̃ and this proves the corollary.

Finally we can give the

Proof of Theorem 6.8. Let ρ : G ↪ GL(V) =: G′ be a representation satisfying (6.1) and consider the induced embedding F̆ ↪ F̆′ := Flag(V). By Proposition 6.2 we have F̆_{b}^{0} = F̆^{an} ∩ F̆′_{b}^{0}. Now the local system \underline{Ṽ}(V) of Corollary 7.5 on F̆′_{b}^{0} pulls back to a local system of ℚ_{p}-vector spaces on F̆_{b}^{0} which has the required property.

Conjecture 7.6. The open subspace F̆ 0 ; b̃ from Corollary 7.5 is the largest open Ĕ-analytic subspace F̆ a ; b̃ of F̆ wa ; b̃ on which a tensor functor \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}F̆ a ; b̃ exists which satisfies (2.8) of Conjecture 2.3.

In particular, we expect that F̆ 0 ; b̃ and \underline{V} from Corollary 7.5 solve the problem posed in Conjecture 2.3 for G = GL(V). Clearly the conjecture follows from Conjecture 6.10 because \underline{V}(V) is a local system as in Conjecture 6.10. We give some evidence for both conjectures in Section 10.

8 PEL period morphisms

Rapoport and Zink also consider period morphisms in the PEL situation with parahoric level structure. They fix data as follows:

Case EL:

Let B be a finite semi-simple algebra over ℚ_{p}.

Let O_{B} be a maximal order in B.

Let V be a finite dimensional B-module.

Let G = GL_{B}(V) as an algebraic group over ℚ_{p}.

Case PEL:

Let B, O_{B}, V be as in case EL.

Let ( . , .) be a non-degenerate alternating ℚ_{p}-bilinear form on V.

Let ∗ : a ↦ a^{∗} be an involution of B which satisfies

(av, w) = (v, a^{∗}w) for all v, w ∈ V.

Let G be the algebraic group over ℚ_{p} whose points with values in a ℚ_{p}-algebra R are

G(R) = { g ∈ GL_{B⊗R}(V_{R}) : ∃ c(g) ∈ R with (gv, gw) = c(g)(v, w) ∀ v, w ∈ V_{R} },

where V_{R} := V ⊗_{ℚ_{p}} R.

In addition they fix an element b ∈ G(K_{0}), a conjugacy class of cocharacters μ : 𝔾_{m} → G and a multichain L of O_{B}-lattices in V [RZ96, Definition 3.4]. They assume that the representation V of G satisfies our condition (6.1) with n = 1. In case PEL they assume in addition that the character c : G → 𝔾_{m} pairs with the slope quasi-cocharacter ν ∈ Hom_{K_{0}}(𝔾_{m}, G) ⊗_{ℤ} ℚ to 1, and that L is self-dual [RZ96, Definition 3.13].

Under these assumptions there is a p-divisible group X over 𝔽_{p}^{alg} whose covariant Dieudonné module is (V_{K_{0}} , b·φ). In particular the group X is equipped with an action i_{X} : B → End(X) ⊗_{ℤ_{p}} ℚ_{p} and, in case PEL a polarization λ : X → X^{∨} with λ^{∨} = −λ, which is uniquely determined up to multiplication by an element of ℚp^{×}; see [RZ96, 3.20]. Let E be the Shimura field, that is the field of definition of the conjugacy class of μ, and let Ĕ = E · K_{0}. Rapoport and Zink consider the following moduli problem.

Definition 8.1 ([RZ96, Definition 3.21]). Let Ğ : Nilp_{O_{Ĕ}} → Sets be the functor which assigns to S ∈ Nilp_{O_{Ĕ}} the following data up to isomorphism.

(a) For each lattice Λ ∈ L a p-divisible group X_{Λ} over S, with an O_{B}-action, i_{Λ} : O_{B} → End X_{Λ}, and

(b) for each lattice Λ ∈ L a quasi-isogeny η_{Λ} : XS̄ → X_{Λ,S̄} which commutes with the action of O_{B}, (here again S̄ is the closed subscheme of S defined by the ideal pO_{S})

such that certain conditions (i)–(v) are satisfied for which we refer to [RZ96, Definition 3.21 and the discussion on pp. 82–88].

Proposition 8.2 ([RZ96, Theorem 3.25]). The functor Ğ: Nilp_{O_{Ĕ}} → Sets is representable by a formal scheme locally formally of finite type over O_{Ĕ}.

The group J(ℚ_{p}) from (2.3) equals the group of quasi-isogenies of X that respect the B-action and the polarization. This group acts on Ğ by γ: (X_{Λ}, η_{Λ})_{Λ} ↦ (X_{Λ}, η_{Λ} ∘ γ^{−1})_{Λ} for γ ∈ J(ℚ_{p}).

The period morphism Ğ^{an} → F̆_{b}^{wa} is defined as follows. Fix a Λ ∈ L, let X_{Λ} be the universal p-divisible group over Ğ, and consider the exact sequence

0 ⟶ (Lie XΛ^{∨})^{∨} ⟶ 𝔻(X_{Λ})Ğ ⟶ Lie X_{Λ} ⟶ 0 .

Then the quasi-isogeny η_{Λ} induces an isomorphism

𝔻(η_{Λ})Ğ^{an} : V ⊗_{ℚ_{p}} OĞ^{an} = 𝔻(X)Ğ^{an} ^{∼} ⟶ 𝔻(X_{Λ})Ğ^{an}

that makes 𝔻(η_{Λ})−1 ; Ğ^{an}(Lie X^{∨})Ğ^{an}^{∨} into a family over Ğ^{an} of B-invariant subspaces of V, which

in case PEL are totally isotropic. These subspaces are classified by F̆^{an} and this defines the period morphism π̆ : Ğ^{an} → F̆^{an}. The whole construction does not dependent on the choice of Λ ∈ L; see [RZ96, 5.16]. The period morphism factors through F̆_{b}^{wa} and by the same argument as in Proposition 7.2 it even factors through F̆_{b}^{0}. We will show in Theorem 8.4 below, that its image equals F̆_{b}^{0}. In Remark 8.5(d) we will give an explanation for the fact noticed by Rapoport and Zink [RZ96, 5.53], that π̆ : Ğ^{an} → F̆_{b}^{wa} is in general not quasi-compact. The period morphism is also equivariant with respect to the J(ℚ_{p})-action on F̆^{an} under which γ ∈ J(ℚ_{p}) ⊂ G(K_{0}) sends μ ∈ F̆^{an} to γμγ^{−1}.

Recall the cohomology class

cl(b, μ) = κ(b) − μ^{#} ∈ H^{1}(ℚ_{p}, G)

and the associated fiber functor ω̃ : \underline{Rep}_{ℚ_{p}}G → (ℚ_{p}-vector spaces) and inner form G̃ of

G described before Corollary 2.6. Let Ṽ := ω̃(V). Since B acts as endomorphisms of the representation V of G, also Ṽ is a B-module. In the PEL case Ṽ is equipped with a non-degenerate alternating ℚ_{p}-bilinear form ( . , .)^{∼} such that

(aṽ, w̃)^{∼} = (ṽ, a^{∗}w̃)^{∼}  for all ṽ, w̃ ∈ Ṽ and all a ∈ B .  (8.1)

Namely, the original form ( . , .) : V ⊗_{ℚ_{p}} V → ℚ_{p} is a morphism from the G-representation V ⊗_{ℚ_{p}} V to the G-representation ℚ_{p} on which G acts through the character c. We let ( . , .)^{∼} be the image of ( . , .) under ω̃. Formula (8.1), being an equality of morphisms Ṽ ⊗_{ℚ_{p}} Ṽ → ℚ_{p}, is the image under ω̃ of the corresponding equality for V. By construction of G̃ we have G̃ ≅ Aut^{⊗}(ω̃) and \underline{Rep}_{ℚ_{p}}G ≅ \underline{Rep}_{ℚ_{p}}G̃; see [DM82, Proposition 2.8 and Theorem 2.11]. Applying ω̃ to the character c : G → 𝔾_{m} we obtain another character which we view as a G̃-representation c̃ : G̃ → 𝔾_{m}. In this way we get identifications of algebraic groups over ℚ_{p}

G̃ = GL_{B}(Ṽ)  in case EL, (8.2)
G̃ = { g̃ ∈ GL_{B}(Ṽ) : (g̃ṽ, g̃w̃)^{∼} = c̃(g̃)(ṽ, w̃)^{∼} ∀ ṽ, w̃ ∈ Ṽ } in case PEL.

Consider a decomposition of B into a product ∏_{i=1}^{r} B_{i} of simple algebras B_{i} ≅ M_{n_{i}}(D_{i}), which are matrix algebras over division algebras D_{i}, in such a way that O_{B} ≅ M_{n_{i}}(O_{D_{i}}). Then any O_{B}-module Λ decomposes accordingly Λ ≅ ⨁_{i=1}^{r} Λ_{i}.

Definition 8.3. In the EL case a multichain of type (L) of O_{B}-lattices in Ṽ is a functor : Λ ↦ Λ̃ from L to the category of O_{B}-lattices in Ṽ such that

(ĩi) If Λ ⊂ Λ′ then Λ̃ ⊂ Λ̃′ and [Λ̃′_{i} : Λ̃_{i}] = [Λ′_{i} : Λ_{i}] for i = 1, …, r.

(ĩii) For any a ∈ B with a^{−1}O_{B}a = O_{B} we have ãΛ = a · Λ̃ .

In the PEL case we speak of a polarized multichain of type (L) and we require in addition that there is a unit ℓ̃∈ ℚp^{×} with

(ṽ) ℓ̃· ˜(Λ^{∨}) = (Λ̃)^{∨} := { ṽ ∈ Ṽ : (ṽ, w̃)^{∼} ∈ ℤ_{p} for all w̃ ∈ Λ̃ } for every Λ ∈ L.

The unusual numbering of our conditions is chosen because the items of our definition correspond to the respective items of [RZ96, Definition 3.21].

For the next theorem again recall from Remark 2.7(a) that with each tensor functor \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}F̆_{b}^{0} one associates a tower of étale covering spaces ĔK̃ of F̆_{b}^{0} for

K̃ ⊂ G̃(ℚ_{p}) compact open. Moreover, we again consider the tower of étale coverings Ğan ; K̃ of Ğ^{an} with its commuting actions of J(ℚ_{p}) by γ: (X_{Λ}, η_{Λ}, αK̃) ↦ (X_{Λ}, η_{Λ} ∘ γ^{−1}, αK̃) for γ ∈ J(ℚ_{p}), and of G̃(ℚ_{p}) by Hecke-correspondences; see Rapoport and Zink [RZ96, 5.32–5.39] and our discussion before Theorem 7.3.

Theorem 8.4. Assume that Ğ^{an} ≠ ∅.

(a) Then the image π̆(Ğ^{an}) of the EL or PEL period morphism π̆ equals the open Ĕanalytic subspace F̆_{b}^{0} ⊂ F̆^{an} from Theorem 6.6.

(b) Fix any lattice Λ ∈ L. Then the rational Tate module V_{p}X_{Λ,Ğ^{an}} of the universal pdivisible group X_{Λ,Ğ^{an}} over Ğ^{an} descends to a local system V of ℚ_{p}-vector spaces on F̆_{b}^{0} which is independent of Λ up to canonical isomorphism. This induces a tensor functor \underline{V} from \underline{Rep}_{ℚ_{p}}G to the category ℚ_{p}-\underline{Loc}F̆_{b}^{0} of local systems of ℚ_{p}-vector spaces

on F̆_{b}^{0} satisfying (2.8) of Conjecture 2.3 and \underline{V}(V) ≅ V.

(c) The tower of Ĕ-analytic spaces (Ğan ; K̃)_{K̃⊂G̃(ℚ_{p})} is canonically isomorphic over F̆_{b}^{0} in a

Hecke equivariant way to the tower of étale covering spaces (ĔK̃)_{K̃⊂G̃(ℚ_{p})} of F̆_{b}^{0} asso-

ciated with the tensor functor \underline{V}. In particular, Ğ^{an} is (non-canonically) isomorphic to the space of (polarized in case PEL) multichains of type (L) inside the local system V (which are defined as in Definition 8.3 using Proposition 2.2).

(d) The tensor functor \underline{V} carries a canonical J(ℚ_{p})-linearization which by Remark 2.7(b) induces an action of J(ℚ_{p}) on the tower (ĔK̃)_{K̃⊂G̃(ℚ_{p})}. The isomorphism from (c) is equivariant for the J(ℚ_{p})-action on both towers.

Remark 8.5. (a) For each connected component Y of F̆_{b}^{0} and each geometric base point μ̄ of Y the tensor functor \underline{V} corresponds to a representation ρ′ : π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p}) by Corollary 2.6. In the course of the proof we show that the composition c̃ ∘ ρ′ with the multiplier c̃ of G̃ equals the cyclotomic character π_{1}^{ét}(Y, μ̄) → Gal(Ĕ^{alg}/Ĕ) ⟶^{χ_{cyc}} ℤp^{×} of the base field Ĕ.

(b) If F/ℚ_{p} is unramified and the Newton polygon and the Hodge polygon of b have no point in common except for the two endpoints, then Chen, Kisin, and Viehmann [CKV13] proved that the image of the period morphism F̆_{b}^{0} is (arcwise) connected; see also [Che11, Lemme 5.1.1.1].

(c) Theorem 8.4(c) implies that Ğan ; K̃ ×F̆_{b}^{0} Y is the étale covering space of Y correspond-

ing to the π_{1}^{ét}(Y, μ̄)-set Isom^{⊗}(ω̃, ωμ̄ ∘ \underline{V})/K̃, which can be identified non-canonically with G̃(ℚ_{p})/K̃; see Remark 2.7. This had been conjectured by Rapoport and Zink [RZ96, 1.37].

(d) The theorem gives a natural explanation for the fact that in general π̆ is not quasicompact, which was discovered in [RZ96, 5.53]. Firstly the fibers of Ğ^{an} over F̆_{b}^{0} are in bijection with a coset of G̃(ℚ_{p}) modulo a compact open subgroup; see the proof of Theorem 8.4(c). In particular the fibers are infinite in general. Secondly, if the inclusion F̆_{b}^{0} ⊂ F̆_{b}^{wa} were quasi-compact then it would be an isomorphism. But Example 6.7 and Theorem 9.3 show that this occurs only in a few low-dimensional cases.

(e) In the case when b is basic, that is, when J is an inner form of G, Kottwitz has formulated a conjecture on how the compactly supported ℓ-adic étale cohomology of the towers in Theorem 8.4(c) decomposes as a representation of G̃(ℚ_{p}) × J(ℚ_{p}) × W_{E}, where W_{E} is the Weil group of E; see [Rap95, Conjecture 5.1].

Before proving the theorem let us mention the following

Conjecture 8.6. The open subspace F̆_{b}^{0} is the largest open Ĕ-analytic subspace F̆_{b}^{a} of F̆_{b}^{wa} on which a tensor functor \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}F̆_{b}^{a} exists which satisfies (2.8) of Conjecture 2.3.

In particular, we expect that F̆_{b}^{0} and \underline{V} from Theorem 8.4(b) solve the problem posed in Conjecture 2.3. Again the conjecture follows from Conjecture 6.10 because \underline{V}(V) is a local system as in Conjecture 6.10. We give some evidence for both conjectures in Section 10. We now turn to the

Proof of Theorem 8.4. (a) 1. By our assumption Ğ^{an} ≠ ∅, there is a finite field extension L/Ĕ and a point x_{0} ∈ Ğ(O_{L}) corresponding to an L-set of p-divisible groups with O_{B}- action {X_{Λ}, i_{Λ}}_{Λ∈L} over O_{L} and O_{B}-equivariant quasi-isogenies η_{Λ} : X_{O_{L}/(p)} → X_{Λ,O_{L}/(p)} satisfying conditions (i)–(v) of [RZ96, Definition 3.21]. In particular η_{Λ′}η_{Λ}^{−1} lifts to a quasiisogeny η_{Λ′,Λ} : X_{Λ} → X_{Λ′} over O_{L} for all Λ, Λ′ ∈ L. Let μ_{0} = π̆(x_{0}) ∈ F̆_{b}^{0} be the image of x_{0}. By Wintenberger’s result [Win97, Corollary to Proposition 4.5.3] the rational Tate modules V_{p}X_{Λ,L^{alg}} are all B-isomorphic to Ṽ via B-isomorphisms δ_{Λ} : V_{p}X_{Λ} ^{∼} ⟶ Ṽ satisfying δ_{Λ} = δ_{Λ′} ∘ V_{p}(η_{Λ′,Λ}). We claim that δ_{Λ}(T_{p}X_{Λ}) =: Λ̃ ⊂ Ṽ forms a (polarized) multichain of type (L) of O_{B}-lattices in Ṽ. Indeed the decomposition O_{B} ≅ ∏_{i=1}^{r} M_{n_{i}}(O_{D_{i}}) induces a decomposition X_{Λ} ≅ ∏_{i=1}^{r} X_{Λ_{i}} of p-divisible groups. If Λ ⊂ Λ′ then [RZ96, Definition 3.21(ii) and (ii bis) from p. 88] implies that η_{Λ′,Λ} is an isogeny whose i-th composition factor η_{Λ′_{i},Λ_{i}} : X_{Λ_{i}} → X_{Λ′_{i}} has height log_{p}[Λ′_{i} : Λ_{i}]. Therefore we obtain Λ̃ ⊂ Λ̃′ and

[Λ̃′_{i} : Λ̃_{i}] = p^{ht(η_{Λ′_{i},Λ_{i}})} = [Λ′_{i} : Λ_{i}] proving (ĩi).

If a ∈ B^{×} satisfies a^{−1}O_{B}a = O_{B} then [RZ96, Definition 3.21(iii)] says that η_{aΛ,Λ} = θ_{a,Λ}∘i_{Λ}(a)^{−1} for an isomorphism θ_{a,Λ} : X_{Λ} → X_{aΛ} of p-divisible groups. Therefore condition (ĩii) follows from

ãΛ = δ_{Λ}V_{p}(η_{Λ,aΛ})(T_{p}X_{aΛ}) = δ_{Λ}V_{p}(i_{Λ}(a) θ_{a,Λ}^{−1})(T_{p}X_{aΛ}) = a · δ_{Λ}(T_{p}X_{Λ}) = a · Λ̃ .

In case PEL there exists by [RZ96, Definition 3.21(v)] a constant ℓ ∈ ℚp^{×} such that for all Λ ∈ L the quasi-isogeny (ηΛ^{∨}^{∨})^{−1} ∘ ℓλ ∘ η_{Λ}^{−1}

\xymatrix{X_{Λ,O_{L}/(p)} \ar[r] & (X_{Λ^{∨},O_{L}/(p)})^{∨} \ar[d]^{(η_{Λ^{∨}})^{∨}} \\ X_{O_{L}/(p)} \ar[u]^{η_{Λ}} \ar[r]^{ℓλ} & XO_{L}/(p)^{∨}}

lifts to an isomorphism p_{Λ} : X_{Λ} ^{∼} ⟶ (X_{Λ^{∨}})^{∨}. Here we obtain the polarization ℓλ : X → X^{∨} from the alternating form ( . , .) on V. That is ( . , .) on V ⊗_{ℚ_{p}} K_{0} = 𝔻(X)_{K_{0}} coincides up to multiplication with a unit in ℚp^{×} with the pairing

𝔻(X)_{K_{0}} ⊗_{K_{0}} 𝔻(X)_{K_{0}} ⟶^{𝔻(ℓλ)⊗ id} 𝔻(X^{∨})_{K_{0}} ⊗_{K_{0}} 𝔻(X)_{K_{0}} ⟶^{𝔻(ψ_{X})_{K_{0}}} 𝔻(𝔾_{m})_{K_{0}} = K_{0}

induced from the pairing ψ_{X} : X^{∨}×X → 𝔾_{m}. Now under the identification δ_{Λ} : V_{p}X_{Λ} ^{∼} ⟶ Ṽ the isomorphism p_{Λ} induces an alternating form

Ṽ ⊗_{ℚ_{p}} Ṽ ⟶^{V_{p}(p_{Λ^{∨}}∘η_{Λ^{∨},Λ})δ_{Λ}^{−1}⊗δ_{Λ}^{−1}} V_{p}XΛ^{∨} ⊗_{ℚ_{p}} V_{p}X_{Λ} ⟶^{V_{p}ψ_{Λ}} V_{p}𝔾_{m,L^{alg}} = ℚ_{p}  (8.3)

which is independent of Λ. Up to multiplication with a unit ℓ̃ ∈ ℚp^{×} this alternating form equals ( . , .)^{∼}, the image of ( . , .) under the fiber functor ω̃ by Wintenberger [Win97, Corollary to Proposition 4.5.3]. Hence (ṽ) follows from

Λ̃^{∨} := δ_{Λ} ∘ V_{p}(η_{Λ,Λ^{∨}})(T_{p}X_{Λ^{∨}})
= δ_{Λ} ∘ V_{p}(η_{Λ,Λ^{∨}} ∘ p_{Λ^{∨}}^{−1})(T_{p}XΛ^{∨})
= { ṽ ∈ Ṽ : ℓ̃· (ṽ, w̃)^{∼} ∈ ℤ_{p} for all w̃ ∈ δ_{Λ}(T_{p}X_{Λ}) = Λ̃ }
= ℓ̃^{−1}(Λ̃)^{∨} .

2. Now in addition to μ_{0} = π̆(x_{0}) let μ ∈ F̆_{b}^{0} be any point. Consider the canonical representation ρ : G ↪ GL(V) =: G′ and the induced embedding F̆_{b}^{0} ↪ F̆_{ρ(b)}^{′0}, μ ↦ ρ ∘ μ into the corresponding open subspace F̆_{ρ(b)}^{′0} for G′. The properties of the latter subspace were discussed in the previous section. Moreover, F̆_{b}^{0} = F̆^{an} ∩ F̆_{ρ(b)}^{′0} by Proposition 6.2. In addition we denote the formal scheme from Proposition 7.1 in this section by Ğ′. By Theorem 7.3 and Faltings’s [Fal10, Theorem 14] there is an admissible affine formal scheme S′ over O_{Ĕ}, a p-divisible group X over S′, and a quasi-isogeny η : X\overline{S′} → X\overline{S′} for \overline{S′} = V(p) ⊂ S′, such that the S′-valued point (X, η) ∈ Ğ′(S′) induces under the period morphism π̆′ : Ğ′^{an} → F̆_{ρ(b)}^{′0} an affinoid subdomain S′^{an} ⊂ F̆_{ρ(b)}^{′0} containing μ. By changing μ_{0} we may assume that S′^{an} also contains μ_{0}. Let S ⊂ S′ be the Zariski closed formal subscheme with S^{an} = F̆_{b}^{0} ×F̆_{ρ(b)}^{′0} S′^{an}. We denote by V_{p}X the π_{1}^{ét}(S^{an}, μ̄_{0})-module V_{p}Xμ̄_{0} at some

fixed geometric base point μ̄_{0} of S^{an} above μ_{0}. By Wintenberger [Win97, Corollary to Proposition 4.5.3] for (G′, ρ(b)) there is an isomorphism δ : V_{p}X ^{∼} ⟶ Ṽ.

Consider the (polarized) multichain of type (L) in Ṽ constructed in part 1 above from an L-set of p-divisible groups with O_{B}-action above x_{0} ∈ Ğ(O_{L}), which we now call \{\overline{X}_{Λ}\}_{Λ∈L}. We claim that there is a projective morphism T → S of admissible formal O_{Ĕ}-schemes with T^{an} → S^{an} a finite étale covering, and for each Λ ∈ L a p-divisible group X_{Λ} over T together with a quasi-isogeny η_{X,Λ} : X_{Λ} → X with δ_{Λ}(T_{p}X_{Λ}) = Λ̃ ⊂ Ṽ for δ_{Λ} := δ ∘ V_{p}(η_{X,Λ}) : V_{p}X_{Λ} ^{∼} ⟶ Ṽ.

Indeed, by using the periodicity p̃Λ = p · Λ̃ which allows to set η_{p^{n}Λ,Λ} := p^{−n} : X_{Λ} → X_{Λ} =: X_{p^{n}Λ}, we only need to check this for finitely many lattices Λ_{ν} ∈ L with Λ̃_{ν} ⊃ δ(T_{p}X). Let T^{an} → S^{an} be the finite étale covering corresponding to the π_{1}^{ét}(S^{an}, μ̄_{0})- set ∏_{ν} δ^{−1}Λ̃_{ν}/T_{p}X. That is, μ̄_{0} lifts to a point of T^{an} and π_{1}^{ét}(T^{an}, μ̄_{0}) acts trivially on ∏_{ν} δ^{−1}Λ̃_{ν}/T_{p}X. This implies the existence of a p-divisible group X_{Λ_{ν}} over T^{an} and a quasi-isogeny η_{Λ_{ν},X} : X → X_{Λ_{ν}} with ker η_{Λ_{ν},X} = (δ^{−1}Λ̃_{ν}/T_{p}X)_{T^{an}}. The family of finite flat subgroup schemes { ker η_{Λ_{ν},X} ⊂ X }_{ν} over T^{an} corresponds to a morphism f from T^{an} to a moduli space of families of finite flat subgroup schemes of X. This moduli space is projective over S. Let T be the scheme theoretic closure of the graph of this morphism f. Then T → S is projective, ker η_{Λ_{ν},X} extends to a finite flat subgroup scheme of X over T, and we set η_{Λ_{ν},X} : X → X_{Λ_{ν}} := X/ ker η_{λ_{ν},X} and η_{X,Λ_{ν}} := η_{Λ_{ν},X}^{−1}. This proves the claim.

We may replace S by T and (X, η) by (X_{Λ}, η_{X,Λ}^{−1} ∘ η) for any Λ. The construction yields that the fiber over μ̄_{0} of X_{Λ} is isomorphic to \overline{X}_{Λ}. This implies that V_{p}X_{Λ} is a B-module and δ_{Λ} is a B-isomorphism.

3. The quasi-isogeny η_{Λ} := η_{X,Λ}^{−1} ∘ η : X\overline{S} → X_{Λ,\overline{S}} induces an action i_{Λ}(a) = η_{Λ} ∘ i_{X}(a) ∘ η_{Λ}^{−1} of a ∈ B as quasi-isogenies on X_{Λ,\overline{S}}. For each a ∈ B there is an integer n such that i_{Λ}(p^{n}a) is even an isogeny. We denote by 𝔻(X_{Λ})_{S} the Lie algebra of the universal vector extension of X_{Λ} over S. It only depends on the reduction X_{Λ,\overline{S}}. We set 𝔻(X_{Λ})_{S^{an}} = 𝔻(X_{Λ})_{S} ⊗_{O_{S}} O_{S^{an}} and denote by 𝔻(η_{Λ})_{S^{an}} : V ⊗_{ℚ_{p}} O_{S^{an}} = 𝔻(X)_{K_{0}} ⊗_{K_{0}} O_{S^{an}} ^{∼} ⟶ 𝔻(X_{Λ})_{S^{an}} the isomorphism induced from η_{Λ} by the crystalline nature of 𝔻(X_{Λ})_{S}. By definition of the period morphism

π̆′

F il^{1}X_{Λ} = 𝔻(X_{Λ})_{S} ∩ (F il^{1}X_{Λ}) ⊗_{O_{S}} O_{S^{an}} = 𝔻(X_{Λ})_{S} ∩ 𝔻(η_{Λ})_{S^{an}}(F il^{1}(V ⊗_{ℚ_{p}} O_{S^{an}}))

for the universal filtration F il^{1}(V ⊗_{ℚ_{p}} O_{S^{an}}). Indeed, the first equation holds by definition and for the second equation the inclusion “⊂” is clear. On the other hand, if m belongs to the right hand side then p^{n}m ∈ F il^{1}X_{Λ} for n ≫ 0. Thus the image m̄ of m inside Lie X_{Λ} = 𝔻(X_{Λ})_{S}/F il^{1}X_{Λ} satisfies p^{n}m̄ = 0. But Lie X_{Λ}, as a locally free O_{S}-module, does not have p-torsion, and so m ∈ F il^{1}X_{Λ}.

Since F il^{1}(V ⊗_{ℚ_{p}} O_{S^{an}}) is B-invariant, i_{Λ}(p^{n}a) lifts to an isogeny of X_{Λ} over S. Hence i_{Λ}(a) := i_{Λ}(p^{n}a) ⊗ p^{−n} ∈ End(X_{Λ}) ⊗_{ℤ_{p}} ℚ_{p}. By construction the B-module structure on V_{p}X_{Λ} induced from V_{p}(i_{Λ}(a)) coincides with the structure considered at the end of part 2 above. For a ∈ O_{B} then a · Λ̃ ⊂ Λ̃ implies V_{p}(i_{Λ}(a))(T_{p}X_{Λ}) ⊂ T_{p}X_{Λ} and therefore i_{Λ}(a) ∈ End(X_{Λ}). In order that (X_{Λ}, i_{Λ}, η_{Λ})_{Λ∈L} is an S-valued point of Ğ we have to verify conditions (i)–(v) of [RZ96, Definition 3.21].

By construction 𝔻(η_{Λ})_{S^{an}} induces an isomorphism of exact sequences of B-modules

[figure: 0; (F il^{1}X_{Λ}) ⊗_{O_{S}} O_{S^{an}}; 𝔻(X_{Λ})_{S^{an}}; Lie X_{Λ} ⊗_{O_{S}} O_{S^{an}}; 0; 0; F il^{1}(V ⊗_{ℚ_{p}} O_{S^{an}}); ∼; 𝔻(X)_{K_{0}} ⊗_{K_{0}} O_{S^{an}}; ∼; 𝔻(η_{Λ})_{S^{an}}; gr^{0}(V ⊗_{ℚ_{p}} O_{S^{an}}); ∼; 0; 0; (V ⊗_{ℚ_{p}} O_{S^{an}})^{μ_{S^{an}}^{univ}(p)=p}; V ⊗_{ℚ_{p}} O_{S^{an}}; (V ⊗_{ℚ_{p}} O_{S^{an}})^{μ_{S^{an}}^{univ}(p)=1}; ∼; 0 ,]

where μ_{S^{an}}^{univ} is the universal cocharacter on S^{an}. Hence det_{S}(a; Lie X_{Λ}) = det_{S^{an}}(a; (V ⊗_{ℚ_{p}} O_{S^{an}})^{μ_{S^{an}}^{univ}(p)=1}) for every a ∈ O_{B}. This implies condition (iv). Note that (iv) further implies (i) by [RZ96, 3.23(c)].

Next, for any Λ ⊂ Λ′ in L our condition Λ̃ ⊂ Λ̃′ implies that V_{p}(η_{Λ′,Λ})(T_{p}X_{Λ}) ⊂ T_{p}X_{Λ′} and so η_{Λ′,Λ} : X_{Λ} → X_{Λ′} is an isogeny. Under the decompositions O_{B} ≅ ∏_{i=1}^{r} M_{n_{i}}(O_{D_{i}}), Λ ≅ ⨁_{i} Λ_{i} and X_{Λ} ≅ ∏_{i=1}^{r} X_{Λ_{i}} it induces an isogeny X_{Λ_{i}} → X_{Λ′_{i}} of height equal to

log_{p}[T_{p}X_{Λ′_{i}} : V_{p}(η_{Λ′,Λ})(T_{p}X_{Λ_{i}})] = log_{p}[Λ̃′_{i} : Λ̃_{i}] = log_{p}[Λ′_{i} : Λ_{i}]. Condition (ii) follows from this by [RZ96, 3.23(d)].

If a ∈ B^{×} satisfies a^{−1}O_{B}a = O_{B} then the quasi-isogeny θ_{a,Λ} = η_{aΛ,Λ}∘i_{Λ}(a) : X_{Λ} → X_{aΛ} satisfies

V_{p}(θ_{a,Λ})(T_{p}X_{Λ}) = V_{p}(θ_{a,Λ}) ∘ δ_{Λ}^{−1}(Λ̃) = V_{p}(η_{aΛ,Λ}) ∘ δ_{Λ}^{−1}(a · Λ̃) = δ_{aΛ}^{−1}(ãΛ) = T_{p}X_{aΛ} .

Hence θ_{a,Λ} : X_{Λ} → X_{aΛ} is an isomorphism and this proves condition (iii).

In case PEL the quasi-isogeny λ : X → X^{∨} induces a quasi-isogeny

\overline{p}_{Λ^{∨}} := (ηΛ^{∨})^{−1} ∘ λ ∘ η_{Λ^{∨}}^{−1} : X_{Λ^{∨},\overline{S}} ⟶ (X_{Λ,\overline{S}})^{∨} .

By the crystalline nature of the functor 𝔻( .)_{S^{an}} we obtain an isomorphism 𝔻(\overline{p}_{Λ^{∨}})_{S^{an}} : 𝔻(X_{Λ^{∨}})_{S^{an}} ^{∼} ⟶ 𝔻(XΛ^{∨})_{S^{an}} ≅ 𝔻(X_{Λ})S^{an}^{∨} where the last isomorphism is induced by the pairing ψ_{Λ}. Since the alternating form ( . , .) on V ⊗_{ℚ_{p}} K_{0} = 𝔻(X)_{K_{0}} coincides with

𝔻(X)_{K_{0}} ⊗_{K_{0}} 𝔻(X)_{K_{0}} ⟶^{𝔻(λ)⊗ id} 𝔻(X^{∨})_{K_{0}} ⊗_{K_{0}} 𝔻(X)_{K_{0}} ⟶^{𝔻(ψ_{X})_{K_{0}}} 𝔻(𝔾_{m})_{K_{0}} = K_{0}

up to multiplication with a unit in ℚp^{×} and since F il^{1}(V ⊗_{ℚ_{p}} O_{S^{an}}) is totally isotropic for ( . , .) we see that

𝔻(\overline{p}_{Λ^{∨}})_{S^{an}}(F il^{1}X_{Λ^{∨}}) ⊗_{O_{S}} O_{S^{an}} = 𝔻(η_{Λ}^{−1})^{∨}𝔻(λ)(F il^{1}(V ⊗_{ℚ_{p}} O_{S^{an}}))

⊂ 𝔻(η_{Λ}^{−1})^{∨}({ h ∈ V^{∨} ⊗_{ℚ_{p}} O_{S^{an}} : h(F il^{1}(V ⊗_{ℚ_{p}} O_{S^{an}})) = 0 })
= { h′ ∈ 𝔻(X_{Λ})S^{an}^{∨} : h′(F il^{1}X_{Λ}) = 0 }
= (F il^{1}XΛ^{∨}) ⊗_{S} O_{S^{an}} .

This shows that \overline{p}_{Λ^{∨}} lifts to a quasi-isogeny p_{Λ^{∨}} : X_{Λ^{∨}} → XΛ^{∨} over S. By part 1 above the quasi-isogeny p_{Λ^{∨}} ∘ η_{Λ^{∨},Λ} : X_{Λ} → XΛ^{∨} induces an alternating form on Ṽ ≅ V_{p}X_{Λ} as in (8.3) which equals ℓ̃· ( . , .)^{∼} for a constant ℓ̃ ∈ ℚp^{×}. This ℓ̃ is also the constant from Definition 8.3(ṽ) of the multichain {Λ̃}_{Λ∈L}. Then

V_{p}(p_{Λ^{∨}})(T_{p}X_{Λ^{∨}}) = V_{p}(\frac{1}{ℓ̃} p_{Λ^{∨}} ∘ η_{Λ^{∨},Λ})δ_{Λ}^{−1}(ℓ̃· Λ̃^{∨})
= \frac{1}{ℓ̃} V_{p}(p_{Λ^{∨}} ∘ η_{Λ^{∨},Λ})δ_{Λ}^{−1}(Λ̃)^{∨}
= \frac{1}{ℓ̃} V_{p}(p_{Λ^{∨}} ∘ η_{Λ^{∨},Λ})δ_{Λ}^{−1}(\{ ṽ ∈ Ṽ : (ṽ, w̃)^{∼} ∈ ℤ_{p} for all w̃ ∈ Λ̃ \})
= T_{p}XΛ^{∨} .

Therefore p_{Λ^{∨}} : X_{Λ^{∨}} → XΛ^{∨} is an isomorphism and this establishes [RZ96, Definition 3.21(v)]. Hence (X_{Λ}, i_{Λ}, η_{Λ}) is a point in Ğ(S) such that the induced morphism S^{an} → Ğ^{an} ⟶^{π̆} F̆_{b}^{0} coincides with the morphism constructed in part 2 above. This shows that μ ∈ F̆_{b}^{0} lies in the image of π̆ and proves (a).

(b) As in the proof of Theorem 7.3(b) the rational Tate module V_{p}X_{Λ} descends to a local system V of ℚ_{p}-vector spaces on the space F̆_{b}^{0}. Consider a connected component Y of F̆_{b}^{0} and a geometric base point μ̄ ∈ Y which lies over an L-valued point μ ∈ Y (L) for a finite field extension L/Ĕ. By Proposition 2.2 the local system V on Y corresponds to a representation ρ′ : π_{1}^{ét}(Y, μ̄) → GL(Vμ̄) ≅ GL(Ṽ) where we identify V_{p}X_{Λ} ≅ Ṽ as in part 1 of (a) above. There is even an isomorphism of Gal(L^{alg}/L)-representations

V_{μ} = V_{p}X_{Λ,O_{L}} ≅ V_{cris}(𝔻(X_{Λ,O_{L}}), F il^{1}X_{Λ,O_{L}}) ≅ V_{cris}(\underline{D}_{b,μ}(V)) .  (8.4)

In order to prove that V induces a tensor functor and (8.4) is a tensor isomorphism it suffices by Corollary 2.6 to show that ρ′ factors through G̃(ℚ_{p}) ⊂ GL(Ṽ), which in turn follows from the explicit description of G̃ given in (8.2). Indeed, since the universal X_{Λ} over Ğ carries an action of O_{B} the image of π_{1}^{ét}(Y, μ̄) in GL(Ṽ) commutes with B and this already proves the claim in case EL. In case PEL, diagram (8.3) shows that the action of γ ∈ π_{1}^{ét}(Y, μ̄) on Ṽ satisfies (γ · ṽ, γ · w̃)^{∼} = χ_{cyc}(γ) · (ṽ, w̃)^{∼} for all ṽ, w̃ ∈ Ṽ, where χ_{cyc} : π_{1}^{ét}(Y, μ̄) → Gal(Ĕ^{alg}/Ĕ) ⟶^{χ_{cyc}} ℤp^{×} is the cyclotomic character of the base field Ĕ.

In particular, ρ′ factors through a morphism ρ′ : π_{1}^{ét}(Y, μ̄) → G̃(ℚ_{p}) with c̃ ∘ ρ′ = χ_{cyc}, and by Corollary 2.6 this induces the desired tensor functor \underline{V} : \underline{Rep}_{ℚ_{p}}G → ℚ_{p}-\underline{Loc}_{Y} with \underline{V}(V) = V satisfying (2.9). Moreover, (8.4) is compatible with the action of B on both sides and, in the PEL case, the alternating pairings V_{p}ψ_{Λ} and V_{cris}(𝔻(ψ_{X})_{K_{0}}); see (8.3). Hence (8.4) induces an isomorphism of tensor functors on \underline{Rep}_{ℚ_{p}}G and also (2.8) of Conjecture 2.3 holds at the fixed μ. But the construction is independent of μ. So it holds on all connected components Y of F̆_{b}^{0} and this establishes (b) and Remark 8.5(a).

(c) The argument of Theorem 7.3(c) applies literally to prove the first part of (c). For the statement about Ğ^{an} let K̃ ⊂ G̃(ℚ_{p}) be the stabilizer of the (polarized) multichain (Λ̃)_{Λ∈L} of type (L) in Ṽ constructed in part 1 above. The construction of parts 2 and 3 yields an isomorphism Ğ^{an} ^{∼} ⟶ Ğan ; K̃ ≅ ĔK̃ where ĔK̃ is the space of (polarized) multichains of type (L) in V.

(d) is proved by the same argument as Theorem 7.3(d).

9 When does weakly admissible imply admissible ?

In this section we approach the question for which groups G and which elements b ∈ G(K_{0}) as in (6.1) “weakly admissible implies admissible”, that is F̆_{b}^{wa} = F̆_{b}^{0}. Since we do not yet have a good group theoretic understanding of the phenomenon we only treat the case G = GL_{n} comprehensively. We begin with the following observations

Lemma 9.1. Let G = G_{1} ×G_{2} be a product of two reductive groups over ℚ_{p} and b = (b_{1}, b_{2}) such that each b_{i} ∈ G_{i}(K_{0}) satisfies (6.1) for the same integer n. If F̆_{G_{1},b_{1}}^{wa} ≠ F̆_{G_{1},b_{1}}^{0} and F̆_{G_{2},b_{2}}^{wa} ≠ ∅ then F̆_{G,b}^{wa} ≠ F̆_{G,b}^{0}.

Proof. Let μ_{1} ∈ F̆_{G_{1},b_{1}}^{wa} ∖F̆_{G_{1},b_{1}}^{0} be an L_{1}-valued point and μ_{2} ∈ F̆_{G_{2},b_{2}}^{wa}(L_{2}) for complete field extensions L_{1}/Ĕ_{1} and L_{2}/Ĕ_{2} with [L_{2} : Ĕ_{2}] < ∞. Let L be some compositum of L_{1} and L_{2} in some algebraic closure of L_{1} and set μ = (μ_{1}, μ_{2}) : 𝔾_{m} → G. Let ρ_{i} : G_{i} ↪ GL(V_{i}) be representations satisfying (6.1) for n. Then ρ : G ↪ GL(V_{1} ⊕ V_{2}) likewise satisfies (6.1) for n. Consider the filtered isocrystals \underline{D}_{i} = \underline{D}_{b_{i},μ_{i}}(V_{i}) and

\underline{D} = \underline{D}_{1} ⊕ \underline{D}_{2} = \underline{D}_{b,μ}(V_{1} ⊕ V_{2})

over L which all three are weakly admissible. Then

M(\underline{D}) = M(\underline{D}_{1}) ⊕ M(\underline{D}_{2}) ≅ M̸(0, 1) ⊗_{ℚ_{p}} (V_{1} ⊕ V_{2})

because M(\underline{D}_{1}) ≅ M̸(0, 1) ⊗_{ℚ_{p}} V_{1}. This shows that μ ∈ F̆_{G,b}^{wa} ∖ F̆_{G,b}^{0} as desired.

The next lemma shows that the question is invariant under dualization.

Lemma 9.2. Let G_{i} = GL_{n} and b_{i} ∈ GL_{n}(K_{0}) for i = 1, 2 such that the isocrystals \underline{D}_{i} = (K_{0}^{n} , b_{i}·φ) satisfy (6.1). Assume that

(a) \underline{D}_{2} ≅ \underline{D}_{1} ⊗ 1l(r) or

(b) \underline{D}_{2} ≅ Hom(\underline{D}_{1}, 1l(r))

holds for some r ∈ ℤ (see Lemma 5.3). Then F̆_{G_{1},b_{1}}^{wa} = F̆_{G_{1},b_{1}}^{0} if and only if F̆_{G_{2},b_{2}}^{wa} = F̆_{G_{2},b_{2}}^{0}.

Proof. By symmetry it suffices to prove one direction. So assume that F̆_{G_{2},b_{2}}^{wa} = F̆_{G_{2},b_{2}}^{0} and let μ_{1} ∈ F̆_{G_{1},b_{1}}^{wa}. Since the filtered isocrystal \underline{D}_{1} = (K_{0}^{n}, b_{1}·φ, F il_{μ_{1}}^{•}) is weakly admissible, also the filtered isocrystal \underline{D}_{2} := \underline{D}_{1}⊗1l(r) in case (a), respectively \underline{D}_{2} := Hom(\underline{D}_{1}, 1l(r)) in case (b) is weakly admissible and corresponds to a point μ_{2} ∈ F̆_{G_{2},b_{2}}^{wa}. Then our assumption implies M(\underline{D}_{2}) ≅ M(0, 1)^{n}. Lemma 5.3 and Proposition 5.4 yield in case

(a) M(\underline{D}_{1}) ≅ M(\underline{D}_{2}) ≅ M(0, 1)^{n}, respectively

(b) M(\underline{D}_{1}) ≅ M(\underline{D}2^{∨} ⊗ 1l(r)) ≅ M(\underline{D}2^{∨}) ≅(M(0, 1)^{n})^{∨}≅ M(0, 1)^{n},

and therefore μ_{1} ∈ F̆_{G_{1},b_{1}}^{0} as desired.

Next we state the main theorem of this section.

Theorem 9.3. Let G = GL_{n}, let b ∈ GL_{n}(K_{0}) be such that the isocrystal (K_{0}^{n}, b · φ) has Newton slopes in the interval [0, 1], and let {μ_{0}} be the conjugacy class for which the standard representation ℚ_{p}^{n} of G satisfies (6.1). Then the equality F̆_{b}^{wa} = F̆_{b}^{0} holds if and only if the Newton slopes of b are

(a) (1^{(h_{1})}, \frac{1}{h}^{(h)}, 0^{(h_{0})}) for h_{1}, h, h_{0} ∈ ℕ_{0}, or

(b) (1^{(h_{1})}, \frac{h−1}{h}^{(h)}, 0^{(h_{0})}) for h_{1}, h, h_{0} ∈ ℕ_{0}, or

(c) (1^{(h_{1})}, \frac{1}{2}^{(4)}, 0^{(h_{0})}) for h_{1}, h_{0} ∈ ℕ_{0}.

In particular F̆_{b}^{wa} = F̆_{b}^{0} for n ≤ 4. (Here s^{(m)} means that the slope s ∈ ℚ occurs with multiplicity m ∈ ℕ_{0}. If m = 0 this slope does not occur. The sum of all multiplicities is n.)

Before proving the theorem we derive the following

Corollary 9.4. Let (G, b, {μ}) be arbitrary such that there is a faithful representation ρ : G ↪ GL(V) as in (6.1) with dim_{ℚ_{p}} V ≤ 4. Then F̆_{b}^{wa} = F̆_{b}^{0}.

Proof. This follows from the theorem and the fact that F̆_{G,b}^{0} = F̆_{G,b}^{wa} ∩ F̆_{GL(V),ρ(b)}^{0}, see (6.2) and Proposition 6.2.

Remark 9.5. 1. The case of Theorem 9.3 where h = 0 and h_{1} = h_{0} was discussed by Rapoport and Zink [RZ96, 5.51]. In this case the flag variety is the Grassmannian Grass(h_{0}, 2h_{0}), and F̆_{b}^{wa} is the big cell of subspaces transversal to the sub-isocrystal of slope zero. It is isomorphic to affine space 𝔸^{h_{0}^{2}}. The period morphism was constructed by Dwork (compare [Kat73]) and, according to [RZ96, Proposition 5.52], is given componentwise as the p-adic logarithm. By the surjectivity of the latter, the image F̆_{b}^{0} of the period morphism equals F̆_{b}^{wa}.

2. Let us now assume that h_{1} = h_{0} = 0 in Theorem 9.3.

Then case (b) (and its dual case (a)) correspond to the Lubin-Tate deformation space [LT65] and were studied by Gross and Hopkins [HG94a, HG94b]. In these cases the flag variety F̆ equals ℙ^{h−1}, the space of hyperplanes (respectively lines) in ℚ_{p}^{⊕h}, the Rapoport-Zink space Ğ is ∐_{ℤ} Spf W (𝔽_{p}^{alg})[[t_{1}, …, t_{h−1}]], and the period morphism has image F̆_{b}^{0} = F̆_{b}^{wa} = (ℙ^{h−1})^{an} by [HG94a, HG94b]; see also [RZ96, 5.50]. In this case our Theorem 7.3 was proved by [dJ95, §§1,7] who also determines the image of the representation ρ : π_{1}^{ét}(F̆_{b}^{1}, μ̄) → GL_{h}(ℚ_{p}) corresponding to the tensor functor \underline{V}.

The third alternative, case (c) in Theorem 9.3 comes as a surprise and the author has no interpretation for it so far.

3. Among examples for Corollary 9.4 is the EL case where B is the quaternion algebra over ℚ_{p}, where V is a free B-module of rank one, and G = (B^{opp})^{×}; see Section 8. We present B as

B = ℚ_{p^{2}}[Π] with Π^{2} = p, and Πx = φ(x)Π

and let b = p^{−1}Π ∈ G(K_{0}). This case was studied by Drinfeld [Dri76]. Here the flag variety is ℙ^{1}, and F̆_{b}^{wa} is the base change Ω×_{ℚ_{p}} K_{0} of Drinfeld’s upper half-plane Ω = ℙ^{1} ∖ℙ^{1}(ℚ_{p}); see [RZ96, 1.44–1.46]. The Rapoport-Zink space Ğ is the disjoint union indexed by ℤ of Deligne’s integral model of Ω ×_{ℚ_{p}} K_{0}. In particular each connected component of Ğ^{an} maps isomorphically onto F̆_{b}^{wa} and hence F̆_{b}^{0} = F̆_{b}^{wa}; see [RZ96, 3.45–3.77 and 5.48–5.49]. This case directly generalizes to the situation where B is a central division algebra of dimension e and Hasse invariant \frac{1}{e}.

The rest of this section is devoted to the

Proof of Theorem 9.3. We first prove that the conditions (a), (b), and (c) are sufficient. Let μ ∈ F̆^{an} ∖ F̆_{b}^{0}. We must show that μ ∉ F̆_{b}^{wa}. We set M_{μ} = M(\underline{D}_{b,μ}(ℚ_{p}^{n})) and D = D(\underline{D}_{b,μ}(ℚ_{p}^{n})) (Definition 5.2), and consider the decompositions of D and M_{μ} from Theorem 4.3. We write M_{μ} = M_{+} ⊕ M_{0} ⊕ M_{−} such that the weights of all components of M_{+} (respectively M_{0}, respectively M_{−}) are positive (respectively zero, respectively negative). Note that M_{−} ≠ (0) since M_{μ} ≅ M̸(0, 1)^{n} and deg M_{μ} = 0 by Theorem 5.5 and Lemma 6.1. Likewise we write D = M(1, 1)^{⊕h_{1}} ⊕ D_{+} ⊕ M(0, 1)^{⊕h_{0}} such that the weights of all components of D_{+} lie strictly between 0 and 1. Then the φ-module t^{−1}D = M(0, 1)^{⊕h_{1}} ⊕ t^{−1}D_{+} ⊕ M(−1, 1)^{⊕h_{0}}. We consider the inclusions D ↪ M_{μ} ↪ t^{−1}D whose composition is the natural inclusion D ⊂ t^{−1}D, and the induced morphisms M(0, 1)^{⊕h_{0}} → M_{0} → M(0, 1)^{⊕h_{1}} whose composition is the zero map. Since End_{φ}(M(0, 1)) = ℚ_{p} by Proposition 4.12(c), we can decompose the φ-modules M_{μ} = M_{+} ⊕ M_{0}^{1} ⊕ M_{0}^{2} ⊕ M_{0}^{3} ⊕ M_{−}, and D = M(1, 1)^{⊕h̃_{1}} ⊕ M(1, 1)^{⊕h′_{1}} ⊕ D_{+} ⊕ M(0, 1)^{⊕h′_{0}} ⊕ M(0, 1)^{⊕h̃_{0}} such that with respect to appropriate bases of all these φ-modules (and the induced basis for t^{−1}D) the inclusions are given by block matrices

M_{μ} ↪ t^{−1}D M_{+} M_{0}^{1} M_{0}^{2} M_{0}^{3} M_{−}
M(0, 1)^{⊕h̃_{1}} A_{1} 0  0  0  0
M(0, 1)^{⊕h′_{1}} A_{2} Id 0  0  0
t^{−1}D_{+}  D E_{1} E_{2} E_{3} F
M(−1, 1)^{⊕h′_{0}} G_{1} H_{11} H_{12} H_{13} I_{1}
M(−1, 1)^{⊕h̃_{0}} G_{2} H_{21} H_{22} H_{23} I_{2}

and

D ↪ M_{μ} M(1, 1)^{⊕h̃_{1}} M(1, 1)^{⊕h′_{1}} D_{+} M(0, 1)^{⊕h′_{0}} M(0, 1)^{⊕h̃_{0}}
M_{+}  a_{1}  a_{2}  c  0  0
M_{0}^{1}  d_{11}  d_{12}  e_{1}  0  0
M_{0}^{2}  d_{21}  d_{22}  e_{2}  0  0
M_{0}^{3}  d_{31}  d_{32}  e_{3}  Id  0
M_{−}  f_{1}  f_{2}  g  i_{1}  i_{2}

We put r_{+} := rk M_{+} , r_{0}^{i} := rk M_{0}^{i} , r_{−} := rk M_{−}, and h := rk D_{+}. Here and in the sequel superscripts are indices and do not mean powers. We must have

r_{0}^{1} = h′_{1} ,  r_{0}^{3} = h′_{0} ,  h̃_{0} ≤ r_{−} ,  and  r_{0}^{2} + r_{0}^{3} + r_{−} ≤ h + h_{0}  (9.1)

since the maps M(0, 1)^{⊕h̃_{0}} ↪ M_{−} and M_{0}^{2} ⊕ M_{0}^{3} ⊕ M_{−} ↪ t^{−1}D_{+} ⊕ M(−1, 1)^{⊕h_{0}} are injective. The inclusion D ↪ t^{−1}D is represented by the product of the two big block matrices which hence equals

t Id_{n} = ( \begin{matrix}A_{1}a_{1} & A_{1}a_{2} & A_{1}c & 0 & 0 \\ A_{2}a_{1} + d_{11} & A_{2}a_{2} + d_{12} & A_{2}c + e_{1} & 0 & 0 \\ & & & E_{3} + F i_{1} & F i_{2} \\ & & & H_{13} + I_{1}i_{1} & I_{1}i_{2} \\ & & & H_{23} + I_{2}i_{1} & I_{2}i_{2} \end{matrix} )

This implies A_{1}c = 0 , A_{2}c + e_{1} = 0 , E_{3} + F i_{1} = 0 , and F i_{2} = 0. Considering the entries of these matrices as elements of the fraction field Q of the integral domain B̃_{rig}^{†} we have dim_{Q} ker A_{1} ≥ rk c and dim_{Q} ker F ≥ rk i_{2} = h̃_{0}. Thus

h̃_{1} = rk A_{1} ≤ r_{+} − rk c  and  r_{−} = rk F + dim_{Q} ker F ≥ h̃_{0} + rk F . (9.2)

Now we distinguish several cases.

\underline{Case c = 0 :} This implies e_{1} = 0 and we obtain inclusions

D′ := D_{+} ⊕ M(0, 1)^{⊕h_{0}} ↪ M_{0}^{2} ⊕ M_{0}^{3} ⊕ M_{−} ↪ t^{−1}D_{+} ⊕ M(−1, 1)^{⊕h_{0}}

Therefore D′ ⊂ D comes from a sub-isocrystal \underline{D}′ ⊂ \underline{D} with M(\underline{D}′) ⊃ M_{0}^{2} ⊕ M_{0}^{3} ⊕ M_{−}. This implies t_{N}(\underline{D}′) − t_{H}(\underline{D}′) = deg M(\underline{D}′) ≤ deg(M_{0}^{2} ⊕ M_{0}^{3} ⊕ M_{−}) < 0 by Theorem 5.5 and [Ked05, Lemma 3.4.2]. So μ is not weakly admissible, μ ∉ F̆_{b}^{wa}.

\underline{Case F = 0 :} This implies E_{3} = 0 and we obtain inclusions

D′ := M(0, 1)^{⊕h_{0}} ↪ M_{0}^{3} ⊕ M_{−} ↪ M(−1, 1)^{⊕h_{0}}

Therefore D′ ⊂ D comes from a sub-isocrystal \underline{D}′ ⊂ \underline{D} with M(\underline{D}′) ⊃ M_{0}^{3} ⊕ M_{−}. This again implies t_{N}(\underline{D}′) − t_{H}(\underline{D}′) < 0 by Theorem 5.5 and [Ked05, Lemma 3.4.2], and so μ is not weakly admissible, μ ∉ F̆_{b}^{wa}.

\underline{Case c ≠ 0 and rk c = h :} Then (9.2) implies h̃_{1} + h ≤ r_{+} and

h̃_{0} = n − h′_{1} − h̃_{1} − h − h′_{0} ≥ n − r_{0}^{1} − r_{+} − r_{0}^{3} = r_{0}^{2} + r_{−} ≥ r_{−} ≥ h̃_{0} ,

implies r_{−} = h̃_{0}. Since t^{−1}I_{2}i_{2} = Id_{r_{−}} we have i_{2} ∈ GL_{r_{−}}(Q). Thus F i_{2} = 0 yields F = 0 in which case we just saw that μ ∉ F̆_{b}^{wa}.

\underline{Case F ≠ 0 and rk F = h :} Then (9.2) implies h̃_{0} + h ≤ r_{−} and

h_{0} + h = h′_{0} + h̃_{0} + h ≤ r_{0}^{3} + r_{−} ≤ r_{0}^{2} + r_{0}^{3} + r_{−} ≤ h_{0} + h ,

implies h_{0} + h = r_{0}^{2} + r_{0}^{3} + r_{−} and h̃_{1} = r_{+}. Since t^{−1}A_{1}a_{1} = Id_{r_{+}} we have A_{1} ∈ GL_{r_{+}}(Q) and hence A_{1}c = 0 yields c = 0. This case was treated above.

\underline{Case c ≠ 0 and F ≠ 0 :} To treat this case we use the special form D_{+} has under the conditions (a), (b), and (c).

Under condition (a) D_{+} ≅ M(1, h). The φ-module which is the image of c : D_{+} → M_{+} is isomorphic to ⨁_{i} M(s_{i}, n_{i}). Since it is a quotient of D_{+} and contained in M_{+} we have \frac{1}{h} ≥ \frac{s_{i}}{n_{i}} > 0, whence n_{i} ≥ s_{i}h ≥ h for all i. This means rk c = h and in fact, we are in a case treated above.

Under condition (b) D_{+} ≅ M(h − 1, h). The image of F : M_{−} → t^{−1}D_{+} is isomorphic to ⨁_{i} M(s′_{i}, n′_{i}). Since it is a quotient of M_{−} and contained in t^{−1}D_{+} ≅ M(−1, h) we have 0 > \frac{s′_{i}}{n′_{i}} ≥ \frac{−1}{h}, whence n′_{i} ≥ −s′_{i}h ≥ h for all i. This means rk F = h and we are again in a case treated above.

Under condition (c) D_{+} ≅ M(1, 2)^{⊕2}. Again the image of c : D_{+} → M_{+} is isomorphic to ⨁_{i} M(s_{i}, n_{i}) with \frac{1}{2} ≥ \frac{s_{i}}{n_{i}} > 0, whence n_{i} ≥ 2s_{i} ≥ 2 and rk c ≥ 2. Furthermore the

image of F : M_{−} → t^{−1}D_{+} is isomorphic to ⨁_{i} M(s′_{i}, n′_{i}) with 0 > \frac{s′_{i}}{n′_{i}} ≥ \frac{−1}{2}, whence n′_{i} ≥ −2s′_{i} ≥ 2 and rk F ≥ 2. Then

n = h_{0} + 4 + h_{1} ≤ h′_{0} + h̃_{0} + rk F + rk c + h̃_{1} + h′_{1} ≤ r_{0}^{3} + r_{−} + r_{+} + r_{0}^{1} = n − r_{0}^{2} ≤ n

yields r_{0}^{2} = 0 , rk c = 2 = rk F and r_{−} = h̃_{0} + rk F = h̃_{0} + 2. This implies n′_{i} ≤ 2, whence n′_{i} = 2, s′_{i} = −1 and im(F) ≅ M(−1, 2). We may decompose t^{−1}D_{+} = t^{−1}D_{+}^{1} ⊕ t^{−1}D_{+}^{2} with im(F) = t^{−1}D_{+}^{2}. With respect to a basis suited to this decomposition we write F = \binom{0}{F_{2}} , E_{3} = \binom{E_{31}}{E_{32}} , c = (c_{1}|c_{2}) and e_{1} = (e_{11}|e_{12}). Since the zero map

M_{−} \underrightarrow{\binom{0}{F_{2}}} t^{−1}D_{+} ⟶^{(c_{1}|c_{2})} t^{−1}M_{+}

equals c_{2}F_{2} and rk F_{2} = 2 we find c_{2} = 0. Then \binom{E_{31}}{E_{32}} + \binom{0}{F_{2}}i_{1} = 0 implies E_{31} = 0 and A_{2}(c_{1}|c_{2}) + (e_{11}|e_{12}) = 0 implies e_{12} = 0. We obtain inclusions

D′ := D_{+}^{2} ⊕ M(0, 1)^{⊕h_{0}} ↪ M_{0}^{3} ⊕ M_{−} ↪ t^{−1}D_{+}^{2} ⊕ M(−1, 1)^{⊕h_{0}}

The φ-submodule D_{+}^{2} is inherited from a sub-isocrystal \underline{D}_{+}^{2} := (K_{0}^{n}, b·φ) ∩ D_{+}^{2} of the isocrystal \underline{D}_{+} = (K_{0}^{n} , b·φ) ∩ D_{+} because all slopes of im(F) and t^{−1}D_{+} are equal, and End_{φ}(M(−1, 2)) equals the central division algebra over ℚ_{p} of dimension 4 and Hasse invariant \frac{1}{2} by Proposition 4.12. Therefore D′ ⊂ D comes from a sub-isocrystal \underline{D}′ ⊂ \underline{D} with M(\underline{D}′) ⊃ M_{0}^{3} ⊕ M_{−}. This implies t_{N}(\underline{D}′) − t_{H}(\underline{D}′) < 0 by Theorem 5.5 and [Ked05, Lemma 3.4.2], and so μ is not weakly admissible, μ ∉ F̆_{b}^{wa}.

Altogether we have proved that under conditions (a), (b), and (c) the equality F̆_{b}^{wa} = F̆_{b}^{0} holds, that is “weakly admissible implies admissible”.

To prove the converse we assume that conditions (a), (b), or (c) are not satisfied. By Lemma 9.1 we only need to find a direct summand of the isocrystal (K_{0}^{n} , b·φ) for which “weakly admissible does not imply admissible”. Considering simple sub-isocrystals it hence suffices to treat the cases where D = D(\underline{D}) has the following form 1. D ≅ M(c, h) and t^{−1}D ≅ M(−d, h) for c, h ∈ ℕ_{>0} relatively prime, d = h − c, with

2 ≤ c, d ≤ h − 2 and h ≥ 5.

2. D ≅ M(1, h_{1}) ⊕ M(1, h_{2}) with 2 ≤ h_{1} ≤ h_{2} and 2 < h_{2}.

3. D ≅ M(h_{1} − 1, h_{1}) ⊕ M(h_{2} − 1, h_{2}) with 2 ≤ h_{1} ≤ h_{2} and 2 < h_{2}.

4. D ≅ M(1, h_{1}) ⊕ M(h_{2} − 1, h_{2}) with 3 ≤ h_{1}, h_{2}.

5. D ≅ M(1, 2)^{⊕3}.

Since the isocrystals in cases 2. and 3. are dual to each other, case 3. follows from case 2. by Lemma 9.2. Also note that a special instance of case 1. was discussed in Example 6.7. However, the beginning of our argument treats cases 1., 2., 4., and 5. simultaneously. We set h := rk D, c := t_{N}(\underline{D}) = deg D, and d = h − c. In case 1. the conditions on c, d, h imply cd > h and we define e = ⌊\frac{cd}{h}⌋ ≥ 1 such that \frac{−e}{c} ≥ \frac{−d}{h}. In the other three cases we set e = 1. By Proposition 4.12 there exists 0 ≠ A ∈ Hom_{φ}(M(−e, c), t^{−1}D), such that depending on the case

Case 1. the map A : M(−e, c) → t^{−1}D is arbitrary but non-zero.

Case 2. the map A : M(−e, c) = M(−1, 2) → t^{−1}D ≅ M(1−h_{1}, h_{1})⊕M(1−h_{2}, h_{2}) induces non-zero maps A_{1} and A_{2} into the first, resp. second summand of t^{−1}D. If h_{1} = h_{2} > 2 we require in addition that A_{1}, A_{2} are linearly independent over the central division algebra End_{φ}(M(1 − h_{1}, h_{1})) of dimension (h_{1})^{2} over ℚ_{p}. This is possible since dim_{ℚ_{p}} Hom_{φ}(M(−1, 2), M(1 − h_{1}, h_{1})) = ∞ in this case by Proposition 4.12.

Case 4. the map A : M(−e, c) = M(−1, h_{2}) → t^{−1}D ≅ M(1 − h_{1}, h_{1}) ⊕ M(−1, h_{2}) induces a non-zero map into the first summand of t^{−1}D and is the identity onto the second summand.

Case 5. the map A : M(−e, c) = M(−1, 3) → t^{−1}D ≅ M(−1, 2)^{⊕3} induces maps A_{i} into the i-th summand of t^{−1}D which are linearly independent over the central division algebra End_{φ}(M(−1, 2)) of dimension 4 over ℚ_{p}. Again this is possible since dim_{ℚ_{p}} Hom_{φ}(M(−1, 3), M(−1, 2)) = ∞ by Proposition 4.12.

Since both M(−e, c) and t^{−1}D are represented over B̃^{]0,1]}(ℂ_{p}), we may by Proposition 4.9 choose a K_{0}-basis of \underline{D} and consider A as an (h × c)-matrix with entries in B̃^{]0,1]}. There is a μ defined over ℂ_{p} in the conjugacy class {μ_{0}}, such that the c-dimensional space F il_{μ}^{1}ℂ_{p}^{n} contains the columns of the matrix θ_{t^{−1}D}(A). By Definition 5.2 (compare Example 6.7) the non-zero map M(−e, c) → t^{−1}D factors through M_{μ} for this μ, and therefore M_{μ} ≅ M(0, 1)^{dim D} and μ ∉ F̆_{b}^{0}. It remains to show that μ ∈ F̆_{b}^{wa}. We distinguish the cases

1. In this case \underline{D} is simple and hence F̆_{b}^{wa} = F̆^{an}. This implies μ ∈ F̆_{b}^{wa}.

2. Assume that there is a sub-isocrystal (0) ≠ \underline{D}′ ⊊ \underline{D} that contradicts weak admissibility. Then t^{−1}D′ := t^{−1}\underline{D}′⊗_{K_{0}} B̃_{rig}^{†} is the kernel of a non-zero map E : t^{−1}D → M(1−h_{i}, h_{i}) for i = 1 or i = 2. If μ ∉ F̆_{b}^{wa} then

1 = t_{N}(\underline{D}′) < t_{H}(\underline{D}′) = dim_{ℂ_{p}}(D′_{ℂ_{p}} ∩ F il_{μ}^{1}D_{ℂ_{p}}) ≤ dim_{ℂ_{p}} F il_{μ}^{1}D_{ℂ_{p}} = 2 ,

and this would imply F il_{μ}^{1}D_{ℂ_{p}} ⊂ D′_{ℂ_{p}}, whence θ_{M(1−h_{i},h_{i})}(EA) = 0. Thus EA = T · B for some map B ∈ Hom_{φ}(M(−1, 2) , M(1, h_{i}))) by Corollary 4.11. Then Proposition 4.12 shows that B = 0, whence EA = 0. However, this is not the case by our assumptions on A, since for h_{1} ≠ h_{2} the map E is the projection onto the summand M(1 − h_{i}, h_{i}) of t^{−1}D. Likewise for h_{1} = h_{2} we have E = (E_{1}|E_{2}) : t^{−1}D → M(1 − h_{1}, h_{1}) with E_{i} ∈ End_{φ}(M(1 − h_{1}, h_{1})) and E_{1}A_{1} + E_{2}A_{2} = 0 contradicts the linear independence of A_{1}, A_{2} over End_{φ}(M(1 − h_{1}, h_{1})). Therefore μ ∈ F̆_{b}^{wa}.

4. In this case A = \binom{A_{1}}{Id_{h_{2}}} with 0 ≠ A_{1} ∈ Hom_{φ}(M(−1, h_{2}) , M(1 − h_{1}, h_{1})), and

θ_{t^{−1}D}(A) = \binom{θ(A_{1})}{Id_{h_{2}}} ≠ \binom{0}{Id_{h_{2}}} because θ_{M(1−h_{1},h_{1})}(A_{1}) = 0 would by Corollary 4.11 imply that A_{1} = T · B for 0 ≠ B ∈ Hom_{φ}(M(−1, h_{2}) , M(1, h_{1})) which is impossible by Proposition 4.12. In particular, θ_{t^{−1}D}(A) has rank c and F il_{μ}^{1}D_{ℂ_{p}} equals im(θ_{t^{−1}D}(A)).

There are exactly two sub-isocrystals (0) ≠ \underline{D}′ ⊊ \underline{D} of \underline{D}. For the first one we have D′ := \underline{D}′ ⊗_{K_{0}} B̃_{rig}^{†} = M(1, h_{1}) and consequently t_{N}(\underline{D}′) = 1. Furthermore, we have D′_{ℂ_{p}} ∩ F il_{μ}^{1}D_{ℂ_{p}} = \{\binom{v}{0} ∈ im \binom{θ(A_{1})}{Id_{h_{2}}}\} = (0) and t_{H}(\underline{D}′) = 0. For the second sub-isocrystal we have t^{−1}D′ = M(−1, h_{2}) and t_{N}(\underline{D}′) = h_{2} − 1. We claim that t_{H}(\underline{D}′) ≤ t_{N}(\underline{D}′). Namely, if

h_{2} − 1 = t_{N}(\underline{D}′) < t_{H}(\underline{D}′) = dim_{ℂ_{p}}(D′_{ℂ_{p}} ∩ F il_{μ}^{1}D_{ℂ_{p}}) ≤ dim_{ℂ_{p}} F il_{μ}^{1}D_{ℂ_{p}} = h_{2}

then im(θ_{t^{−1}D}(A)) = F il_{μ}^{1}D_{ℂ_{p}} = D′_{ℂ_{p}} = im \binom{0}{Id_{h_{2}}} in contradiction to θ_{M(1−h_{1},h_{1})}(A_{1}) ≠ 0. This proves that μ is weakly admissible.

5. Here the sub-isocrystals (0) ≠ \underline{D}′ ⊊ \underline{D} are of two kinds. For the first kind t^{−1}D′ has rank 4 and is the kernel of a non-zero map E = (E_{1}|E_{2}|E_{3}) : t^{−1}D → M(−1, 2) for E_{i} ∈ End_{φ}(M(−1, 2)). Then t_{N}(\underline{D}′) = 2 and we claim that t_{H}(\underline{D}′) ≤ 2. Indeed

2 < t_{H}(\underline{D}′) = dim_{ℂ_{p}}(D′_{ℂ_{p}} ∩ F il_{μ}^{1}D_{ℂ_{p}}) ≤ dim_{ℂ_{p}} F il_{μ}^{1}D_{ℂ_{p}} = 3

would imply that F il_{μ}^{1}D_{ℂ_{p}} ⊂ D′_{ℂ_{p}} and θ_{M(−1,2)}(EA) = 0. Thus EA = T · B for some morphism B ∈ Hom_{φ}(M(−1, 3) , M(1, 2)) = (0) and this implies E_{1}A_{1} + E_{2}A_{2} + E_{3}A_{3} = 0 in contradiction to the linear independence of A_{1}, A_{2}, A_{3} over End_{φ}(M(−1, 2)).

For the second kind of sub-isocrystal t^{−1}D′ ≅ M(−1, 2) and t_{N}(\underline{D}′) = 1. Again we claim that t_{H}(\underline{D}′) ≤ 1. Indeed

1 < t_{H}(\underline{D}′) = dim_{ℂ_{p}}(D′_{ℂ_{p}} ∩ F il_{μ}^{1}D_{ℂ_{p}}) ≤ dim_{ℂ_{p}} D′_{ℂ_{p}} = 2

implies that D′_{ℂ_{p}} ⊂ F il_{μ}^{1}D_{ℂ_{p}} and hence t^{−1}D′ ⊂ M_{μ} ⊂ t^{−1}D = M(−1, 2)^{⊕3}. Therefore M_{μ} ≅ M(−1, 2) ⊕ ⨁_{i} M(s_{i}, n_{i}) for −\frac{1}{2} ≤ \frac{s_{i}}{n_{i}} ≤ \frac{1}{2}. We had defined μ by constructing a morphism M(−1, 3) → M_{μ}. If the induced morphism M(−1, 3) → M(s_{i}, n_{i}) were nonzero for some i, then \frac{s_{i}}{n_{i}} ≤ −\frac{1}{3} and s_{i} ≤ −1 and n_{i} ≥ −2s_{i} ≥ 2. The constraints on the remaining \frac{s_{i}}{n_{i}} imply that deg M_{μ} ≤ −1, a contradiction to Theorem 5.5 and Lemma 6.1. Therefore A : M(−1, 3) → t^{−1}D factors through t^{−1}D′. But t^{−1}D′ lies in the kernel of an appropriate morphism 0 ≠ (E_{1}, E_{2}, E_{3}) : t^{−1}D → M(−1, 2) for E_{i} ∈ End_{φ}(M(−1, 2)). Again this means E_{1}A_{1} + E_{2}A_{2} + E_{3}A_{3} = 0 in contradiction to the linear independence of the A_{i}. This shows that t_{H}(\underline{D}′) ≤ t_{N}(\underline{D}′) and so μ is weakly admissible.

Therefore we have established in all cases that μ ∈ F̆_{b}^{wa} ∖ F̆_{b}^{0} and the theorem is proved.

10 Conjectures

As stated in Conjectures 6.4 and 6.5 we expect that the open subspace F̆_{b}^{0} ⊂ F̆_{b}^{wa} is arcwise connected and satisfies F̆_{b}^{0}(L) = F̆_{b}^{wa}(L) if the value group of L is finitely generated. This belief comes from the function field analog [Har11], [Har09, §§5.3 and 6.2] of the theory developed here. In that analog we were able to prove the assertions corresponding to these conjectures; see [Har11, Theorems 2.5.3 and 3.2.5].

Unfortunately we have little more to support Conjecture 6.4. However, the proof of Theorem 7.3 and the computations in Example 6.7 and Theorem 9.3 suggest that the fields L with F̆_{b}^{0}(L) ⊊ F̆_{b}^{wa}(L) must contain elements of the form

∑_{j=0}^{c−1} p^{j} ∑_{ν∈ℤ} p^{cν}u_{j}^{p^{−eν}}

for 0 < c ≤ e and u_{j} ∈ C, |u_{j}| < 1. Maybe one could show that fields containing such elements cannot have finitely generated value group.

Evidence for the connectedness of F̆_{b}^{0}

We suggest a strategy to prove that F̆_{b}^{0} is arcwise connected by noting that this follows from Conjecture 6.4. In part (a) of the following proposition we also add the proof of a fact mentioned on page 6.

Proposition 10.1. (a) There exists an arcwise connected paracompact open Ĕ-analytic subspace F̆_{b}^{wa} of F̆^{an} whose associated rigid analytic space is the period space (F̆_{b}^{wa})^{rig}.

(b) If (G, b, {μ}) satisfies (6.1) and Conjecture 6.4 is true then F̆_{b}^{0} is arcwise connected.

(c) If (G, b, {μ}) satisfies (6.1) and F̆_{b}^{0} is connected then any open Ĕ-analytic subspace F̆_{b}^{a} with F̆_{b}^{0} ⊂ F̆_{b}^{a} ⊂ F̆_{b}^{wa} is also arcwise connected.

Proof of Proposition 10.1. (a) Since F̆^{an} has a finite covering by affine spaces, all its open Ĕ-analytic subspaces are paracompact by Lemma A.3. From its construction in [RZ96, Proposition 1.34] the period space (F̆_{b}^{wa})^{rig} possesses an admissible covering {X_{i}^{rig}}_{i∈ℕ} by admissible subsets X_{i}^{rig} ⊂ F̆^{rig} such that F̆^{rig} ∖ X_{i}^{rig} is a finite union ⋃_{j} Sp B_{i,j} of affinoid subdomains and X_{i}^{rig} ⊂ X_{i+1}^{rig}. The X_{i}^{rig} correspond to open Ĕ-analytic subspaces X_{i} := F̆^{an} ∖⋃_{j} M(B_{i,j}) of F̆^{an}. Moreover, it follows from [RZ96, Proposition 1.34] that the closure \overline{X}_{i} of X_{i} in F̆^{an} is a finite union of Ĕ-affinoid subdomains and \overline{X}_{i} ⊂ X_{i+1}. Thus the union F̆_{b}^{wa} := ⋃_{i∈ℕ} X_{i} is an open Ĕ-analytic subspace of F̆^{an} whose associated rigid analytic space equals (F̆^{wa})^{rig}. (Use [Beh93, Lemma 1.6.2] to see that the Grothendieck topology induced from F̆_{b}^{wa} on { μ ∈ F̆_{b}^{wa} : H(μ)/Ĕ is finite } coincides with the Grothendieck topology of (F̆_{b}^{wa})^{rig}.)

To prove the connectedness of F̆_{b}^{wa} note that every Ĕ-analytic space is locally arcwise connected by [Beh90, Theorem 3.2.1]. The flag variety F̆^{an} has a finite covering by affine spaces, hence a countable affinoid covering by polydiscs U ≅ M(Ĕ⟨\frac{x_{1}}{ζ} , …, \frac{x_{n}}{ζ} ⟩) for varying ζ ∈ Ĕ. Since the points x ∈ F̆_{b}^{wa} with H(x) finite over Ĕ lie dense in F̆_{b}^{wa} by [Beh90, Proposition 2.1.15] it suffices to exhibit for every such point x ∈ F̆_{b}^{wa} ∩ U a continuous map α from the compact interval [0, |ζ|] into F̆_{b}^{wa} ∩ U such that α(0) = x and α(|ζ|) is the point corresponding to the Gauß norm on U.

So let x ∈ F̆_{b}^{wa}∩U with H(x) finite over Ĕ be the point with coordinates x_{i} = c_{i} ∈ H(x), |c_{i}| ≤ |ζ|. To define α : [0, |ζ|] → F̆_{b}^{wa} ∩ U we expand each f ∈ Ĕ⟨\frac{x_{1}}{ζ} , …, \frac{x_{n}}{ζ} ⟩ around x

f = ∑_{\underline{i}∈ℕ_{0}^{n}} a\underline{i} (x_{1} − c_{1})^{i_{1}} ⋯ (x_{n} − c_{n})^{i_{n}} with a\underline{i} ∈ Ĕ(c_{1}, …, c_{n}) .

For every element t ∈ [0, |ζ|] we obtain an analytic point P = P (x, t) in U by setting

|f|_{P} := sup\{ |a\underline{i}| t^{i_{1}+...+i_{n}} : \underline{i} ∈ ℕ_{0}^{n} \} .

If x is fixed it is easy to see that this defines a continuous and injective map α : [0, |ζ|] → U, t ↦ P (x, t) with α(0) = x and α(|ζ|) being the point corresponding to the Gauß norm on U. It remains to show that α(t) lies in F̆_{b}^{wa}. For t > 0 the prime ideal ker | . |_{α(t)} := { f ∈ Ĕ⟨\frac{x_{1}}{ζ} , …, \frac{x_{n}}{ζ} ⟩ : |f|_{α(t)} = 0 \} corresponding to α(t) is the zero ideal. Since by construction in [RZ96, Proposition 1.36] the set U ∖ F̆_{b}^{wa} is a union of Zariski closed subsets of U and since it does not contain the point x, we obtain α(t) ∈ F̆_{b}^{wa}. This implies that F̆_{b}^{wa} is arcwise connected.

(b) To prove the connectedness of F̆_{b}^{0} under the assumption that Conjecture 6.4 is true, we show that even α([0, |ζ|]) ⊂ F̆_{b}^{0} ∩ U. Namely, the value group of H(α(t)) is generated by |Ĕ(c_{1}, …, c_{n})^{×}| and t and hence is finitely generated. So Conjecture 6.4 will imply α(t) ∈ F̆_{b}^{0} ∩ U and so F̆_{b}^{0} is arcwise connected.

(c) To prove the connectedness of F̆_{b}^{a} we note that F̆_{b}^{a}(L) = F̆_{b}^{wa}(L) for all finite extensions L/Ĕ because F̆_{b}^{0}(L) = F̆_{b}^{wa}(L). Since the points x ∈ F̆_{b}^{a} with H(x) finite over Ĕ lie dense in F̆_{b}^{a} by [Beh90, Proposition 2.1.15], F̆_{b}^{0} is a dense connected subset of F̆_{b}^{a}.

Evidence for the maximality of F̆_{b}^{0}

We conjectured in Conjecture 6.10 that the open Ĕ-analytic subspace F̆_{b}^{0} ⊂ F̆_{b}^{wa} and the local system V from Theorem 6.8 are maximal. From this also Conjectures 7.6 and 8.6 would follow. For this maximality it remains to prove that the local system does not extend to a larger open subspace F̆_{b}^{0} ⊊ Y ⊂ F̆_{b}^{wa}. This is suggested by the following results of Andreatta and Brinon [And06, AB08, AB10]. Assume there exists a local system V of ℚ_{p}-vector spaces on Y as in Theorem 6.8 and let Ỹ be the space of ℤ_{p}-lattices inside V.

Consider a morphism f : U → Ỹ of Ĕ-analytic spaces such that U possesses an affine formal model Spf R that is étale over Spf W (𝔽_{p}^{alg})⟨T_{1}, T_{1}^{−1}, …, T_{d}, T_{d}^{−1}⟩. (This hypothesis on U can even be weakened slightly; see the conditions in [And06, AB08].) The pullback of V to U corresponds by Proposition 2.2 to a representation ρ_{V} : π_{1}^{ét}(U, x̄) → GL(V)(ℚ_{p}) which stabilizes a ℤ_{p}-lattice in V = Vx̄. Thus ρ_{V} factors through

π_{1}^{ét}(U, x̄) → π_{1}^{alg}(U, x̄) → GL(V)(ℤ_{p}) ⊂ GL(V)(ℚ_{p}),

where π_{1}^{alg}(U, x̄) is the algebraic fundamental group classifying finite étale coverings; see [dJ95]. By [And06, Theorem 7.11] ρ_{V} corresponds to an étale (φ, Γ)-module M over A_{R} which is overconvergent by [AB08]. If μ ∈ U is an analytic point and C is the completion of an algebraic closure of H(μ) the fiber M_{μ} ⊗ B̃(C) at μ is therefore of the form M_{μ}^{†} ⊗_{B̃^{†}(C)} B̃(C) for a φ-module M_{μ}^{†} over B̃^{†}(C). By [Ked05, Proposition 5.5.1] the φ-

module M_{μ}^{†} ⊗_{B̃^{†}(C)} B̃_{rig}^{†}(C) over B̃_{rig}^{†}(C) is isomorphic to M(0, 1) ⊗_{ℚ_{p}} V. Due to (2.8) we

have M_{μ}^{†} ⊗_{B̃^{†}(C)} B̃_{rig}^{†}(C) ≅ M((V_{K_{0}}, b·φ, F il_{f(μ)}^{•}V_{H(μ)})). Thus the image of U in Y must be contained in F̆_{b}^{0}.

However, at present this approach does not prove Conjecture 6.10 because unfortunately Y cannot be covered by Ĕ-analytic spaces U allowed by [And06, AB08]. For instance consider on an annulus Y_{1} = M(Ĕ⟨T, \frac{p}{T} ⟩) the point y_{1} with | ∑_{i} a_{i}T^{i}|_{y} = sup\{|a_{i}| |p|^{ri} : i ∈ ℤ} for r ∈ [0, 1] ∖ ℚ. Furthermore, consider a point y_{2} with H(y_{2}) = ℂ_{p} in a small polydisc Y_{2} = M(Ĕ⟨\frac{T_{1}}{ζ} , …, \frac{T_{n}}{ζ} ⟩) for ζ ∈ Ĕ, 0 < |ζ| ≪ 1 (such points exist; see [Har11, Example A.2.2 (d)]). If Y contains Y_{1} × Y_{2} then the point y = (y_{1}, y_{2}) ∈ Y does not lie in the image of any U as above. So this method does not yield y ∈ F̆_{b}^{0}. Also Conjecture 6.4 does not give y ∈ F̆_{b}^{0} because the value group |H(y)^{×}| ⊃ |H(y_{2})^{×}| ≅ ℚ is not finitely generated.

Remark 10.2. Due to these restrictions of Andreatta’s and Brinon’s theory [And06, AB08], Kedlaya uses a general new foundation of relative p-adic Hodge theory [KL12, KL13] to obtain the generalizations of our results, which he announced in [Ked10].

A Berkovich’s Analytic Spaces

Let O_{F} be a (not necessarily discrete) complete valuation ring of rank one and let F be its fraction field. We briefly recall Berkovich’s [Beh90, Beh93] theory of F -analytic spaces. Let B be an affinoid F -algebra in the sense of [BGR84, Chapter 6] with F -Banach norm | . |. Berkovich calls these algebras strictly F -affinoid.

Definition A.1. An analytic point x of B is a semi-norm | . |_{x} : B → ℝ_{≥0} which satisfies:

(a) |f + g|_{x} ≤ max{ |f|_{x}, |g|_{x} } for all f, g ∈ B,

(b) |fg|_{x} = |f|_{x} |g|_{x} for all f, g ∈ B,

(c) |λ|_{x} = |λ| for all λ ∈ F,

(d) | . |_{x} : B → ℝ_{≥0} is continuous with respect to the norm | . | on B.

The set of all analytic points of B is denoted M(B). On M(B) one considers the coarsest topology such that for every f ∈ B the map M(B) → ℝ_{≥0} given by x ↦ |f|_{x} is continuous. Equipped with this topology, M(B) is a compact Hausdorff space; see [Beh90, Theorem 1.2.1]. Such a space is called a strictly F -affinoid space.

Every morphism α : B → B′ of affinoid F -algebras is automatically continuous and hence induces a continuous morphism M(α) : M(B′) → M(B) by mapping the semi-norm B′ → ℝ_{≥0} to the composition B → B′ → ℝ_{≥0}. By definition the M(α) are the morphisms in the category of strictly F -affinoid spaces. In particular, for an affinoid subdomain Sp B′ ⊂ Sp B this morphism identifies M(B′) with a closed subset of M(B).

For every analytic point x ∈ M(B) we let ker | . |_{x} := { b ∈ B : |b|_{x} = 0 }. It is a prime ideal in B. We define the (complete) residue field of x as the completion with respect to | . |_{x} of the fraction field of B/ ker | . |_{x}. It will be denoted H(x). There is a natural continuous homomorphism B → H(x) of F -algebras. Conversely let L be a complete extension of F, by which we mean a field extension of F equipped with an absolute value | . | : L → ℝ_{≥0} which restricts on F to the norm of F such that L is complete with respect to | . |. Any continuous F -algebra homomorphism B → L defines on B a semi-norm which is an analytic point.

In [Beh93, §1.2] Berkovich defines the category of strictly F -analytic spaces. These spaces are topological spaces which admit an atlas with strictly F -affinoid charts.

The spaces M(B) for affinoid F -algebras B are examples for strictly F -analytic spaces. Other examples arise from schemes Y which are locally of finite type over F, see [Beh90, §3.4]. Namely if Y = 𝔸_{F}^{n} is affine n-space over F the associated strictly F -analytic space Y^{an} = (𝔸_{F}^{n})^{an} consists of all semi-norms on the polynomial ring F [y_{1}, …, y_{n}] as in Definition A.1 (a)–(c). The topology on (𝔸_{F}^{n})^{an} is the coarsest topology such that for all f ∈ F [y_{1}, …, y_{n}] the map (𝔸_{F}^{n})^{an} → ℝ_{≥0}, x ↦ |f|_{x} is continuous. The space (𝔸_{F}^{n})^{an} is the union of the increasing sequence of compact polydiscs M(F ⟨\frac{y_{1}}{ζ^{m}} , …, \frac{y_{n}}{ζ^{m}} ⟩) of radii (|ζ^{m}|, …, |ζ^{m}|) for m ∈ ℕ_{0} where ζ ∈ F has |ζ| > 1. If Y ⊂ 𝔸_{F}^{n} is a closed subscheme of affine n-space with coherent ideal sheaf J, the ideal sheaf J O_{(𝔸_{F}^{n})^{an}} defines a closed strictly F -analytic subspace Y^{an} of (𝔸_{F}^{n})^{an}. Finally, if Y is arbitrary and {Y_{i}}_{i} is a covering of Y by affine open subschemes then one can glue the associated strictly F -affinoid spaces Y_{i}^{an} to the F -analytic space Y^{an}. Moreover Y^{an} is Hausdorff if and only if the scheme Y is separated, see [Beh90, Theorems 3.4.1 and 3.4.8].

The relation between strictly F -analytic spaces, rigid analytic spaces, and formal schemes is as follows. To every strictly F -analytic space X which is Hausdorff one can associate a quasi-separated rigid analytic space

X^{rig} := { x ∈ X : H(x) is a finite extension of F },

see [Beh93, §1.6]. Recall that a rigid analytic space is called quasi-separated if the intersection of any two affinoid subdomains is a finite union of affinoid subdomains. To describe the subcategories on which the functor X ↦ X^{rig} is an equivalence we need the following terminology. A topological Hausdorff space is called paracompact if every open covering {U_{i}}_{i} has a locally finite refinement {V_{j}}_{j}, where locally finite means that every point has a neighborhood which meets only finitely many of the V_{j}. On the other hand an admissible covering of a rigid analytic space is said to be of finite type if every member of the covering meets only finitely many of the other members. A rigid analytic space over F is called quasiparacompact if it possesses an admissible affinoid covering of finite type. Similarly we define the notions of finite type and quasi-paracompact also for (an open covering of) an admissible formal O_{F} -scheme in the sense of Raynaud [Ray74]; see also [BL93a, BL93b, Bos08].

Theorem A.2. The following three categories are equivalent:

(a) the category of paracompact strictly F -analytic spaces,

(b) the category of quasi-separated quasi-paracompact rigid analytic spaces over F, and

(c) the category of quasi-paracompact admissible formal O_{F} -schemes, localized by admissible formal blowing-ups.

Proof. It is shown in [Beh93, Theorem 1.6.1] that X ↦ X^{rig} is an equivalence between (a) and (b). The equivalence of (c) with (b) is due to Raynaud. See [Bos08, Theorem 2.8/3] for a proof.

Regarding paracompactness the following result was proved in [Har11, Lemma A.2.6].

REFERENCES 52

Lemma A.3. Let F be discretely valued with countable residue field and let X be a strictly F -analytic space. Assume that X is Hausdorff and admits a countable covering by strictly F -affinoid spaces. Then every open subset of X is a paracompact strictly F -analytic space.

This applies in particular if X = Y^{an} for a separated scheme Y of finite type over F.

We will also need the following well known fact which we were unable to find in the literature.

Proposition A.4. Let f : X → Y be an étale morphism of F -analytic spaces ([Beh93, §3.3]) such that f is bijective on C-valued points for any algebraically closed complete extension C of F. Then f is an isomorphism.

Proof. Let x ∈ X and y = f(x) ∈ Y. Since f is quasi-finite there are open neighborhoods V = V_{x} ⊂ X of x and U = U_{y} ⊂ Y of y with f|_{V} : V → U finite étale. Since f is open by [Beh93, Proposition 3.2.7] we may replace U by f(V). For any affinoid M(A) ⊂ U the preimage (f|_{V})^{−1}(M(A)) = M(B) is affinoid with B finite flat over A by [Beh93, Proposition 3.2.3]. Hence B is finite locally free since A is noetherian by [Beh90, Proposition 2.1.3]. For any C-valued point of M(A) the fiber M(B ⊗_{A} C) → M(C) is finite étale by [Beh93, Corollary 3.3.8] and bijective by our assumption. Since C is algebraically closed, B ⊗_{A} C = C and so rk_{A} B = 1. Therefore the inverse map (f|_{V})^{−1} : U → V exists by [Beh93, Proposition 1.2.15(i)]. Now our assumption on bijectivity on C-valued points implies that the local inverses (f|_{V})^{−1} : U ^{∼} ⟶ V glue by [Beh93, Proposition 1.3.2] to the global inverse of f.

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Urs Hartl University of Muenster Institute of Mathematics Einsteinstr. 62 D – 48149 Muenster Germany http://www.math.uni-muenster.de/u/urs.hartl/