THE UNIVERSAL FAMILY OF SEMI-STABLE p-ADIC GALOIS REPRESENTATIONS

URS HARTL AND EUGEN HELLMANN

Abstract. Let K be a finite field extension of ℚ_{p} and let 𝒢_{K} be its absolute Galois group. We construct the universal family of semi-stable 𝒢_{K}-representations in ℚ_{p}-algebras and the moduli space of such representations. It is an Artin stack in adic spaces locally of finite type over ℚ_{p} in the sense of Huber.

Contents

1. Introduction 1 2. Families of (φ, N)-modules with Hodge-Pink lattice 4 3. Moduli spaces for (φ, N)-modules with Hodge-Pink lattice 10 4. Vector bundles on the open unit disc 17 5. Weak admissibility 22 6. The étale locus 26 7. Sheaves of period rings and the admissible locus 31 8. The universal semi-stable representation 34 9. The morphism to the adjoint quotient 38 References 41

1. Introduction

The study of arithmetic families of p-adic Hodge structures and their associated Galois representations was begun by one of us in [Hel13], where a universal family of filtered φ-modules was constructed and, building on this, a universal family of crystalline representations with Hodge-Tate weights in {0, 1}. The approach is based on Kisin’s integral p-adic Hodge theory cf. [Kis06]. In the present article we generalize this result to the case of semi-stable representations of the absolute Galois group 𝒢_{K} = Gal(\overline{K}/K) of a finite extension K of ℚ_{p} with arbitrary (fixed) Hodge-Tate weights. If the Hodge-Tate weights do not just lie in {0, 1}, Kisin’s integral theory does not describe 𝒢_{K}-stable ℤ_{p}-lattices in a crystalline resp. semistable 𝒢_{K}-representation but all 𝒢_{K_{∞}} = Gal(\overline{K}/K_{∞})-stable ℤ_{p}-lattices, where K_{∞} is a certain Kummer extension of K appearing in [Kis06]. This leads us to consider not only filtered (φ, N)-modules but rather (φ, N)-modules together with a so called Hodge-Pink lattice. We describe a moduli space (actually a stack) of (φ, N)-modules with Hodge-Pink lattice. This stack turns out to be a vector bundle over a space of filtered (φ, N)-modules. The original space of filtered (φ, N)-modules can be recovered as a section defined by a certain transversality condition in this vector bundle. The inspiration to work with Hodge-Pink lattices instead of filtrations is taken from the analogous theory over function fields; see [Pin97, GL11, Har11]. It was already applied to Kisin’s integral p-adic Hodge theory by Genestier and Lafforgue [GL12] in the absolute case for φ-modules over ℚ_{p}.

We can define a notion of weak admissibility for (φ, N)-modules with Hodge-Pink lattice and show that being weakly admissible is an open condition in the set up of adic spaces generalizing the corresponding result for filtered φ-modules in [Hel13]. Following the method of [Kis06] and [Hel13] we further cut out an open subspace over which an integral structure for the (φ, N)-modules with Hodge-Pink lattice exists and an open subspace over which a family of 𝒢_{K_{∞}} = Gal(\overline{K}/K_{∞})-representation exists. If we restrict ourselves to the subspace of filtered (φ, N)-modules one can promote this family of 𝒢_{K_{∞}}-representations to the universal family of semi-stable 𝒢_{K}-representations.

It is worth pointing out that, unlike in the case of filtered (φ, N)-modules, the Hodge-Tate weights can jump in a family of (φ, N)-modules with Hodge-Pink lattice, and likewise in a family of 𝒢_{K_{∞}}- representations (where we define the Hodge-Tate weights to be those of the corresponding Hodge-Pink lattice). As the Hodge-Tate weights of a family of 𝒢_{K}-representations vary p-adically continuously, and hence cannot jump, this is certainly an obstruction which forces us to restrict ourselves to the subspace of fixed Hodge-Tate weights in order to construct a family of 𝒢_{K}-representations.

We describe our results in more detail. Let K be a finite extension of ℚ_{p} with absolute Galois group 𝒢_{K} and maximal unramified sub-extension K_{0}. Let Frob_{p} be the p-Frobenius on K_{0}. We consider families of (φ, N)-modules over ℚ_{p}-schemes X, that is finite locally free O_{X} ⊗_{ℚ_{p}} K_{0}-modules D together with a φ := id ⊗ Frob_{p}-linear automorphism Φ and a linear monodromy operator N: D → D satisfying the usual relation NΦ = pΦN. Choosing locally on X a basis of D and considering Φ ∈ GL_{d}(O_{X} ⊗_{ℚ_{p}} K_{0}) and N ∈ Mat_{d×d}(O_{X} ⊗_{ℚ_{p}} K_{0}) as matrices, the condition NΦ = pΦN cuts out a closed subscheme P_{K_{0},d} ⊂ Res_{K_{0}/ℚ_{p}} GL_{d} ×_{ℚ_{p}} Res_{K_{0}/ℚ_{p}} Mat_{d×d}. We can describe the geometry of this scheme as follows.

Theorem 3.2. The scheme P_{K_{0},d} is equi-dimensional of dimension [K_{0} : ℚ_{p}] d^{2}. It is reduced, Cohen-Macaulay and generically smooth over ℚ_{p}. Its irreducible components are indexed by the possible Jordan types of the (necessarily nilpotent) monodromy operator N.

Further we consider families of (φ, N)-modules D with a filtration F^{•} on D ⊗_{K_{0}} K and more generally families of (φ, N)-modules with a Hodge-Pink lattice 𝔮, see Definition 2.5 for the precise definitions. Given a cocharacter μ of the algebraic group Res_{K/ℚ_{p}} GL_{d,K} (or more precisely a cocharacter of the Weil restriction of the diagonal torus which is dominant with respect to the Weil restriction of the upper triangular matrices) we define a notion of a filtration F^{•} resp. a Hodge-Pink lattice with constant Hodge polygon equal to μ, and a notion of boundedness by μ for a Hodge-Pink lattice 𝔮. With μ is associated a reflex field E_{μ} which is a finite extension of ℚ_{p}.

Theorem 3.6.

(a) The stack ℋ_{φ,N,⪯μ} parametrizing rank d families of (φ, N)-modules with Hodge-Pink lattice bounded by μ on the category of E_{μ}-schemes is an Artin stack. It is equi-dimensional and generically smooth. Its dimension can be explicitly described in terms of the cocharacter μ and its irreducible components are indexed by the possible Jordan types of the (nilpotent) monodromy operator.

(b) The stack ℋ_{φ,N,μ} parametrizing rank d families of (φ, N)-modules with Hodge-Pink lattice with constant Hodge polygon equal to μ, is an open and dense substack of ℋ_{φ,N,⪯μ}. Further it is reduced and Cohen-Macaulay. It admits a canonical map to the stack 𝒟_{φ,N,μ} of filtered (φ, N)-modules with filtration of type μ. This map is representable by a vector bundle.

If we restrict ourselves to the case of vanishing monodromy, i.e. the case N = 0, we cut out a single irreducible component ℋ_{φ,⪯μ} ⊂ ℋ_{φ,N,⪯μ} and similarly for the other stacks in the theorem. Following [Hel13] we consider the above stacks also as stacks on the category of adic spaces locally of finite type over ℚ_{p}, i.e. we consider the adification ℋ_{φ,N,⪯μ}^{ad}, etc. Passing from ℚ_{p}-schemes to adic spaces allows us to generalize Kisin’s comparison between filtered (φ, N)-modules and vector bundles on the open unit disc (together with certain additional structures). To do so we need to fix a uniformizer π of K as well as its minimal polynomial E(u) over K_{0}.

Theorem 4.6. For every adic space X locally of finite type over ℚ_{p} there is a natural equivalence of categories between the category of (φ, N)-modules with Hodge-Pink lattice over X and the category of (φ, N_{∇})-modules over X, i.e. the category of vector bundles M on the product of X with the open unit disc 𝕌 over K_{0} together with a semi-linear map Φ_{M}: M → M that is an isomorphism away from X × {E(u) = 0} ⊂ X × 𝕌 and a differential operator N_{∇}^{M} satisfying

N_{∇}^{M} ∘ Φ_{M} ∘ φ = p \frac{E(u)}{E(0)} · Φ_{M} ∘ φ ∘ N_{∇}^{M}.

Theorem 4.9. The differential operator N_{∇}^{M} defines a canonical meromorphic connection on the vector bundle M. The closed substack ℋ_{φ,N,μ}^{∇} ⊂ ℋ_{φ,N,μ}^{ad} where this connection is holomorphic coincides with the zero section of the vector bundle ℋ_{φ,N,μ}^{ad} → 𝒟_{φ,N,μ}^{ad}.

Similarly to the case of filtered φ-modules in [Hel13] there is a notion of weak admissibility for families of (φ, N)-modules with Hodge-Pink lattice over an adic space. We show that weak admissibility is an open condition.

Theorem 5.6. Let μ be a cocharacter as above, then the groupoid

X ↦ {(D, Φ, N, 𝔮) ∈ ℋ_{φ,N,⪯μ}(X) | D ⊗ κ(x) is weakly admissible for all x ∈ X}

is an open substack ℋ_{φ,N,⪯μ}^{ad,wa} of ℋ_{φ,N,⪯μ}^{ad}.

Following the construction in [Hel13] we construct an open substack ℋ_{φ,N,⪯μ}^{ad,int} ⊂ ℋ_{φ,N,⪯μ}^{ad,wa} where an integral model for the (φ, N_{∇})-module over the open unit disc exists. Here integral means with respect to the ring of integers W in K_{0}. Dealing with Hodge-Pink lattices instead of filtrations makes is possible to generalize the period morphism of [PR09, § 5] beyond the miniscule case. That is, we consider a stack Ĉ_{⪯μ,N,K} in the category of formal schemes over Spf O_{E_{μ}} whose R-valued points parameterize tuples (𝔐, Φ, N) where 𝔐 is a finite locally free (R ⊗_{ℤ_{p}} W)[[u]] module, Φ is a semi-linear morphism Φ: 𝔐 → 𝔐 which is an isomorphism away from E(u) = 0, whose behavior at E(u) is controlled in terms of μ, and N is an endomorphism of 𝔐/u𝔐 satisfying NΦ = pΦN; see after Remark 6.10 for the precise definition.

Given a p-adic formal scheme X over Spf O_{E_{μ}} we construct a period morphism

Π(X): Ĉ_{⪯μ,N,K}(X) ⟶ ℋ_{φ,N,⪯μ}^{ad}(X^{ad})

and the substack ℋ_{φ,N,⪯μ}^{ad,int} will serve as the image of this morphism in the following sense.

Corollary 6.12. Let X be an adic space locally of finite type over the reflex field E_{μ} of μ and let f: X → ℋ_{φ,N,⪯μ}^{ad} be a morphism defined by (D, Φ, N, 𝔮). Then f factors over ℋ_{φ,N,⪯μ}^{ad,int} if and only if there exists an fpqc-covering (U_{i} → X)_{i∈I} and formal models U_{i} of U_{i} together with (𝔐_{i}, Φ_{i}) ∈ Ĉ_{⪯μ,N,K}(U_{i}) such that Π(U_{i})(𝔐_{i}, Φ_{i}) = (D, Φ, N, 𝔮)|_{U_{i}}.

Finally we go back to Galois representations. We prove that there is a canonical open subspace ℋ_{φ,N,⪯μ}^{red,ad,adm} of the reduced space underlying ℋ_{φ,N,⪯μ}^{ad,int} which carries a family of 𝒢_{K_{∞}}-representations. This family is universal in a sense made precise in the body of the article. Roughly this means that a morphism f: X → ℋ_{φ,N,⪯μ}^{ad,int} defined by some (𝔐, Φ, N) over a formal model X of X factors over ℋ_{φ,N,⪯μ}^{ad,adm} if and only there exists a family of 𝒢_{K_{∞}}-representations E on X such that the φ-module of E, in the sense of Fontaine, is (up to inverting p) given by the p-adic completion of (𝔐, Φ)[1/u]. For a finite extension L of E_{μ}, Kisin’s theory implies that we have an equality

ℋ_{φ,N,⪯μ}^{red,ad,adm}(L) = ℋ_{φ,N,⪯μ}^{ad,int}(L) = ℋ_{φ,N,⪯μ}^{ad,wa}(L)

of L-valued points.

If we restrict ourselves to the case of filtrations, i.e. to the above mentioned zero section in the vector bundle ℋ_{φ,N,μ}^{ad} → 𝒟_{φ,N,μ}^{ad} we may extend this family of 𝒢_{K_{∞}}-representations to a family of semi-stable 𝒢_{K}-representations which turns out to be the universal family of semi-stable representations (with fixed Hodge-Tate weights).

Theorem 8.8. There is an open substack 𝒟_{φ,N,μ}^{ad,adm} ⊂ 𝒟_{φ,N,μ}^{ad} over which there exists a family E of semistable 𝒢_{K}-representations such that D_{st}(E) = (D, Φ, N, F^{•}) is the restriction of the universal family of filtered (φ, N)-modules on 𝒟_{φ,N,μ}^{ad} to 𝒟_{φ,N,μ}^{ad,adm}. This family is universal in the following sense: Let X be an adic space locally of finite type over the reflex field E_{μ} of μ, and let E′ be a family of semi-stable 𝒢_{K}-representations on X with constant Hodge polygon equal to μ. Then there is a unique morphism f: X → 𝒟_{φ,N,μ}^{ad,adm} such that E′ ≅ f^{∗}E as families of 𝒢_{K}-representations.

The corresponding result for crystalline 𝒢_{K}-representations with constant Hodge polygon equal to μ, whose moduli space is 𝒟_{φ,μ}^{ad,adm}, is formulated and proved in Corollary 8.9.

Notations: Let K be a finite field extension of the p-adic numbers ℚ_{p} and fix an algebraic closure \overline{K} of K. We write ℂ_{p} for the p-adic completion of \overline{K} and let 𝒢_{K} = Gal(\overline{K}/K) be the absolute Galois group of K. Let K̃ be the Galois closure of K inside \overline{K}. Let K_{0} be the maximal unramified subfield of K and W its ring of integers. Set f := [K_{0} : ℚ_{p}], and let Frob_{p} be the Frobenius automorphism of K_{0} which induces the p-power map on the residue field of K_{0}. We fix once and for all a uniformizer π of K and its minimal polynomial E(u) = mipo_{π/K_{0}}(u) ∈ W [u] over K_{0}. It is an Eisenstein polynomial, and K = K_{0}[u]/(E(u)). We choose a compatible system π_{n} of p^{n}-th roots of π in \overline{K} and write K_{∞} for the field obtained from K by adjoining all π_{n}.

Acknowledgements. The second author acknowledges support of the DFG (German Research Foundation) in form of SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” and in form of a Forschungsstipendium He 6753/1-1. Both authors were also supported by SFB 878 “Groups, Geometry & Actions” of the DFG. We would further like to thank A. Mézard, M. Rapoport, T. Richarz and P. Scholze for helpful discussions.

2. Families of (φ, N)-modules with Hodge-Pink lattice

Let R be a ℚ_{p}-algebra and consider the endomorphism φ := id_{R} ⊗ Frob_{p} of R⊗_{ℚ_{p}} K_{0}. For an R⊗_{ℚ_{p}} K_{0}- module M we set φ^{∗}M := M ⊗_{R⊗_{ℚp}K_{0}, φ} R ⊗_{ℚ_{p}} K_{0}. Similar notation is applied to morphisms between R ⊗_{ℚ_{p}} K_{0}-modules. We let φ^{∗}: M → φ^{∗}M be the φ-semi-linear map with φ^{∗}(m) = m ⊗ 1.

We introduce the rings

𝔹_{R}^{+} := lim_{⟵_{i}}(R ⊗_{ℚ_{p}} K_{0}[u])/(E(u)^{i}) and 𝔹_{R} := 𝔹_{R}^{+}[\frac{1}{E(u)}].

In a certain sense 𝔹_{ℚ_{p}}^{+} and 𝔹_{ℚ_{p}} are the analogs of Fontaine’s rings B_{dR}^{+} and B_{dR} in Kisin’s theory [Kis06] of p-adic Galois representation. By Cohen’s structure theorem [Ser79, Theorem II.4.2] the ring 𝔹_{ℚ_{p}}^{+} = lim_{⟵} K_{0}[u]/(E(u)^{i}) is isomorphic to K[[t]] under a map sending t to \frac{E(u)}{E(0)}. The rings 𝔹_{R}^{+} and 𝔹_{R} are

relative versions over R, and are isomorphic to (R ⊗_{ℚ_{p}} K)[[t]], respectively (R ⊗_{ℚ_{p}} K)[[t]][\frac{1}{t}]. We extend φ to R ⊗_{ℚ_{p}} K_{0}[u] by requiring φ(u) = u^{p} and we define φ^{n}(𝔹_{R}^{+}) := lim_{⟵ i}(R ⊗_{ℚ_{p}} K_{0}[u])/(φ^{n}(E(u))^{i}).

Note that we may also identify φ^{n}(𝔹_{R}^{+}) with lim_{⟵}(R ⊗_{ℚ_{p}} K(π_{n})[u])/(1 − \frac{u}{π_{n}})^{i} = (R ⊗_{ℚ_{p}} K(π_{n}))[[1 − \frac{u}{π_{n}}]]

under the assignment \frac{E(u)}{E(0)} ↦ 1 − \frac{u}{π_{n}}; compare [Kis06, (1.1.1)]. We extend these rings to sheaves of rings φ^{n}(𝔹_{X}^{+}) := φ^{n}(𝔹_{O_{X}}^{+} ) on ℚ_{p}-schemes X or adic spaces X ∈ Ad_{ℚ_{p}}^{lft}. Here Ad_{ℚ_{p}}^{lft} denotes the category of adic spaces locally of finite type, see [Hub94] for example.

Remark 2.1. Note that φ^{n}(𝔹_{R}^{+}) is not a subring of 𝔹_{R}^{+}. If X = Spa(R, R^{∘}) is an affinoid adic space of finite type over ℚ_{p} one should think of φ^{n}(𝔹_{R}^{+}) as the completion of the structure sheaf on X × 𝕌 along the section defined by φ^{n}(E(u)) ∈ 𝕌. Here 𝕌 denotes the open unit disc over K_{0}.

Definition 2.2. (a) A φ-module (D, Φ) over R consists of a locally free R ⊗_{ℚ_{p}} K_{0}-module D of finite rank, and an R ⊗_{ℚ_{p}} K_{0}-linear isomorphism Φ: φ^{∗}D ^{∼} ⟶ D. A morphism α: (D, Φ) → (D̃, Φ̃) of φ-modules is an R ⊗_{ℚ_{p}} K_{0}-homomorphism α: D → D̃ with α ∘ Φ = Φ̃ ∘ φ^{∗}α.

(b) A (φ, N)-module (D, Φ, N) over R consists of a φ-module (D, Φ) over R and an R ⊗_{ℚ_{p}} K_{0}-linear endomorphism N: D → D satisfying N ∘ Φ = p · Φ ∘ φ^{∗}N. A morphism α: (D, Φ, N) → (D̃, Φ̃, Ñ) of (φ, N)-modules is morphism of φ-modules with α ∘ N = Ñ ∘ α.

The rank of D over R ⊗_{ℚ_{p}} K_{0} is called the rank of (D, Φ), resp. (D, Φ, N).

Every φ-module over R can be viewed as a (φ, N)-module with N = 0.

Lemma 2.3. (a) Every φ-module (D, Φ) over R is Zariski locally on Spec R free over R ⊗_{ℚ_{p}} K_{0}. (b) The endomorphism N of a (φ, N)-module over R is automatically nilpotent.

Proof. (a) Let 𝔪 ⊂ R be a maximal ideal. Then R/𝔪 ⊗_{ℚ_{p}} K_{0} is a direct product of fields which are transitively permuted by Gal(K_{0}/ℚ_{p}). The existence of the isomorphism Φ implies that D ⊗_{R} R/𝔪 is free over R/𝔪 ⊗_{ℚ_{p}} K_{0}. Now the assertion follows by Nakayama’s lemma.

(b) By (a) we may locally on R write N as a matrix with entries in R ⊗_{ℚ_{p}} K_{0}. Set d := rk D. If the entries of the d-th power N^{d} lie in Rad(0) ⊗_{ℚ_{p}} K_{0}, where Rad(0) = ⋂_{𝔭⊂R prime} 𝔭 is the nil-radical, then N is nilpotent. Thus we may check the assertion in L = Frac(R/𝔭)^{alg} for all primes 𝔭 ⊂ R. We replace R by L. Then D = ∏ V_{ψ} splits up into a direct product of d-dimensional L-vector spaces indexed by the embeddings ψ: K_{0} ↪ L. For every fixed embedding ψ the f-th power Φ^{f} restricts to an endomorphism Φ_{ψ} of V_{ψ} satisfying NΦ_{ψ} = p^{f}Φ_{ψ}N. If V (λ, Φ_{ψ}) denotes the generalized eigenspace for some λ ∈ L^{×}, then N maps V (λ, Φ_{ψ}) to V (p^{f}λ, Φ_{ψ}) and hence N is nilpotent, as there are only finitely many non-zero eigenspaces. This implies that N^{d} = 0. □

Remark 2.4. If R is even a K_{0}-algebra, we can decompose R ⊗_{ℚ_{p}} K_{0} ≅ ∏_{i∈ℤ/fℤ} R where the i-th factor is given by the map R⊗_{ℚ_{p}} K_{0} → R, a⊗b ↦ a Frob_{p}^{−i}(b) for a ∈ R, b ∈ K_{0}. For a (φ, N)-module over R we obtain corresponding decompositions D = ∏_{i} D_{i} and φ^{∗}D = ∏_{i}(φ^{∗}D)_{i} with (φ^{∗}D)_{i} = D_{i−1}, and hence also Φ = (Φ_{i}: D_{i−1} ^{∼} ⟶ D_{i})_{i} and N = (N_{i}: D_{i} → D_{i})_{i} with p Φ_{i} ∘N_{i−1} = N_{i} ∘Φ_{i}, because (φ^{∗}N)_{i} = N_{i−1}. If we set Ψ_{i} := Φ_{i} ∘ … ∘ Φ_{1} = (Φ ∘ φ^{∗}Φ ∘ … ∘ φ^{(i−1)∗}Φ)_{i}: D_{0} = (φ^{i∗}D)_{i} ^{∼} ⟶ D_{i} then p^{i} Ψ_{i} ∘ N_{0} = N_{i} ∘ Ψ_{i} for all i, and Ψ_{f} = (Φ^{f})_{0}. There is an isomorphism of (φ, N)-modules over R

(2.1) ( id_{D_{0}}, Ψ_{1}, …, Ψ_{f−1}): (∏_{i} D_{0}, ((Φ^{f})_{0}, id_{D_{0}}, …, id_{D_{0}}), (p^{i}N_{0})_{i}) ^{∼} ⟶ (∏_{i} D_{i}, (Φ_{i})_{i}, (N_{i})_{i}) .

Thus (D, Φ, N) is uniquely determined by (D_{0}, (Φ^{f})_{0}, N_{0}) satisfying p^{f}(Φ^{f})_{0} ∘ N_{0} = N_{0} ∘ (Φ^{f})_{0}. Further note that under this isomorphism Φ^{f} on (D, Φ, N) corresponds to ((Φ^{f})_{0}, …, (Φ^{f})_{0}) on the left hand side.

Definition 2.5. (a) A K-filtered (φ, N)-module (D, Φ, N, F^{•}) over R consists of a (φ, N)-module (D, Φ, N) over R together with a decreasing separated and exhaustive ℤ-filtration F^{•} on D_{K} := D ⊗_{K_{0}} K by R ⊗_{ℚ_{p}} K-submodules such that gr_{F}^{i} D_{K} := F^{i}D_{K}/F^{i+1}D_{K} is locally free as an Rmodule for all i. A morphism α: (D, Φ, N, F^{•}) → (D̃, Φ̃, Ñ, F̃^{•}) is a morphism of (φ, N)-modules with (α ⊗ id)(F^{i}D_{K}) ⊂ F̃^{i}D̃_{K}.

(b) A (φ, N)-module with Hodge-Pink lattice (D, Φ, N, 𝔮) over R consists of a (φ, N)-module (D, Φ, N) over R together with a 𝔹_{R}^{+}-lattice 𝔮 ⊂ D ⊗_{R⊗K_{0}} 𝔹_{R}. This means that 𝔮 is a finitely generated 𝔹_{R}^{+}-submodule, which is a direct summand as R-module satisfying 𝔹_{R} ·𝔮 = D ⊗_{R⊗K_{0}} 𝔹_{R}. We call 𝔮 the Hodge-Pink lattice of (D, Φ, N, 𝔮). A morphism α: (D, Φ, N, 𝔮) → (D̃, Φ̃, Ñ, 𝔮̃) is a morphism of (φ, N)-modules with (α ⊗ id)(𝔮) ⊂ 𝔮̃.

For every (φ, N)-module with Hodge-Pink lattice (D, Φ, N, 𝔮) over R we also consider the tautological 𝔹_{R}^{+}-lattice 𝔭 := D ⊗_{R⊗K_{0}} 𝔹_{R}^{+}.

Lemma 2.6. Let 𝔮 ⊂ D ⊗_{R⊗K_{0}} 𝔹_{R} be a 𝔹_{R}^{+}-submodule. Then 𝔮 is a 𝔹_{R}^{+}-lattice if and only if E(u)^{n}𝔭 ⊂ 𝔮 ⊂ E(u)^{−m}𝔭 for all n, m ≫ 0 and for any (some) such n, m the quotients E(u)^{−m}𝔭/𝔮 and 𝔮/E(u)^{n}𝔭 are finite locally free R-modules.

If this is the case then étale locally on Spec R the 𝔹_{R}^{+}-module 𝔮 is free of the same rank as 𝔭.

Proof. The assertion E(u)^{n}𝔭 ⊂ 𝔮 ⊂ E(u)^{−m}𝔭 for all n, m ≫ 0 is equivalent to 𝔹_{R}·𝔮 = D⊗_{R⊗K_{0}} 𝔹_{R} when 𝔮 is finitely generated. Consider such n, m. If 𝔮 is a 𝔹_{R}^{+}-lattice, hence a direct summand of D⊗_{R⊗K_{0}} 𝔹_{R} there is an R-linear section of the projection D ⊗_{R⊗K_{0}} 𝔹_{R} ↠ (D ⊗_{R⊗K_{0}} 𝔹_{R})/𝔮. The composition of this section with the inclusion E(u)^{−m}𝔭/𝔮 ↪ (D ⊗_{R⊗K_{0}} 𝔹_{R})/𝔮 factors through E(u)^{−m}𝔭 and realizes E(u)^{−m}𝔭/𝔮 as a direct summand of the R-module E(u)^{−m}𝔭/E(u)^{n}𝔭 which is locally free by Lemma 2.3(a). This shows that E(u)^{−m}𝔭/E(u)^{n}𝔭 ≅ (E(u)^{−m}𝔭/𝔮)⊕(𝔮/E(u)^{n}𝔭) and both E(u)^{−m}𝔭/𝔮 and 𝔮/E(u)^{n}𝔭 are finite locally free R-modules.

Conversely any isomorphism E(u)^{−m}𝔭/E(u)^{n}𝔭 ≅ (E(u)^{−m}𝔭/𝔮)⊕(𝔮/E(u)^{n}𝔭) together with the decomposition D⊗_{R⊗K_{0}} 𝔹_{R} ≅ (E(u)^{n}𝔭) ⊕ (E(u)^{−m}𝔭/E(u)^{n}𝔭) ⊕ (D⊗_{R⊗K_{0}} 𝔹_{R})/E(u)^{−m}𝔭 realizes 𝔮 as a direct summand of D ⊗_{R⊗K_{0}} 𝔹_{R}. Since E(u)^{n}𝔭 is finitely generated over 𝔹_{R}^{+} and 𝔮/E(u)^{n}𝔭 is finitely generated over R, also 𝔮 is finitely generated over 𝔹_{R}^{+}, hence a 𝔹_{R}^{+}-lattice.

To prove the local freeness of 𝔮 we may work locally on R and assume by Lemma 2.3(a) that 𝔭 is free over 𝔹_{R}^{+}, say of rank d, and 𝔮/E(u)^{n}𝔭 and E(u)^{−m}𝔭/𝔮 are free over R. There is a noetherian subring R̃ of R and a short exact sequence

(2.2)  \xymatrix{0 \ar[r] & Q̃ \ar[r] & P̃ \ar[r] & Ñ \ar[r] & 0}

6 U. HARTL, E. HELLMANN

of 𝔹+ ; R̃-modules which are free R̃-modules, such that the tensor product of (2.2) with R over R̃ is isomorphic to

(2.3)  \xymatrix{0 \ar[r] & 𝔮/E(u)^{n}𝔭 \ar[r] & E(u)^{−m}𝔭/E(u)^{n}𝔭 \ar[r] & E(u)^{−m}𝔭/𝔮 \ar[r] & 0 .}

Indeed, we can take R̃ as the finitely generated ℚ_{p}-algebra containing all the coefficients appearing in matrix representations of the maps in (2.3) and the action of K_{0}[u]/(E(u))^{m+n}.

Let 𝔭̃ be a free 𝔹+ ; R̃-module of rank d, fix an isomorphism E(u)^{−m}𝔭̃ ⊗_{𝔹+ ; R̃} 𝔹+ ; R̃/(E(u))^{m+n} ^{∼} ⟶ P̃ and let the 𝔹+ ; R̃-module 𝔮̃ be defined by the exact sequence

(2.4)  \xymatrix{0 \ar[r] & 𝔮̃ \ar[r] & E(u)^{−m}𝔭̃ \ar[r] & Ñ \ar[r] & 0 .}

Since 𝔹+ ; R̃ ≅ (R̃ ⊗_{ℚ_{p}} K)[[t]] is noetherian, 𝔮̃ is finitely generated. Consider a maximal ideal 𝔪 ⊂ 𝔹+ ; R̃. Since t ∈ 𝔪, it maps to a maximal ideal 𝔫 of R̃. Since 𝔫 is finitely generated, 𝔹+ ; R̃ ⊗R̃ R̃/𝔫 ≅ (R̃/𝔫 ⊗_{ℚ_{p}} K)[[t]] and this is a direct product of discrete valuation rings. Thus 𝔮̃ ⊗R̃ R̃/𝔫 is locally free of rank d by the elementary divisor theorem. Since this holds for all 𝔪, [EGA, IV_{3}, Theorem 11.3.10] implies that 𝔮̃ is a projective 𝔹+ ; R̃-module and by [EGA, I_{new}, Proposition 10.10.8.6] it is locally on Spec R̃ ⊗_{ℚ_{p}} K free over 𝔹+ ; R̃. Let \{ψ: K ↪ \overline{ℚ}_{p}\} be the set of all ℚ_{p}-homomorphisms and let K̃ be the compositum of all ψ(K) inside \overline{ℚ}_{p}. Then R̃ → R̃ ⊗_{ℚ_{p}} K̃ is finite étale and the pullback of 𝔮̃ under this base change is locally on Spec R̃ ⊗_{ℚ_{p}} K̃ ⊗_{ℚ_{p}} K free over 𝔹_{R̃⊗_{ℚp} K̃}^{+}. Since Spec R̃ ⊗_{ℚ_{p}} K̃ ⊗_{ℚ_{p}} K = ∐_{ψ} Spec R̃ ⊗_{ℚ_{p}} K̃ it follows that the pullback of 𝔮̃ is already locally on Spec R̃ ⊗_{ℚ_{p}} K̃ free over 𝔹_{R̃⊗_{ℚp} K̃}^{+}.

To finish the proof it remains to show that 𝔮̃ ⊗_{𝔹+ ; R̃} 𝔹_{R}^{+} ≅ 𝔮. Tensoring (2.4) with 𝔹_{R}^{+} over 𝔹+ ; R̃ we obtain the top row in the diagram

[figure: 0; Tor_{1}^{𝔹+ ; R̃}(Ñ, 𝔹_{R}^{+}); 𝔮̃ ⊗_{𝔹+ ; R̃} 𝔹_{R}^{+}; E(u)^{−m}𝔭; Ñ ⊗_{𝔹+ ; R̃} 𝔹_{R}^{+}; ≅; 0; 0; 𝔮; E(u)^{−m}𝔭; E(u)^{−m}𝔭/𝔮; 0 .]

Abbreviate ℓ := m + n. Since the functor Ñ ⊗_{𝔹+ ; R̃} _{•} equals the composition of the functors (𝔹+ ; R̃/t^{ℓ}) ⊗_{𝔹+ ; R̃} _{•} followed by Ñ ⊗_{𝔹+ ; R̃/t^{ℓ}} _{•}, the Tor_{1}-module on the left can be computed from a change of rings spectral sequence and its associated 5-term sequence of low degrees

… ⟶ Tor_{1}^{𝔹+ ; R̃}(𝔹+ ; R̃/t^{ℓ}, 𝔹_{R}^{+}) ⊗_{𝔹+ ; R̃/t^{ℓ}} Ñ ⟶ Tor_{1}^{𝔹+ ; R̃}(Ñ, 𝔹_{R}^{+}) ⟶ Tor_{1}^{𝔹+ ; R̃/t^{ℓ}}(Ñ, 𝔹_{R}^{+}/t^{ℓ}) ⟶ 0 .

The right term in this sequence is zero because Tor_{1}^{𝔹+ ; R̃/t^{ℓ}}(Ñ, 𝔹_{R}^{+}/t^{ℓ}) = TorR̃ ; 1(Ñ, R) and Ñ is flat over R̃. The left term is zero because t^{ℓ} is a non-zero-divisor both in 𝔹+ ; R̃ and 𝔹_{R}^{+}. This shows that Tor_{1}^{𝔹+ ; R̃}(Ñ, 𝔹_{R}^{+}) = 0 and proves the lemma. □ Remark 2.7. (1) Let R = L be a field and let (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice over L. The Hodge-Pink lattice 𝔮 gives rise to a K-filtration F_{𝔮}^{•} as follows. Consider the natural projection

𝔭 ↠ 𝔭/E(u)𝔭 = D ⊗_{R⊗K_{0}} 𝔹_{R}^{+}/(E(u)) = D ⊗_{R⊗K_{0}} R ⊗_{ℚ_{p}} K = D_{K}

and let F_{𝔮}^{i}D_{K} be the image of 𝔭 ∩ E(u)^{i}𝔮 in D_{K} for all i ∈ ℤ, that is

F_{𝔮}^{i}D_{K} := (𝔭 ∩ E(u)^{i}𝔮)/(E(u)𝔭 ∩ E(u)^{i}𝔮) .

Since L is a field (D, Φ, N, F_{𝔮}^{•}) is a K-filtered (φ, N)-module over L. Note that this functor does not exist for general R, because gr_{F_{𝔮}}^{i}D_{K} will not be locally free over R in general. This is related to the fact that the Hodge polygon of F_{𝔮}^{•} is locally constant on R whereas the Hodge polygon of 𝔮 is only semi-continuous; see Remark 2.11 below.

(2) However, for general R consider the category of (φ, N)-modules with Hodge-Pink lattice (D, Φ, N, 𝔮) over R, such that 𝔭 ⊂ 𝔮 ⊂ E(u)^{−1}𝔭. This category is equivalent to the category of K-filtered (φ, N)- modules (D, Φ, N, F^{•}) over R with F^{0}D_{K} = D_{K} and F^{2} = 0. Namely, defining F_{𝔮}^{•} as in (1) we obtain

gr_{F_{𝔮}}^{i} D_{K} ≅ \{ \begin{matrix}E(u)^{−1}𝔭/𝔮 for & i = 0 , \\ 𝔮/𝔭  for & i = 1 , \\ 0  for & i ≠ 0, 1 , \end{matrix}

and so (D, Φ, N, F_{𝔮}^{•}) is a K-filtered (φ, N)-module by Lemma 2.6. Conversely, 𝔮 equals the preimage of

F_{𝔮}^{1}D_{K} under the morphism E(u)^{−1}𝔭 ⟶^{·E(u)} 𝔭 ↠ D_{K} and this defines the inverse functor.

(3) Now let (D, Φ, N, F^{•}) be a K-filtered (φ, N)-module over R. Using that 𝔹_{R}^{+} = (R ⊗_{ℚ_{p}} K)[[t]] is an R ⊗_{ℚ_{p}} K-algebra, we can define the Hodge-Pink lattice

𝔮 := 𝔮(F^{•}) := ∑_{i∈ℤ} E(u)^{−i}(F^{i}D_{K}) ⊗_{R⊗K} 𝔹_{R}^{+}.

It satisfies F_{𝔮}^{•} = F^{•}. Using Lemma 2.6 one easily finds that 𝔮(F^{•}) is indeed a 𝔹_{R}^{+}-lattice.

Example 2.8. The K-filtered (φ, N)-modules over R = ℚ_{p} which correspond to the cyclotomic character χ_{cyc} : 𝒢_{K} → ℤp^{×} are D_{st}(χ_{cyc}) = (K_{0}, Φ = p^{−1}, N = 0, F^{•}) with F^{−1} = K ⊋ F^{0} = (0) and its dual D_{st}^{∗}(χ_{cyc}) = (K_{0}, Φ = p, N = 0, F^{•}) with F^{1} = K ⊋ F^{2} = (0); see Definition 8.1 and Formula (8.2) below. For both there exists a unique Hodge-Pink lattice which induces the filtration. On D_{st}(χ_{cyc}) it is 𝔮 = E(u)𝔭 and on D_{st}^{∗}(χ_{cyc}) it is 𝔮 = E(u)^{−1}𝔭.

We want to introduce Hodge weights and Hodge polygons. Let d > 0, let B ⊂ GL_{d} be the Borel subgroup of upper triangular matrices and let T ⊂ B be the maximal torus consisting of the diagonal matrices. Let G̃ := Res_{K/ℚ_{p}} GL_{d,K}, B̃ = Res_{K/ℚ_{p}} B and T̃ := Res_{K/ℚ_{p}} T be the Weil restrictions. We consider cocharacters

(2.5)  μ: 𝔾_{m,\overline{ℚ}_{p}} ⟶ T̃_{\overline{ℚ}_{p}}

which are dominant with respect to the Borel B̃ of G̃. In other words on \overline{ℚ}_{p}-valued points the cocharacter

μ: \overline{ℚ}_{p}^{×} ⟶ ∏_{ψ: K→\overline{ℚ}_{p}} T (\overline{ℚ}_{p}) ,

where ψ runs over all ℚ_{p}-homomorphisms ψ: K → \overline{ℚ}_{p}, is given by cocharacters

μ_{ψ}: x ↦ diag(x^{μ_{ψ,1}}, …, x^{μ_{ψ,d}})

for some integers μ_{ψ,j} ∈ ℤ with μ_{ψ,j} ≥ μ_{ψ,j+1}. We define the reflex field E_{μ} of μ as the fixed field in \overline{ℚ}_{p} of { σ ∈ 𝒢_{ℚ_{p}}: μ_{σψ,j} = μ_{ψ,j} ∀ j, ψ }. It is a finite extension of ℚ_{p} which is contained in the compositum K̃ of all ψ(K) inside \overline{ℚ}_{p}. For each j the locally constant function ψ ↦ μ_{ψ,j} on Spec K̃ ⊗_{ℚ_{p}} K ≅ ∐_{ψ: K→K̃} Spec K̃ descends to a ℤ-valued function μ_{j} on Spec E_{μ} ⊗_{ℚ_{p}} K, because μ_{j} is constant on the

fibers of Spec K̃ ⊗_{ℚ_{p}} K → Spec E_{μ} ⊗_{ℚ_{p}} K. In particular, the cocharacter μ is defined over E_{μ}. If R is an E_{μ}-algebra we also view μ_{j} as a locally constant ℤ-valued function on Spec R ⊗_{ℚ_{p}} K.

Construction 2.9. Let \underline{D} = (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice of rank d over a field extension L of ℚ_{p}. By Lemma 2.3(a) the L ⊗_{ℚ_{p}} K_{0}-module D is free. Since L ⊗_{ℚ_{p}} K is a product of fields, 𝔹_{L}^{+} = (L ⊗_{ℚ_{p}} K)[[t]] is a product of discrete valuation rings and 𝔮 is a free 𝔹_{L}^{+}-module of rank d. We choose bases of D and 𝔮. Then the inclusion 𝔮 ⊂ D ⊗_{L⊗K_{0}} 𝔹_{L} is given by an element γ of GL_{d}(𝔹_{L}) = G̃(L((t))). By the Cartan decomposition for G̃ there is a uniquely determined dominant cocharacter μ_{L}: 𝔾_{m,L} → T̃_{L} over L with γ ∈ G̃(L[[t]])μ_{L}(t)^{−1}G̃(L[[t]]). This cocharacter is independent of the chosen bases. If L contains K̃, it is defined over K̃ because T̃ splits over K̃. In this case we view it as an element of X_{∗}(TK̃)_{dom} and denote it by μ_{\underline{D}}(Spec L). It has the following explicit description. Under the decomposition L ⊗_{ℚ_{p}} K = ∏_{ψ: K→K̃} L we have G̃(L((t))) = ∏_{ψ} GL_{d}(L((t))), μ_{L} = (μ_{ψ})_{ψ}, and γ ∈ ∏_{ψ} GL_{d}(L[[t]])μ_{ψ}(t)^{−1} GL_{d}(L[[t]]). The t^{−μ_{ψ,1}}, …, t^{−μ_{ψ,d}} are the elementary divisors of the ψcomponent 𝔮_{ψ} of 𝔮 with respect to 𝔭. That is, there is an L[[t]]-basis (v_{ψ,1}, …, v_{ψ,d}) of the ψ-component 𝔭_{ψ} of 𝔭 such that (t^{−μ_{ψ,1}} v_{ψ,1}, …, t^{−μ_{ψ,d}} v_{ψ,d}) is an L[[t]]-basis of 𝔮_{ψ}.

Let (D, Φ, N, F_{𝔮}^{•}) be the K-filtered (φ, N)-module associated with \underline{D} by Remark 2.7(1). Then F_{𝔮}^{i}D_{K,ψ} = ⟨v_{ψ,j}: i − μ_{ψ,j} ≤ 0⟩_{L} and

dim_{L} gr_{F_{𝔮}}^{i}D_{K,ψ} = #{j: i − μ_{ψ,j} = 0} .

More generally, for a K-filtered (φ, N)-module (D, Φ, N, F^{•}) over a field extension L of K̃ we consider the decomposition D_{K} = ∏_{ψ} D_{K,ψ} and define the integers μ_{ψ,1} ≥ … ≥ μ_{ψ,d} by the formula

dim_{L} gr_{F}^{i} D_{K,ψ} = #{j: μ_{ψ,j} = i} .

We define the cocharacter μ_{(D,Φ,N,F^{•})}(Spec L) := (μ_{ψ})_{ψ} and view it as an element of X_{∗}(TK̃)_{dom}.

Definition 2.10. (a) Let R be a K̃-algebra and consider the decomposition R ⊗_{ℚ_{p}} K = ∏_{ψ:K↪K̃} R. Let \underline{D} be a (φ, N)-module with Hodge-Pink lattice (respectively a K-filtered (φ, N)-module) of rank d over R. For every point s ∈ Spec R we consider the base change s^{∗}\underline{D} of \underline{D} to κ(s). We call the cocharacter μ_{\underline{D}}(s) := μ_{s^{∗}\underline{D}}(Spec κ(s)) from Construction 2.9 the Hodge polygon of \underline{D} at s and we consider μ_{\underline{D}} as a function μ_{\underline{D}} : Spec R → X_{∗}(TK̃)_{dom}. The integers −μ_{ψ,j}(s) are called the Hodge weights of \underline{D} at s.

Now let μ : \overline{ℚ}_{p}^{×} → T̃(\overline{ℚ}_{p}) be a dominant cocharacter as in (2.5), let E_{μ} denote the reflex field of μ, and let R be an E_{μ}-algebra.

(b) Let \underline{D} be a (φ, N)-module with Hodge-Pink lattice (respectively a K-filtered (φ, N)-module) of rank d over R. We say that \underline{D} has constant Hodge polygon equal to μ if μ_{\underline{D}}(s) = μ for every point s ∈ Spec(R ⊗_{E_{μ}} K̃).

(c) Let \underline{D} = (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice over Spec R. We say that \underline{D} has Hodge polygon bounded by μ if

⋀_{𝔹_{R}^{+}}^{j} 𝔮 ⊂ E(u)^{−μ_{1}−...−μ_{j}} · ⋀_{𝔹_{R}^{+}}^{j} 𝔭

for all j = 1, …, d with equality for j = d, where the μ_{i} are the ℤ-valued functions on Spec R⊗_{ℚ_{p}} K determined by μ; see the discussion before Construction 2.9.

Equivalently the condition of being bounded by μ can be described as follows: Over K̃ the cocharacter μ is described by a decreasing sequence of integers μ_{ψ,1} ≥ ⋯ ≥ μ_{ψ,d} for every ℚ_{p}-embedding ψ : K ↪ \overline{ℚ}_{p}. Let R′ = R ⊗_{E_{μ}} K̃, then R′ ⊗_{ℚ_{p}} K ≅ ∏_{ψ:K→K̃} R′_{ψ} with each R′_{ψ} = R′ under the isomorphism a ⊗ b ↦

(aψ(b))_{ψ}, where ψ : K → R′ is given via the embedding into the second factor of R′ = R⊗_{E_{μ}} K̃. Especially we view R′_{ψ} as a K-algebra via ψ. Under this isomorphism D ⊗_{R⊗K_{0}} 𝔹_{R′} =: 𝔭_{R′}[\frac{1}{t}] decomposes into a product ∏_{ψ} 𝔭_{R′}[\frac{1}{t}]_{ψ}, where 𝔭_{R′}[\frac{1}{t}]_{ψ} is a free R′_{ψ}[[t]][\frac{1}{t}]-module and the 𝔹_{R′}^{+}-lattice 𝔭_{R′} ⊂ 𝔭_{R′}[\frac{1}{t}] decomposes into a product of R′_{ψ}[[t]]-lattices 𝔭_{R′,ψ} ⊂ 𝔭_{R′}[\frac{1}{t}]_{ψ}.

Further, under the isomorphism D ⊗_{R⊗K_{0}} 𝔹_{R′} ≅ ∏_{ψ} 𝔭_{R′}[\frac{1}{t}]_{ψ} the Hodge-Pink lattice 𝔮_{R′} = 𝔮 ⊗_{R} R′ decomposes into a product 𝔮_{R′} = ∏_{ψ} 𝔮_{R′,ψ}, where 𝔮_{R′,ψ} is an R′_{ψ}[[t]]-lattice in 𝔭_{R′}[\frac{1}{t}]_{ψ}. Then the condition of being bounded by μ is equivalent to

(2.6)  ⋀_{𝔹_{R′}^{+}}^{j} 𝔮_{R′,ψ} ⊂ E(u)^{−μ_{ψ,1}−...−μ_{ψ,j}} · ⋀_{𝔹_{R′}^{+}}^{j} 𝔭_{R′,ψ}

for all ψ and all j = 1, …, d with equality for j = d.

Note that by Cramer’s rule (e.g. [Bou70, III.8.6, Formulas (21) and (22)]) the condition of Definition 2.10(c), respectively (2.6) is equivalent to

⋀_{𝔹_{R}^{+}}^{j} 𝔭 ⊂ E(u)^{μ_{d−j+1}+...+μ_{d}} · ⋀_{𝔹_{R}^{+}}^{j} 𝔮 ,

respectively

(2.7)  ⋀_{𝔹_{R′}^{+}}^{j} 𝔭_{R′,ψ} ⊂ E(u)^{μ_{ψ,d−j+1}+...+μ_{ψ,d}} · ⋀_{𝔹_{R′}^{+}}^{j} 𝔮_{R′,ψ}

for all j = 1, …, d with equality for j = d.

Remark 2.11. The Hodge polygon of a K-filtered (φ, N)-module (D, Φ, N, F^{•}) is locally constant on R, because gr_{F}^{i} D_{K,ψ} is locally free over R as a direct summand of the locally free R-module gr_{F}^{i} D_{K}.

In contrast, the Hodge polygon of a (φ, N)-module \underline{D} with Hodge-Pink lattice over R is not locally constant in general. Nevertheless, for any cocharacter μ as in (2.5) the set of points s ∈ Spec R such that μ_{\underline{D}}(s) ⪯ μ in the Bruhat order, is closed in Spec R. This is a consequence of the next

Proposition 2.12. Let μ ∈ X_{∗}(TK̃)_{dom} be a dominant cocharacter with reflex field E_{μ} and let R be an E_{μ}-algebra. Let \underline{D} = (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice of rank d over R.

(a) The condition that \underline{D} has Hodge polygon bounded by μ is representable by a finitely presented closed immersion (Spec R)_{⪯μ} ↪ Spec R.

(b) If R is reduced then \underline{D} has Hodge polygon bounded by μ if and only if for all points s ∈ Spec R⊗_{E_{μ}} K̃ we have μ_{\underline{D}}(s) ⪯ μ in the Bruhat order, that is, for all ψ the vector μ_{ψ} − μ_{\underline{D}}(s)_{ψ} ∈ ℤ^{d} is a nonnegative linear combination of the positive coroots α̌_{j} = (…, 0, 1, −1, 0, …) having the “ 1” as j-th entry.

(c) Let μ′ be another dominant cocharacter such that μ′ ⪯ μ in the Bruhat order. Let E_{μ′} denote its reflex field and let E = E_{μ}E_{μ′} ⊂ K̃ be the composite field. Assume that R is an E-algebra, then (Spec R)_{⪯μ′} ↪ (Spec R)_{⪯μ} as closed subschemes of Spec R.

Proof. (a) By Lemma 2.6 we find a large positive integer n such that E(u)^{n}𝔭 ⊂ 𝔮 ⊂ E(u)^{−n}𝔭. This implies ∧^{j}𝔮 ⊂ E(u)^{−jn} ∧^{j} 𝔭 for all j and ∧^{d}𝔭 ⊂ E(u)^{−dn} ∧^{d} 𝔮. Viewing μ_{j}: Spec R ⊗_{ℚ_{p}} K → ℤ as locally constant function as in the discussion before Construction 2.9, we consider the modules over 𝔹_{R}^{+} ≅ (R ⊗_{ℚ_{p}} K)[[t]]

(2.8)  M_{0} := E(u)^{−dn} ∧^{d} 𝔮 / E(u)^{μ_{1}+...+μ_{d}} · ∧^{d}𝔮 and
M_{j} := E(u)^{−jn} ∧^{j} 𝔭 / E(u)^{−μ_{1}−...−μ_{j}} · ∧^{j}𝔭 for 1 ≤ j ≤ d .

As R-modules they are finite locally free. Then \underline{D} has Hodge polygon bounded by μ if and only if for all j = 1, …, d all generators of ∧^{j}𝔮 are mapped to zero in M_{j} and all generators of ∧^{d}𝔭 are mapped to zero in M_{0}. Since M := M_{0} ⊕ … ⊕ M_{d} is finite locally free over R, this condition is represented by a finitely presented closed immersion into Spec R by [EGA, I_{new}, Lemma 9.7.9.1].

(b) If R is reduced then also the étale R-algebra R′ := R⊗_{E_{μ}} K̃ is reduced and R ↪ R′ ↪ ∏_{s∈Spec R′} κ(s) is injective. Therefore also M ↪ M ⊗_{R} (∏_{s∈Spec R′} κ(s)) is injective. So \underline{D} has Hodge polygon bounded by μ if and only if this holds for the pullbacks s^{∗}\underline{D} to Spec κ(s) at all points s ∈ Spec R′. By definition of μ′ := μ_{\underline{D}}(s) there is a κ(s)[[t]]-basis (v_{ψ,1}, …, v_{ψ,d}) of the ψ-component (s^{∗}𝔭)_{ψ} of s^{∗}𝔭 such that (t^{−μ′_{ψ,1}} v_{ψ,1}, …, t^{−μ′_{ψ,d}} v_{ψ,d}) is a κ(s)[[t]]-basis of (s^{∗}𝔮)_{ψ}. Therefore condition (2.6) holds if and only if μ_{ψ,1} + … + μ_{ψ,j} ≥ μ′_{ψ,1} + … + μ′_{ψ,j} for all ψ and j with equality for j = d. One easily checks that this is equivalent to μ′ ⪯ μ.

(c) Again μ′ ⪯ μ implies μ_{ψ,1} + … + μ_{ψ,j} ≥ μ′_{ψ,1} + … + μ′_{ψ,j} for all ψ and j with equality for j = d. We view μ_{j}, μ′_{j} as locally constant ℤ-valued functions on Spec E ⊗_{ℚ_{p}} K. Then μ_{1} + … + μ_{j} ≥ μ′_{1} + … + μ′_{j} for all j with equality for j = d. In terms of (2.8) the R-modules M_{j} for μ are quotients of the R-modules M′_{j} for μ′ with M′_{0} = M_{0}. Therefore (Spec R)_{⪯μ′} ↪ Spec R factors through (Spec R)_{⪯μ}. □

Remark 2.13. The reader should note that μ′ ⪯ μ does not imply a relation between E_{μ′} and E_{μ} as can be seen from the following example. Let d = 2 and [K : ℚ_{p}] = 2 and {ψ : K ↪ K̃} = Gal(K/ℚ_{p}) = {ψ_{1}, ψ_{2}}. Consider the three cocharacters μ, μ′, μ′′ given by μ_{ψ_{1}} = (2, 0), μ_{ψ_{2}} = (2, 0) and μ′_{ψ_{1}} = (2, 0), μ′_{ψ_{2}} = (1, 1) and μ′′_{ψ_{1}} = (1, 1), μ′′_{ψ_{2}} = (1, 1). Then μ′′ ⪯ μ′ ⪯ μ. On the other hand we find E_{μ} = E_{μ′′} = ℚ_{p} and E_{μ′} = K̃ = K.

Remark 2.14. In Definition 2.10(a) we assumed that R is a K̃-algebra to obtain a well defined Hodge polygon μ_{\underline{D}}(s) ∈ X_{∗}(T̃K̃). In Definition 2.10(b) we can lower the ground field over which R is defined to E_{μ} because Gal(K̃/E_{μ}) fixes μ. The ground field cannot be lowered further, as one sees from the following

Proposition 2.15. Let \underline{D} be a (φ, N)-module with Hodge-Pink lattice (respectively a K-filtered (φ, N)- module) of rank d over a field L such that μ_{\underline{D}}(s) = μ for all points s ∈ Spec L ⊗_{ℚ_{p}} K̃. Then there is a canonical inclusion of the reflex field E_{μ} ↪ L.

Proof. Since every K-filtered (φ, N)-module arises from a (φ, N)-module with Hodge-Pink lattice as in Remark 2.7(3), it suffices to treat the case where \underline{D} is a (φ, N)-module with Hodge-Pink lattice. We consider the decomposition L̃ := L ⊗_{ℚ_{p}} K̃ = ∏_{s∈Spec L̃} κ(s) and for each s we denote by α_{s} : L ↪ κ(s)

and β_{s} : K̃ ↪ κ(s) the induced inclusions. Let μ_{L} : 𝔾_{m,L} → T̃_{L} be the cocharacter over L associated with \underline{D} in Construction 2.9. The assumption of the proposition means that α_{s}(μ_{L}) = β_{s}(μ) for all s. The Galois group G := Gal(K̃, ℚ_{p}) acts on L̃. The Galois group Gal(κ(s)/α_{s}(L)) can be identified with the decomposition group G_{s} := {σ ∈ G : σ(s) = s} under the monomorphism Gal(κ(s)/α_{s}(L)) ↪ G, τ ↦ β_{s}^{−1} ∘ τ|_{β_{s}(K̃)} ∘ β_{s}. Since μ_{L} is defined over L, each τ ∈ Gal(κ(s)/α_{s}(L)) satisfies τ(α_{s}(μ_{L})) = α_{s}(μ_{L}), and hence (β_{s}^{−1} ∘ τ|_{β_{s}(K̃)} ∘ β_{s})(μ) = μ. By definition of the reflex field E_{μ} this implies that β_{s}^{−1} ∘ τ|_{β_{s}(K̃)} ∘ β_{s} ∈

Gal(K̃/E_{μ}) and τ|_{β_{s}(E_{μ})} = id. So β_{s}(E_{μ}) ⊂ α_{s}(L) and we get an inclusion α_{s}^{−1}β_{s} : E_{μ} ↪ L. To see that this is independent of s choose a σ ∈ G with σ(s) = s̃. Then αs̃ = σ ∘ α_{s} and βs̃ = σ ∘ β_{s}. □

3. Moduli spaces for (φ, N)-modules with Hodge-Pink lattice

We will introduce and study moduli spaces for the objects introduced in Chapter 2. Proposition 2.15 suggests to work over the reflex field.

Definition 3.1. Let μ be a cocharacter as in (2.5) and let E_{μ} be its reflex field. We define fpqc-stacks 𝒟_{φ,N,μ}, resp. ℋ_{φ,N,⪯μ}, resp. ℋ_{φ,N,μ} on the category of E_{μ}-schemes. For an affine E_{μ}-scheme Spec R

(a) the groupoid 𝒟_{φ,N,μ}(Spec R) consists of K-filtered (φ, N)-modules (D, Φ, N, F^{•}) over R of rank d with constant Hodge polygon equal to μ.

(b) the groupoid ℋ_{φ,N,⪯μ}(Spec R) consists of (φ, N)-modules with Hodge-Pink lattice (D, Φ, N, 𝔮) over R of rank d with Hodge polygon bounded by μ.

(c) the groupoid ℋ_{φ,N,μ}(Spec R) consists of (φ, N)-modules with Hodge-Pink lattice (D, Φ, N, 𝔮) over

R of rank d with Hodge polygon bounded by μ and constant equal to μ. Let 𝒟_{φ,μ} ⊂ 𝒟_{φ,N,μ}, resp. ℋ_{φ,⪯μ} ⊂ ℋ_{φ,N,⪯μ}, resp. ℋ_{φ,μ} ⊂ ℋ_{φ,N,μ} be the closed substacks on which N is zero. They classify φ-modules with K-filtration, resp. Hodge-Pink lattice and the corresponding condition on the Hodge polygon.

We are going to show that these stacks are Artin stacks of finite type over E_{μ}.

Locally on Spec R we may choose an isomorphism D ≅ (R ⊗_{ℚ_{p}} K_{0})^{d} by Lemma 2.3(a). Then Φ and N correspond to matrices Φ ∈ GL_{d}(R ⊗_{ℚ_{p}} K_{0}) = (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})(R) and N ∈ Mat_{d×d}(R ⊗_{ℚ_{p}} K_{0}) = (Res_{K_{0}/ℚ_{p}} Mat_{d×d})(R). The relation Φ φ^{∗}N = p N Φ is represented by a closed subscheme

P_{K_{0},d} ⊂ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}}) ×_{Spec ℚ_{p}} (Res_{K_{0}/ℚ_{p}} Mat_{d×d}) .

Theorem 3.2. (a) The ℚ_{p}-scheme P_{K_{0},d} is reduced, Cohen-Macaulay, generically smooth and equidimensional of dimension fd^{2}. In the notation of Remark 2.4 the matrix (Φ^{f})_{0} has no multiple eigenvalues at the generic points of the irreducible components of P_{K_{0},d}.

(b) The generic points of P_{K_{0},d} are in bijection with the partitions d = k_{1} + … + k_{m} for integers m and 1 ≤ k_{1} ≤ … ≤ k_{m}. To such a partition corresponds the generic point at which the suitably ordered eigenvalues λ_{1}, …, λ_{d} of (Φ^{f})_{0} satisfy p^{f}λ_{i} = λ_{j} if and only if j = i + 1 and i ∉ {k_{1}, k_{1} + k_{2}, …, k_{1} + … + k_{m}}. Equivalently to such a partition corresponds the generic point at which the nilpotent endomorphism N_{0}, in the notation of 2.4, has Jordan canonical form with m Jordan blocks of size k_{1}, …, k_{m}.

For the proof we will need the following lemma.

Lemma 3.3. Let r_{1}, …, r_{n} be integers with r_{1} + … + r_{n} ≥ n. Then ∑_{i=1}^{n} r_{i}^{2} − ∑_{i=1}^{n−1} r_{i} r_{i+1} > 1, except for the case when r_{1} = … = r_{n} = 1.

Proof. We multiply the claimed inequality with 2 and write it as r_{1}^{2} + ∑_{i=1}^{n−1}(r_{i} − r_{i+1})^{2} + r_{n}^{2} > 2. There are the following three critical cases

(a) ∑_{i}(r_{i} − r_{i+1})^{2} = 0,

(b) ∑_{i}(r_{i} − r_{i+1})^{2} = 1,

(c) ∑_{i}(r_{i} − r_{i+1})^{2} = 2.

In case (a) we have r_{1} = … = r_{n}. Since r_{1} = … = r_{n} = 1 was excluded and r_{1} ≤ 0 contradicts r_{1} + … + r_{n} ≥ n, we have r_{1}^{2} + r_{n}^{2} > 2.

In case (b) there is exactly one index 1 ≤ i < n with r_{1} = … = r_{i} ≠ r_{i+1} = … = r_{n} and |r_{i} −r_{i+1}| = 1. If r_{1} ≠ 0 ≠ r_{n} then r_{1}^{2} + ∑_{i=1}^{n−1}(r_{i} − r_{i+1})^{2} + r_{n}^{2} > 2. On the other hand, if r_{1} = ±1 and r_{n} = 0, then ∑_{i} r_{i} = ±i < n. And if r_{1} = 0 and r_{n} = ±1, then ∑_{i} r_{i} = ±(n − i) < n. Both are contradictions.

In case (c) there are exactly two indices 1 ≤ i < j < n with r_{1} = … = r_{i} and r_{i+1} = … = r_{j} and r_{j+1} = … = r_{n}, as well as |r_{i}−r_{i+1}| = 1 = |r_{j}−r_{j+1}|. If in addition r_{1} = r_{n} = 0 then ∑_{i} r_{i} = ±(j−i) < n, which is a contradiction. Therefore r_{1}^{2} + r_{n}^{2} > 0 and r_{1}^{2} + ∑_{i=1}^{n+1}(r_{i} − r_{i+1})^{2} + r_{n}^{2} > 2 as desired. □

Proof of Theorem 3.2. 1. By [EGA, IV_{2}, Proposition 6.5.3, Corollaires 6.3.5(ii), 6.1.2, and IV_{4}, Proposition 17.7.1] the statement may be checked after the finite étale base change Spec K_{0} → Spec ℚ_{p}. We will use throughout that after this base change, Remark 2.4 allows to decompose Φ = (Φ_{i})_{i} and N = (N_{i})_{i} such that p Φ_{i} ∘ N_{i} = N_{i+1} ∘ Φ_{i}.

2. We first prove that all irreducible components of P_{K_{0},d} have dimension greater or equal to fd^{2}. Sending (Φ, N) to the entries of the matrices Φ_{i}, N_{i} embeds P_{K_{0},d} ×_{ℚ_{p}} K_{0} into affine space 𝔸_{K_{0}}^{2fd^{2}} as a locally closed subscheme cut out by the fd^{2} equations p Φ_{i} ∘ N_{i} = N_{i+1} ∘ Φ_{i} for i = 0, …, f − 1. Therefore the codimension of P_{K_{0},d} ×_{ℚ_{p}} K_{0} in 𝔸_{K_{0}}^{2fd^{2}} is less or equal to fd^{2} by Krull’s principal ideal theorem [Eis95, Theorem 10.2], and all irreducible components of P_{K_{0},d} have dimension greater or equal to fd^{2} by [Eis95, Corollary 13.4].

3. We next prove the assertion on the generic points. Let y = (Φ, N) be the generic point of an irreducible component Y of P_{K_{0},d}. After passing to an algebraic closure L of κ(y) we may use Remark 2.4 to find a base change matrix S ∈ GL_{d}(L ⊗_{ℚ_{p}} K_{0}) such that S^{−1}Φ φ(S) = ((Φ^{f})_{0}, Id_{d}, …, Id_{d}) and (Φ^{f})_{0} is a block diagonal matrix in Jordan canonical form

(Φ^{f})_{0} = ( J_{1} J_{r} ) with J_{i} = ( ρ_{i} 1 1 ρ_{i} ) and N_{0} = ( N_{11} N_{1r} N_{r1} N_{rr} ) .

Note that a priory some of the ρ_{i} can be equal. Let s_{i} be the size of the Jordan block J_{i}. Then N_{ij} is an s_{i} × s_{j}-matrix. The condition p^{f}(Φ^{f})_{0} ∘ N_{0} = N_{0} ∘ (Φ^{f})_{0} is equivalent to p^{f}J_{i} N_{ij} = N_{ij}J_{j} for all i, j. It yields N_{ij} = (0) for p^{f}ρ_{i} ≠ ρ_{j}. By renumbering the J_{i} we may assume that N_{ij} ≠ (0) implies i < j. We set N_{ij} = (n_{μ,ν}^{(ij)})_{μ=1...s_{i}, ν=1...s_{j}}. When p^{f}ρ_{i} = ρ_{j} it follows from

( p^{f}n_{2,1}  p^{f}n_{2,s_{j}} p^{f}n_{s_{i},1} p^{f}n_{s_{i},s_{j}} 0  0 ) = p^{f}(J_{i} − ρ_{i})N_{ij} = N_{ij}(J_{j} − ρ_{j}) = ( 0 n_{1,1}  n_{1,s_{j}−1} 0 n_{s_{i},1} n_{s_{i},s_{j}−1} )

that p^{f}n_{μ,ν}^{(ij)} = n_{μ−1,ν−1}^{(ij)} for all μ, ν ≥ 2 and n_{μ,ν}^{(ij)} = 0 whenever μ − ν > min{0, s_{i} − s_{j}}. We set s := max{s_{i}}. The assertion of the theorem says that s = 1 and that all ρ_{i} are pairwise different.

First assume that s > 1. We exhibit a morphism Spec L[z, z^{−1}] → P_{K_{0},d} which sends the point {z = 1} to y and the generic point Spec L(z) to a point at which the maximal size of the Jordan blocks is strictly less than s. Since y was a generic point of P_{K_{0},d} this is impossible. The morphism Spec L[z, z^{−1}] → P_{K_{0},d} is given by matrices S̃, (Φ̃^{f})_{0} and Ñ_{0} as follows. We set S̃ := S. For all i with s_{i} = s we set

J̃_{i} := ( ρ_{i} 1 ρ_{i} 1 zρ_{i} )

and for all i with s_{i} < s we set J̃_{i} := J_{i}. When p^{f}ρ_{i} ≠ ρ_{j} we set Ñ_{ij} := (0). To define Ñ_{ij} when p^{f}ρ_{i} = ρ_{j}, and hence i < j, we distinguish the following cases

(a) If s_{i}, s_{j} < s we set Ñ_{ij} = N_{ij}.

(b) If s_{i} = s > s_{j} we set Ñ_{ij} = N_{ij}.

(c) If s_{i} < s = s_{j} we set Ñ_{ij} = (ñ_{μ,ν}^{(ij)})_{μ,ν} with ñ_{μ,s_{j}}^{(ij)} := n_{μ,s_{j}}^{(ij)} for all μ, with ñ_{μ,ν}^{(ij)} := 0 whenever

μ > ν + s_{i} − s_{j} + 1, and with ñ_{μ,ν}^{(ij)} := n_{μ,ν}^{(ij)} + (1 − z)p^{(s_{j}−1−ν)f} · ρ_{j} · n_{μ−ν+s_{j}−1, s_{j}}^{(ij)} for ν < s_{j} and μ ≤ ν + s_{i} − s_{j} + 1.

(d) If s_{i} = s_{j} = s we set Ñ_{ij} = (ñ_{μ,ν}^{(ij)})_{μ,ν} with ñ_{μ,s}^{(ij)} := n_{μ,s}^{(ij)} for all μ, with ñ_{μ,ν}^{(ij)} := 0 whenever μ > ν,

and with ñ_{μ,ν}^{(ij)} := n_{μ,ν}^{(ij)} + (1 − z)p^{(s−1−ν)f} · ρ_{j} · n_{μ−ν+s−1, s}^{(ij)} for all μ ≤ ν < s.

We have to check that p^{f}J̃_{i} Ñ_{ij} = Ñ_{ij}J̃_{j} for all i, j with p^{f}ρ_{i} = ρ_{j}. In case (a) this is obvious and in case (b) it follows from the fact that the bottom row of N_{ij} is zero. For case (c) we compute

p^{f}(J̃_{i} − ρ_{i})Ñ_{ij} = ( p^{f} ñ_{2,1}  p^{f} ñ_{2,s_{j}} p^{f} ñ_{s_{i},1} p^{f} ñ_{s_{i},s_{j}} 0  0 ) =

= ( 0 ñ_{1,1}  ñ_{1,s_{j}−2} ñ_{1,s_{j}−1} + (z − 1)ρ_{j} ñ_{1,s_{j}} 0 ñ_{s_{i},1}  ñ_{s_{i},s_{j}−2} ñ_{s_{i},s_{j}−1} + (z − 1)ρ_{j} ñ_{s_{i},s_{j}} ) = Ñ_{ij}(J̃_{j} − ρ_{j}) .

Finally for case (d) we compute

p^{f}(J̃_{i} − ρ_{i})Ñ_{ij} = ( p^{f} ñ_{2,1} p^{f} ñ_{2,s−1}  p^{f} ñ_{2,s} p^{f} ñ_{s,1} p^{f} ñ_{s,s−1}  p^{f} ñ_{s,s} 0  0  (z − 1)p^{f}ρ_{i} ñ_{s,s} ) =

= ( 0 ñ_{1,1}  ñ_{1,s−2} ñ_{1,s−1} + (z − 1)ρ_{j} ñ_{1,s} 0 ñ_{s,1}  ñ_{s,s−2} ñ_{s,s−1} + (z − 1)ρ_{j} ñ_{s,s} ) = Ñ_{ij}(J̃_{j} − ρ_{j}) .

Altogether this defines the desired morphism Spec L[z, z^{−1}] → P_{K_{0},d}.

So we have shown that s = 1 at the generic point y and that (Φ^{f})_{0} is a diagonal matrix. We still have to show that all diagonal entries are pairwise different. For this purpose we rewrite (Φ^{f})_{0} and N_{0} as

(Φ^{f})_{0} = ( λ_{1} Id_{r_{1}} λ_{n} Id_{r_{n}} ) and N_{0} = ( M_{11} M_{1n} M_{n1} M_{nn} ) .

We denote the multiplicity of the eigenvalue λ_{i} by r_{i} ≥ 1. Then M_{ij} is an r_{i} × r_{j}-matrix. By renumbering the λ_{i} we may assume that there are indices 0 = l_{0} < l_{1} < … < l_{m} = d such that p^{f}λ_{i} = λ_{j} if and only if j = i + 1 and i ∉ {l_{1}, …, l_{m}}.

We compute dim Y = trdeg_{ℚ_{p}} κ(y) = trdeg_{ℚ_{p}} L as follows. The eigenvalues λ_{l_{1}}, …, λ_{l_{m}} contribute at most the summand m to trdeg_{ℚ_{p}} L.

The matrix S ∈ GL_{d}(L ⊗_{ℚ_{p}} K_{0}) is determined only up to multiplication on the right with an element of the φ-centralizer C(L) := { S ∈ GL_{d}(L ⊗_{ℚ_{p}} K_{0}): S((Φ^{f})_{0}, Id_{d}, …, Id_{d}) = ((Φ^{f})_{0}, Id_{d}, …, Id_{d})φ(S) }

of ((Φ^{f})_{0}, Id_{d}, …, Id_{d}). Writing S = (S_{0}, …, S_{f−1}) this condition implies that S_{i} =^{!} (φ(S))_{i} := S_{i−1} for

i = 1, …, f − 1 and S_{0}(Φ^{f})_{0} =^{!} (Φ^{f})_{0}(φ(S))_{0} := (Φ^{f})_{0}S_{f−1} = (Φ^{f})_{0}S_{0}. Therefore C has dimension ∑_{i} r_{i}^{2} and the entries of S ∈ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})/C contribute at most the summand fd^{2} − ∑_{i} r_{i}^{2} to trdeg_{ℚ_{p}} L.

The condition p^{f}(Φ^{f})_{0} ∘ N_{0} = N_{0} ∘ (Φ^{f})_{0} is equivalent to p^{f}λ_{i} M_{ij} = λ_{j}M_{ij} for all i, j. This implies that there is no condition on M_{ij} when j = i + 1 and i ∉ {l_{1}, …, l_{m}}, and that all other M_{ij} are zero. So the entries of the M_{ij} contribute at most the summand ∑_{i∉{l_{1},...,l_{m}}} r_{i} r_{i+1} to trdeg_{ℚ_{p}} L.

Adding all summands and comparing with our estimate in part 2 above, we obtain

fd^{2} ≤ dim Y = trdeg_{ℚ_{p}} κ(y) ≤ m + fd^{2} − ∑_{i=1}^{n} r_{i}^{2} + ∑_{i∉{l_{1},...,l_{m}}} r_{i} r_{i+1}

= fd^{2} + ∑_{ν=0}^{m−1}(1 − ∑_{i=1+l_{ν}}^{l_{ν+1}} r_{i}^{2} + ∑_{i=1+l_{ν}}^{l_{ν+1}−1} r_{i} r_{i+1}) .

By Lemma 3.3 the parentheses are zero when all r_{i} = 1, and negative otherwise. So we have proved that r_{1} = … = r_{n} = 1. In other words, all diagonal entries of (Φ^{f})_{0} are pairwise different. Let k_{ν} := l_{ν} − l_{ν−1} for ν = 1, …, m. Then the generic point y corresponds to the partition d = k_{1} + … + k_{m} under the description of the generic points in the theorem. As we have noticed above the 1 × 1 matrices M_{ij} vanish at y unless j = i + 1 and i ∉ {l_{1}, …, l_{m}} and in the latter case we must have M_{ij}(y) ≠ 0. This implies the claim on the Jordan type of N_{0} at the generic points of the irreducible components.

Moreover, it follows that dim Y = fd^{2} for all irreducible components Y of P_{K_{0},d}. By [Eis95, Proposition 18.13] this also implies that P_{K_{0},d} ×_{ℚ_{p}} K_{0} is Cohen-Macaulay.

4. It remains to show that P_{K_{0},d} is generically smooth over ℚ_{p}. From this it follows that it is reduced, because it is Cohen-Macaulay. Let again y be the generic point of an irreducible component of P_{K_{0},d}×_{ℚ_{p}}K_{0} and let L be an algebraic closure of κ(y). As above, Remark 2.4 allows us to change the basis over L and assume that Φ = ((Φ_{0}^{f}), id, …, id) and N = (p^{i}N_{0})_{i} with (Φ^{f})_{0} = diag(λ_{1}, …, λ_{d}) and λ_{i} ≠ λ_{j} for all i ≠ j. We write F^{(0)} := (Φ^{f})_{0} and N_{0}^{(0)} := N_{0} = (n_{ij})_{ij}. The condition N_{0}^{(0)}F^{(0)} = p^{f}F^{(0)}N_{0}^{(0)} implies that n_{ij} = 0 if p^{f}λ_{i} ≠ λ_{j}. And conversely n_{ij} ≠ 0 if p^{f}λ_{i} = λ_{j} by our explicit description of N_{0} at y above.

We claim that for every n ≥ 1, any deformation (F^{(n−1)}, N_{0}^{(n−1)}) ∈ P_{K_{0},d}(L[ε]/ε^{n}) of (F^{(0)}, N_{0}^{(0)}) can be lifted further to (F^{(n)}, N_{0}^{(n)}) ∈ P_{K_{0},d}(L[ε]/ε^{n+1}). This implies that P_{K_{0},d} is smooth at y, as it follows that any tangent vector O_{P_{K_{0},d},y} → L[ε]/ε^{2} comes from a map O_{P_{K_{0},d},y} → L[[ε]] and hence the image of

Spec ( ⨁_{i≥0}(𝔪_{P_{K_{0},d},y}^{i}/𝔪_{P_{K_{0},d},y}^{i+1})) → Spec ( Sym^{•}(𝔪_{P_{K_{0},d},y}/𝔪_{P_{K_{0},d},y}^{2}))

contains any tangent vector. This means that the tangent cone at y equals the tangent space, and hence by [Mum99, III, §4 Definition 2 and Corollary 1] P_{K_{0},d} is smooth at y. Let us take any deformation (F̃, Ñ_{0}) ∈ GL_{d}(L[ε]/ε^{n+1}) × Mat_{d×d}(L[ε]/ε^{n+1}) of (F^{(n−1)}, N_{0}^{(n−1)}). Then we have Ñ_{0}F̃ − p^{f} F̃Ñ_{0} ∈ ε^{n}Mat_{d×d}(L). Changing (F̃, Ñ_{0}) to (F^{(n)}, N_{0}^{(n)}) = (F̃ + ε^{n}F′, Ñ_{0} + ε^{n}N′_{0}) with F′, N′_{0} ∈ Mat_{d×d}(L) we find

N_{0}^{(n)}F^{(n)} − p^{f}F^{(n)}N_{0}^{(n)} = (Ñ_{0}F̃ − p^{f} F̃Ñ_{0}) + ε^{n}(N_{0}^{(0)}F′ + N′_{0}F^{(0)} − p^{f}(F′N_{0}^{(0)} + F^{(0)}N′_{0}))

and hence it suffices to show that the map h : Mat_{d×d}(L) × Mat_{d×d}(L) → Mat_{d×d}(L) given by

h : (F′, N′_{0}) ⟼ (N′_{0}F^{(0)} − p^{f}F^{(0)}N′_{0}) + (N_{0}^{(0)}F′ − p^{f}F′N_{0}^{(0)})

is surjective. For this purpose let N′_{0} = (b_{ij})_{ij} and F′ = diag(a_{1}, …, a_{d}). Then we find that

h_{ij}(F′, N′_{0}) := h(F′, N′_{0})_{ij} = (λ_{j} − p^{f}λ_{i})b_{ij} + a_{j} n_{ij} − p^{f}a_{i} n_{ij} .

Whenever λ_{j} − p^{f}λ_{i} ∈ L^{×} and hence n_{ij} = 0 we obtain the surjectivity of h_{ij}. By permuting the indices we may assume that n_{ij} ≠ 0 implies j = i + 1. Treating every Jordan block of N_{0}^{(0)} separately we may further assume that n_{i,i+1} ≠ 0 for all i. It then follows that we have

h_{i,i+1}(F′, N′_{0}) = (a_{i+1} − p^{f}a_{i})n_{i,i+1}

which suffices to see that the map h is surjective. □

Remark 3.4. The scheme P_{K_{0},d} is in general not normal. For example if K = ℚ_{p} and d = 2 then P_{ℚ_{p},2} has two generic points. In one of them Φ has eigenvalues λ, pλ and N ≠ 0. In the other Φ has eigenvalues λ_{1}, λ_{2} with λ_{j} ≠ pλ_{i} for all i, j and N = 0. Both irreducible components meet in the codimension one point where λ_{2} = pλ_{1} and N = 0.

Let Δ denote the set of simple roots (defined over \overline{ℚ}_{p}) of G̃ := Res_{K/ℚ_{p}} GL_{d,K} with respect to the Borel subgroup B̃ and denote by Δ_{μ} ⊂ Δ the set of all simple roots α such that ⟨α, μ⟩ = 0. Here ⟨−, −⟩ is the canonical pairing between characters and cocharacters. We write P_{μ} for the parabolic subgroup of G̃ containing B̃ and corresponding to Δ_{μ} ⊂ Δ. This parabolic subgroup is defined over E_{μ}, and the quotient by this parabolic is a projective E_{μ}-variety

(3.1)  Flag_{K,d,μ} = G̃_{E_{μ}}/P_{μ}

representing the functor

R ↦ {filtrations F^{•} of R ⊗_{ℚ_{p}} K^{⊕d} with constant Hodge polygon equal to μ}

Thus 𝒟_{φ,N,μ} and 𝒟_{φ,μ} are isomorphic to the stack quotients

𝒟_{φ,N,μ} ≅ (P_{K_{0},d} ×_{Spec ℚ_{p}} Flag_{K,d,μ}) / (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}  and
𝒟_{φ,μ} ≅ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ×_{Spec ℚ_{p}} Flag_{K,d,μ}) / (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}

where g ∈ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} acts on (Φ, N, F^{•}) ∈ P_{K_{0},d} ×_{Spec ℚ_{p}} Flag_{K,d,μ} by

(Φ, N, F^{•}) ⟼ (g^{−1}Φ φ(g) , g^{−1}Ng , g^{−1}F^{•}) .

We next describe the moduli space for the Hodge-Pink lattice 𝔮. Fix the integers m = max{ μ_{ψ,1} | ψ : K → K̃ } and n = max{ −μ_{ψ,d} | ψ : K → K̃ }. Then by Cramer’s rule E(u)^{n}𝔭 ⊂ 𝔮 ⊂ E(u)^{−m}𝔭. So 𝔮 is determined by the epimorphism

(3.2)  pr: R ⊗_{ℚ_{p}} (K[t]/t^{m+n})^{⊕d} ^{∼} ⟶ E(u)^{−m}𝔭/E(u)^{n}𝔭 ↠ E(u)^{−m}𝔭/𝔮

which is induced by choosing an isomorphism D ≅ (R ⊗_{ℚ_{p}} K_{0})^{d} locally on R. The quotient E(u)^{−m}𝔭/𝔮 is a finite locally free R-module and of finite presentation over R ⊗_{ℚ_{p}} K[t]/t^{m+n} by Lemma 2.6. Therefore it is an R-valued point of Grothendieck’s Quot-scheme Quot_{O^{d} | K[t]/t^{m+n} | ℚ_{p}}; see [FGA, n^{∘}221, Theorem 3.1] or [AK80, Theorem 2.6]. This Quot-scheme is projective over ℚ_{p}. The boundedness by μ is represented by a closed subscheme Q_{K,d,⪯μ} of Quot_{O^{d} | K[t]/t^{m+n} | ℚ_{p}} ×_{Spec ℚ_{p}} Spec E_{μ} according to Proposition 2.12(a). Thus ℋ_{φ,N,⪯μ} and ℋ_{φ,⪯μ} are isomorphic to the stack quotients

ℋ_{φ,N,⪯μ} ≅ (P_{K_{0},d} ×_{Spec ℚ_{p}} Q_{K,d,⪯μ}) / (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}  and
ℋ_{φ,⪯μ} ≅ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ×_{Spec ℚ_{p}} Q_{K,d,⪯μ}) / (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}

where g ∈ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} acts on (Φ, N, pr) ∈ P_{K_{0},d} ×_{Spec ℚ_{p}} Q_{K,d,⪯μ} with pr from (3.2) by

(Φ, N, pr) ⟼ (g^{−1}Φ φ(g) , g^{−1}Ng , pr ∘ (g ⊗_{ℚ_{p}} id_{ℚ_{p}[t]/t^{m+n}})) .

Let Q_{K,d,μ} be the complement in Q_{K,d,⪯μ} of the image of ⋃_{μ′≺μ} Q_{K,d,⪯μ′} ×_{Spec E_{μ′}} Spec K̃ under the finite étale projection Q_{K,d,⪯μ} ×_{Spec E_{μ}} Spec K̃ → Q_{K,d,⪯μ}. Here the union is taken over all dominant cocharacters μ′ : 𝔾_{m,\overline{ℚ}_{p}} → T̃_{\overline{ℚ}_{p}} which are strictly less than μ in the Bruhat order; see Proposition 2.12(b). Since there are only finitely many such μ′ the scheme Q_{K,d,μ} is an open subscheme of Q_{K,d,⪯μ} and quasiprojective over E_{μ}. By Proposition 2.12(a) the stacks ℋ_{φ,N,μ} ⊂ ℋ_{φ,N,⪯μ} and ℋ_{φ,μ} ⊂ ℋ_{φ,⪯μ} are therefore open substacks and isomorphic to the stack quotients

ℋ_{φ,N,μ} ≅ (P_{K_{0},d} ×_{Spec ℚ_{p}} Q_{K,d,μ}) / (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}  and
ℋ_{φ,μ} ≅ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ×_{Spec ℚ_{p}} Q_{K,d,μ}) / (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} .

There is another description of Q_{K,d,⪯μ} in terms of the affine Grassmannian. Consider the infinite dimensional affine group schemes L^{+}GL_{d} and L^{+}G̃ over ℚ_{p}, and the sheaves LGL_{d} and LG̃ for the fpqc-topology on ℚ_{p} whose sections over a ℚ_{p}-algebra R are given by

L^{+}GL_{d}(R) = GL_{d}(R[[t]]) ,
L^{+}G̃(R) = G̃(R[[t]]) = GL_{d}(R ⊗_{ℚ_{p}} K[[t]]) = GL_{d}(𝔹_{R}^{+}) ,
LGL_{d}(R) = GL_{d}(R[[t]][\frac{1}{t}]) ,
LG̃(R) = G̃(R[[t]][\frac{1}{t}]) = GL_{d}(R ⊗_{ℚ_{p}} K[[t]][\frac{1}{t}]) = GL_{d}(𝔹_{R}) .

L^{+}GL_{d} and L^{+}G̃ are called the group of positive loops, and LGL_{d} and LG̃ are called the loop group of GL_{d}, resp. G̃. The affine Grassmannian of GL_{d}, resp. G̃ is the quotient sheaf for the fppf-topology on ℚ_{p}

Gr_{GL_{d}} := LGL_{d} /L^{+}GL_{d} ,  resp.  GrG̃ := LG̃/L^{+}G̃ .

They are ind-schemes over ℚ_{p} which are ind-projective; see [BD, §4.5], [BL94], [LS97], [HV11]. We set Gr_{GL_{d},K̃} := Gr_{GL_{d}} ×_{Spec ℚ_{p}} Spec K̃. Then there are morphisms

(3.3)  Q_{K,d,⪯μ} ⟶ GrG̃ ×_{Spec ℚ_{p}} Spec E_{μ} =: Gr_{G̃,E_{μ}}  and
Q_{K,d,⪯μ} ×_{Spec E_{μ}} Spec K̃ ⟶ ∏_{ψ:K→K̃} Gr_{GL_{d},K̃} ,

which are defined as follows. Let 𝔮 ⊂ (𝔹_{Q_{K,d,⪯μ}})^{⊕d} be the universal Hodge-Pink lattice over Q_{K,d,⪯μ}. Then by Lemma 2.6 there is an étale covering f : Spec R → Q_{K,d,⪯μ} such that f^{∗}𝔮 is free over 𝔹_{R}^{+}. With respect to a basis of f^{∗}𝔮 the equality 𝔹_{R} · f^{∗}𝔮 = D ⊗_{R⊗K_{0}} 𝔹_{R} corresponds to a matrix A ∈ GL_{d}(𝔹_{R}) = LG̃(R). The image of A in GrG̃(R) is independent of the basis and by étale descend defines the first factor of the map Q_{K,d,⪯μ} → GrG̃ ×_{Spec ℚ_{p}} Spec E_{μ}. The base change of this map along the finite étale morphism Spec K̃ → Spec E_{μ} defines the second map in (3.3), using the splitting G̃ ×_{ℚ_{p}} K̃ = ∏_{ψ} GL_{d,K̃} which

induces similar splittings for L^{+}G̃, LG̃, and GrG̃.

The boundedness by μ is represented by closed ind-subschemes

Gr_{G̃,E_{μ}}^{⪯μ}  and  Gr_{G̃,E_{μ}}^{⪯μ} ×_{Spec E_{μ}} Spec K̃ = ∏_{ψ} Gr_{GL_{d},K̃}^{⪯μ_{ψ}}

of Gr_{G̃,E_{μ}}, resp. ∏_{ψ} Gr_{GL_{d},K̃} through which the maps (3.3) factor. Conversely the universal matrix A over

LG̃ defines a 𝔹_{LG̃}^{+}-lattice 𝔮 = A · (𝔹_{LG̃}^{+})^{d}. Its restriction to Gr_{G̃,E_{μ}}^{⪯μ} has Hodge polygon bounded by μ and

corresponds to the inverses of the maps (3.3). This yields canonical isomorphisms Q_{K,d,⪯μ} ≅ Gr_{G̃,E_{μ}}^{⪯μ} and Q_{K,d,⪯μ} ×_{Spec E_{μ}} Spec K̃ ≅ ∏_{ψ} Gr_{GL_{d},K̃}^{⪯μ_{ψ}}. These isomorphisms restrict to isomorphisms of open

subschemes Q_{K,d,μ} ≅ Gr_{G̃,E_{μ}}^{μ} and Q_{K,d,μ} ×_{Spec E_{μ}} Spec K̃ ≅ ∏_{ψ} Gr_{GL_{d},K̃}^{μ_{ψ}}.

In view of [HV11, §4], especially Lemma 4.3, the boundedness by μ on ∏_{ψ} Gr_{GL_{d},K̃}^{⪯μ_{ψ}} can be phrased in terms of Weyl module representations of GL_{d,K̃}. In this formulation it was proved by Varshavsky [Var04, Proposition A.9] that Gr_{GL_{d},K̃}^{μ_{ψ}} is reduced. Therefore this locally closed subscheme is determined by its underlying set of points. Reasoning with the elementary divisor theorem as in Construction 2.9 shows that Gr_{GL_{d},K̃}^{μ_{ψ}} is equal to the locally closed Schubert cell L^{+}GL_{d,K̃} · μ_{ψ}(t)^{−1} ·L^{+}GL_{d,K̃} / L^{+}GL_{d,K̃} and is

a homogeneous space under L^{+}GL_{d,K̃}. This description descends to Q_{K,d,μ} and shows that the latter is

reduced and isomorphic to the locally closed Schubert cell L^{+}G̃_{E_{μ}} · μ(t)^{−1} · L^{+}G̃_{E_{μ}} / L^{+}G̃_{E_{μ}} which is a homogeneous space under L^{+}G̃_{E_{μ}} := L^{+}G̃ ×_{Spec ℚ_{p}} Spec E_{μ}.

These homogeneous spaces can be described more explicitly. Set

S_{GL_{d},μ_{ψ}} := L^{+}GL_{d,K̃} ∩ μ_{ψ}(t)^{−1} · L^{+}GL_{d,K̃} · μ_{ψ}(t) ⊂ L^{+}GL_{d,K̃}  and
S_{G̃,μ} := L^{+}G̃_{E_{μ}} ∩ μ(t)^{−1} · L^{+}G̃_{E_{μ}} · μ(t) ⊂ L^{+}G̃_{E_{μ}} .

These are closed subgroup schemes and the homogeneous spaces are isomorphic to the quotients

L^{+}GL_{d,K̃} /S_{GL_{d},μ_{ψ}}  ^{∼} ⟶ L^{+}GL_{d,K̃} · μ_{ψ}(t)^{−1} ·L^{+}GL_{d,K̃} / L^{+}GL_{d,K̃}  and
L^{+}G̃_{E_{μ}}/S_{G̃,μ} ^{∼} ⟶ L^{+}G̃_{E_{μ}} · μ(t)^{−1} · L^{+}G̃_{E_{μ}} / L^{+}G̃_{E_{μ}} ≅ Q_{K,d,μ} .

Consider the closed normal subgroup L^{++}G̃_{E_{μ}}(R) := { A ∈ L^{+}G̃_{E_{μ}}(R) : A ≡ 1 mod t }. Then the parabolic subgroup P_{μ} from (3.1) equals

P_{μ} = S_{G̃,μ} · L^{++}G̃_{E_{μ}} / L^{++}G̃_{E_{μ}} ⊂ L^{+}G̃_{E_{μ}} / L^{++}G̃_{E_{μ}} = G̃_{E_{μ}}

and this yields a morphism

(3.4)  Q_{K,d,μ} = L^{+}G̃_{E_{μ}}/S_{G̃,μ} ↠ L^{+}G̃_{E_{μ}}/S_{G̃,μ} · L^{++}G̃_{E_{μ}} = G̃_{E_{μ}}/P_{μ} = Flag_{K,d,μ} ,

with fibers isomorphic to S_{G̃,μ} · L^{++}G̃_{E_{μ}}/S_{G̃,μ}. The latter is an affine space because we may consider the

base change from E_{μ} to K̃ and the decomposition

(S_{G̃,μ} · L^{++}G̃_{E_{μ}}/S_{G̃,μ}) ×_{Spec E_{μ}} Spec K̃ = ∏_{ψ}(S_{GL_{d},μ_{ψ}} · L^{++} GL_{d,K̃} /S_{GL_{d},μ_{ψ}}).

Each component is an affine space whose R-valued points are in bijection with the matrices

(\begin{matrix}1 \\ a_{21} & 1 \\ ⋮ & ... & ... \\ a_{d1} & … & a_{d,d−1} & 1 \end{matrix})

where a_{ij} ∈ ⨁_{k=1}^{μ_{ψ,j}−μ_{ψ,i}−1} t^{k}R. The Galois group Gal(K̃/E_{μ}) canonically identifies the components with the same values for μ_{ψ}. Therefore S_{G̃,μ} · L^{++}G̃_{E_{μ}}/S_{G̃,μ} is an affine space.

We show that Q_{K,d,μ} is a geometric vector bundle over Flag_{K,d,μ} by exhibiting its zero section. The projection L^{+}G̃_{E_{μ}} ↠ G̃_{E_{μ}} has a section given on R-valued points by the map G̃_{E_{μ}}(R) → L^{+}G̃_{E_{μ}}(R) = G̃_{E_{μ}}(R[[t]]) induced from the natural inclusion R ↪ R[[t]]. Since L^{+}P_{μ} ⊂ S_{G̃,μ} by definition of P_{μ}, this section induces a section

Flag_{K,d,μ} ⟶ L^{+}G̃_{E_{μ}}/L^{+}P_{μ} ⟶ L^{+}G̃_{E_{μ}}/S_{G̃,μ} = Q_{K,d,μ} .

This is the zero section of the geometric vector bundle Q_{K,d,μ} over Flag_{K,d,μ}. Using lattices the section coincides with the map F^{•} ↦ 𝔮(F^{•}) defined in Remark 2.7(3) and the projection Q_{K,d,μ} → Flag_{K,d,μ} coincides with the map 𝔮 ↦ F_{𝔮}^{•} from Remark 2.7(1). Let us summarize.

Proposition 3.5. (a) Q_{K,d,⪯μ} is projective over E_{μ} of dimension ∑_{ψ,j}(d + 1 − 2j)μ_{ψ,j} and contains

Q_{K,d,μ} as dense open subscheme. Both schemes are irreducible.

(b) Q_{K,d,μ} is smooth over E_{μ} and isomorphic to the homogeneous space L^{+}G̃_{E_{μ}}/S_{G̃,μ} which is a geometric vector bundle over Flag_{K,d,μ}.

Proof. Everything was proved above, except the formula for the dimension and the density of Q_{K,d,μ} which follow from [BD, 4.5.8 and 4.5.12]. The irreducibility of Q_{K,d,⪯μ} is a consequence of the density statement. □

Theorem 3.6. (a) The moduli stacks 𝒟_{φ,N,μ}, 𝒟_{φ,μ}, ℋ_{φ,N,⪯μ}, ℋ_{φ,⪯μ}, ℋ_{φ,N,μ} and ℋ_{φ,μ} are noether-

ian Artin stacks of finite type over E_{μ}.

(b) The stack ℋ_{φ,N,μ} is a dense open substack of ℋ_{φ,N,⪯μ} and projects onto 𝒟_{φ,N,μ}. The morphism ℋ_{φ,N,μ} → 𝒟_{φ,N,μ} has a section and is relatively representable by a vector bundle.

(c) The stack ℋ_{φ,μ} is a dense open substack of ℋ_{φ,⪯μ} and projects onto 𝒟_{φ,μ}. The morphism ℋ_{φ,μ} → 𝒟_{φ,μ} has a section and is relatively representable by a vector bundle.

(d) The stacks ℋ_{φ,⪯μ}, ℋ_{φ,μ} are irreducible of dimension ∑_{ψ,j}(d + 1 − 2j)μ_{ψ,j}, and 𝒟_{φ,μ} is irreducible of dimension ∑_{ψ} #{(i, j) : μ_{ψ,i} > μ_{ψ,j}}. The stacks ℋ_{φ,μ} and 𝒟_{φ,μ} are smooth over E_{μ}.

(e) The stacks ℋ_{φ,N,⪯μ}, ℋ_{φ,N,μ} are equi-dimensional of dimension ∑_{ψ,j}(d + 1 − 2j)μ_{ψ,j}, and 𝒟_{φ,N,μ} is equi-dimensional of dimension ∑_{ψ} #{(i, j) : μ_{ψ,i} > μ_{ψ,j}}. The stacks ℋ_{φ,N,μ} and 𝒟_{φ,N,μ} are reduced, Cohen-Macaulay and generically smooth over E_{μ}. The irreducible components of ℋ_{φ,N,⪯μ}, ℋ_{φ,N,μ} and 𝒟_{φ,N,μ} are indexed by the possible Jordan types of the nilpotent endomorphism N.

Proof. (a) The stacks are quotients of noetherian schemes of finite type over E_{μ} by the action of the smooth group scheme (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} and hence are noetherian Artin stacks of finite type by [LM00, 4.6.1, 4.7.1, 4.14].

(b) and (c) follow from the corresponding statements for Q_{K,d,μ} in Proposition 3.5.

(d) The covering spaces

Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ×_{Spec ℚ_{p}} Q_{K,d,⪯μ}, resp.
Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ×_{Spec ℚ_{p}} Q_{K,d,μ}, resp.
Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ×_{Spec ℚ_{p}} Flag_{K,d,μ}

of these stacks are irreducible because Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} is geometrically irreducible. This implies the irreducibility of the stacks. The formulas for the dimension follow from [LM00, pp. 98f] and Proposition 3.5, respectively the well known dimension formula for partial flag varieties. The smoothness follows from the smoothness of Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ×_{Spec ℚ_{p}} Q_{K,d,μ}, resp. Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ×_{Spec ℚ_{p}} Flag_{K,d,μ} by [LM00, 4.14].

(e) As in (d) these results are direct consequences of the corresponding results on the covering spaces, which follow from Theorem 3.2. We only need to convince ourselves that the action of (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} does not identify irreducible components of P_{K_{0},d}. However this follows from the fact that the Jordan canonical forms of the nilpotent endomorphism N at two distinct generic points y_{1} and y_{2} of P_{K_{0},d} are distinct by the description in Theorem 3.2. □

Remark 3.7. These stacks are not separated. Namely, let \underline{D}, \underline{D}′ be two (φ, N)-modules with Hodge-Pink lattice (respectively two K-filtered (φ, N)-modules) over R. Then Isom(\underline{D}, \underline{D}′) is representable by an algebraic space, separated and of finite type over R; see [LM00, Lemme 4.2]. The above stacks are separated over E_{μ} if and only if all these algebraic spaces Isom(\underline{D}, \underline{D}′) are proper. This is not the case in general. For example let R be a discrete valuation ring with fraction field L, let D = D′ = R ⊗_{ℚ_{p}} K_{0}^{d} with Φ = id and N = 0. Then every element f ∈ L is an automorphism of D ⊗_{R} L, compatible with Φ and N. However, it extends to an automorphism of D only if f ∈ R^{×}.

4. Vector bundles on the open unit disc

In [Kis06] Kisin related K-filtered (φ, N)-modules over ℚ_{p} to vector bundles on the open unit disc. This was generalized in [Hel13, §5] to families of K-filtered φ-modules with Hodge-Tate weights 0 and −1. In this section we generalize it to arbitrary families of (φ, N)-modules with Hodge-Pink lattice. For this purpose we work in the category Ad_{ℚ_{p}}^{lft} of adic spaces locally of finite type over Spa(ℚ_{p}, ℤ_{p}); see [Hub93, Hub94, Hub96] and [Hel13, §2.2]. Since the stacks 𝒟_{φ,μ}, 𝒟_{φ,N,μ}, ℋ_{φ,⪯μ}, ℋ_{φ,N,⪯μ}, ℋ_{φ,μ} and ℋ_{φ,N,μ} are quotients of quasi-projective schemes over E_{μ} they give rise to stacks on Ad_{E_{μ}}^{lft} which we denote by ℋ_{φ,N,μ}^{ad}, etc.

For 0 ≤ r < 1 we write 𝔹_{[0,r]} for the closed disc of radius r over K_{0} in the category of adic spaces and denote by

𝕌 = lim_{\overrightarrow{r→1}} 𝔹_{[0,r]}

the open unit disc. This is an open subspace of the closed unit disc (which is not identified with the set of all points x in the closed unit disc with |x| < 1 in the adic setting). In the following we will always write u for the coordinate function on 𝔹_{[0,r]} and 𝕌, i.e. we view

B^{[0,r]} := Γ(𝔹_{[0,r]}, O_{𝔹_{[0,r]}})

and B^{[0,1)} := Γ(𝕌, O_{𝕌}) as sub-rings of K_{0}[[u]].

Let X ∈ Ad_{ℚ_{p}}^{lft} be an adic space over ℚ_{p} and write

ℬ_{X}^{[0,r]} = O_{X} ˆ⊗_{ℚ_{p}}B^{[0,r]} = pr_{X,∗} O_{X×𝔹_{[0,r]}}
ℬ_{X}^{[0,1)} = O_{X} ˆ⊗_{ℚ_{p}}B^{[0,1)} = pr_{X,∗} O_{X×𝕌}

for the sheafified versions of the rings B^{[0,r]} and B^{[0,1)} where pr_{X} is the projection onto X. These are sheaves of topological O_{X}-algebras on X.

We introduce the function

(4.1)  λ := ∏_{n≥0} φ^{n}(E(u)/E(0)) ∈ B^{[0,1)} .

and the differential operator N_{∇} := −uλ\frac{d}{du} : B_{[0,1)} → B_{[0,1)}. For any adic space X ∈ Ad_{ℚ_{p}}^{lft} we view λ as

a section of ℬ_{X}^{[0,1)} and N_{∇} as a differential operator on ℬ_{X}^{[0,1)}. The Frobenius φ on O_{X} ⊗_{ℚ_{p}} K_{0}[u] extends to a Frobenius endomorphism of ℬ_{X}^{[0,1)} again denoted by φ by means of φ(u) = u^{p}. These operators satisfy the relation

(4.2)  N_{∇} φ = p \frac{E(u)}{E(0)} · φ N_{∇} .

Definition 4.1. A (φ, N_{∇})-module (M, Φ_{M}, N_{∇}^{M}) over an adic space X ∈ Ad_{ℚ_{p}}^{lft} consists of a locally free sheaf M of finite rank on X ×_{ℚ_{p}} 𝕌, a differential operator N_{∇}^{M} : M → M[\frac{1}{λ}] over N_{∇}, that is N_{∇}^{M}(fm) = −uλ\frac{df}{du} · m + f · N_{∇}^{M}(m) for all sections f of O_{X×_{ℚp}𝕌} and m of M, and an O_{X×_{ℚp}𝕌}-linear

isomorphism Φ_{M} : (φ^{∗}M)[\frac{1}{E(u)}] ^{∼} ⟶ M[\frac{1}{E(u)}], satisfying N_{∇}^{M} ∘ Φ_{M} ∘ φ = p \frac{E(u)}{E(0)} · Φ_{M} ∘ φ ∘ N_{∇}^{M}.

A morphism α : (M, Φ_{M}, N_{∇}^{M}) → (N , Φ_{N} , N_{∇}^{N}) between (φ, N_{∇})-modules over X is a morphism α : M → N of sheaves satisfying α ∘ Φ_{M} = Φ_{N} ∘ φ^{∗}(α) and N_{∇}^{N} ∘ α = α ∘ N_{∇}^{M}.

Remark 4.2. (1) Note that it is not clear whether a (φ, N_{∇})-module M is locally on X free over X × 𝕌 and hence it is not clear whether pr_{X,∗} M is locally on X a free ℬ_{X}^{[0,1)}-module. However it follows from [KPX12, Proposition 2.1.15] that pr_{X,∗} M is a finitely presented ℬ_{X}^{[0,1)}-module.

(2) The differential operator N_{∇}^{M} can be equivalently described as a connection ∇_{M} : M → M ⊗ u^{−1}Ω_{X×𝕌/X}^{1}[\frac{1}{λ}] when we set ∇_{M}(m) := −\frac{1}{λ}N_{∇}^{M}(m) ⊗ \frac{du}{u}. Then N_{∇}^{M} is recovered as the composition of ∇_{M} followed by the map u^{−1}Ω_{X×𝕌/X}^{1}[\frac{1}{λ}] → M, du ↦ −uλ.

Let X ∈ Ad_{ℚ_{p}}^{lft} be an adic space. We will show that the category of (φ, N_{∇})-modules over X is equivalent to the category of (φ, N)-modules with Hodge-Pink lattice over X by defining two mutually quasi-inverse functors \underline{M} and \underline{D}.

To define \underline{M} let \underline{D} = (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice over X. We denote by pr : X ×_{ℚ_{p}} 𝕌 → X ×_{ℚ_{p}} K_{0} the projection and set (D, Φ_{D}) := pr^{∗}(D, Φ). Then

pr_{X,∗}(D, Φ_{D}) = (D, Φ) ⊗_{(O_{X}⊗K_{0})} ℬ_{X}^{[0,1)}.

We choose a 𝔹_{O_{X}}^{+} -automorphism η_{\underline{D}} of 𝔭 := D ⊗_{O_{X}⊗K_{0}} 𝔹_{O_{X}}^{+} and we let ι_{0} : D ↪ D ⊗_{O_{X}⊗K_{0}} 𝔹_{O_{X}}^{+} be

the embedding obtained as the composition of the natural inclusion D ⊗_{O_{X}⊗K_{0}} ℬ_{X}^{[0,1)} ↪ D ⊗_{O_{X}⊗K_{0}} 𝔹_{O_{X}}^{+} composed with the automorphism η_{\underline{D}}. Here we follow Kisin [Kis06, § 1.2] and choose

η_{\underline{D}} : D ⊗_{O_{X}⊗K_{0}} 𝔹_{O_{X}}^{+} ^{∼} ⟶ D ⊗_{O_{X}⊗K_{0}} 𝔹_{O_{X}}^{+} ,
(4.3)  d_{0} ⊗ f ⟼ ∑_{i} N^{i}(d_{0}) ⊗ \frac{(−1)^{i}}{i!} log(1 − \frac{E(u)}{E(0)})^{i} · f .

Remark 4.3. (1) Actually, Kisin introduces a formal variable ℓ_{u} over ℬ_{X}^{[0,1)} which formally acts like log u. He extends φ to ℬ_{X}^{[0,1)}[ℓ_{u}] via φ(ℓ_{u}) = p ℓ_{u}, extends N_{∇} to a derivation on ℬ_{X}^{[0,1)}[ℓ_{u}] via N_{∇}(ℓ_{u}) = −λ, and defines N as the ℬ_{X}^{[0,1)}-linear derivation on ℬ_{X}^{[0,1)}[ℓ_{u}] that acts as the differentiation of the formal variable ℓ_{u}. Under the Φ-equivariant identification

[figure: D[ℓ_{u}]^{N=0} := \{ ∑_{i=0}^{<∞} d_{i}ℓ_{u}^{i} : d_{i} ∈ D with N(∑_{i} d_{i}ℓ_{u}^{i}) = 0 \}; D; ≅; ∑_{i=0}^{<∞} \frac{(−1)^{i}}{i!} N^{i}(d_{0})ℓ_{u}^{i}; d_{0}]

Kisin’s map ι_{0} : D[ℓ_{u}]^{N=0} ⊗_{O_{X}⊗K_{0}} ℬ_{X}^{[0,1)} ↪ 𝔭 , ∑_{i} d_{i}ℓ_{u}^{i} ⊗ f ↦ ∑_{i} d_{i} ⊗ f · (log \frac{u}{π})^{i} corresponds to our ι_{0}, because we identify \frac{E(u)}{E(0)} with 1 − \frac{u}{π}.

(2) Instead of the above η_{\underline{D}} one could also choose η_{\underline{D}} = id_{𝔭}. This would lead to a few changes which we will comment on in Remark 4.11. Note that our η_{\underline{D}} from (4.3) is different from id_{𝔭} if N ≠ 0.

For all n ≥ 0 we now consider the map

pr_{X,∗} D[\frac{1}{λ}] ⟶^{Φ_{D}^{−j}} pr_{X,∗} φ^{j∗}(D[\frac{1}{λ}]) = pr_{X,∗} D[\frac{1}{λ}] ⊗_{ℬ_{X}^{[0,1)}, φ^{j}} ℬ_{X}^{[0,1)} ⟶^{φ^{j∗}ι_{0}} 𝔭[\frac{1}{E(u)}] ⊗_{𝔹_{O_{X}}^{+}} φ^{j}(𝔹_{O_{X}}^{+} )

where we write φ^{j∗}ι_{0} for ι_{0} ⊗ id. We set

(4.4)  pr_{X,∗} M := \{ m ∈ pr_{X,∗} D[\frac{1}{λ}] : φ^{j∗}ι_{0} ∘ Φ_{D}^{−j}(m) ∈ 𝔮 ⊗_{𝔹_{O_{X}}^{+}} φ^{j}(𝔹_{O_{X}}^{+} ) for all j ≥ 0 \} .

and we let M be the induced sheaf on X ×_{ℚ_{p}} 𝕌. Since λ = \frac{E(u)}{E(0)}φ(λ) the isomorphism Φ_{D} induces an

isomorphism Φ_{M} : (φ^{∗}M)[\frac{1}{E(u)}] ^{∼} ⟶ M[\frac{1}{E(u)}].

We want to show that D and M are locally free sheaves of finite rank on X ×_{ℚ_{p}} 𝕌. For D this follows from D|_{X×𝔹_{[0,r]}} = D ⊗_{(O_{X}⊗K_{0})} O_{X×𝔹_{[0,r]}}. We work on a covering of X by affinoids Y = Spa(A, A^{+}). Let h ∈ ℤ be such that 𝔮 ⊂ E(u)^{−h}𝔭 on Y and let n be maximal such that φ^{n}(E(u)) is not a unit in B_{[0,r]}, that is, such that r^{p^{n}} ≥ |π|. Then M|_{Y ×𝔹_{[0,r]}} is defined by the exact sequence

0 ⟶ M|_{Y ×𝔹_{[0,r]}} ⟶ λ^{−h}D|_{Y ×𝔹_{[0,r]}} \underrightarrow{⨁_{j=0}^{n} φ^{j∗}ι_{0}∘Φ_{D}^{−j}} ⨁_{j=0}^{n}(E(u)^{−h}𝔭/𝔮) ⊗_{𝔹_{A}^{+}, φ^{j}} φ^{j}(𝔹_{A}^{+}) ⟶ 0 .

The A ⊗_{ℚ_{p}} K_{0}[u]-module E(u)^{−h}𝔭/𝔮 is locally free over A, say of rank k. The endomorphism φ : K_{0}[u] → K_{0}[u] makes the target K_{0}[u] into a free module of rank p over the source K_{0}[u]. Therefore (E(u)^{−h}𝔭/𝔮) ⊗_{𝔹_{A}^{+}, φ^{j}} φ^{j}(𝔹_{A}^{+}) is locally free over A of rank p^{j}k. Since the affinoid algebra A is noetherian and B_{[0,r]} is a principal ideal domain by [Laz62, Corollary of Proposition 4] also Γ(Y ×𝔹_{[0,r]}, O_{Y ×𝔹_{[0,r]}}) = Aˆ⊗_{ℚ_{p}}B_{[0,r]} is noetherian. So Γ(Y × 𝔹_{[0,r]}, M) is finitely generated over Aˆ⊗_{ℚ_{p}}B_{[0,r]} and flat over A. The residue field of each maximal ideal 𝔪 ⊂ Aˆ⊗_{ℚ_{p}}B_{[0,r]} is finite over ℚ_{p} by [BGR84, Corollary 6.1.2/3]. Therefore 𝔫 = 𝔪 ∩ A is a maximal ideal of A. By the elementary divisor theorem A/𝔫 ⊗_{A} Γ(Y × 𝔹_{[0,r]}, M) is free over the product of principal ideal domains A/𝔫 ⊗_{ℚ_{p}} B_{[0,r]}. Therefore Γ(Y × 𝔹_{[0,r]}, M) is locally free of rank d over Aˆ⊗_{ℚ_{p}}B_{[0,r]} by [EGA, IV_{3}, Theorem 11.3.10]. This shows that M is a locally free sheaf of rank d on X ×_{ℚ_{p}} 𝕌.

We equip M with a differential operator N_{∇}^{M} over N_{∇}. On λ^{−h}D = D ⊗_{(O_{X}⊗K_{0})} λ^{−h}ℬ_{X}^{[0,1)} we have the differential operator N_{∇}^{D} := N ⊗ λ + id_{D} ⊗N_{∇}

(4.5)  \xymatrix{λ^{−h}D \ar[r]^{N⊗λ+ id_{D} ⊗N_{∇}} & λ^{−h}D \\ d ⊗ λ^{−h}f \ar@{|->}[r] & N(d) ⊗ λ^{1−h}f + d ⊗ (huλ^{−h}f \frac{dλ}{du} − uλ^{1−h} \frac{df}{du})}

with d ∈ D and f ∈ ℬ_{X}^{[0,1)}. Its image lies in λ^{−h}D. If E(u)^{n}𝔭 ⊂ 𝔮 ⊂ E(u)^{−h}𝔭 then λ^{n}D ⊂ M ⊂ λ^{−h}D. Thus N_{∇}^{D}(M) ⊂ λ^{−h}D ⊂ λ^{−h−n}M and we let N_{∇}^{M} be the restriction of N_{∇}^{D} to M. The equation N_{∇}^{M} ∘ Φ_{M} ∘ φ = p \frac{E(u)}{E(0)} · Φ_{M} ∘ φ ∘ N_{∇}^{M} is satisfied because it is satisfied on D by (4.5). Therefore we have constructed a (φ, N_{∇})-module \underline{M}(D, Φ, N, 𝔮) := (M, Φ_{M}, N_{∇}^{M}) over X. Note that in terms of Kisin’s description of D ≅ D[ℓ_{u}]^{N=0} ⊗_{(O_{X}⊗K_{0})} ℬ_{X}^{[0,1)} the differential operator N_{∇}^{M} is given as id_{D} ⊗N_{∇}. Example 4.4. The (φ, N)-modules with Hodge-Pink lattice from Example 2.8, corresponding to the cyclotomic character, give rise to the following (φ, N_{∇})-modules of rank 1 over X = Spa(ℚ_{p}, ℤ_{p}). For \underline{D} = (K_{0}, Φ = p^{−1}, N = 0, 𝔮 = E(u)𝔭) we obtain D = (ℬ_{X}^{[0,1)}, Φ_{D} = p^{−1}, N_{∇}) and M = λℬ_{X}^{[0,1)}. On the basis vector λ of M the actions of Φ_{D} and N_{∇} are given by Φ_{D}(φ(λ)) = p^{−1}φ(λ) = \frac{E(0)}{pE(u)} λ and N_{∇}(λ) = −u\frac{dλ}{du} λ. So we find \underline{M}(\underline{D}) ≅ (ℬ_{X}^{[0,1)}, Φ_{M} = \frac{E(0)}{pE(u)}, N_{∇}^{M}) with N_{∇}^{M}(f) = N_{∇}(f) − u\frac{dλ}{du} f. Similarly for \underline{D} = (K_{0}, Φ = p, N = 0, 𝔮 = E(u)^{−1}𝔭) we obtain D = (ℬ_{X}^{[0,1)}, Φ_{D} = p, N_{∇}) and M = λ^{−1}ℬ_{X}^{[0,1)} which leads to \underline{M}(\underline{D}) ≅ (ℬ_{X}^{[0,1)}, Φ_{M} = \frac{pE(u)}{E(0)}, N_{∇}^{M}) with N_{∇}^{M}(f) = N_{∇}(f) + u\frac{dλ}{du} f.

To define the quasi-inverse functor \underline{D} let (M, Φ_{M}, N_{∇}^{M}) be a (φ, N_{∇})-module over X. We denote by e : X ×_{ℚ_{p}} K_{0} → X ×_{ℚ_{p}} 𝕌 the isomorphism x ↦ (x, 0) onto the closed subspace defined by u = 0. Let (D, Φ, N) := e^{∗}(M, Φ_{M}, N_{∇}^{M}). It is a (φ, N)-module over X because N is clearly O_{X} ⊗_{ℚ_{p}} K_{0}-linear and e^{∗}(\frac{E(u)}{E(0)}) = 1 implies N ∘ Φ = p · Φ ∘ φ^{∗}N. By [PR09, Proposition 5.2] there is a unique O_{X×_{ℚp}𝕌}-linear isomorphism

(4.6)  ξ : pr^{∗}D[\frac{1}{λ}] ^{∼} ⟶ M[\frac{1}{λ}]

satisfying ξ ∘ pr^{∗} Φ = Φ_{M} ∘ φ^{∗}ξ and e^{∗}ξ = id_{D}. In particular the composition pr^{∗} Φ ∘ (φ^{∗}ξ)^{−1} = ξ^{−1} ∘ Φ_{M} induces an isomorphism φ^{∗}M ⊗ 𝔹_{O_{X}}^{+} ^{∼} ⟶ D ⊗_{(O_{X}⊗K_{0})} 𝔹_{O_{X}}^{+} = 𝔭 of 𝔹_{O_{X}}^{+} -modules. We set 𝔮 := η_{\underline{D}} ∘ (ξ ⊗ id_{𝔹_{O_{X}}})^{−1}(M ⊗ 𝔹_{O_{X}}^{+} ). Then \underline{D}(M, Φ_{M}, N_{∇}^{M}) := (D, Φ, N, 𝔮) is a (φ, N)-module with Hodge-Pink lattice over X by Lemma 2.6 and the following lemma. Lemma 4.5. Locally on a covering of X by affinoids Y = Spa(A, A^{+}) there exist integers h, n with E(u)^{n}Φ_{M}(φ^{∗}M) ⊂ M ⊂ E(u)^{−h}Φ_{M}(φ^{∗}M) such that the quotients

E(u)^{−h}Φ_{M}(φ^{∗}M)/M and M/E(u)^{n}Φ_{M}(φ^{∗}M)

are finite locally free over A. Proof. The existence of h and n follows from the finiteness of M and φ^{∗}M. Let 𝔪 ⊂ A be a maximal ideal

and set L = A/𝔪. Let |π| < r < 1 and set M̃:= Γ(Y ×_{ℚ_{p}} 𝔹_{[0,r]}, M) and { \atop φ^{∗}M} := Γ(Y ×_{ℚ_{p}} 𝔹_{[0,r]}, φ^{∗}M).

Then M/E(u)^{n}Φ_{M}(φ^{∗}M) ≅ M̃/E(u)^{n}Φ_{M}({ \atop φ^{∗}M}). Consider the exact sequence

\xymatrix{0 \ar[r] & E(u)^{n}{ \atop φ^{∗}M} \ar[r]^{Φ_{M}} & M̃ \ar[r] & M/E(u)^{n}Φ_{M}(φ^{∗}M) \ar[r] & 0}

in which the first map is injective because E(u) is a non-zero-divisor in Aˆ⊗_{ℚ_{p}}B_{[0,r]}. We tensor the sequence with L over A to obtain the exact sequence of L ⊗_{ℚ_{p}} B_{[0,r]}-modules

\xymatrix{0 \ar[r] & T \ar[r] & L ⊗_{A} E(u)^{n}{ \atop φ^{∗}M} \ar[r]^{id_{L} ⊗Φ_{M}} & L ⊗_{A} M̃ \ar[r] & L ⊗_{A} (M/E(u)^{n}Φ_{M}(φ^{∗}M)) \ar[r] & 0}

with T = Tor_{1}^{A}(L, M/E(u)^{n}Φ_{M}(φ^{∗}M)). Since L ⊗_{ℚ_{p}} K_{0} is a product of fields, L ⊗_{ℚ_{p}} B_{[0,r]} is a product of principal ideal domains by [Laz62, Corollary of Proposition 4]. Since E(u)^{n+h} annihilates L ⊗_{A} M/E(u)^{n}Φ_{M}(φ^{∗}M) the latter is a torsion module over L ⊗_{ℚ_{p}} B_{[0,r]}. By the elementary divisor theorem id_{L} ⊗Φ_{M} is injective and T = 0. Since M/E(u)^{n}Φ_{M}(φ^{∗}M) is finite over the noetherian ring A it is locally free by the local criterion of flatness [Eis95, Theorem 6.8]. From the exact sequence

\xymatrix{0 \ar[r] & E(u)^{−h}Φ_{M}(φ^{∗}M)/M \ar[r]^{·E(u)^{n+h}} & M/E(u)^{n+h}M \ar[r] & M/E(u)^{n}Φ_{M}(φ^{∗}M) \ar[r] & 0}

it follows that also E(u)^{−h}Φ_{M}(φ^{∗}M)/M is finite locally free over A. □

Theorem 4.6. For every adic space X ∈ Ad_{ℚ_{p}}^{lft} the functors \underline{M} and \underline{D} constructed above are mutually quasi-inverse equivalences between the category of (φ, N)-modules with Hodge-Pink lattice over X and the category of (φ, N_{∇})-modules over X.

Proof. We must show that the functors are mutually quasi-inverse. To prove one direction let (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice over X and let (M, Φ_{M}, N_{∇}^{M}) = \underline{M}(D, Φ, N, 𝔮). By construction e^{∗}M = D, and under this equality e^{∗}Φ_{M} corresponds to Φ. Since e^{∗}λ = 1, formula (4.5) shows that e^{∗}N_{∇}^{M} corresponds to N on D. By the uniqueness of the map ξ from (4.6), its inverse ξ^{−1} equals the inclusion M ↪ D[\frac{1}{λ}], by which we defined M. This shows that η_{\underline{D}} ∘ (ξ ⊗ id_{𝔹_{O_{X}}})^{−1}(M ⊗ 𝔹_{O_{X}}^{+} ) equals 𝔮 and that \underline{D} ∘ \underline{M} = id.

Conversely let (M, Φ_{M}, N_{∇}^{M}) be a (φ, N_{∇})-module over X and let (D, Φ, N, 𝔮) = \underline{D}(M, Φ_{M}, N_{∇}^{M}). Via the isomorphism ξ from (4.6), M is a φ-submodule of pr^{∗}D[\frac{1}{λ}]. By construction of 𝔮 and

\underline{M}(\underline{D}(M, Φ_{M}, N_{∇}^{M})) ⊂ pr^{∗}D[\frac{1}{λ}],

the latter submodule coincides with M modulo all powers of E(u). Since both submodules have a Frobenius which is an isomorphism outside V(E(u)) they are equal on all of X ×_{ℚ_{p}} 𝕌. It remains to show that N_{∇}^{M} is compatible with N_{∇}^{pr^{∗}D} under the isomorphism ξ : pr^{∗}D[\frac{1}{λ}] ^{∼} ⟶ M[\frac{1}{λ}]. We follow [Kis06, Lemma 1.2.12(3)] and let σ := ξ ∘ N_{∇}^{pr^{∗}D} − N_{∇}^{M} ∘ ξ. Then σ : pr^{∗}D[\frac{1}{λ}] → M[\frac{1}{λ}] is O_{X×_{ℚp}𝕌}-linear and

it suffices to show that σ(D) = 0. By (4.5) both N_{∇}^{pr^{∗}D} and N_{∇}^{M} reduce to N modulo u. Therefore σ(D) ⊂ uM[\frac{1}{λ}]. One checks that σ ∘ Φ_{pr^{∗}D} ∘ φ = p\frac{E(u)}{E(0)} · Φ_{M} ∘ φ ∘ σ and this implies

σ(D) = σ ∘ Φ_{pr^{∗}D}(φ^{∗}D) = p\frac{E(u)}{E(0)} · Φ_{M} ∘ φ^{∗}(uM[\frac{1}{λ}]) ⊂ u^{p}M[\frac{1}{λ}] .

By induction σ(D) ⊂ u^{p^{i}}M[\frac{1}{λ}] for all i and hence σ(D) = 0. This shows that also \underline{M} ∘ \underline{D} is isomorphic to the identity and proves the theorem. □

Corollary 4.7. The stack ℋ_{φ,N,⪯μ}^{ad} is isomorphic to the stack whose groupoid of X-valued points for X ∈ Ad_{E_{μ}}^{lft} consists of (φ, N_{∇})-modules (M, Φ_{M}, N_{∇}^{M}) over X satisfying

⋀_{O_{X×𝕌}}^{j} M ⊂ E(u)^{−μ_{1}−...−μ_{j}} · ⋀_{O_{X×𝕌}}^{j} Φ_{M}(φ^{∗}M)

with equality for j = rk M. Here μ_{i} is viewed as a ℤ-valued function on X ×_{ℚ_{p}} K.

Proof. This follows from the definition of the functor \underline{D}, in particular the definition of the Hodge-Pink lattice. □

Definition 4.8. We define substacks ℋ_{φ,N,μ}^{∇} ⊂ ℋ_{φ,N,μ}^{ad}, resp. ℋ_{φ,N,⪯μ}^{∇} ⊂ ℋ_{φ,N,⪯μ}^{ad}, resp. ℋ_{φ,μ}^{∇} ⊂ ℋ_{φ,μ}^{ad}, resp. ℋ_{φ,⪯μ}^{∇} ⊂ ℋ_{φ,⪯μ}^{ad}. For an adic space X ∈ Ad_{E_{μ}}^{lft} the groupoid ℋ_{φ,N,μ}^{∇}(X) consists of those (D, φ, N, 𝔮) ∈ ℋ_{φ,N,μ}^{ad}(X) for which the associated (φ, N_{∇})-module (M, Φ_{M}, N_{∇}^{M}) satisfies N_{∇}^{M}(M) ⊂ M. The groupoids ℋ_{φ,N,⪯μ}^{∇}(X), resp. ℋ_{φ,μ}^{∇}(X), resp. ℋ_{φ,⪯μ}^{∇}(X) are defined by the same condition. (Note that on the latter two N = 0, but N_{∇}^{M} ≠ 0.)

Theorem 4.9. The substacks ℋ_{φ,N,μ}^{∇} ⊂ ℋ_{φ,N,μ}^{ad}, resp. ℋ_{φ,N,⪯μ}^{∇} ⊂ ℋ_{φ,N,⪯μ}^{ad}, resp. ℋ_{φ,μ}^{∇} ⊂ ℋ_{φ,μ}^{ad}, resp. ℋ_{φ,⪯μ}^{∇} ⊂ ℋ_{φ,⪯μ}^{ad} are Zariski closed substacks. The substack ℋ_{φ,N,μ}^{∇} coincides with the image of the zero section of the vector bundle ℋ_{φ,N,μ}^{ad} → 𝒟_{φ,N,μ}^{ad}.

Remark 4.10. We can consider a family of (φ, N_{∇})-modules over 𝒟_{φ,N,μ}^{ad} namely we pull back the canonical family of (φ, N_{∇})-modules on ℋ_{φ,N,μ}^{ad} along the zero section. Then for x ∈ 𝒟_{φ,N,μ}^{ad}(ℚ_{p}) the fiber of this family at x coincides with the (φ, N_{∇})-module that Kisin [Kis06] associates with the filtered (φ, N)-module defined by x.

Remark 4.11. If instead of the isomorphism η_{\underline{D}} from (4.3) we choose η_{\underline{D}} = id_{𝔭} as in Remark 4.3(2), the above results remain valid, except that ℋ_{φ,N,μ}^{∇} coincides with the image of a different section. This section

is obtained by composing the zero section with the inverse η_{\underline{D}}^{−1} : d ⊗ f ↦ ∑_{i} N^{i}(d) ⊗ \frac{1}{i!} log(1 − \frac{E(u)}{E(0)})^{i} · f of the automorphism η_{\underline{D}}. It sends a filtration F^{•} to η_{\underline{D}}^{−1}(∑_{i∈ℤ} E(u)^{−i}(F^{i}D_{K}) ⊗_{R⊗K} 𝔹_{R}^{+}). Note that both sections coincide on the closed substack 𝒟_{φ,μ}^{ad} where N = 0.

Proof of Theorem 4.9. To prove that the substacks are closed let \underline{D} ∈ ℋ_{φ,N,⪯μ}^{ad}(X) for an adic space X ∈ Ad_{E_{μ}}^{lft} and let (M, Φ_{M}, N_{∇}^{M}) = \underline{M}(\underline{D}) be the associated (φ, N_{∇})-module over X. Locally on X there is an integer h with N_{∇}^{M}(M) ⊂ λ^{−h}M by Lemma 2.6 and the construction of N_{∇}^{M}. The quotients (λ^{−h}M/M)⊗(ℬ_{X}^{[0,1)}/(φ^{n}(E(u))^{h})) are finite locally free as O_{X}-modules for all n ≥ 0. Now the condition N_{∇}^{M}(M) ⊂ M is equivalent to the vanishing of the images under N_{∇}^{M} of a set of generators of M in (λ^{−h}M/M) ⊗ (ℬ_{X}^{[0,1)}/(φ^{n}(E(u))^{h})) for each n ≥ 0. Due to [EGA, I_{new}, Lemma 9.7.9.1] the latter is represented by a Zariski closed subspace of X.

We show that the closed substack ℋ_{φ,N,μ}^{∇} of ℋ_{φ,N,μ}^{ad} coincides with the image of the zero section. Since

N_{∇}^{D} on D := D ⊗_{O_{X}⊗K_{0}} ℬ_{O_{X}}^{[0,1)} induces the differential operator id_{D} ⊗N_{∇} on 𝔭 := D ⊗_{O_{X}⊗K_{0}} 𝔹_{O_{X}}^{+} under the map i_{0} = η_{\underline{D}} ∘ inclusion : D ↪ 𝔭 from (4.3), it follows directly that the image of the zero section is contained in ℋ_{φ,N,μ}^{∇}. To prove the converse we may work on the coverings X := (P_{K_{0},d} ×_{ℚ_{p}} Q_{K,d,μ})^{ad} of ℋ_{φ,N,μ}^{ad} and (P_{K_{0},d} ×_{ℚ_{p}} Flag_{K,d,μ})^{ad} of 𝒟_{φ,N,μ}^{ad} because the zero section and ℋ_{φ,N,μ}^{∇} are both invariant under the action of (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}. We first claim that both have the same underlying topological space. By [BGR84, Corollary 6.1.2/3] this can be checked on L-valued points of X for finite extensions L of E_{μ}. For those it was proved by Kisin [Kis06, Lemma 1.2.12(4)] that the universal Hodge-Pink lattice 𝔮 at L lies in the image of the zero section if the pullback \underline{M} to L of the universal (φ, N_{∇})-module on X has holomorphic N_{∇}^{M}. From this our claim follows.

To prove equality as closed subspaces of X we look at a closed point x ∈ X and its complete local ring Ô_{X,x}. Let 𝔪_{x} ⊂ Ô_{X,x} be the maximal ideal, let I ⊂ Ô_{X,x} be the ideal defining ℋ_{φ,N,μ}^{∇}, and set R_{n} := Ô_{X,x}/(𝔪_{x}^{n} + I). Then R_{n} is a finite dimensional ℚ_{p}-vector space by [BGR84, Corollary 6.1.2/3]. We consider the universal \underline{D}_{R_{n}} = (D, Φ, N, 𝔮) over R_{n} by restriction of scalars from R_{n} to ℚ_{p} as a (φ, N)-module ˜ \underline{D} with Hodge-Pink lattice over ℚ_{p} of rank (dim_{ℚ_{p}} R_{n})(rk_{R_{n}⊗K_{0}} D) = dim_{K_{0}} D. It is equipped with a ring homomorphism R_{n} → End( ˜\underline{D}). Since N_{∇}^{M} is holomorphic on \underline{M}(\underline{D}_{R_{n}}), Kisin [Kis06, Lemma 1.2.12(4)] tells us again that 𝔮 = 𝔮(F^{•}) for the filtration F^{•} = F_{𝔮}^{•} from Remark 2.7. This shows that the ideal J defining the zero section in X vanishes in R_{n}. Since this holds for all n, the ideals I and J are equal in Ô_{X,x}. As x was arbitrary, they coincide on all of X and this proves the theorem. □

5. Weak admissibility

Similar to the case of filtrations, one can define a notion of weak admissibility for (φ, N)-modules with Hodge-Pink lattice and develop a Harder-Narasimhan formalism. Compare also [Hel11, § 2] for the following. Recall that f = [K_{0} : ℚ_{p}] and e = [K : K_{0}].

Definition 5.1. Let L be a field with a valuation v_{L} : L → Γ_{L} ∪ {0} in the sense of Huber, see [Hub93, § 2, Definition] and set Γ_{L}^{ℚ} := Γ_{L} ⊗_{ℤ} ℚ.

(i) Let \underline{D} = (D, Φ, N) be a (φ, N)-module over L. Then define

t_{N}(\underline{D}) := v_{L}(det_{L} Φ^{f})^{1/f^{2}} ∈ Γ_{L}^{ℚ}.

If L ⊃ K_{0} we are in the situation of Remark 2.4 and have t_{N}(\underline{D}) = v_{L}(det_{L}(Φ^{f})_{0})^{1/f}.

(ii) Let \underline{D} = (D, Φ, N, F^{•}) be a K-filtered (φ, N)-module over L. Then

t_{H}(\underline{D}) := \frac{1}{ef} ∑_{i∈ℤ} i dim_{L}(F^{i}D_{K}/F^{i+1}D_{K}) ∈ ℚ.

(iii) Let \underline{D} = (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice of rank d over L. Then we set

t_{H}(\underline{D}) := \frac{1}{ef} (dim_{L}(𝔮/t^{n}𝔭) − dim_{L}(𝔭/t^{n}𝔭)) = \frac{1}{ef} dim_{L}(𝔮/t^{n}𝔭) − n rk \underline{D} ∈ ℚ

for n ≫ 0, which is independent of n whenever t^{n}𝔭 ⊂ 𝔮. If L is an extension of K̃ and (μ_{ψ})_{ψ} = μ_{\underline{D}}(Spec L) is the Hodge polygon of \underline{D}, see Definition 2.10, then t_{H}(\underline{D}) := \frac{1}{ef} ∑_{ψ} μ_{ψ,1}+…+μ_{ψ,d}. If the ψ-component 𝔮_{ψ} satisfies ∧^{d}𝔮_{ψ} = t^{−h_{ψ}} ∧^{d} 𝔭_{ψ} then t_{H}(\underline{D}) = \frac{1}{ef} ∑_{ψ} h_{ψ}. Moreover t_{H}(\underline{D}) = t_{H}(D, Φ, N, F_{𝔮}^{•}). (iv) Let \underline{D} be a (φ, N)-module with Hodge-Pink lattice (resp. a K-filtered (φ, N)-module) over L. Then its slope is defined to be

λ(\underline{D}) := (v_{L}(p)^{t_{H}(\underline{D})} · t_{N}(\underline{D})^{−1})^{1/d} ∈ Γ_{L}^{ℚ}.

Definition 5.2. (i) A (φ, N)-module with Hodge-Pink lattice \underline{D} = (D, Φ, N, 𝔮) over a field L endowed with a valuation is called semi-stable if λ(\underline{D}′) ≥ λ(\underline{D}) for all \underline{D}′ = (D′, Φ|_{φ^{∗}D′}, N|_{D′}, 𝔮 ∩ D′ ⊗_{L⊗_{ℚp}K_{0}} 𝔹_{L}) where D′ ⊂ D is a free L ⊗_{ℚ_{p}} K_{0}-submodule stable under Φ and N. (ii) A K-filtered (φ, N)-module \underline{D} = (D, Φ, N, F^{•}) over L is called semi-stable if λ(\underline{D}′) ≥ λ(\underline{D}) for all \underline{D}′ = (D′, Φ|_{φ^{∗}D′}, N|_{D′}, F^{•} ∩ D′_{K}) where D′ ⊂ D is a free L ⊗_{ℚ_{p}} K_{0}-submodule stable under Φ and N. (iii) A (φ, N)-module with Hodge-Pink lattice (resp. a K-filtered (φ, N)-module) is called weakly admissible if it is semi-stable of slope 1.

Lemma 5.3. Let (D, Φ, N, F^{•}) be a K-filtered (φ, N)-module over a valued field L and let (D, Φ, N, 𝔮) denote the (φ, N)-module with Hodge-Pink lattice associated to (D, Φ, N, F^{•}) by the zero section F^{•} ↦ 𝔮 = 𝔮(F^{•}) of Remark 2.7(3). Then (D, Φ, N, F^{•}) is weakly admissible if and only if (D, Φ, N, 𝔮) is.

Proof. It is obvious from the definitions that t_{H}(D, Φ, N, F^{•}) = t_{H}(D, Φ, N, 𝔮(F^{•})). Further we have to test on the same sub-objects D′ ⊂ D. Hence the claim follows from the fact

𝔮(F^{•} ∩ D′_{K}) = 𝔮(F^{•}) ∩ D′ ⊗_{L⊗_{ℚp}K_{0}} 𝔹_{L},

which is obvious from the description of 𝔮(−) in Remark 2.7(3) by choosing an L ⊗_{ℚ_{p}} K-basis of D_{K} adapted to the submodules F^{i}D′_{K} and F^{i}D_{K}. □

Proposition 5.4. Let (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice defined over some valued field L. Then there is a unique Harder-Narasimhan filtration

0 = D_{0} ⊂ D_{1} ⊂ ⋯ ⊂ D_{r} = D

of (D, Φ, N, 𝔮), by free L ⊗_{ℚ_{p}} K_{0}-submodules stable under Φ and N such that the subquotients D_{i}/D_{i−1} with their induced Hodge-Pink lattice are semi-stable of slope λ_{i} ∈ Γ_{L} ⊗ ℚ and λ_{1} < λ_{2} < ⋯ < λ_{r}.

Proof. This is the usual Harder-Narasimhan formalism, see [FF, 3] for a fairly general exposition. See also [Hel11, Proposition 2.19]. □

Corollary 5.5. Let (D, Φ, N, 𝔮) be a (φ, N)-module with Hodge-Pink lattice over L and let L′ be an extension of L with valuation v_{L′} extending the valuation v_{L}. Then (D, Φ, N, 𝔮) is weakly admissible if and only if (D′, Φ′, N′, 𝔮′) = (D ⊗_{L} L′, Φ ⊗ id, N ⊗ id, 𝔮 ⊗_{L} L′) is weakly admissible.

Proof. This is similar to [Hel11, Corollary 2.22].

If (D′, Φ′, N′, 𝔮′) is weakly admissible, then every (Φ, N)-stable subobject D_{1} ⊂ D defines a (Φ′, N′)- stable subobject D′_{1} = D_{1} ⊗_{L} L′ of D′ such that

λ(D_{1}, Φ|_{φ^{∗}D_{1}}, N|_{D_{1}}, 𝔮 ∩ D_{1} ⊗_{L⊗K_{0}} 𝔹_{L}) = λ(D′_{1}, Φ′|_{φ^{∗}D′_{1}}, N′|_{D′_{1}}, 𝔮′ ∩ D′_{1} ⊗_{L′⊗K_{0}} 𝔹_{L′}).

It follows that (D, Φ, N, 𝔮) is weakly admissible, as (D′, Φ′, N′, 𝔮′) is.

Now assume that (D, Φ, N, 𝔮) is weakly admissible. We may reduce to the case where L′ is finitely generated over L and Aut(L′/L) is large enough. As in the proof of [Hel11, Corollary 2.22] one shows that the action of Aut(L′/L) preserves the slope of Φ′-stable sub-objects of D′. Hence the Harder-Narasimhan filtration of D′ descends to D. As D is weakly admissible, this filtration can only have one step. □

Theorem 5.6. Let μ be a cocharacter as in (2.5) with reflex field E_{μ}. Then the groupoid

X ↦ {(D, Φ, N, 𝔮) ∈ ℋ_{φ,N,⪯μ}(X) | D ⊗ κ(x) is weakly admissible for all x ∈ X}

is an open substack ℋ_{φ,N,⪯μ}^{ad,wa} of ℋ_{φ,N,⪯μ}^{ad} on the category of adic spaces locally of finite type over E_{μ}.

Proof. This is similar to the proof of [Hel13, Theorem 4.1].

It follows from Corollary 5.5 that ℋ_{φ,N,⪯μ}^{ad,wa} is indeed a stack, i.e. weak admissibility may be checked over an fpqc-covering. Hence it suffices to show that the weakly admissible locus is open in

X_{μ} := P_{K_{0},d} ×_{ℚ_{p}} Q_{K,d,⪯μ}.

Let us denote by Z_{i} the projective P_{K_{0},d}-scheme whose S-valued points are given by pairs (x, U) with x = (g, N) ∈ P_{K_{0},d}(S) ⊂ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})×(Res_{K_{0}/ℚ_{p}} Mat_{d×d}) and an O_{S}⊗K_{0}-subspace U ⊂ O_{S}⊗K_{0}^{⊕d} which is locally on S free of rank i, a direct summand as O_{S}-module, and stable under the action of Φ_{g} = g·φ and N. This is a closed subscheme of the product P_{K_{0},d}×_{ℚ_{p}}Quot_{O^{d} | K_{0} | ℚ_{p}} (where Quot_{O^{d} | K_{0} | ℚ_{p}} is Grothendieck’s Quot-scheme which is projective over ℚ_{p}; see [FGA, n^{∘}221, Theorem 3.1] or [AK80, Theorem 2.6]), cut out by the invariance conditions under Φ_{g} and N. Further write f_{i} ∈ Γ(Z_{i}, O_{Z_{i}}) for the global section defined by

f_{i}(g, U) = det(g · φ)^{f}|_{U} = det(g · φ(g) · . . . · φ^{f−1}(g))^{f}|_{U},

where f = [K_{0} : ℚ_{p}], and where the determinant is the determinant as O_{Z_{i}}-modules. Write U for the pullback of the universal (Φ, N)-invariant subspace on Z_{i} to the product Z_{i} × Q_{K,d,⪯μ}, write 𝔮 for the pullback of the universal 𝔹^{+}-lattice on Q_{K,d,⪯μ} to Z_{i} × Q_{K,d,⪯μ}, and write 𝔭 = (𝔹^{+})^{⊕d} for the pullback of the tautological 𝔹^{+}-lattice D ⊗ 𝔹^{+} on P_{K_{0},d} to Z_{i} × Q_{K,d,⪯μ}. Fix integers n, h with t^{n}𝔭 ⊂ 𝔮 ⊂ t^{−h}𝔭 and consider the complex of finite locally free sheaves on Z_{i} × Q_{K,d,⪯μ}

P_{•} : P_{1} := t^{−h}𝔭/t^{n}𝔭 ⟶^{δ} t^{−h}𝔭/𝔮 ⊕ (D/U ⊗ t^{−h}𝔹^{+}/t^{n}𝔹^{+}) =: P_{0}

given by the canonical projection D ↠ D/U in the second summand. Let T_{1} be the functor from the category of quasi-coherent sheaves on Z_{i} × Q_{K,d,⪯μ} to itself defined by

T_{1} : M ⟼ ker(δ ⊗ id_{M} : P_{1} ⊗ M → P_{0} ⊗ M) .

If M = κ(y) for a point y = (g_{y}, N_{y}, U_{y}, 𝔮_{y}) ∈ Z_{i} × Q_{K,d,⪯μ} then T_{1}(κ(y)) = (𝔮_{y} ∩ 𝔭_{i,y}[\frac{1}{t}])/t^{n}𝔭_{i,y} where we write 𝔭_{i,y} := U_{y} ⊗_{κ(y)⊗K_{0}} 𝔹_{κ(y)}^{+}. We consider the function

h_{i} : Z_{i} × Q_{K,d,⪯μ} ⟶ ℚ,
(5.1)  y ⟼ \frac{1}{ef} dim_{κ(y)} T_{1}(κ(y)) − ni = t_{H}(U_{y}, g_{y}( id ⊗φ)|_{U_{y}}, N|_{U_{y}}, 𝔮_{y} ∩ 𝔭_{i,y}[\frac{1}{t}]) .

We write Z_{i}^{ad} resp. Q_{K,d,⪯μ}^{ad} for the adic spaces associated to the varieties Z_{i} and Q_{K,d,⪯μ}. Similarly we write h_{i}^{ad} for the function on the adic spaces Z_{i}^{ad} × Q_{K,d,⪯μ}^{ad} defined by the same formula as in (5.1). By semi-continuity [EGA, III_{2}, Théorème 7.6.9], the sets

Y_{i,m} = {y ∈ Z_{i}^{ad} × Q_{K,d,⪯μ}^{ad} | h_{i}(y) ≥ m}

are closed and hence proper over X_{μ}^{ad} = P_{K_{0},d}^{ad} ×_{ℚ_{p}} Q_{K,d,≤μ}^{ad}. We write

pr_{i,m} : Y_{i,m} ⟶ X_{μ}^{ad}

for the canonical, proper projection.

If we write X_{0} ⊂ X_{μ}^{ad} for the open subset of all (D, Φ, N, 𝔮) such that λ(D, Φ, N, 𝔮) = 1, then

(5.2)  X_{0} ∖ X_{μ}^{wa} = X_{0} ∩ ⋃_{i,m} pr_{i,m} ({y ∈ Y_{i,m} | v_{y}(f_{i}) > v_{y}(p)^{f^{2}m}}),

where the union runs over 1 ≤ i ≤ d − 1 and m ∈ ℤ. Indeed: Let x = (D, Φ, N, 𝔮) be an L-valued point of X_{0}, then any proper (Φ, N)-stable subspace of D′ ⊂ D defines (for some 1 ≤ i ≤ d−1) a point y = (D′, 𝔮) of Z_{i} × Q_{K,d,≤μ} mapping to x. This subspace violates the weak admissibility condition if and only if

v_{y}(f_{i}) = t_{N}(D′, Φ|_{φ^{∗}D′})^{f^{2}} > v_{y}(p)^{f^{2} t_{H}(D′,Φ,𝔮∩(D′⊗𝔹_{κ(x)}))} = v_{y}(p)^{f^{2}h_{i}(y)},

and hence (5.2) follows. On the other hand the union

⋃_{i,m} pr_{i,m} ({y ∈ Y_{i,m} | v_{y}(f_{i}) > v_{y}(p)^{f^{2}m}})

is a finite union, because Y_{i,m} = ∅ for m > hi and Y_{i,m} = Z_{i}^{ad} × Q_{K,d,⪯μ}^{ad} for m ≤ −ni. Therefore the union is closed by properness of the map pr_{i,m} and the definition of the topology on an adic space. The theorem follows from this. □

We define sub-groupoids ℋ_{φ,⪯μ}^{ad,wa} ⊂ ℋ_{φ,⪯μ}^{ad}, resp. ℋ_{φ,N,μ}^{ad,wa} ⊂ ℋ_{φ,N,μ}^{ad}, resp. ℋ_{φ,μ}^{ad,wa} ⊂ ℋ_{φ,μ}^{ad}, resp. 𝒟_{φ,N,μ}^{ad,wa} ⊂ 𝒟_{φ,N,μ}^{ad}, resp. 𝒟_{φ,μ}^{ad,wa} ⊂ 𝒟_{φ,μ}^{ad} as follows: Given an adic space X and (D, Φ, N, 𝔮) ∈ ℋ_{φ,⪯μ}^{ad}, we say that (D, Φ, N, 𝔮) ∈ ℋ_{φ,⪯μ}^{ad,wa} if and only if (D, Φ, N, 𝔮) ⊗ κ(x) is weakly admissible for all points x ∈ X. The other sub-groupoids are defined in the same manner.

Corollary 5.7. The sub-groupoids ℋ_{φ,⪯μ}^{ad,wa} ⊂ ℋ_{φ,⪯μ}^{ad}, resp. ℋ_{φ,N,μ}^{ad,wa} ⊂ ℋ_{φ,N,μ}^{ad}, resp. ℋ_{φ,μ}^{ad,wa} ⊂ ℋ_{φ,μ}^{ad}, resp. 𝒟_{φ,N,μ}^{ad,wa} ⊂ 𝒟_{φ,N,μ}^{ad}, resp. 𝒟_{φ,μ}^{ad,wa} ⊂ 𝒟_{φ,μ}^{ad} are open substacks.

Proof. This follows by pulling back ℋ_{φ,N,⪯μ}^{ad,wa} ⊂ ℋ_{φ,N,⪯μ}^{ad} along the morphisms ℋ_{φ,⪯μ}^{ad} → ℋ_{φ,N,⪯μ}^{ad}, resp. ℋ_{φ,N,μ}^{ad} → ℋ_{φ,N,⪯μ}^{ad}, resp. ℋ_{φ,μ}^{ad} → ℋ_{φ,N,⪯μ}^{ad}, resp. 𝒟_{φ,N,μ}^{ad} → ℋ_{φ,N,⪯μ}^{ad}, resp. 𝒟_{φ,μ}^{ad} → ℋ_{φ,N,⪯μ}^{ad}. Here we use the fact that the zero sections 𝒟_{φ,N,μ}^{ad} → ℋ_{φ,N,⪯μ}^{ad} and 𝒟_{φ,μ}^{ad} → ℋ_{φ,N,⪯μ}^{ad} preserve weak admissibility by Lemma 5.3. □

Remark 5.8. Note that the projection

pr : ℋ_{φ,N,μ}^{ad} ⟶ 𝒟_{φ,N,μ}^{ad}

does not preserve weak admissibility. We always have pr^{−1}(𝒟_{φ,N,μ}^{ad,wa}) ⊂ ℋ_{φ,N,μ}^{ad,wa} and hence especially any section of the vector bundle ℋ_{φ,N,μ}^{ad} → 𝒟_{φ,N,μ}^{ad} maps the weakly admissible locus to the weakly admissible locus.

Indeed, let \underline{D} = (D, Φ, N, 𝔮) be a point of ℋ_{φ,N,μ}^{ad} over a field L whose image (D, Φ, N, F_{𝔮}^{•}) in 𝒟_{φ,N,μ}^{ad} is weakly admissible. Let D′ ⊂ D be an L ⊗_{ℚ_{p}} K_{0}-submodule which is stable under Φ and N, and set 𝔭′ := D′ ⊗_{L⊗K_{0}} 𝔹_{L}^{+}. Then 𝔮′ := 𝔮 ∩ 𝔭′[\frac{1}{t}] satisfies t^{i}𝔮′ ∩ 𝔭′ ⊂ t^{i}𝔮 ∩ 𝔭 and F_{𝔮′}^{i}D′_{K} ⊂ F_{𝔮}^{i}D_{K} ∩ D′_{K}. This implies

t_{H}(D′, Φ|_{φ^{∗}D′}, N|_{D′}, 𝔮′) = t_{H}(D′, Φ|_{φ^{∗}D′}, N|_{D′}, F_{𝔮′}^{•}) ≤ t_{H}(D′, Φ|_{φ^{∗}D′}, N|_{D′}, F_{𝔮}^{•} ∩ D′_{K})

with equality for D′ = D, and

v_{L}(p)^{t_{H}(D′,Φ|_{φ^{∗}D′},N|_{D′},𝔮′)} ≥ v_{L}(p)^{t_{H}(D′,Φ|_{φ^{∗}D′},N|_{D′},F_{𝔮}^{•}∩D′_{K})} ≥ t_{N}(D′, Φ|_{φ^{∗}D′})

with equality for D′ = D, because (D, Φ, N, F_{𝔮}^{•}) is weakly admissible. Therefore also \underline{D} is weakly admissible. This proves that pr^{−1}(𝒟_{φ,N,μ}^{ad,wa}) ⊂ ℋ_{φ,N,μ}^{ad,wa}. However, in general this inclusion is strict as can be seen from the following example.

Example 5.9. Let K = K_{0} = ℚ_{p}, d = 2 and μ = (2, 0). We consider points \underline{D} = (D, Φ, N, F^{•}) in 𝒟_{φ,N,μ}^{ad} over a field L with Φ = p Id_{2} and N = 0. The filtration is of the form D = F^{0}D ⊃ F^{1}D = F^{2}D = ( \begin{matrix}u \\ v \end{matrix} ) · L ⊃ F^{3}D = (0). None of these points is weakly admissible, because the subspace D′ = ( \begin{matrix}u \\ v \end{matrix} ) · L ⊂ D has t_{N}(\underline{D}′) = v_{L}(p) and F^{2}D′ = D′, whence t_{H}(\underline{D}′) = 2 and λ(\underline{D}′) = v_{L}(p) < 1.

The preimage of such a point in ℋ_{φ,N,μ}^{ad} is given by a Hodge-Pink lattice 𝔮 with 𝔭 ⊂ 𝔮 ⊂ t^{−2}𝔭 with Hodge weights 0 and −2. This means that 𝔮 = 𝔭+( \begin{matrix}u+tu′ \\ v+tv′ \end{matrix} )·t^{−2}𝔹_{L}^{+}. If the vectors ( \begin{matrix}u \\ v \end{matrix} ) and ( \begin{matrix}u′ \\ v′ \end{matrix} ) are linearly dependent then \underline{D} = (D, Φ, N, 𝔮) is not weakly admissible, because the subspace D′ = ( \begin{matrix}u \\ v \end{matrix} ) · L ⊂ D has t_{N}(\underline{D}′) = v_{L}(p) and 𝔮′ := 𝔮 ∩ D′ ⊗_{L} 𝔹_{L} = t^{−2}D′ ⊗_{L} 𝔹_{L}^{+}, whence t_{H}(\underline{D}′) = 2 and λ(\underline{D}′) = v_{L}(p) < 1.

On the other hand, if the vectors ( \begin{matrix}u \\ v \end{matrix} ) and ( \begin{matrix}u′ \\ v′ \end{matrix} ) are linearly independent then \underline{D} = (D, Φ, N, 𝔮) is weakly admissible, because then 𝔮′ ⊂ t^{−1}D′ ⊗_{L} 𝔹_{L}^{+} for any subspace D′ = ( \begin{matrix}a \\ b \end{matrix} ) · L ⊂ D, whence t_{N}(\underline{D}′) = v_{L}(p),

t_{H}(\underline{D}′) ≤ 1 and λ(\underline{D}′) ≥ 1. Indeed, ( \begin{matrix}a \\ b \end{matrix} ) · t^{−2} ∈ 𝔮′ would imply that ( \begin{matrix}a \\ b \end{matrix} ) · t^{−2} ≡ ( \begin{matrix}u+tu′ \\ v+tv′ \end{matrix} ) · t^{−2} · (c + tc′) ≡

c( \begin{matrix}u \\ v \end{matrix} ) · t^{−2} + (c′( \begin{matrix}u \\ v \end{matrix} ) + c( \begin{matrix}u′ \\ v′ \end{matrix} ))t^{−1} mod 𝔭 for c, c′ ∈ L. This implies ( \begin{matrix}a \\ b \end{matrix} ) = c( \begin{matrix}u \\ v \end{matrix} ) and c′( \begin{matrix}u \\ v \end{matrix} ) + c( \begin{matrix}u′ \\ v′ \end{matrix} ) = 0

contradicting the linear independence.

Thus the weakly admissible locus ℋ_{φ,μ}^{ad,wa} in the fiber of ℋ_{φ,N,μ} over the point (Φ, N) = (p Id_{2}, 0) in P_{ℚ_{p},2} equals the complement of the zero section, while this fiber in 𝒟_{φ,N,μ}^{ad,wa} is empty; see also Lemma 5.3.

We end this section by remarking that the weakly admissible locus is determined by the rigid analytic points, i.e. those points of an adic space whose residue field is a finite extension of ℚ_{p}.

Lemma 5.10. Let X be an adic space locally of finite type over E_{μ} and let f : X → ℋ_{φ,N,⪯μ}^{ad} be a morphism defined by a (φ, N)-module with Hodge-Pink lattice \underline{D}. Then f factors over ℋ_{φ,N,⪯μ}^{ad,wa} if and only if \underline{D} ⊗ κ(x) is weakly admissible for all rigid analytic points x ∈ X. Proof. One implication is obvious and the other one is an easy application of the maximum modulus principle. It is proven along the same lines as [Hel13, Proposition 4.3]. □

Remark 5.11. The analogous statements for the stacks ℋ_{φ,⪯μ}^{ad,wa} ⊂ ℋ_{φ,⪯μ}^{ad}, resp. ℋ_{φ,N,μ}^{ad,wa} ⊂ ℋ_{φ,N,μ}^{ad}, resp. ℋ_{φ,μ}^{ad,wa} ⊂ ℋ_{φ,μ}^{ad}, resp. 𝒟_{φ,N,μ}^{ad,wa} ⊂ 𝒟_{φ,N,μ}^{ad}, resp. 𝒟_{φ,μ}^{ad,wa} ⊂ 𝒟_{φ,μ}^{ad} are also true and are a direct consequence of their construction.

6. The étale locus

Let us denote by 𝔹_{[r,s]} the closed annulus over K_{0} of inner radius r and outer radius s for some r, s ∈ [0, 1) ∩ p^{ℚ}. For an adic space X ∈ Ad_{ℚ_{p}}^{lft} we write

𝒜_{X}^{[0,1)} = pr_{X,∗} O_{X×𝕌}^{+} ⊂ ℬ_{X}^{[0,1)} = pr_{X,∗} O_{X×𝕌}
𝒜_{X}^{[r,s]} = pr_{X,∗} O_{X×𝔹_{[r,s]}}^{+} ⊂ ℬ_{X}^{[r,s]} = pr_{X,∗} O_{X×𝔹_{[r,s]}}

The Frobenius φ on ℬ_{X}^{[0,1)} restricts to a ring homomorphism φ on 𝒜_{X}^{[0,1)}. For this section we adapt the notation from [Hel13] and write r_{i} = r^{1/p^{i}}. Then φ restricts to a homomorphism

φ : ℬ_{X}^{[r,s]} ⟶ ℬ_{X}^{[r_{1},s_{1}]}.

Lemma 6.1. If A is a reduced p-adically complete ℤ_{p}-algebra topologically of finite type, then

Γ(Spa(A[1/p], A), 𝒜_{Spa(A[1/p],A)}^{[0,1)}) = (A ⊗_{ℤ_{p}} W)[[u]].

Proof. Let X = Spa(A[1/p], A). We need to compute the global sections of Γ(X ×𝕌, O_{X×𝕌}^{+}). By definition we have

Γ(X × 𝕌, O_{X×𝕌}^{+}) = lim_{⟵ r} Γ(X × 𝔹_{[0,r]}, O_{X×𝕌}^{+}) ⊂ lim_{⟵ r} Γ(X × 𝔹_{[0,r]}, O_{X×𝕌}) ⊂ (A ⊗_{ℤ_{p}} K_{0})[[u]].

The inclusion (A ⊗_{ℤ_{p}} W)[[u]] ⊂ Γ(X × 𝕌, O_{X×𝕌}) is obvious. Let

f = ∑_{i} f_{i}u^{i} ∈ Γ(X × 𝕌, O_{X×𝕌}^{+}) ⊂ (A ⊗_{ℤ_{p}} K_{0})[[u]].

Then f_{i}(x) ∈ Γ(X ⊗_{ℤ_{p}} W, O_{X⊗W}^{+} ) = A ⊗_{ℤ_{p}} W ⊂ A ⊗_{ℤ_{p}} K_{0} for all i. The claim follows from this. □

Definition 6.2. A φ-module of finite height over 𝒜_{X}^{[0,1)} is an 𝒜_{X}^{[0,1)}-module 𝔐 which is locally on X free of finite rank over 𝒜_{X}^{[0,1)} together with an injective morphism Φ : φ^{∗}𝔐 → 𝔐 of 𝒜_{X}^{[0,1)}-modules such that coker Φ is killed by some power of E(u) ∈ W [[u]] ⊂ 𝒜_{X}^{[0,1)}.

Inspired by Example 4.4 we define the (φ, N_{∇})-module ℬ_{X}^{[0,1)}(1) over X to be (ℬ_{X}^{[0,1)}, Φ_{M} = \frac{pE(u)}{E(0)}, N_{∇}^{M}) with N_{∇}^{M}(f) = N_{∇}(f)+u\frac{dλ}{du} f. For an integer n ∈ ℤ we set ℬ_{X}^{[0,1)}(n) := ℬ_{X}^{[0,1)}(1)^{⊗n} = (ℬ_{X}^{[0,1)}, (\frac{pE(u)}{E(0)})^{n}, N_{∇}^{M}) with N_{∇}^{M}(f) = N_{∇}(f) + nu\frac{dλ}{du} f. Given a (φ, N_{∇})-module (M, Φ_{M}) on X we write (M, Φ_{M})(n) for the twist M ⊗_{ℬ_{X}^{[0,1)}} ℬ_{X}^{[0,1)}(n). Note that \frac{p}{E(0)} ∈ W^{×} since E(u) is an Eisenstein polynomial. Thus for n ≥ 0

we have an obvious integral model 𝒜_{X}^{[0,1)}(n) for ℬ^{[0,1)}(n) which is a φ-module of finite height over 𝒜_{X}^{[0,1)} (by forgetting the N_{∇}-action). Further we write A^{[0,1)}(n) = 𝒜_{Spa(ℚ_{p},ℤ_{p})}(n) for the W [[u]]-module of rank 1 with basis e on which Φ acts via Φ(e) = (\frac{E(u)}{pE(0)})^{n}e.

Definition 6.3. Let (M, Φ_{M}, N_{∇}^{M}) be a (φ, N_{∇})-module over an adic space X ∈ Ad_{ℚ_{p}}^{lft}.

(i) The module M is called étale if there exists an fpqc-covering (U_{i} → X), an integer n ≥ 0 and φ-modules (𝔐_{i}, Φ_{𝔐_{i}}) of finite height over 𝒜_{U_{i}}^{[0,1)} such that

(M, Φ_{M})(n)|_{U_{i}} = (𝔐_{i}, Φ_{𝔐_{i}}) ⊗_{𝒜_{U_{i}}^{[0,1)}} ℬ_{U_{i}}^{[0,1)}.

(ii) Let x ∈ X, then M is called étale at x if there exists an integer n ≥ 0 and a (κ(x)^{+} ⊗_{ℤ_{p}} W)[[u]]-lattice 𝔐 ⊂ M(n) ⊗ κ(x) such that E(u)^{h}𝔐 ⊂ Φ_{M(n)}(φ^{∗}𝔐) ⊂ 𝔐 for some integer h ≥ 0.

Theorem 6.4. Let X be an adic space locally of finite type over ℚ_{p} and let (M, Φ) be a (φ, N_{∇})-module. Then the subset

X^{int} = {x ∈ X | M is étale at x}

is open and the restriction M|_{X^{int}} is étale.

This is similar to the proof of [Hel13, Theorem 7.6]. However, we need to make a few generalizations as we cannot rely on a reduced universal case. Given an affinoid algebra A and r, s ∈ [0, 1) ∩ p^{ℚ} we write

B_{A}^{[r,s]} = Γ(Spa(A, A^{∘}), ℬ_{Spa(A,A^{∘})}^{[r,s]}) = Aˆ⊗_{ℚ_{p}}B^{[r,s]} = A_{W} ⟨T/s, r/T ⟩,
A_{A}^{[r,s]} = Γ(Spa(A, A^{∘}), 𝒜_{Spa(A,A^{∘})}^{[r,s]}) = A^{∘} ˆ⊗_{ℤ_{p}}A^{[r,s]} = A_{W}^{∘} ⟨T/s, r/T ⟩.

The following is the analogue of [Hel13, Theorem 6.9] in the non-reduced case.

Theorem 6.5. Let X be an adic space locally of finite type over ℚ_{p} and let N be a family of free φ-modules of rank d over ℬ_{X}^{[r,r_{2}]}. Assume that there exists x ∈ X and an 𝒜_{X}^{[r,r_{2}]} ⊗ κ(x)^{∘}-lattice N_{x} ⊂ N ⊗ κ(x) such that Φ induces an isomorphism

(6.1)  Φ : φ^{∗}(N_{x} ⊗_{𝒜_{X}^{[r,r_{2}]}} 𝒜_{X}^{[r,r_{1}]}) ^{∼} ⟶ N_{x} ⊗_{𝒜_{X}^{[r,r_{2}]}} 𝒜_{X}^{[r_{1},r_{2}]}.

Then there exists an open neighborhood U ⊂ X of x and a locally free 𝒜_{U}^{[r,r_{2}]}-submodule N ⊂ N of rank d such that

N ⊗ κ(x)^{∘} = N_{x}
Φ(φ^{∗}N|_{U×𝔹_{[r,r_{1}]}}) = N|_{U×𝔹_{[r_{1},r_{2}]}}
N ⊗_{𝒜_{U}^{[r,r_{2}]}} ℬ_{U}^{[r,r_{2}]} = N |_{U}.

Proof. We may assume that X = Spa(A, A^{∘}) is affinoid and we may choose a Banach norm || · || and a ℤ_{p}-subalgebra A^{+} = {x ∈ A | ||x|| ≤ 1} ⊂ A^{∘} such that A = A^{+}[1/p] and X = Spa(A, A^{+}) = Spa(A, A^{∘}).

Choose a basis \underline{e}_{x} of N_{x} and denote by D_{0} ∈ GL_{d}(𝒜_{X}^{[r,r_{2}]} ⊗ κ(x)^{∘}) the matrix of Φ in this basis. After shrinking X if necessary we may lift the matrix D_{0} to a matrix D with coefficients in Γ(X, 𝒜_{X}^{[r,r_{2}]}). Localizing further we may assume that D is invertible over Γ(X, 𝒜_{X}^{[r,r_{2}]}), as we only need to ensure that the inverse of its determinant has coefficients a_{i} ∈ A^{+}, i.e. ||a_{i}|| ≤ 1 for some Banach norm ||·|| corresponding to A^{+}. However, this is clear for all but finitely many coefficients and hence we may localize further to ensure that the finitely many remaining coefficients are small, as this is true at x.

Fixing a basis \underline{b} of N we denote by S ∈ GL_{d}(B_{A}^{[r,r]}) the matrix of Φ in this basis. Further we denote by V a lift of the change of basis matrix from the basis \underline{e}_{x} to the basis \underline{b} mod x. From now on the proof is the same as the proof of [Hel13, Theorem 6.9]. □

Proposition 6.6. Let X = Spa (A, A^{+}) be an affinoid adic space of finite type over ℚ_{p}. Let r > |π| with r ∈ p^{ℚ} and set r_{i} = r^{1/p^{i}}. Let M_{r} be a free vector bundle on X × 𝔹_{[0,r_{2}]} together with an injection

Φ : φ^{∗}(M_{r}|_{X×𝔹_{[0,r_{1}]}}) ⟶ M_{r}

with cokernel supported at the point defined by E(u) Assume that there exists a free A_{A}^{[r,r_{2}]} = A^{+}⟨T/r_{2}, r/T ⟩ submodule

N_{r} ⊂ N_{r} := M_{r} ⊗_{B_{A}^{[0,r_{2}]}} B_{A}^{[r,r_{2}]}

of rank d, containing a basis of N_{r} such that

Φ(φ^{∗}(N_{r} ⊗_{A_{A}^{[r,r_{2}]}} A_{A}^{[r,r_{1}]})) = N_{r} ⊗_{A_{A}^{[r,r_{2}]}} A_{A}^{[r_{1},r_{2}]}.

Then fpqc-locally on X there exists a free A_{A}^{[0,r_{2}]}-submodule M_{r} ⊂ M_{r} of rank d, containing a basis of M_{r} such that

(6.2)  Φ : φ^{∗}(M_{r} ⊗_{A_{A}^{[0,r_{2}]}} A_{A}^{[0,r_{1}]}) ⟶ M_{r}

is injective with cokernel killed by E(u).

Proof. This is the generalization of [Hel13, Proposition 7.7] to our context. We also write M_{r} for the global sections of the vector bundle. Write M′_{r} = M_{r} ∩ N_{r} ⊂ N_{r}. This is an A^{+}⟨T/r_{2}⟩-module. Further we set

M_{r} = (M′_{r} ⊗_{A_{A}^{[0,r_{2}]}} A_{A}^{[r,r_{2}]}) ∩ M′_{r}[\frac{1}{p}] ⊂ N_{r}.

Then M_{r} is a finitely generated A^{+}⟨T/r_{2}⟩-module as the ring is noetherian. First we need to make some modification in order to assure that M_{r} is flat. Let Y = Spf W ⟨T/r_{2}⟩ denote the formal model of 𝔹_{[0,r_{2}]} and let Y′ = Spf W ⟨T/r_{2}, r/T ⟩ denote the formal model of 𝔹_{[r,r_{2}]}. By [BL93, Theorem 4.1] there exists a blow-up X̃ of Spf(A) such that the strict transform M̃_{r} of M_{r} in X̃ × Y is flat over X̃. We write M_{r,X̃}

(resp. N_{r,X̃}) for the pullback of M_{r} (resp. N_{r}) to the generic fiber of X̃ × Y (resp. to X̃ × Y′). If we set

M̃′_{r} = M_{r,X̃} ∩ N_{r,X̃} then one easily finds

M̃_{r} = (M̃′_{r} ⊗_{𝒜 [0,r_{2}] ; X̃} 𝒜 [r,r_{2}] ; X̃) ∩ M̃′_{r}[\frac{1}{p}].

If follows that M̃_{r} is stable under Φ. Further, as M̃_{r} is flat, it has no p-power torsion and hence we find that the formation (M_{r,X̃}, N_{r,X̃}) ↦ M̃_{r} commutes with base change Spf O ↪ X̃ for any finite flat ℤ_{p}-algebra O, compare the proof of [Hel13, Proposition 7.7]. Especially this pullback is free over O ⊗_{ℤ_{p}} W ⟨T/r_{2}⟩ and the cokernel of Φ is annihilated by E(u)^{k_{μ}} for some k_{μ} ≫ 0 depending only on the Hodge polygon μ (for an arbitrary finite flat ℤ_{p}-algebra O this follows by forgetting the O-structure and only considering the ℤ_{p}-structure).

It follows that the restriction of M̃_{r} to the reduced special fiber X̃_{0} of X̃ is locally free over X̃_{0} ×𝔸^{1} and hence, as in the proof of [Hel13, Proposition 7.7] we may locally lift a basis and find that M̃_{r} is locally on X̃ free over X̃ × Y.

It is only left to show that E(u)^{k_{μ}} coker Φ = 0 over X̃. To do so we may localize and assume that X̃ is affine. By abuse of language we denote it again by Spf A^{+} and write N = E(u)^{k_{μ}} coker Φ. If I denotes the ideal of nilpotent elements in A^{+}, we need to show that the multiplication I ⊗_{A^{+}} N → N is the zero map. Then the claim follows from the above. However for some k ≫ 0 we know that I^{k} ⊗_{A^{+}} N → N is the zero map, as I is nilpotent. Then N = N/I^{k} and the multiplication map I^{k−1} ⊗_{A^{+}} N → N factors over I^{k−1}/I^{k} ⊗_{A^{+}} N → N and this is a map of finitely generated A^{+}/I-modules which vanishes after pulling back to a quotient A^{+}/I → O_{L} onto the ring of integers in some finite extension L of ℚ_{p}. This can be seen as follows. The map on this pull back is induced by the pullback of the multiplication to a quotient of A^{+} which is finite flat over ℤ_{p}, and where N is known to vanish by the above. It follows that I^{k−1} ⊗_{A^{+}} N → N is the zero map and by descending induction we find that I acts trivial on N. □

Proof of Theorem 6.4. Fix some r > |π| and re-define

X^{int} = {x ∈ X | M|_{X×𝔹_{[r,r_{2}]}} ⊗ κ(x) is étale in the sense of (6.1)}.

By Theorem 6.5 this subset is open and we need to show that the restriction M|_{X^{int}} is étale. Then is follows directly that X^{int} coincides with the characterization in the theorem, as the notion of being étale at points may be checked fpqc-locally by [Hel13, Proposition 6.14].

However Proposition 6.6 provides (locally on X^{int}) an integral model 𝔐_{[0,r_{2}]} over X × 𝔹_{[0,r_{2}]}. Now we can glue 𝔐_{[0,r_{2}]} and φ^{∗}𝔐_{[0,r_{2}]} over X ×𝔹_{[r_{2},r_{3}]} and hence extend 𝔐_{[0,r_{2}]} to a model 𝔐_{[0,r_{3}]} over X ×𝔹_{[0,r_{3}]}. Proceeding by induction we get a model 𝔐 on X × 𝕌 and [Hel13, Proposition 6.5] guarantees that 𝔐 is locally in X free over 𝒜_{X}^{[0,1)}. Hence it is the desired étale model. □

Corollary 6.7. Let μ be a cocharacter as in (2.5) with reflex field E_{μ}. Then there is an open substack ℋ_{φ,N,⪯μ}^{ad,int} ⊂ ℋ_{φ,N,⪯μ}^{ad} such that f : X → ℋ_{φ,N,⪯μ}^{ad} factors over ℋ_{φ,N,⪯μ}^{ad,int} if and only if the family (M, Φ_{M}, N_{∇}^{M}) defined by f and \underline{M} is étale.

Proof. Let \underline{M}(\underline{D}) be the universal (φ, N_{∇})-module over ℋ_{φ,N,⪯μ}^{ad}. By Theorem 6.4 the set ℋ_{φ,N,⪯μ}^{ad,int} := \{x ∈ ℋ_{φ,N,⪯μ}^{ad} : \underline{M}(\underline{D}) is étale at x\} is open and above it \underline{M}(\underline{D}) is étale. If f factors over ℋ_{φ,N,⪯μ}^{ad,int} then (M, Φ_{M}, N_{∇}^{M}) is the pullback of the universal \underline{M}(\underline{D}) and hence is étale. Conversely if (M, Φ_{M}, N_{∇}^{M}) is étale, then it is étale at all points and f factors over ℋ_{φ,N,⪯μ}^{ad,int}, because the notion of being étale at points may be checked fpqc-locally by [Hel13, Proposition 6.14]. □

Proposition 6.8. Let L be a finite extension of E_{μ}, then ℋ_{φ,N,⪯μ}^{ad,int}(L) = ℋ_{φ,N,⪯μ}^{ad,wa}(L) and hence ℋ_{φ,N,⪯μ}^{ad,int} ⊂ ℋ_{φ,N,⪯μ}^{ad,wa}.

Proof. We show that being weakly admissible translates into being pure of slope zero over the Robba ring (in the sense of [Ked08]) under the equivalence of categories from Theorem 4.6. However, the proof is the same as in [Kis06, Theorem 1.3.8]. One easily verifies that the functor \underline{M} preserves the slope and that the slope filtration on the base change of \underline{M}(D, Φ, N, 𝔮) to the Robba ring extends to all of \underline{M}(D, Φ, N, 𝔮). Compare [Kis06, Proposition 1.3.7].

As in [Hel13, Theorem 7.6 (ii)] the second part is now a consequence of the fact that ℋ_{φ,N,⪯μ}^{ad,wa} ⊂ ℋ_{φ,N,⪯μ}^{ad} is the maximal open subspace whose rigid analytic points are exactly the weakly admissible ones, see Lemma 5.10. □

In the miniscule case Pappas and Rapoport [PR09, 5.b] define a period morphism from a stack of integral data to a stack of filtered φ-modules as follows. Let d > 0 and let μ : 𝔾_{m,\overline{ℚ}_{p}} → T̃_{\overline{ℚ}_{p}} be a

cocharacter as in (2.5). Pappas and Rapoport [PR09, 3.d] define an fpqc-stack Ĉ_{μ,K} on the category Nil_{O_{Eμ}} of schemes over the ring of integers O_{E_{μ}} of E_{μ} on which p is locally nilpotent. If R is an O_{E_{μ}}- algebra, we set R_{W} = R ⊗_{ℤ_{p}} W and denote by φ : R_{W} ((u)) → R_{W} ((u)) the ring homomorphism that is the identity on R, the p-Frobenius on W and that maps u to u^{p}. Now the R-valued points of the stack Ĉ_{⪯μ,K} are given by a subset

Ĉ_{⪯μ,K}(R) ⊂ {𝔐, Φ: φ^{∗}𝔐[1/u] ^{∼} ⟶ 𝔐[1/u]}

where 𝔐 is an R_{W} [[u]] = (R ⊗_{ℤ_{p}} W)[[u]]-module that is fpqc-locally on Spec R free as an R_{W} [[u]]-module of rank d. This subset is cut out by a condition prescribing the relative position of Φ(φ^{∗}𝔐) with respect to 𝔐 at the locus E(u) = 0 in terms of the cocharacter μ, see [PR09, 3.c,d] for the precise definition.

Remark 6.9. In [PR09] the stack Ĉ_{⪯μ,K} is only defined in the case of a minuscule cocharacter and is named Ĉ_{μ,K} in loc. cit. However, we can use the same definition, using the local model M_{μ,K}^{loc}, for a general cocharacter. This yields an integral version of the condition “being bounded by μ”. In fact this boundedness condition is the defining condition of the generic fiber of the local model M_{μ,K}^{loc} and then M_{μ,K}^{loc} is defined by taking the flat closure (inside some affine Grassmannian).

If μ is miniscule, this condition implies that E(u)𝔐 ⊂ Φ(φ^{∗}𝔐) ⊂ 𝔐.

Now let μ be arbitrary and let R be a p-adically complete O_{E_{μ}}-algebra topologically of finite type over O_{E_{μ}} and let (𝔐, Φ) ∈ Ĉ_{μ,K}(Spf R). The construction of section 4 associates to

(6.3)  (M, Φ_{M}) = (𝔐, Φ) ⊗_{R_{W} [[u]]} ℬ_{Spa(R[1/p],R)}^{[0,1)}

a K-filtered φ-module over Spa(R[1/p], R). Given a formal scheme X locally topologically of finite type over O_{E_{μ}}, this yields a period functor

Π(X) : Ĉ_{μ,K}(X) ⟶ 𝒟_{φ,μ}^{ad}(X^{rig}),

where X^{rig} denotes the generic fiber of the formal scheme X in the sense of rigid geometry (or in the sense of adic spaces). If μ is not miniscule, this construction breaks down, as the family of vector bundles on the open unit disc defined by (6.3) is not necessarily associated to a filtered φ-module: the monodromy operator N_{∇}^{M} is not necessarily holomorphic.

If we use φ-modules with Hodge-Pink lattices instead of filtered φ-modules we can overcome this problem. Using section 4, the same construction as in [PR09] gives a period functor

(6.4)  Π(X) : Ĉ_{⪯μ,K}(X) ⟶ ℋ_{φ,⪯μ}^{ad}(X^{rig}).

When X = Spf O_{L} for a finite field extension L of E_{μ}, it was shown by Genestier and Lafforgue [GL12, Théorème 0.6] that Π(Spf O_{L}) ⊗_{ℤ_{p}} ℚ_{p} is fully faithful, and surjective onto ℋ_{φ,⪯μ}^{ad,wa}(L) = ℋ_{φ,⪯μ}^{ad,int}(L).

Remark 6.10. From the point of view of Galois representations it is not surprising that we can not define a general period morphism using filtered φ-modules. If R is finite over O_{E_{μ}}, then the points of Ĉ_{μ,K}(R) correspond to 𝒢_{K_{∞}}-representations rather than to 𝒢_{K}-representations. This also explains why the target of the period map is ℋ_{φ,μ}^{ad} instead of ℋ_{φ,N,μ}^{ad}: the 𝒢_{K_{∞}}-representation does not see the monodromy.

If we want to take the monodromy into account we have to consider a stack Ĉ_{⪯μ,N,K} whose X -valued points are given by (𝔐, Φ, N) with (𝔐, Φ) ∈ Ĉ_{⪯μ,K}(X) and N : 𝔐/u𝔐 → 𝔐/u𝔐 satisfying

(6.5)  N ∘ \overline{Φ}(n) = p · \overline{Φ}(n) ∘ N.

Here (𝔐(n), Φ(n)) = (𝔐, Φ)⊗_{W [[u]]}A^{[0,1)}(n) is the twist of (𝔐, Φ) with the object A^{[0,1)}(n) defined before Definition 6.3 and n ≫ 0 is some integer such that Φ(n)(φ^{∗}𝔐) ⊂ 𝔐 and \overline{Φ} denotes the reduction of Φ modulo u. Note that given μ we may choose an n like that for all (𝔐, Φ) ∈ Ĉ_{⪯μ,K}(X) and the map \overline{Φ} (and hence the equation (6.5)) makes sense after this twist. Further the condition defined by (6.5) is independent of the chosen n.

Remark 6.11. (i) Using (2.7) we observe that if μ_{ψ,d} ≥ 0 for all ψ, and if L is a finite extension of E_{μ}, a Spf O_{L}-valued point of the stack Ĉ_{⪯μ,N,K} gives rise to an object of the category Mod_{/𝔖}^{φ,N} in the sense of Kisin [Kis06, (1.3.12)]. We only use the twist in order to define the stack in the general case (i.e. if Φ(φ^{∗}𝔐) is not contained in 𝔐). Kisin’s definition takes place in the generic fiber. However, we can not use this as a good definition as our stack is defined for p-power torsion objects. (ii) Note that we do not know much about the stack Ĉ_{⪯μ,N,K} and its definition is rather ad hoc. Especially we doubt that it is flat over Spf ℤ_{p}. This means that there is no reason to expect that we can reconstruct Kisin’s semi-stable deformation rings [Kis08] by using a similar construction as in [PR09, § 4].

In this general case described above we obtain a similar period morphism

(6.6)  Ĉ_{⪯μ,N,K}(X) ⟶ ℋ_{φ,N,⪯μ}^{ad}(X^{rig}).

As in [Hel13, Theorem 7.8] the above allows us to determine the image of the period morphism. Recall that a valued field (L, v_{L}) over ℚ_{p} is called of p-adic type if it is complete, topologically finitely generated over ℚ_{p} and if for all f_{1}, …, f_{m} ∈ L the closure of ℚ_{p}[f_{1} …, f_{m}] inside L is a Tate algebra, i.e. the quotient of some ℚ_{p}⟨T_{1}, …, T_{m′}⟩.

Corollary 6.12. The substack ℋ_{φ,N,⪯μ}^{ad,int} is the image of the period morphism (6.6) in the following sense:

(i) If X is a p-adic formal scheme and (𝔐, Φ, N) ∈ Ĉ_{⪯μ,N,K}(X), then Π(X)(𝔐, Φ, N) ∈ ℋ_{φ,N,⪯μ}^{ad,int}(X^{rig}). (ii) Let L be a field of p-adic type over E_{μ} and (D, Φ, N, 𝔮) ∈ ℋ_{φ,N,⪯μ}(L). Then there exists (𝔐, Φ, N) ∈ Ĉ_{⪯μ,N,K}(Spf L^{+}) such that Π(Spf L^{+})(𝔐, Φ, N) = (D, Φ, N, 𝔮) if and only if

\underline{M}(D) = 𝔐 ⊗_{L_{W}^{+} [[u]]} ℬ_{L}^{[0,1)}.

is étale, if and only if Spa(L, L^{+}) → ℋ_{φ,N,⪯μ}^{ad} factors over ℋ_{φ,N,⪯μ}^{ad,int}. (iii) Let X ∈ Ad_{E_{μ}}^{lft} and let f : X → ℋ_{φ,N,⪯μ}^{ad} be a morphism defined by (D, Φ, N, 𝔮). Then f factors over ℋ_{φ,N,⪯μ}^{ad,int} if and only if there exists a fpqc-covering (U_{i} → X)_{i∈I} and formal models U_{i} of U_{i} together with (𝔐_{i}, Φ_{i}, N) ∈ Ĉ_{⪯μ,N,K}(U_{i}) such that Π(U_{i})(𝔐_{i}, Φ_{i}, N) = (D, Φ, N, 𝔮)|_{U_{i}}.

Remark 6.13. If we consider the period morphism without monodromy, then we obtain a similar characterization of the stack ℋ_{φ,⪯μ}^{ad,int} ⊂ ℋ_{φ,⪯μ}^{ad} as the image of the period morphism (6.4).

7. Sheaves of period rings and the admissible locus

We recall the definition of some sheafified period rings from [Hel13]. Let R = lim_{⟵} O_{ℂ_{p}}/pO_{ℂ_{p}} be the inverse limit with transition maps given by the p-th power. Given a reduced p-adically complete ℤ_{p}-algebra A^{+} topologically of finite type, we define

A^{+} ˆ⊗W (R) = lim_{⟵ i} A^{+} ˆ⊗_{ℤ_{p}}W_{i}(R),

where the completed tensor product on the right hand side means completion with respect to the canonical topology on the truncated Witt vectors W_{i}(R) and the discrete topology on A^{+}/p^{i}A^{+}.

If X is a reduced adic space locally of finite type over ℚ_{p}, then there are sheaves O_{X}^{+} ˆ⊗W (R) and O_{X} ˆ⊗W (R) whose sections over an affinoid open U = Spa(A, A^{+}) ⊂ X are given by

Γ(U, O_{X}^{+} ˆ⊗W (R)) = A^{+} ˆ⊗_{ℤ_{p}}W (R)
Γ(U, O_{X} ˆ⊗W (R)) = (A^{+} ˆ⊗_{ℤ_{p}}W (R))[\frac{1}{p}].

In the same fashion we can define sheaves of topological rings O_{X}^{+} ˆ⊗W (Frac R) and O_{X} ˆ⊗W (Frac R).

Let A^{[0,1)} = W [[u]] and let A denote the p-adic completion of W ((u)). Further let B = A[1/p]. There are embeddings of A^{[0,1)}, A and B into W (Frac R) sending u to the Teichmüller representative [\underline{π}] ∈ W (R) of the element \underline{π} = (π_{n})_{n} ∈ R. We write Ã for the ring of integers in the maximal unramified extension B̃ of B inside W (Frac R)[1/p]. Finally we set Ã^{[0,1)} = Ã ∩ W (R) ⊂ W (Frac R). All these rings come along with a Frobenius endomorphism φ which is induced by the canonical Frobenius on W (Frac R). Note that all these rings have a canonical topology induced from the one on W (Frac R).

We define sheafified versions of these rings as follows, compare [Hel13, 8.1]. Let X be a reduced adic

space locally of finite type over ℚ_{p}. We define the sheaves 𝒜_{X}, resp. 𝒜̃_{X}, resp. 𝒜̃_{X}^{[0,1)} by specifying their

sections on open affinoids U = Spa(A, A^{+}) ⊂ X: we define Γ(U, 𝒜_{X}), resp. Γ(U, 𝒜̃_{X}), resp. Γ(U, 𝒜̃_{X}^{[0,1)}) to be the closure of A^{+} ⊗_{ℤ_{p}} A, resp. A^{+} ⊗_{ℤ_{p}} Ã, resp. A^{+} ⊗_{ℤ_{p}} Ã^{[0,1)} in Γ(U, O_{X}^{+} ˆ⊗W (Frac R)).

Further we consider the rational analogues ℬ_{X} and ℬ̃_{X} of these sheaves given by inverting p in 𝒜_{X} and 𝒜̃_{X}. We can consider these sheaves also on non-reduces spaces by locally embedding X into a reduced space Y and restricting the corresponding sheaves from Y to X.

Finally we recall the construction of the sheaf O_{X} ˆ⊗B_{cris} from [Hel13, 8.1]. For a reduced adic space X the map θ : W (R) → O_{ℂ_{p}} given by [(x, x^{1/p}, x^{1/p^{2}}, …)] ↦ x extends to an O_{X}^{+}-linear map

θ_{X} : O_{X}^{+} ˆ⊗W (R) → O_{X}^{+} ˆ⊗O_{ℂ_{p}}.

We define O_{X}^{+} ˆ⊗A_{cris} to be the p-adic completion of the divided power envelope of O_{X}^{+} ˆ⊗W (R) with respect to the kernel of θ_{X}. Finally we set

O_{X} ˆ⊗B_{cris}^{+} = (O_{X}^{+} ˆ⊗A_{cris})[1/p],
O_{X} ˆ⊗B_{cris} = (O_{X} ˆ⊗B_{cris}^{+})[1/t],
O_{X} ˆ⊗B_{st} = (O_{X} ˆ⊗B_{cris})[ℓ_{u}].

Here t = log[(1, ε_{1}, ε_{2}, …)] ∈ B_{cris} is the period of the cyclotomic character (where (ε_{i}) is a compatible system of p^{i}-th roots of unity) and ℓ_{u} is an indeterminate thought of as a formal logarithm of [\underline{π}].

Remark 7.1. The indeterminate ℓ_{u} considered here is the same indeterminate as in section 2.2.(b) and we identify both indeterminates. That is, the inclusion B^{[0,1)} ⊂ B_{cris}^{+} given by u ↦ [\underline{π}] will be extended to B^{[0,1)}[ℓ_{u}] ↪ B_{st}^{+} by means of ℓ_{u} ↦ ℓ_{u} and similar for the sheafified versions.

On O_{X} ˆ⊗B_{cris} there is a canonical Frobenius φ induced by the Frobenius on O_{X}^{+} ˆ⊗W (R). This endomorphism extends to a morphism

φ : O_{X} ˆ⊗B_{st} ⟶ O_{X} ˆ⊗B_{st},

where φ(ℓ_{u}) = pℓ_{u}. Further N = \frac{d}{dℓ_{u}} defines an endomorphism of O_{X} ˆ⊗B_{st} which satisfies Nφ = pφN.

Finally the continuous 𝒢_{K}-action on O_{X}^{+} ˆ⊗W (R) extends to O_{X} ˆ⊗B_{cris} and we further extend this action to O_{X} ˆ⊗B_{st} by means of γ · ℓ_{u} = ℓ_{u} + c(γ)t, where c : 𝒢_{K} → ℤ_{p} is defined by γ(π_{n}) = π_{n} · (ε_{n})^{c(γ)} for all n ≥ 0.

The following proposition summarizes the basic properties of (some of) these sheaves. It will allow us to define ℤ-filtrations Fil^{i}(O_{X} ˆ⊗B_{cris}) resp. Fil^{i}(O_{X} ˆ⊗B_{st}) on O_{X} ˆ⊗B_{cris} resp. O_{X} ˆ⊗B_{st} as follows. Given i ∈ ℤ and a reduced adic space X, a section f ∈ Γ(X, O_{X} ˆ⊗B_{cris}) lies in Γ(X, Fil^{i}(O_{X} ˆ⊗B_{cris})), if f(x) ∈ Fil^{i} B_{cris} ⊗_{ℚ_{p}} κ(x) for all rigid analytic points x ∈ X. Here Fil^{i} B_{cris} is the usual filtration on B_{cris} induced by restricting the t-adic filtration on Fontaine’s ring B_{dR} to B_{cris}. A similar construction also applies to the filtration on O_{X} ˆ⊗B_{st}. As above we extend our definition of the filtrations to all adic spaces locally of finite type over ℚ_{p} by locally embedding a non-reduced space into a reduced adic space and restricting the filtration along the ideal defining the closed immersion.

Proposition 7.2. Let X be an adic space locally of finite type over ℚ_{p}.

(i) For a rigid analytic point x ∈ X, we have

ℬ̃_{X} ⊗ κ(x) = B̃ ⊗_{ℚ_{p}} κ(x)
(O_{X} ˆ⊗B_{cris}) ⊗ κ(x) = B_{cris} ⊗_{ℚ_{p}} κ(x)
(O_{X} ˆ⊗B_{st}) ⊗ κ(x) = B_{st} ⊗_{ℚ_{p}} κ(x),

where the fiber R ⊗ κ(x) of a sheaf R of O_{X}-algebras at a point x ∈ X is given by the quotient of the stalk R_{x} by the ideal 𝔪_{X,x}R_{x} generated by the maximal ideal 𝔪_{X,x} ⊂ O_{X,x}. (ii) Let f : X → Y be a finite morphism with Y reduced. The natural maps

Γ(X, ℬ̃_{X} ) ⟶ ∏ B̃ ⊗_{ℚ_{p}} Γ(f^{−1}(y), O_{X})
Γ(X, O_{X} ˆ⊗B_{cris}) ⟶ ∏ B_{cris} ⊗_{ℚ_{p}} Γ(f^{−1}(y), O_{X})
Γ(X, O_{X} ˆ⊗B_{st}) ⟶ ∏ B_{st} ⊗_{ℚ_{p}} Γ(f^{−1}(y), O_{X})

are injective. Here the product runs over all rigid analytic points y ∈ Y. (iii) Let g ∈ Γ(X, ℬ̃_{X} ) such that

g(y) ∈ Γ(f^{−1}(y), O_{X}) ⊂ B̃ ⊗_{ℚ_{p}} Γ(f^{−1}(y), O_{X})

for all rigid analytic points y ∈ Y. Then g ∈ O_{X} ⊂ ℬ̃_{X} . Especially ℬ̃_{X}^{Φ=id} = O_{X}. The same holds true for O_{X} ˆ⊗B_{cris} and O_{X} ˆ⊗B_{st}. Especially

Fil^{0} (O_{X} ˆ⊗B_{cris})^{Φ=id} = O_{X} and
Fil^{0} (O_{X} ˆ⊗B_{st})^{Φ=id,N=0} = O_{X}.

(iv) Further one has

(O_{X} ˆ⊗B_{cris})^{𝒢_{K}} = (O_{X} ˆ⊗B_{st})^{𝒢_{K}} = O_{X} ⊗_{ℚ_{p}} K_{0}

Proof. The statements concerning ℬ̃_{X} and O_{X} ˆ⊗B_{cris} are contained in [Hel13, Lemma 8.6, Corollary 8.8] resp. [Hel13, Lemma 8.11, Corollary 8.13]. To prove the statements concerning O_{X} ˆ⊗B_{st}, note that for every open subset U ⊂ X, every element f ∈ Γ(U, O_{X} ˆ⊗B_{st}) is a polynomial with coefficients in Γ(U, O_{X} ˆ⊗B_{cris}). The statements follow from this. For the last two statements note that we can always choose a finite morphism X → Y to a reduced space Y locally on X. □

Definition 7.3. Let 𝒢 denote a topological group. A family of 𝒢 -representations on an adic space X consists of a vector bundle E on X together with an O_{X}-linear action of the group 𝒢 on E which is continuous for the topologies on the sections Γ(−, E). This definition extends to the category of stacks on Ad_{ℚ_{p}}^{lft}.

Definition 7.4. Let X be an adic space locally of finite type over ℚ_{p}.

(i) A φ-module over 𝒜_{X} is an 𝒜_{X}-module M which is locally on X free of finite rank over 𝒜_{X} together with an isomorphism Φ : φ^{∗}M ^{∼} ⟶ M. (ii) A φ-module over ℬ_{X} is an ℬ_{X}-module M which is locally on X free of finite rank over ℬ_{X} together with an isomorphism Φ : φ^{∗}M ^{∼} ⟶ M. (iii) A φ-module M over ℬ_{X} is called étale if it is locally on X of the form N ⊗_{𝒜_{X}} ℬ_{X} for a φ-module N over 𝒜_{X}.

The following theorem summarizes results of [Hel13] which are needed in the sequel.

Theorem 7.5. Let X be a reduced adic space locally of finite type over ℚ_{p} and let (N , Φ) be an étale φ-module of rank d over ℬ_{X}.

(i) The set

X^{adm} = {x ∈ X | dim_{κ(x)}((N ⊗_{ℬ_{X}} ℬ̃_{X} ) ⊗ κ(x))^{Φ=id} = d} ⊂ X

is an open subspace and

V = (N ⊗ ℬ̃_{X} )^{Φ=id}

is a family of 𝒢_{K_{∞}}-representations on X^{adm}. (ii) If f : Y → X is a morphism in Ad^{lft} and if (N_{Y} , Φ_{Y}) denotes the pullback of (N , Φ) along f, then Y^{adm} = f^{−1}(X^{adm}) and

(N_{Y} ⊗ ℬ̃_{Y} )^{Φ=id} = (f|_{Y^{adm}})^{∗}V

as families of 𝒢_{K_{∞}}-representations on Y^{adm}. (iii) If (𝔐, Φ) is a φ-module of finite height over 𝒜_{X}^{[0,1)} as in Definition 6.2 and (N , Φ) = (𝔐, Φ) ⊗_{𝒜_{X}^{[0,1)}} ℬ_{X}, then

U = X^{adm} = {x ∈ X | rk_{κ(x)^{+}} Hom_{𝒜_{X}^{[0,1)}⊗κ(x),Φ} (𝔐 ⊗ κ(x), 𝒜̃_{X}^{[0,1)} ⊗ κ(x)) = d}

and

ℋ om_{𝒜_{U}^{[0,1)},Φ} (𝔐|_{U}, 𝒜̃_{U}^{[0,1)}) ⊗_{ℤ_{p}} ℚ_{p} = ℋ om_{ℬ_{U} ,Φ} (𝔐|_{U} ⊗_{𝒜_{U}^{[0,1)}} ℬ_{U}, ℬ̃_{U} )

as families of 𝒢_{K_{∞}}-representations on U = X^{adm}.

Proof. This is a summary of [Hel13, Proposition 8.20, Corollary 8.21, Proposition 8.22 and Proposition 8.23]. □

Given a cocharacter μ as in (2.5), the stack ℋ_{φ,N,⪯μ} is the stack quotient of P_{K_{0},d} ×_{Spec ℚ_{p}} Q_{K,d,μ} by the action of the reductive group (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}. Let us denote by ℋ_{φ,N,⪯μ}^{red} the quotient of the reduced subscheme underlying P_{K_{0},d} ×_{Spec ℚ_{p}} Q_{K,d,⪯μ} by the induced action of (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}. Recall that P_{K_{0},d} ×_{Spec ℚ_{p}} Q_{K,d,μ} is reduced, hence this modification will not be necessary if we restrict to the case where the Hodge type is fixed by μ.

Corollary 7.6. There is an open substack ℋ_{φ,N,⪯μ}^{red,ad,adm} ⊂ ℋ_{φ,N,⪯μ}^{red,ad,int} and a family E of 𝒢_{K_{∞}}-representa-tions on ℋ_{φ,N,⪯μ}^{red,ad,adm} such that

E = (\underline{M}(D, Φ, N, 𝔮) ⊗_{ℬ_{X}^{[0,1)}} ℬ̃_{X} )^{Φ= id},

where (D, Φ, N, 𝔮) denotes the restriction of the universal family on ℋ_{φ,N,⪯μ}^{red,ad}. This subspace is maximal in the following sense: If X is a reduced adic space and if \underline{D}′ is a (φ, N)-module with Hodge-Pink lattice over X with Hodge polygon bounded by μ, then the induced map f : X → ℋ_{φ,N,⪯μ}^{ad} factors over ℋ_{φ,N,⪯μ}^{red,ad,adm} if and only if X = X^{adm} with respect to the family

\underline{M}(\underline{D}′) ⊗_{ℬ_{X}^{[0,1)}} ℬ̃.

In this case there is a canonical isomorphism of 𝒢_{K_{∞}}-representations

f^{∗}E = (\underline{M}(\underline{D}′) ⊗_{ℬ_{X}^{[0,1)}} ℬ̃)^{Φ= id}.

If L is a finite extension of E_{μ}, then ℋ_{φ,N,⪯μ}^{red,ad,adm}(L) = ℋ_{φ,N,⪯μ}^{red,ad,int}(L)

Proof. Let us write X_{⪯μ} = (P_{K_{0},d} ×_{Spec ℚ_{p}} Q_{K,d,⪯μ})^{red,ad} for the moment. Further we denote the pullback of the universal family of vector bundles on the open unit disc to X_{⪯μ} by (M, Φ, N_{∇}^{M}) = \underline{M}(D, Φ, N, 𝔮). Locally on X_{⪯μ}^{int} there exists a φ-module of finite height 𝔐 inside (M, Φ), at least after a Tate twist. It follows that 𝔐 ⊗_{𝒜_{X}^{[0,1)}} 𝒜 is étale and we may apply the above theorem. Then X_{⪯μ}^{adm} ⊂ X_{⪯μ} is invariant under the action of (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} and hence its quotient by this group is an open substack ℋ_{φ,N,⪯μ}^{red,ad,adm} ⊂ ℋ_{φ,N,⪯μ}^{red,ad}. Further

(M ⊗_{ℬ_{X{adm \atop μ}}^{[0,1)}} ℬ̃_{X_{μ}^{adm}} )^{Φ= id}

is a (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}}-equivariant vector bundle with 𝒢_{K_{∞}}-action on X_{⪯μ}^{adm}. Hence it defines a family of 𝒢_{K_{∞}}-representations on ℋ_{φ,N,⪯μ}^{red,ad,adm}.

The second statement is local on X and hence, after locally choosing a basis of D, we can locally lift the morphism f : X → ℋ_{φ,N,⪯μ}^{red,ad} to a morphism f′ : X → X_{⪯μ} such that the pullback of (D, Φ, N, 𝔮) on X_{⪯μ} along f′ is isomorphic to \underline{D}′. Now the claim follows from Theorem 7.5 (ii). □

8. The universal semi-stable representation

In this section we want to construct a semi-stable 𝒢_{K}-representation out of the 𝒢_{K_{∞}}-representation on ℋ_{φ,N,⪯μ}^{red,ad,adm} from Corollary 7.6. This will be possible only on a part of ℋ_{φ,N,⪯μ}^{red,ad,adm}. First of all we need to restrict to the open subspace where the Hodge polygon is constant. This can be seen as follows. Let E be a family of 𝒢_{K}-representations on an adic space X. It follows from [BC08, § 4.1] that the (generalized) Hodge-Tate weights vary continuously on X. Namely, they are the eigenvalues of Sen’s operator Θ_{Sen} constructed in [BC08, before Remark 4.1.3]. The characteristic polynomial of Θ_{Sen} has coefficients in O_{X} ⊗_{ℚ_{p}} K. However, as E is semi-stable the Hodge-Tate weights of E ⊗ κ(x) are integers for all x ∈ X by [BC08, Corollaire 5.1.2] and hence the Hodge-Tate weights and the Hodge polygon are locally constant on X.

Secondly, Kisin [Kis06, Theorem 0.1 and Corollary 1.3.15] showed that the universal étale (φ, N_{∇})- module \underline{M} on ℋ_{φ,N,⪯μ}^{ad,int} from Corollary 6.7 can come from a semi-stable 𝒢_{K}-representation only if the connection ∇ has logarithmic singularities, which is equivalent to N_{∇}^{M} being holomorphic; see Remark 4.2(2). Therefore we have to restrict further to the closed subspace ℋ_{φ,N,μ}^{∇} ∩ ℋ_{φ,N,μ}^{ad,adm} of ℋ_{φ,N,μ}^{ad,adm} which is isomorphic to 𝒟_{φ,N,μ}^{ad,adm}. Here 𝒟_{φ,N,μ}^{ad,adm} ⊂ 𝒟_{φ,N,μ}^{ad} is the admissible locus with respect to the family defined in Remark 4.10.

Definition 8.1. Let E be a family of 𝒢_{K}-representations of rank d on an adic space X locally of finite type over ℚ_{p}.

(i) The family E is called crystalline if

D_{cris}(E) := (E ⊗_{O_{X}} (O_{X} ˆ⊗B_{cris}))^{𝒢_{K}}

is locally on X free of rank d as an O_{X} ⊗_{ℚ_{p}} K_{0}-module and if the descending ℤ-filtration Fil^{•} D_{cris}(E)_{K} of D_{cris}(E)_{K} has locally free subquotients. (ii) The family E is called semi-stable if

D_{st}(E) := (E ⊗_{O_{X}} (O_{X} ˆ⊗B_{st}))^{𝒢_{K}}

is locally on X free of rank d as an O_{X} ⊗_{ℚ_{p}} K_{0}-module and if the descending ℤ-filtration Fil^{•} D_{st}(E)_{K} of D_{st}(E)_{K} has locally free subquotients. (iii) Let μ be a cocharacter as in (2.5), let E_{μ} be its reflex field, and let X be an adic space locally of finite type over E_{μ}. We say that a crystalline, resp. semi-stable 𝒢_{K}-representation E over X has constant Hodge polygon equal to μ if the K-filtered φ-module D_{cris}(E), resp. D_{st}(E) over X has this property.

It is obvious from the definition that D_{st} defines a functor from the category of semi-stable representations with constant Hodge polygon μ over an adic space X to the category of K-filtered (φ, N)-modules over X with constant Hodge polygon μ and similarly for crystalline representations.

Remark 8.2. As in [Hel13, Lemma 8.15] one easily sees that the property of being crystalline oder semi-stable is local for the fpqc-topology.

Lemma 8.3. (i) Let E be a family of crystalline representations on X. Then the canonical map

f : D_{cris}(E) ⊗_{O_{X}⊗K_{0}} (O_{X} ˆ⊗B_{cris}) ⟶ E ⊗_{O_{X}} (O_{X} ˆ⊗B_{cris})

is an isomorphism. Further one has

D_{cris}(E) ⊗ κ(x) ≅ D_{cris}(E ⊗ κ(x))
(Fil^{i} D_{cris}(E)) ⊗ κ(x) ≅ Fil^{i} D_{cris}(E ⊗ κ(x))

for all rigid analytic points x ∈ X. Especially the representations E ⊗ κ(x) are crystalline for all rigid points x ∈ X. (ii) Let E be a family of semi-stable representations on X. Then the canonical map

(8.1)  f : D_{st}(E) ⊗_{O_{X}⊗K_{0}} (O_{X} ˆ⊗B_{st}) ⟶ E ⊗_{O_{X}} (O_{X} ˆ⊗B_{st})

is an isomorphism. Further one has

D_{st}(E) ⊗ κ(x) ≅ D_{st}(E ⊗ κ(x))
(Fil^{i} D_{st}(E)) ⊗ κ(x) ≅ Fil^{i} D_{st}(E ⊗ κ(x))

for all rigid analytic points x ∈ X. Especially the representations E ⊗ κ(x) are semi-stable for all rigid points x ∈ X.

Proof. We only prove the second part of the lemma. The claim D_{st}(E) ⊗ κ(x) ≅ D_{st}(E ⊗ κ(x)) follows from [BC08, Théorème 6.3.2] and hence the representations E ⊗ κ(x) are semi-stable. Further the map (8.1) is a morphism between free O_{X} ˆ⊗B_{st}-modules of the same rank and D_{st}(E) ⊗ κ(x) ≅ D_{st}(E ⊗ κ(x)) implies that f ⊗ κ(x) is an isomorphism for all rigid analytic points x ∈ X. Now Proposition 7.2 implies that (8.1) is an isomorphism. It remains to prove (Fil^{i} D_{st}(E)) ⊗ κ(x) ≅ Fil^{i} D_{st}(E ⊗ κ(x)). This may be checked after base change to some finite extension L of K containing the Galois closure K̃ of K, i.e. we change X to X_{L}. Given an embedding ψ : K → L we write D_{ψ} = D ⊗_{O_{X}⊗K_{0}, id ⊗ψ} O_{X} and we write Fil^{•} D_{ψ} for the corresponding ℤ-filtration of D_{ψ}. Then Fil^{i} D_{ψ} is locally on X a direct summand of O_{X} and it suffices to prove that Fil^{•}(D_{ψ} ⊗ κ(x)) = (Fil^{•} D_{ψ}) ⊗ κ(x).

Let us write i for the integer such that Fil^{i}(D_{ψ} ⊗κ(x)) = D_{ψ} ⊗κ(x) for all x ∈ X and Fil^{i+1} D_{ψ} ⊗κ(x) ≠ D for at least one x ∈ X. By Definition 8.1 it follows that dim_{κ(x)} Fil^{i}(D_{ψ} ⊗ κ(x))/ Fil^{i+1}(D_{ψ} ⊗ κ(x)) is independent of x ∈ X. As the filtration on O_{X} ˆ⊗B_{st} is defined using the filtrations on (O_{X} ˆ⊗B_{st})⊗κ(x) = B_{st} ⊗_{ℚ_{p}} κ(x) for all rigid analytic points x ∈ X we find that f ∈ D_{ψ} ⊂ E ⊗_{O_{X}} (O_{X} ˆ⊗B_{st}) lies in Fil^{j}(E ⊗_{O_{X}} (O_{X} ˆ⊗B_{st})) if and only if

f(x) ∈ Fil^{j} ((E ⊗ κ(x)) ⊗_{κ(x)} (B_{st} ⊗_{ℚ_{p}} κ(x)))

for all rigid analytic points x. It follows that D = ⋯ = Fil^{i−1} D_{ψ} = Fil^{i} D_{ψ}. Further it follows that

(Fil^{i} D_{ψ}/ Fil^{i+1} D_{ψ}) ⊗ κ(x) ⟶ Fil^{i}(D_{ψ} ⊗ κ(x))/ Fil^{i+1}(D_{ψ} ⊗ κ(x))

is surjective for all x ∈ X. However, the construction implies that this map is generically on X an isomorphism and hence it is an isomorphism everywhere as both sides have the same dimension for all x ∈ X. It follows that (Fil^{i+1} D_{ψ}) ⊗ κ(x) = Fil^{i+1}(D_{ψ} ⊗ κ(x)) and the claim follows by descending induction. □

Lemma 8.4. Let E be a family of G_{K}-representations on an adic space X locally of finite type over ℚ_{p}. Then E is crystalline if and only it is semi-stable and the monodromy N on D_{st}(E) vanishes. In this case we have D_{st}(E) = D_{cris}(E) as subobjects of E ⊗_{O_{X}} (O_{X} ˆ⊗B_{st}).

Proof. Let us assume that E is crystalline. Then by the preceding lemma we obtain an isomorphism

D_{cris}(E) ⊗_{O_{X}⊗K_{0}} (O_{X} ˆ⊗B_{cris}) ⟶ E ⊗_{O_{X}} (O_{X} ˆ⊗B_{cris})

compatible with the action of φ and G_{K} (and compatible with filtrations). Tensorizing this isomorphism over B_{cris} with B_{st} and taking N to be the induced monodromy action on both sides we obtain an isomorphism

D_{cris}(E) ⊗_{O_{X}⊗K_{0}} (O_{X} ˆ⊗B_{st}) ⟶ E ⊗_{O_{X}} (O_{X} ˆ⊗B_{st})

compatible with the actions of φ, N and G_{K} (and compatible with the filtrations). Taking G_{K}-invariants on both sides we find that D_{cris}(E) ≅ D_{st}(E) compatible with the actions of φ and N (and compatible with the filtrations).

Conversely assume that E is semi-stable with vanishing monodromy. As the isomorphism (8.1) is equivariant for the action of N the claim follows after taking N = 0 on both sides. □

Similar to the functors D_{cris}(−) and D_{st}(−) we also consider their contravariant variants

(8.2)  D_{cris}^{∗} : E ⟼ ℋ om_{O_{X}[𝒢_{K}]}(E, O_{X} ˆ⊗B_{cris}) and
D_{st}^{∗} : E ⟼ ℋ om_{O_{X}[𝒢_{K}]}(E, O_{X} ˆ⊗B_{st}).

An easy calculation shows that D_{cris}(E^{∨}) ≅ D_{cris}^{∗}(E) and similar for semi-stable representations.

Lemma 8.5. Let X be an adic space locally of finite type over ℚ_{p} and let E, E_{1} and E_{2} be families of semi-stable representations.

(i) The canonical morphism

E ⟶ V_{st}(D_{st}(E)) = Fil^{0}(D_{st}(E) ⊗_{O_{X}⊗_{ℚp}K_{0}} (O_{X} ˆ⊗B_{st}))^{φ= id,N=0}

is an isomorphism. (ii) One has E_{1} ≅ E_{2} if and only if D_{st}(E_{1}) ≅ D_{st}(E_{2}).

Proof. The first part follows from Fil^{0}(O_{X} ˆ⊗B_{st})^{φ= id,N=0} = O_{X} and the lemma above. The second part is an easy consequence. □

Proposition 8.6. Let X be a reduced adic space locally of finite type over E_{μ} and let \underline{D} = (D, Φ, N, F^{•}) ∈ 𝒟_{φ,N,μ}^{an,adm}(X). Then

ℋ om_{𝒜_{X}^{[0,1)},Φ} (𝔐, 𝒜̃_{X}^{[0,1)}) ⊗_{ℤ_{p}} ℚ_{p} → ℋ om_{Φ,N,Fil}(\underline{D}, O_{X} ˆ⊗B_{st}) =: E

is an isomorphism for every φ-module (𝔐, Φ) of finite height over 𝒜_{X}^{[0,1)} such that 𝔐 ⊗_{𝒜_{X}^{[0,1)}} ℬ_{X}^{[0,1)} = \underline{M}(\underline{D}). Further E is a vector bundle of rank d on X and the 𝒢_{K}-action on B_{st} makes E into a family of semi-stable 𝒢_{K}-representations such that there is a canonical isomorphism of filtered (φ, N)-modules

D_{st}^{∗}(E) ≅ (D, Φ, N, F^{•}).

Proof. As in the proof of [Hel13, Proposition 8.24], the proof follows the one of [Kis06, Proposition 2.1.5]. If we write M = 𝔐 ⊗_{𝒜_{X}^{[0,1)}} ℬ_{X}^{[0,1)}, then the morphism

ℋ om_{𝒜_{X}^{[0,1)},Φ} (𝔐, 𝒜̃_{X}^{[0,1)}) ⊗_{ℤ_{p}} ℚ_{p} ⟶ ℋ om_{Φ,N,Fil}(\underline{D}, O_{X} ˆ⊗B_{st})

is given by the composition of injections

ℋ om_{𝒜_{X}^{[0,1)},Φ} (𝔐, 𝒜̃_{X}^{[0,1)}) ⊗_{ℤ_{p}} ℚ_{p} ↪ ℋ om_{𝒜_{X}^{[0,1)},Φ} (𝔐, O_{X} ˆ⊗B_{st})
↪ ℋ om_{ℬ_{X}^{[0,1)},Φ} (M, O_{X} ˆ⊗B_{st})
↪ ℋ om_{ℬ_{X}^{[0,1)},Φ} (\underline{D} ⊗_{K_{0}} ℬ_{X}^{[0,1)}, O_{X} ˆ⊗B_{st})
= ℋ om_{Φ}(\underline{D}, O_{X} ˆ⊗B_{st}),

compare [Kis06, Proposition 2.1.5] and [Hel13, Proposition 8.24] for the definition of these maps. The image of the above injections is contained in

ℋ om_{Φ,N,Fil}(\underline{D}, O_{X} ˆ⊗B_{st}) = E,

because this is true at all rigid analytic points by [Kis06, Proposition 2.1.5] and X is reduced. Now

E′ = ℋ om_{𝒜_{X}^{[0,1)},Φ} (𝔐, 𝒜̃_{X}^{[0,1)}) ⊗_{ℤ_{p}} ℚ_{p}

is by Theorem 7.5 a vector bundle of rank d on X. We claim that the inclusion

(8.3)  E′ ⊗_{O_{X}} (O_{X} ˆ⊗B_{st}) ⊂ ℋ om_{O_{X}} (\underline{D}, O_{X} ˆ⊗B_{st})

is an equality at all rigid analytic points x ∈ X. First one has

E′ ⊗ κ(x) ≅ Hom_{𝒜_{X}^{[0,1)}⊗κ(x),Φ} (𝔐 ⊗ κ(x), 𝒜̃_{X}^{[0,1)} ⊗ κ(x)) ⊗_{ℤ_{p}} ℚ_{p}
ℋ om_{O_{X}} (\underline{D}, O_{X} ˆ⊗B_{st}) ⊗ κ(x) ≅ Hom_{κ(x)}(\underline{D} ⊗ κ(x), B_{st} ⊗_{ℚ_{p}} κ(x)) = Hom_{ℚ_{p}}(\underline{D} ⊗ κ(x), B_{st}).

Finally the inclusion (8.3) is defined by canonical maps and hence the induced map

Hom_{𝒜_{X}^{[0,1)}⊗κ(x),Φ} (𝔐 ⊗ κ(x), 𝒜̃_{X}^{[0,1)} ⊗ κ(x)) ⊗_{ℤ_{p}} ℚ_{p} ⟶ Hom_{ℚ_{p}}(\underline{D} ⊗ κ(x), B_{st})

again is the canonical map and hence an isomorphism by [Kis06, Proposition 2.1.5]. After choosing bases, the inclusion (8.3) is given by a d × d-matrix with entries in O_{X} ˆ⊗B_{st}. This matrix is invertible at all rigid analytic points in X and hence it is an isomorphism by Proposition 7.2. It follows that E′ = E, as both sides are given by Fil^{0}(−)^{Φ=id,N=0} on

E′ ⊗_{O_{X}} (O_{X} ˆ⊗B_{st}) = ℋ om_{O_{X}} (\underline{D}, O_{X} ˆ⊗B_{st}),

where Φ and N act on the left hand side by acting on B_{st} and the filtration on the left hand side is induced from the filtration on B_{st}, i.e. E′ is equipped with the trivial filtration and with the monodromy N = 0. Here we use that Fil^{0}(O_{X} ˆ⊗B_{st})^{Φ=id,N=0} = O_{X} by Proposition 7.2. Finally we have D_{st}^{∗}(E) ≅ \underline{D} as the canonical pairing

(E ⊗_{O_{X}} (O_{X} ˆ⊗B_{st})) ⊗_{O_{X}⊗B_{st}} (D ⊗_{O_{X}⊗K_{0}} (O_{X} ˆ⊗B_{st})) ⟶ O_{X} ˆ⊗B_{st}

given by f ⊗d ↦ f(d) is a 𝒢_{K}-equivariant paring of free O_{X} ˆ⊗B_{st}-modules of the same rank which induces a perfect pairing at all rigid points. Hence, using Proposition 7.2 again, we see that the pairing itself is perfect and the claim follows. □

Recall that the stack 𝒟_{φ,N,μ}^{ad} is the quotient of the adic space X_{μ} associated to P_{K_{0},d} × Flag_{K,d,μ} by the action of the group (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} and consider the open subspace X_{μ}^{adm} ⊂ X_{μ}^{int} ⊂ X_{μ}. This subset is stable under the action of (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} and we write 𝒟_{φ,N,μ}^{ad,adm} for the quotient of X_{μ}^{adm} by this action.

Proposition 8.7. Let μ be a cocharacter as in (2.5) and let E_{μ} be its reflex field. Let X be a reduced adic space locally of finite type over E_{μ} and let E be a family of semi-stable 𝒢_{K}-representations on X with constant Hodge polygon equal to μ. Then the morphism

X ⟶ 𝒟_{φ,N,μ}^{ad}

induced by the K-filtered (φ, N)-module D_{st}(E) factors over 𝒟_{φ,N,μ}^{ad,adm}.

Proof. Write D_{st}(E) = (D, Φ, N, F^{•}) = \underline{D} for the associated filtered (φ, N)-module on X. After twisting with a high enough power of the cyclotomic character, we may assume that all Hodge weights −μ_{ψ,j} lie between −h and 0 for some h ≫ 0. Hence we may consider the inclusions

ℬ_{X}^{[0,1)}(E ⊗_{O_{X}} O_{X} ˆ⊗B_{st})^{𝒢_{K}} ⊂ M = \underline{M}(\underline{D})
⊂ λ^{−h}ℬ_{X}^{[0,1)}(E ⊗_{O_{X}} O_{X} ˆ⊗B_{st})^{𝒢_{K}} ⊂ E ⊗_{O_{X}} O_{X} ˆ⊗B_{st},

where λ is the element defined in (4.1). Here ℬ_{X}^{[0,1)}(E ⊗_{O_{X}} O_{X} ˆ⊗B_{st})^{𝒢_{K}} is the sub-ℬ_{X}^{[0,1)}-module of E ⊗_{O_{X}} O_{X} ˆ⊗B_{st} generated by (E ⊗_{O_{X}} O_{X} ˆ⊗B_{st})^{𝒢_{K}} and so on.

Now the 𝒢_{K_{∞}}-action on M ⊂ E ˆ⊗_{ℚ_{p}}B_{st} is trivial, as 𝒢_{K_{∞}} acts trivially on ℬ_{X}^{[0,1)}, hence especially on λ, and on (E ⊗_{O_{X}} O_{X} ˆ⊗B_{st})^{𝒢_{K}}. This claim now implies that

M ⊗_{ℬ_{X}^{[0,1)}} ℬ_{X}^{[r,s]} ⊂ (E⊗_{O_{X}} ℬ̃_{X}^{[r,s]} )^{𝒢_{K∞}}

for some 0 < r < r^{1/p^{2}} ≤ s < 1 near the boundary and where

ℬ̃_{X}^{[r,s]} = 𝒜̃_{X}^{[0,1)} ˆ⊗_{𝒜_{X}^{[0,1)}} ℬ_{X}^{[r,s]}.

Now

M ⊗_{ℬ_{X}^{[0,1)}} ℬ̃_{X}^{[r,s]} = E ⊗_{O_{X}} ℬ̃_{X}^{[r,s]} ,

as the cokernel vanishes at all rigid analytic points by an analogous argument as in the proof of Proposition 8.6 and as X is reduced. Now the proof runs exactly as the proof of [Hel13] Proposition 8.27 and Proposition 8.28. □

Theorem 8.8. There is a family E of semi-stable 𝒢_{K}-representations on 𝒟_{φ,N,μ}^{ad,adm} such that D_{st}(E) = (D, Φ, N, F^{•}) is the universal family of filtered (φ, N)-modules on 𝒟_{φ,N,μ}^{ad,adm}. This family is universal in the following sense: Let X be an adic space locally of finite type over E_{μ} and let E′ be a family of semistable 𝒢_{K}-representations on X with constant Hodge polygon equal to μ. Then there is a unique morphism f : X → 𝒟_{φ,N,μ}^{ad,adm} such that E′ ≅ f^{∗}E as families of 𝒢_{K}-representations.

Proof. The existence of the family E follows by applying Proposition 8.6 to the family (𝔐, Φ) of φ-modules of finite height over 𝒜^{[0,1)} on

Y = (P_{K_{0},d} × Flag_{K,d,μ})^{ad,adm}.

This family exists locally on open subsets U ⊂ Y and the semi-stable 𝒢_{K}-representation on U is canonically contained in ℋ om_{O_{X}⊗K_{0}}(D, O_{X} ˆ⊗B_{st}). Hence they glue together to give rise to the claimed family E on Y. By functoriality of ℋ om_{Φ,N,Fil}(−, O_{X} ˆ⊗B_{st}) this vector bundle is equivariant for the action of (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} and hence defines the desired family of semi-stable 𝒢_{K}-representations on 𝒟_{φ,N,μ}^{ad,adm}. Further the isomorphism D_{st}(E) ≅ (D, Φ, N, F^{•}) on Y is by construction equivariant under (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})_{E_{μ}} and hence descends to 𝒟_{φ,N,μ}^{ad,adm}.

Now let X be as above. The K-filtered (φ, N)-module D_{st}(E′) defines a morphism f : X → 𝒟_{φ,N,μ}^{ad}. This map factors over 𝒟_{φ,N,μ}^{ad,adm} by Proposition 8.7 as factoring over an open subspace may be check on the reduced space underlying X. Further we have isomorphisms D_{st}(E′) ≅ f^{∗}D_{st}(E) ≅ D_{st}(f^{∗}E). Now the claim follows from Lemma 8.5. □

Corollary 8.9. There is a family E of crystalline 𝒢_{K}-representations on 𝒟_{φ,μ}^{ad,adm} such that D_{crys}(E) = (D, Φ, F^{•}) is the universal family of filtered φ-modules on 𝒟_{φ,μ}^{ad,adm}. This family is universal in the following sense: Let X be an adic space locally of finite type over E_{μ} and let E′ be a family of crystalline 𝒢_{K}- representations on X with constant Hodge polygon μ. Then there is a unique morphism f : X → 𝒟_{φ,μ}^{ad,adm} such that E′ ≅ f^{∗}E as families of 𝒢_{K}-representations.

Proof. This is a direct consequence of Lemma 8.4. □

9. The morphism to the adjoint quotient

As in [Hel11, § 4] we consider the adjoint quotient A/𝔖_{d} where A ⊂ GL_{d,ℚ_{p}} is the diagonal torus and 𝔖_{d} is the finite Weyl group of GL_{d}. Under the morphism c : A → 𝔸_{ℚ_{p}}^{d−1} ×_{ℚ_{p}} 𝔾_{m,ℚ_{p}} which maps an element g of A to the coefficients c_{1}, …, c_{d} of its characteristic polynomial χ_{g} = X^{d} + c_{1}X^{d−1} + … + c_{d}, the adjoint quotient A/𝔖_{d} is isomorphic to 𝔸_{ℚ_{p}}^{d−1} ×_{ℚ_{p}} 𝔾_{m,ℚ_{p}} = Spec ℚ_{p}[c_{1}, …, c_{d}, c_{d}^{−1}]. Recall from [Hel11, § 4] that there is a morphism

(9.1)  Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} ⟶ A/𝔖_{d}

which is invariant under φ-conjugation on the source. It is defined on R-valued points by sending b ∈ (Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}})(R) = GL_{d}(R⊗_{ℚ_{p}} K_{0}) to the characteristic polynomial of (b·φ)^{f} = b·φ(b)·. . .·φ^{(f−1)}(b) where f = [K_{0} : ℚ_{p}]. This characteristic polynomial actually has coefficients in R, because it is invariant under φ, as can be seen from the formula φ(b · φ)^{f} = b^{−1} · (b · φ)^{f} · φ^{f}(b) = b^{−1} · (b · φ)^{f} · b. Since Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} acts on itself by φ-conjugation via (g, b) ↦ g^{−1}b φ(g) and (g^{−1}b φ(g) φ)^{f} = g^{−1}·(b·φ)^{f} ·g the map (9.1) is invariant under φ-conjugation.

Let μ be a cocharacter as in (2.5), let E_{μ} be its reflex field, and set (A/𝔖_{d})_{E_{μ}} := A/𝔖_{d} ×_{ℚ_{p}} E_{μ}. By projecting to Res_{K_{0}/ℚ_{p}} GL_{d,K_{0}} we may extend β to morphisms

[figure: P_{K_{0},d} ×_{ℚ_{p}} Q_{K,d,⪯μ}; α̃; (A/𝔖_{d})_{E_{μ}}; ℋ_{φ,N,⪯μ}; α; (A/𝔖_{d})_{E_{μ}} .]

We further obtain morphisms to (A/𝔖_{d})_{E_{μ}} from the locally closed substacks ℋ_{φ,⪯μ}, ℋ_{φ,N,μ}, ℋ_{φ,μ}, 𝒟_{φ,N,μ}, and 𝒟_{φ,μ}, which we likewise denote by α. Here we view 𝒟_{φ,N,μ} and 𝒟_{φ,μ} as substacks of ℋ_{φ,N,μ} via the zero section from Remark 2.7(3). We also consider the adification of these morphisms.

Theorem 9.1. Let μ be a cocharacter as in (2.5), let E_{μ} be its reflex field and let x ∈ (A/𝔖_{d})_{E_{μ}}^{ad}. Then there exists an open subscheme X of α^{−1}(x) such that the weakly admissible locus in the fiber over x is given by

α^{−1}(x)^{wa} = X^{ad} .

Proof. This is similar to the proof of [Hel11, Theorem 4.1]. Let x = (c_{1}, …, c_{d}) ∈ κ(x)^{d−1} × κ(x)^{×} and let v_{x} denote the (multiplicative) valuation on κ(x). First note that

c_{d} = det_{κ(x)⊗_{ℚp}K_{0}}(b · φ)^{f} = det_{κ(x)}((b · φ)^{f})^{1/f}

and hence α^{−1}(x)^{wa} = ∅ unless

v_{x}(c_{d})^{−1/f} · v_{x}(p)^{\frac{1}{ef} ∑_{ψ,j} μ_{ψ,j}} = λ(\underline{D}) = 1.

In the following we will assume that this condition is satisfied. We now revert to the notation of the proof of Theorem 5.6. In particular we consider the projective P_{K_{0},d}-schemes Z_{i}, the global sections f_{i} ∈ Γ(Z_{i}, O_{Z_{i}}), the functions h_{i}, the closed subsets

Y_{i,m} = {y ∈ Z_{i}^{ad} × Q_{K,d,⪯μ}^{ad} | h_{i}(y) ≥ m}

and the proper projections pr_{i,m} : Y_{i,m} → P_{K_{0},d} ×_{ℚ_{p}} Q_{K,d,⪯μ}. This time

S_{i,m} = {y = (g_{y}, N_{y}, U_{y}, 𝔮_{y}) ∈ Y_{i,m} ×_{(P_{K_{0},d}×Q_{K,d,⪯μ})} α^{−1}(x) | v_{y}(f_{i}(g_{y}, U_{y})) > v_{y}(p)^{f^{2}m}}

is a union of connected components of Y_{i,m} ×_{(P_{K_{0},d}×Q_{K,d,⪯μ})} α^{−1}(x), hence a closed subscheme and not just a closed adic subspace. This can be seen as follows: Let λ_{1}, …, λ_{d} denote the zeros of the polynomial

X^{d} + c_{1}X^{d−1} + ⋯ + c_{d−1}X + c_{d}.

Then every possible value of the f_{i} is a product of some of the λ_{i} and hence f_{i} can take only finitely many values. As in the proof of Theorem 5.6

α^{−1}(x)^{wa} = α^{−1}(x) ∖ ⋃_{i,m} pr_{i,m}(S_{i,m}),

where the union runs over 1 ≤ i ≤ d − 1 and m ∈ ℤ. So α^{−1}(x)^{wa} is an open subscheme of α^{−1}(x). □

Corollary 9.2. Let x ∈ (A/𝔖_{d})_{E}^{ad} and consider the 2-fiber product

\xymatrix{α^{−1}(x)^{wa} \ar[r] \ar[dd] & ℋ_{φ,N,⪯μ}^{ad,wa} \\  & α \ar[d] \\ x \ar[r] & (A/W)_{E}^{ad}.}

Then there exists an Artin stack in schemes 𝔄 over the field κ(x) which is an open substack of α^{−1}(x), such that α^{−1}(x)^{wa} = 𝔄^{ad}. The same is true for ℋ_{φ,⪯μ}, ℋ_{φ,N,μ}, ℋ_{φ,μ}, 𝒟_{φ,N,μ}, and 𝒟_{φ,μ}.

Proof. This is an immediate consequence of Theorem 9.1 and the proof of Corollary 5.7. □

We also determine the image of the weakly admissible locus in the adjoint quotient.

Theorem 9.3. The image of ℋ_{φ,N,⪯μ}^{ad,wa} (and ℋ_{φ,⪯μ}^{ad,wa}, ℋ_{φ,N,μ}^{ad,wa}, ℋ_{φ,μ}^{ad,wa}, 𝒟_{φ,N,μ}^{ad,wa}, and 𝒟_{φ,μ}^{ad,wa}) under the morphism(s) α is equal to the affinoid subdomain (9.2)

\{ c = (c_{1}, …, c_{d}) ∈ (A/𝔖_{d})_{E_{μ}}^{ad} | v_{c}(c_{i}) ≤ v_{c}(p)\frac{1}{e} ∑_{ψ}(μ_{ψ,d} + … + μ_{ψ,d+1−i}) with equality for i = d \} ,

where v_{c} is the (multiplicative) valuation of the adic point c with v_{c}(p) < 1.

Remark 9.4. (1) The subset described in (9.2) is really an affinoid subdomain. Indeed the adjoint quotient (A/𝔖_{d})_{E_{μ}}^{ad} is (admissibly) covered by the (admissible) open affinoid rigid spaces (resp. adic spaces) X_{M} = {c = (c_{1}, …, c_{d}) ∈ (A/𝔖_{d})_{E_{μ}}^{ad} | v_{c}(c_{i}) ≤ p^{M}, for all i and v_{c}(v_{d}) ≥ −p^{M}} and the subspace (9.2) is easily seen to be a Laurent subdomain of each of these X_{M} for M ≫ 0.

(2) The morphisms α forget the Hodge-Pink lattice 𝔮 (or the K-filtration F^{•}) and in general their fibers contain infinitely many weakly admissible points.

(3) Like in [Hel11, Proposition 5.2] the affinoid subdomain of Theorem 9.3 can be described as the closed Newton stratum of the coweight (−\frac{1}{e} ∑_{ψ} μ_{ψ,d} ≥ … ≥ −\frac{1}{e} ∑_{ψ} μ_{ψ,1}) of A. By this we mean that the \overline{ℚ}_{p}- valued points (i.e. the rigid analytic points) of 9.2 coincide with the points of the corresponding Newton stratum in the sense of Kottwitz. In [Hel11] the claim is made for all points of the corresponding Berkovich space. In the set up of adic spaces we can not rely in Kottwitz definition of a Newton stratum for all points of the adic space, as the valuations are not necessarily rank one valuations, i.e. the value group is not necessarily a subgroup of the real numbers. Especially the Newton strata do not cover the adic space (A/𝔖_{d})^{ad}.

(4) Our affinoid subdomain equals 𝔖_{d}\T′_{ξ} from [BS, Corollary 2.5], where ξ is associated with the cocharacter ξ̃:= (−μ − (0, 1, …, d − 1))_{dom} ∈ X_{∗}(T̃). In this way our theorem generalizes [BS, Proposition 3.2].

Before we prove the theorem we note the following

Lemma 9.5. Set l_{i} := \frac{1}{ef} ∑_{ψ}(μ_{ψ,d} + … + μ_{ψ,d+1−i}). Then l_{i} equals the number l_{i} defined in [Hel11, Formula (5.2) on p. 988]. If \underline{D} = (D, Φ, N, 𝔮) is a (φ, N)-module with Hodge-Pink lattice over a field L ⊃ E_{μ} whose Hodge polygon is bounded by μ and if \underline{D}′ = (D′, Φ|_{φ^{∗}D′}, N|_{D′}, 𝔮 ∩ D′ ⊗_{L⊗K_{0}} 𝔹_{L}) ⊂ \underline{D} for a free L ⊗_{ℚ_{p}} K_{0}-submodule D′ ⊂ D of rank i which is stable under Φ and N, then t_{H}(\underline{D}′) ≥ l_{i}.

Proof. The number l_{i} in [Hel11, (5.2)] was defined as follows. Write {μ_{ψ,1}, …, μ_{ψ,d}} = {x_{ψ,1}, …, x_{ψ,r}} with x_{ψ,j} > x_{ψ,j+1}. Let n_{ψ,j} := max{k : μ_{ψ,k} ≥ x_{ψ,j}}. In particular n_{ψ,r} = d and μ_{ψ,n_{ψ,j}} ≥ x_{ψ,j}. For 0 ≤ i ≤ d let m_{ψ,j}(i) := max{0, n_{ψ,j} + i − d}. So m_{ψ,j}(0) = 0 for all j and m_{ψ,r}(i) = i. It follows that n_{ψ,j} ≥ d − i if and only if μ_{ψ,d−i} ≥ x_{ψ,j}. Now l_{i} was defined in [Hel11, (5.2)] as

l_{i} = \frac{1}{ef} ∑_{ψ}(∑_{j=1}^{r−1}(x_{ψ,j} − x_{ψ,j+1})m_{ψ,j}(i) + x_{ψ,r}m_{ψ,r}(i)).

We compute l_{i+1} − l_{i} = \frac{1}{ef} ∑_{ψ} (∑_{j=1}^{r−1}(x_{ψ,j} − x_{ψ,j+1})(m_{ψ,j}(i + 1) − m_{ψ,j}(i)) + x_{ψ,r}). The difference m_{ψ,j}(i + 1) − m_{ψ,j}(i) is 1 if n_{ψ,j} + i − d ≥ 0, that is if x_{ψ,j} ≤ μ_{ψ,d−i}. Else m_{ψ,j}(i + 1) − m_{ψ,j}(i) is 0. Therefore l_{i+1} − l_{i} = \frac{1}{ef} ∑_{ψ} μ_{ψ,d−i} and l_{0} = 0 implies that l_{i} = \frac{1}{ef} ∑_{ψ}(μ_{ψ,d} + … + μ_{ψ,d+1−i}).

To prove the second assertion let s ∈ Spec L⊗_{E_{μ}} K̃ be a point and let μ′ = μ_{\underline{D}}(s) be the Hodge polygon of \underline{D} at s. Then μ_{ψ,d} + … + μ_{ψ,d+1−i} ≤ μ′_{ψ,d} + … + μ′_{ψ,d+1−i} for all ψ and all i by Proposition 2.12(b). We let 𝔭_{ψ} be the ψ-component of s^{∗}𝔭 := s^{∗}D ⊗_{κ(s)⊗K_{0}} 𝔹_{κ(s)}^{+} and 𝔭′_{ψ} be the ψ-component of s^{∗}𝔭′ := s^{∗}D′ ⊗_{κ(s)⊗K_{0}} 𝔹_{κ(s)}^{+}. By definition of the Hodge polygon, see Construction 2.9, we can choose a κ(s)[[t]]-

basis (v_{ψ,1}, …, v_{ψ,d}) of 𝔭_{ψ} such that (t^{−μ′_{ψ,1}} v_{ψ,1}, …, t^{−μ′_{ψ,d}} v_{ψ,d}) is a κ(s)[[t]]-basis of the ψ-component 𝔮_{ψ} of s^{∗}𝔮. Since dim_{κ(s)((t))}(𝔭′_{ψ}[\frac{1}{t}] ∩ ⟨v_{ψ,1}, …, v_{ψ,n}⟩_{κ(s)((t))}) ≥ n + i − d for all n, we can find a κ(s)[[t]]-basis (v′_{ψ,1}, …, v′_{ψ,i}) of 𝔭′_{ψ} with v′_{ψ,j} ∈ ⟨v_{ψ,1}, …, v_{ψ,d+j−i}⟩_{κ(s)[[t]]}. Namely, for each j we let v̄′_{j} be an element of (𝔭′_{ψ} ∩ ⟨v_{ψ,1}, …, v_{ψ,d+j−i}⟩_{κ(s)[[t]]})/⟨v′_{ψ,1}, …, v′_{ψ,j−1}⟩_{κ(s)[[t]]} which generates a non-zero saturated κ(s)[[t]]- submodule, and we let v′_{ψ,j} ∈ 𝔭′_{ψ} ∩⟨v_{ψ,1}, …, v_{ψ,d+j−i}⟩_{κ(s)[[t]]} be a lift of v̄′_{ψ,j}. Then (v′_{ψ,1}, …, v′_{ψ,j}) is linearly independent over κ(s)((t)) and generates a saturated κ(s)[[t]]-submodule of 𝔭′_{ψ}. Using this basis we see

that t^{−μ′_{ψ,d+j−i}} · v′_{ψ,j} ∈ 𝔮_{ψ} ∩ 𝔭′_{ψ}[\frac{1}{t}]. This implies that t_{H}(\underline{D}′) ≥ \frac{1}{ef} ∑_{ψ} μ′_{ψ,d} + … + μ′_{ψ,d+1−i} ≥ l_{i}. □

Proof of Theorem 9.3. We consider the embedding of 𝒟_{φ,μ} into ℋ_{φ,μ} via the zero section. Under this section 𝒟_{φ,μ}^{ad,wa} is contained in ℋ_{φ,N,⪯μ}^{ad,wa}, ℋ_{φ,⪯μ}^{ad,wa}, ℋ_{φ,N,μ}^{ad,wa}, ℋ_{φ,μ}^{ad,wa}, and 𝒟_{φ,N,μ}^{ad,wa} by Lemma 5.3 or Remark 5.8. Conversely they are all contained in ℋ_{φ,N,⪯μ}^{ad,wa}. Moreover, these inclusions are compatible with the morphisms α to (A/𝔖_{d})_{E_{μ}}.

We first claim that the affinoid subdomain is contained in the image of the weakly admissible locus for all these stacks. By the above it suffices to prove the claim for 𝒟_{φ,μ}^{ad,wa}. In this case the claim follows from [Hel11, Theorem 5.5 and Proposition 5.2] using Lemma 9.5. Note that in loc. cit. only Berkovich’s analytic points are treated, but the given argument works verbatim also for adic points.

Conversely let c = (c_{1}, …, c_{d}) be an L-valued point of (A/𝔖_{d})_{E_{μ}}^{ad} which lies in the image of the

weakly admissible locus of one of these stacks. By the above it lies in the image of ℋ_{φ,N,⪯μ}^{ad,wa}. So let \underline{D} ∈ ℋ_{φ,N,⪯μ}^{ad,wa}(L′) for a field extension L′/L, such that \underline{D} maps to c. By extending the field L′ further we may assume that K_{0} ⊂ L′ and that X^{d} + c_{1}X^{d−1} + … + c_{d} = ∏_{j=1}^{d}(X − λ_{j}) splits into linear factors with λ_{j} ∈ L′. We claim that v_{L′}(∏_{j∈I} λ_{j}) ≤ v_{L′}(p)^{fl_{i}} for all subsets I ⊂ {1, …, d} of cardinality i. By Lemma 9.5 this implies that c lies in our affinoid subdomain.

To prove the claim we use Remark 2.4. Then X^{d} + c_{1}X^{d−1} + … + c_{d} is the characteristic polynomial of the L′-endomorphism (Φ^{f})_{0} of D_{0} and t_{N}(\underline{D}) = v_{L′}(det_{L′}(Φ^{f})_{0})^{1/f}. We write (Φ^{f})_{0} in Jordan canonical form and observe that N_{0} maps the generalized eigenspace of (Φ^{f})_{0} with eigenvalue λ_{j} into the one with eigenvalue p^{−f}λ_{j}. If I ⊂ {1, …, d} is a subset with cardinality i this allows us to find an i-dimensional L′-subspace D′_{0} ⊂ D_{0} which is stable under (Φ^{f})_{0} and N_{0} such that the eigenvalues of (Φ^{f})_{0} on D′_{0} are of the form (p^{−n_{j}}λ_{j} : j ∈ I) for suitable n_{j} ∈ ℤ_{≥0}. We let D′ ⊂ D be the (φ, N)-submodule corresponding to D′_{0} ⊂ D_{0} under Remark 2.4. Then

v_{L′}( ∏_{j∈I} λ_{j}) ≤ v_{L′}( ∏_{j∈I} p^{−n_{j}}λ_{j}) = v_{L′}(det_{L′}(Φ^{f})_{0}|_{D′_{0}}) = t_{N}(\underline{D}′)^{f} ≤ v_{L′}(p)^{f t_{H}(\underline{D}′)} ≤ v_{L′}(p)^{fl_{i}}

by the weak admissibility of \underline{D} and by Lemma 9.5. This proves the theorem. □

References

[AK80] A. Altman, S. Kleiman: Compactifying the Picard scheme, Adv. in Math. 35 (1980), no. 1, 50–112.

[BL94] A. Beauville, Y. Laszlo: Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385–419; also available as arXiv:alg-geom/9309003.

[BL93] S. Bosch, W. Lütkebohmert: Formal and Rigid Geometry II. Flattening Techniques, Math. Ann. 296 (1993), 403–429.

[BD] A. Beilinson, V. Drinfeld: Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint on http://www.math.uchicago.edu/∼mitya/langlands.html.

[BC08] L. Berger, P. Colmez: Familles de représentations de de Rham et monodromie p-adique, in Représentations padiques de groupes p-adiques I: représentations galoisiennes et (φ, Γ)-modules, Astérisque 319 (2008), 303–337; also available at http://www.math.jussieu.fr/∼colmez.

[BS] C. Breuil, P. Schneider, First steps towards p-adic Langlands functoriality, J. reine angew. Math. 610 (2007), 149–180; also available as arXiv:math.NT/0603.5499.

[Bou70] N. Bourbaki: Algèbre, Chapitres 1 à 3, Hermann, Paris 1970.

[BC09] O. Brinon, B. Conrad: p-adic Hodge theory, notes from the CMI summer school 2009; also available at http://math.stanford.edu/∼conrad.

[BGR84] S. Bosch, U. Güntzer, R. Remmert: Non-Archimedean Analysis, Grundlehren 261, Springer-Verlag, Berlin etc. 1984.

[BLR90] S. Bosch, W. Lütkebohmert, M. Raynaud: Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 21, Springer-Verlag, Berlin, 1990.

[CF00] P. Colmez, J.-M. Fontaine: Construction des représentations p-adiques semi-stables, Invent. Math. 140 (2000), no. 1, 1–43.

[DOR10] J-F. Dat, S. Orlik, M. Rapoport: Period domains over finite and p-adic fields, Cambridge Tracts in Math. 183, Cambridge University Press 2010.

[Eis95] D. Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry, GTM 150, Springer-Verlag, Berlin etc. 1995.

[FF] L. Fargues, J.-M. Fontaine: Courbes et fibrés vectoriels en théorie de Hodge p-adique, preprint, 2010, available at http://www.math.jussieu.fr/∼fargues.

[FGA] A. Grothendieck: Fondements de la Géométrie Algébrique, Extraits du Séminaire Bourbaki 1957–1962, Secrétariat mathématique, Paris 1962; also available at http://www.numdam.org/numdam-bin/recherche?au=Grothendieck.

[EGA] A. Grothendieck: Élements de Géométrie Algébrique, Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32, Bures-Sur-Yvette, 1960–1967; see also Grundlehren 166, Springer-Verlag, Berlin etc. 1971; also available at http://www.numdam.org/numdam-bin/recherche?au=Grothendieck.

[GL11] A. Genestier, V. Lafforgue: Théorie de Fontaine en égales charactéristiques, Ann. Sci. École Norm. Supér. 44 (2011), no. 2, 263–360; also available at http://www.math.jussieu.fr/∼vlafforg/.

[GL12] A. Genestier, V. Lafforgue: Structures de Hodge-Pink pour les φ/𝔖-modules de Breuil et Kisin, Compos. Math. 148 (2012), no. 3, 751–789; also available a http://www.math.jussieu.fr/∼vlafforg/.

[Har11] U. Hartl: Period Spaces for Hodge Structures in Equal Characteristic, Annals of Math. 173, n. 3 (2011), 1241–1358; also available as arXiv:math.NT/0511686.

[HV11] U. Hartl, E. Viehmann: The Newton stratification on deformations of local G-shtukas, J. reine angew. Math. (Crelle) 656 (2011), 87–129; also available as arXiv:0810.0821.

[Hel11] E. Hellmann: On families of weakly admissible filtered φ-modules and the adjoint quotient of GL_{d}, Documenta Math. 16 (2011), 969–991, also available as arXiv:1102.0119.

[Hel13] E. Hellmann: On arithmetic families of filtered φ-modules and crystalline representations, to appear in J. Inst. Math. Jussieu; also available as arXiv:1010.4577.

[Hub93] R. Huber: Continuous valuations, Math. Z. 212 (1993), 445–447.

[Hub94] R. Huber: A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), 513–551.

[Hub96] R. Huber: Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Math., E30, Friedr. Vieweg & Sohn, Braunschweig, 1996.

[Ked08] K. Kedlaya: Slope filtrations for relative Frobenius, in Représentations p-adiques I: représentations galoisiennes et (φ, Γ)-modules, Asterisque 319 (2008), 259–301; also available as arXiv:math/0609272.

[KL10] K. Kedlaya, R. Liu: On families of (φ, Γ)-modules, Algebra and Number Theory 4 (2010), no. 7, 943–967; also available as arXiv:0812.0112.

[Kis06] M. Kisin: Crystalline representations and F -crystals, in Algebraic geometry and number theory, pp. 459–496, Progress in Math. 253, Birkhäuser Boston, Boston MA, 2006; also available at www.math.harvard.edu/∼kisin.

[Kis08] M. Kisin: Potentially semi-stable deformation rings, J. of the American Math. Soc. 21 (2008) no. 2, 513–546; also available at www.math.harvard.edu/∼kisin.

[KPX12] K. Kedlaya, J. Pottharst, L. Xiao: Cohomology of arithmetic families of (φ, Γ)-modules, preprint 2012, available as arXiv:1203.5718

[LS97] Y. Laszlo, C. Sorger: The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 4, 499–525, also available as arXiv:alg-geom/9507002.

[LM00] G. Laumon, L. Moret-Bailly: Champs algébriques, Ergebnisse 39, Springer-Verlag, Berlin etc. 2000.

[Laz62] M. Lazard: Les zéros des fonctions analytiques d’une variable sur un corps valué complet, Inst. Hautes Études Sci. Publ. Math. 14 (1962), 47–75; also available at http://www.numdam.org/item?id=PMIHES 1962 14 47 0.

[Liu] R. Liu: Slope filtrations in families, to appear in J. Inst. Math. Jussieu; also available as arXiv:0809.0331.

[Mum99] D. Mumford: The Red book of Varieties and Schemes, second edition, Lecture Notes in Math. 1358, Springer-Verlag 1999.

[Pin97] R. Pink: Hodge Structures over Function Fields, Preprint 1997; available at http://www.math.ethz.ch/∼pinkri/.

[PR09] G. Pappas, M. Rapoport: Φ-modules and coefficient spaces, Moscow Math. J. 9 no. 3 (2009), 625–664, also available as arXiv:0811.1170.

[Ser79] J.-P. Serre: Local fields, GTM 67, Springer-Verlag, New York-Berlin, 1979.

[Var04] Y. Varshavsky: Moduli spaces of principal F -bundles, Selecta Math. (N.S.) 10 (2004), no. 1, 131–166, also available as arXiv:math/0205130.

Urs Hartl Universität Münster Mathematisches Institut Einsteinstr. 62 D – 48149 Münster Germany www.math.uni-muenster.de/u/urs.hartl/ Eugen Hellmann Universität Bonn Mathematisches Institut Endenicher Allee 60 D – 53115 Bonn Germany E-mail: hellmann@math.uni-bonn.de