On F-Crystalline Representations

Bryden Cais, Tong Liu^{1}

Received: May 5, 2015 Revised: November 26, 2015

Communicated by Don Blasius

Abstract. We extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, for a finite and totally ramified extension F/ℚ_{p}, and an arbitrary finite extension K/F, we construct a general class of infinite and totally wildly ramified extensions K_{∞}/K so that the functor V ↦ V |_{G_{K∞}} is fully-faithfull on the category of F - crystalline representations V. We also establish a new classification of F -Barsotti-Tate groups via Kisin modules of height 1 which allows more general lifts of Frobenius.

2010 Mathematics Subject Classification: Primary 14F30,14L05 Keywords and Phrases: F -crystalline representations, Kisin modules

1. Introduction

Let k be a perfect field of characteristic p with ring of Witt vectors W := W (k), write K_{0} := W [1/p] and let K/K_{0} be a finite and totally ramified extension. We fix an algebraic closure \overline{K} of K and set G_{K} := Gal(\overline{K}/K). The theory of Kisin modules and its variants, pioneered by Breuil in [Bre98] and later developed by Kisin [Kis06], provides a powerful set of tools for understanding Galois-stable ℤ_{p}-lattices in ℚ_{p}-valued semistable G_{K}-representations, and has been a key ingredient in many recent advances (e.g. [Kis08], [Kis09a], [Kis09b]). Throughout this theory, the non-Galois “Kummer” extension K_{∞}/K—obtained by adjoining to K a compatible system of choices {π_{n}}_{n≥1} of p^{n}-th roots of a uniformizer π_{0} in K—plays central role. The theory of Kisin modules closely parallels Berger’s work [Ber02], in which the cyclotomic extension of K replaces K_{∞}, and can be thought of as a “K_{∞}-analogue” of the theory of Wach modules developed by Wach [Wac96], Colmez [Col99] and Berger [Ber04]. Along these lines, Kisin and Ren [KR09] generalized the theory of Wach modules to allow the cyclotomic extension of K to be replaced by an arbitrary Lubin–Tate extension. This paper grew out of a desire to better understand the role of K_{∞} in the theories of Breuil and Kisin and related work, and is an attempt to realize Kisin modules and the modules of Wach and Kisin–Ren as “specializations” of a more general theory. To describe our main results, we first fix some notation. Let F ⊆ K be a subfield which is finite over ℚ_{p} with residue field k_{F} of cardinality q = p^{s}. Choose a power series

1The second author is partially supported by a Sloan fellowship and NSF grant DMS-1406926.

f(u) := a_{1}u + a_{2}u^{2} + ⋯ ∈ O_{F} [[u]]

with f(u) ≡ u^{q} mod 𝔪_{F} and a uniformizer π of K with monic minimal polynomial E := E(u) over F_{0} := K_{0} · F. We set π_{0} := π and we choose \underline{π} := \{π_{n}\}_{n≥1} with π_{n} ∈ \overline{K} satisfying f(π_{n}) = π_{n−1} for n ≥ 1. The extension K\underline{π} := ⋃_{n≥0} K(π_{n}) (called a Frobenius iterate extension in [CD15]) is an infinite and totally wildly ramified extension of K which need not be Galois. We set G\underline{π} := Gal(\overline{K}/K\underline{π}). Define 𝔖 := W [[u]] and put 𝔖_{F} = O_{F} ⊗_{W (k_{F})} 𝔖 = O_{F_{0}}[[u]]. We equip 𝔖_{F} with the (unique continuous) Frobenius endomorphism φ which acts on W (k) by the canonical q-power Witt-vector Frobenius, acts as the identity on O_{F}, and sends u to f(u). A Kisin module of E-height r is a finite free 𝔖_{F} -module 𝔐 endowed with φ-semilinear endomorphism φ_{𝔐} : 𝔐 → 𝔐 whose linearization 1 ⊗ φ : φ^{∗}𝔐 → 𝔐 has cokernel killed by E^{r}. When F = ℚ_{p} and f(u) = u^{p} (which we refer to as the classical situation in the following), Kisin’s theory [Kis06] attaches to any G_{K_{∞}}-stable ℤ_{p}-lattice T in a semistable G_{K}-representation V with Hodge–Tate weights in {0, …, r} a unique Kisin module 𝔐 of height r satisfying T ≃ T_{𝔖}(𝔐) (see §3.3 for the definition of T_{𝔖}). Using this association, Kisin proves that the restriction functor V → V |_{G_{K∞}} is fully faithful when restricted to the category of crystalline representations, and that the category of Barsotti–Tate groups over O_{K} is anti-equivalent to the category of Kisin modules of height 1. In this paper, we extend much of the framework of [Kis06] to allow general F and f(u), though for simplicity we will restrict ourselves to the case that q = p, or equivalently that F/ℚ_{p} is totally ramified, and that f(u) is a monic degree-p polynomial. When we extend our coefficients from ℚ_{p} to F, we must further restrict ourselves to studying F -crystalline representations, which are defined following ([KR09]): Let V be a finite dimensional F -vector space with continuous F -linear action of G_{K}. If V is crystalline (when viewed as a ℚ_{p}- representation) then D_{dR}(V) is naturally an F ⊗_{ℚ_{p}} K-module and one has a decomposition D_{dR}(V) = ∏_{𝔪} D_{dR}(V)_{𝔪}, with 𝔪 running over the maximal ideals of F ⊗_{ℚ_{p}} K. We say that V is F -crystalline if the induced filtration on D_{dR}(V)_{𝔪} is trivial unless 𝔪 corresponds to the canonical inclusion F ⊂ K.

On F -Crystalline Representations 225

Theorem 1.0.1. Let V be an F -crystalline representation with Hodge-Tate weights in \{0, …, r\} and T ⊂ V a G\underline{π}-stable O_{F} -lattice. Then there exists a Kisin module 𝔐 of E-height r satisfying T_{𝔖}(𝔐) ≃ T.

Writing v_{F} for the normalized valuation of \overline{K} with v_{F} (F) = ℤ, apart from the classical situation f(u) = u^{p} of Kisin, the above theorem is also known when v_{F} (a_{1}) = 1, which corresponds to the Lubin–Tate cases covered by the work of [KR09]. An important point of our formalism is that 𝔐 may in general not be unique for a fixed lattice T: our general construction produces as special cases the φ-modules over 𝔖_{F} which occur in the theory of Wach modules and its generalizations [KR09], so without the additional action of a Lubin–Tate group Γ, one indeed does not expect these Kisin modules to be uniquely determined; (cf. Example 3.3.7). This is of course quite different from the classical situation. Nonetheless, we prove the following version of Kisin’s “full-faithfulness” result. Writing Rep_{F}^{F -cris,r}(G) for the category of F -crystalline representations with Hodge-Tate weights in \{0, …, r\} and Rep_{F} (G\underline{π}) for the category of F -linear representations of G\underline{π}, we prove:

Theorem 1.0.2. Assume that φ^{n}(f(u)/u) is not a power of E for all n ≥ 0 and that v_{F} (a_{1}) > r, where f(u) = a_{1}u + a_{2}u^{2} + · · ·. Then the restriction functor Rep_{F}^{F -cris,r}(G) ⇝ Rep_{F} (G\underline{π}) induced by V ↦ V |_{G\underline{π}} is fully faithfull.

Although Beilinson and Tavares Ribeiro [BTR13] have given an almost elementary proof of Theorem 1.0.2 in the classical situation F = ℚ_{p} and f(u) = u^{p}, their argument relies crucially on an explicit description of the Galois closure of K_{∞}/K. For more general F and f, we have no idea what the Galois closure of K\underline{π}/K looks like, and describing it in any explicit way seems to be rather difficult in general. It is natural to ask when two different choices f and f′ of p-power Frobenius lifts and corresponding compatible sequences \underline{π} = \{π_{n}\}_{n} and \underline{π}′ = \{π′_{n}\} in \overline{K} yield the same subfield K\underline{π} = K\underline{π}′ of \overline{K}. We prove that this is rare in the following precise sense: if K\underline{π} = K\underline{π}′, then the lowest degree terms of f and f′ coincide, up to multiplication by a unit in O_{F}; see Proposition 3.1.3. It follows that there are infinitely many distinct K\underline{π} for which Theorem 1.0.2 applies. We also remark that any Frobenius–iterate extension K\underline{π} as above is an infinite and totally wildly ramified strictly APF extension in the sense of Wintenberger [Win83]. We therefore think of Theorem 1.0.2 as confirmation of the philosophy that “crystalline p-adic representations are the p-adic analogue of unramified ℓ-adic representations^{2},” since Theorem 1.0.2 is obvious if “crystalline” is replaced with “unramified” throughout (or equivalently in the special case r = 0). More generally, given F and r ≥ 0, it is natural to ask for a characterization of all infinite and totally wildly ramified strictly APF extensions L/K for which restriction of F -crystalline representations of G_{K} with Hodge–Tate weights in {0, …, r} to G_{L} is fully–faithful. We believe that there should be a deep and rather general phenomenon which deserves further study. While the condition that v_{F} (a_{1}) > r is really essential in Theorem 1.0.2 (see Example 4.5.9), we suspect the conclusion is still valid if we remove the assumption that φ^{n}(f(u)/u) is not a power of E for all n ≥ 0. However, we have only successfully removed this assumption when r = 1, thus generalizing Kisin’s classification of Barsotti–Tate groups:

2This philosophy is perhaps best evinced by the p-adic analogue of the good reduction theorem of Néron–Ogg–Shafarevich, which asserts that an abelian variety A over a p-adic field K has good reduction if and only if its p-adic Tate module V_{p}A is a crystalline representation of G_{K} [CI99, Theorem 1].

Theorem 1.0.3. Assume v_{F} (a_{1}) > 1. Then the category of Kisin modules of height 1 is equivalent to the category of F -Barsotti-Tate groups over O_{K}.

Here, an F -Barsotti-Tate group is a Bartotti–Tate group H over O_{K} with the property that the p-adic Tate module V_{p}(H) = ℚ_{p}⊗_{ℤ_{p}} T_{p}(H) is an F -crystalline representation. We note that when F = ℚ_{p}, Theorem 1.0.3 is proved (by different methods) in [CL14]. Besides providing a natural generalization of Kisin’s work and its variants as well as a deeper understanding of some of the finer properties of crystalline padic Galois representations, we expect that our theory will have applications to the study of potentially Barsotti–Tate representations. More precisely, suppose that T is a finite free O_{F} -linear representation of G_{K} with the property that T |_{G_{K′}} is Barsotti-Tate for some finite extension K′/K. If K′/K is not tamely ramified then it is well-known that it is in general difficult to construct “descent data” for the Kisin module 𝔐 associated to T |_{G_{K′}} in order to study T (see the involved computations in [BCDT01]). However, suppose that we can select f(u) and π_{0} such that K′ ⊆ K(π_{n}) for some n. Then, as in the theory of Kisin–Ren [KR09] (see also [BB10]), we expect the appropriate descent data on 𝔐 to be much easier to construct in this “adapted” situation, and we hope this idea can be used to study the reduction of T. Now let us sketch the ideas involved in proving the above theorems and outline the organization of this paper. For any ℤ_{p}-algebra A, we set A_{F} := A ⊗_{ℤ_{p}} O_{F}. In order to connect 𝔖_{F} to Galois representations, we must first embed 𝔖_{F} as a Frobenius-stable subring of W (R)_{F}, which we do in §2.1 following [CD15]. In the following subsection, we collect some useful properties of this embedding and study some “big rings” inside B_{cris,F}^{+} . Contrary to the classical situation, the Galois closure of K\underline{π} appears in general to be rather mysterious. Nonetheless, in §2.3 we are able to establish some basic results on the G_{K}-conjugates of u ∈ 𝔖_{F} ⊆ W (R)_{F} which are just barely sufficient for the development of our theory. Following Fontaine [Fon90], and making use of the main result of [CD15], in §3 we establish a classification of G\underline{π}-representations via étale φ-modules and Kisin modules. In the end of §3, we apply these considerations to prove that the functor T_{𝔖} is fully faithful under the assumption that φ^{n}(f(u)/u) is not a power of E for any n. The technical heart of this paper is §4. In §4.1, we define F -crystalline representations and attach to each F -crystalline representation V a filtered φ-module D_{cris,F} (V) (we warn the reader that the filtration of D_{cris,F} (V) is slightly different from that of D_{cris}(V)). Following [Kis06], in §4.2 we then associate to D = D_{cris}(V) a φ-module M(D) over 𝔒 (here we use 𝔒 for the analogue of O—the ring of rigid-analytic functions on the open unit disk—in Kisin’s work). A shortcoming in our situation is that we do not in general know how to define a reasonable differential operator N_{∇}, even at the level of the ring 𝔒. Consequently, our M(D) only has a Frobenius structure, in contrast to the classical (and Lubin–Tate) situation in which M(D) is also equipped with a natural N_{∇}-structure. Without such an N_{∇}-structure, there is no way to follow Kisin’s (or Berger’s) original strategy to prove that the scalar extension of M(D) to the Robba ring is pure of slope zero, which is key to showing that there exists a Kisin module 𝔐 such that 𝔒 ⊗_{𝔖_{F}} 𝔐 ≃ M(D). We bypass this difficulty by appealing to the fact that M(D) is known to be pure of slope zero in the classical situation of Kisin as follows: letting a superscript of “c” denote the data in the classical situation and using the fact that both M(D) and M^{c}(D) come from the same D, we prove that B̃_{α} ⊗_{𝔒} M(D) ≃ B̃_{α} ⊗_{𝔒^{c}} M^{c}(D) as φmodules for a certain period ring B̃_{α} that contains the ring B̃_{rig,F}^{+} . It turns out that this isomorphism can be descended to B̃_{rig,F}^{+} . Since Kedlaya’s theory of the slope filtration is unaffected by base change from the Robba ring to B̃_{rig,F}^{+} , it follows that M(D) is of pure slope 0 as this is the case for M^{c}(D) thanks to [Kis06]. With this crucial fact in established, we are then able to prove Theorem 1.0.1 along the same lines as [Kis06]. If our modules came equipped with a natural N_{∇}-structure, the full faithfulness of the functor V ↦ V |_{G\underline{π}} would follow easily from the full faithfulness of T_{𝔖}. But without such a structure, we must instead rely heavily on the existence of a unique φ-equivariant section ξ : D(𝔐) → 𝔒_{α} ⊗ φ^{∗}𝔐 to the projection φ^{∗}𝔐 ↠ φ^{∗}𝔐/uφ^{∗}𝔐, where D(𝔐) = (φ^{∗}𝔐/uφ^{∗}𝔐)[1/p]. The hypothesis v_{F} (a_{1}) > r of Theorem 1.0.2 guarantees the existence and uniqueness of such a section ξ. With these preparations, we finally prove Theorem 1.0.2 in §4.5. In §5, we establish Theorem 1.0.3: the equivalence between the category of Kisin modules of height 1 and the category of F -Barsotti-Tate groups over O_{K}. Here we adapt the ideas of [Liu13b] to prove that the functor 𝔐 ↦ T_{𝔖}(𝔐) is an equivalence between the category of Kisin module of height 1 and the category of G_{K}-stable O_{F} -lattices in F -crystalline representations with Hodge-Tate weights in \{0, 1\}. The key difficulty is to extend the G\underline{π}-action on T_{𝔖}(𝔐) to a G_{K}-action which gives T_{𝔖}(𝔐)[1/p] the structure of an F -crystalline representation. In the classical situation, this is done using the (unique) monodromy operator N on S ⊗_{𝔖} φ^{∗}𝔐 (see §2.2 in [Liu13b]). Here again, we are able to sidestep the existence of a monodromy operator to construct a (unique) G_{K}- action on W (R)_{F} ⊗_{𝔖_{F}} 𝔐 which is compatible with the additional structures (see Lemma 5.1.1), and this is enough for us to extend the given G\underline{π}-action to a G_{K}-action on T_{𝔖}(𝔐). As this paper establishes analogues of many of the results of [Kis06] in our more general context, it is natural ask to what extent the entire theory of [Kis06] can be developed in this setting. To that end, we list several interesting (some quite promising) questions for this program in the last section.

Acknowledgements: It is pleasure to thank Laurent Berger, Kiran Kedlaya and Ruochuan Liu for very helpful conversations and correspondence.

Notation. Throughout this paper, we reserve φ for the Frobenius operator, adding appropriate subscripts as needed for clarity: for example, φ_{𝔐} denotes the Frobenius map on 𝔐. We will always drop these subscripts when there is no danger of confusion. Let S be a ring endowed with a Frobenius lift φ_{S} and M an S-module. We always write φ^{∗}M := S ⊗_{φ_{S},S} M. Note that if φ_{M} : M → M is a φ_{S}-semilinear endomorphism, then 1⊗φ_{M} : φ^{∗}M → M is an S-linear map. We reserve f(u) = u^{p} + a_{p−1}u + ⋯ + a_{1}u ≡ u^{p} mod 𝔪_{F} for the polynomial over O_{F} giving our Frobenius lift φ(u) := f(u) as in the introduction. For any discretely valued subfield E ⊆ \overline{K}, we write v_{E} for the normalized p-adic valuation of \overline{K} with v_{E}(E) = ℤ, and for convenience will simply write v := v_{ℚ_{p}}. If A is a ℤ_{p}-module, we set A_{F} := A ⊗_{ℤ_{p}} O_{F} and A[1/p] := A ⊗_{ℤ_{p}} ℚ_{p}. For simplicity, we put G = G_{K} := Gal(\overline{K}/K) and G\underline{π} := Gal(\overline{K}/K\underline{π}). We write M_{d}(S) for the ring of d×d-matrices over S and I_{d} for the d×d-identity matrix.

2. Period rings

In this section, we introduce and study the various “period rings” which will play a central role in the development of our theory. As in the introduction, we fix a perfect field k of characteristic p with ring of Witt vectors W := W (k), as well as a finite and totally ramified extension K of K_{0} := W [1/p]. Let F be a subfield of K, which is finite and totally ramified over ℚ_{p}, and put F_{0} := K_{0}F ⊂ K. Choose uniformizers π of O_{K} and ϖ of O_{F}, and let E := E(u) ∈ O_{F_{0}}[u] be the monic minimal polynomial of π over F_{0}. We set e := [K : K_{0}], and put e_{0} := [K : F_{0}] and e_{F} := [F : ℚ_{p}]. Fix a polynomial f(u) = u^{p} + a_{p−1}u^{p−1} + ⋯ + a_{1}u ∈ O_{F} [u] satisfying f(u) ≡ u^{p} mod ϖ, and recursively choose π_{n} ∈ \overline{K} with f(π_{n}) = π_{n−1} for n ≥ 1 where π_{0} := π. Set K\underline{π} := ⋃_{n≥0} K(π_{n}) and G\underline{π} := Gal(\overline{K}/K\underline{π}), and recall that for convenience we write G = G_{K} := Gal(\overline{K}/K). Recall that 𝔖 = W [[u]], and that we equip the scalar extension 𝔖_{F} with the semilinear Frobenius endomorphism φ : 𝔖_{F} → 𝔖_{F} which acts on W as the unique lift of the p-power Frobenius map on k, acts trivially on O_{F}, and sends u to f(u). The first step in our classification of F -crystalline G_{K}-representations by Kisin modules over 𝔖_{F} is to realize this ring as a Frobenius stable subring of W (R)_{F}, which we do in the following subsection.

2.1. 𝔖_{F} as a subring of W (R)_{F}. As usual, we put R := \underleftarrow{lim}_{x→x^{p}} O\overline{K}/(p),

equipped with its natural coordinate-wise action of G. It is well-known that the natural reduction map

\underleftarrow{lim}_{x→x^{p}} O\overline{K}/(p) → \underleftarrow{lim}_{x→x^{p}} O\overline{K}/(ϖ)

is an isomorphism, so \{π_{n} mod ϖ\}_{n≥0} defines an element \underline{π} ∈ R. Furthermore, writing ℂ_{K} for the completion of \overline{K}, reduction modulo p yields a multiplicative bijection \underleftarrow{lim}_{x→x^{p}} O_{ℂ_{K}} ≃ R, and for any x ∈ R we write (x^{(n)})_{n≥0} for the p-power compatible sequence in \underleftarrow{lim}_{x→x^{p}} O_{ℂ_{K}} corresponding to x under this identification. We write [x] ∈ W (R) for the Techmüller lift of x ∈ R, and denote by θ : W (R) → O_{ℂ_{K}} the unique lift of the projection R ↠ O_{ℂ_{K}} /(p) which sends ∑_{n} p^{n}[x_{n}] to ∑_{n} p^{n}x^{(0)}. By definition, B_{dR}^{+} is the Ker(θ)-adic completion of W (R)[1/p], so θ naturally extends to B_{dR}^{+}. For any subring B ⊂ B_{dR}^{+}, we define Fil^{i}B := (Kerθ)^{i} ∩ B. There is a canonical section \overline{K} ↪ B_{dR}^{+}, so we may view F as a subring of B_{dR}^{+}, and in this way we obtain embeddings W (R)_{F} ↪ B_{cris,F}^{+} ↪ B_{dR}^{+}. Define θ_{F} := θ|_{W (R)_{F}}. One checks that W (R)_{F} is ϖ-adically complete and that every element of W (R)_{F} has the form ∑_{n≥0}[a_{n}]ϖ^{n} with a_{n} ∈ R. The map θ_{F} carries

∑_{n≥0}[a_{n}]ϖ^{n} to ∑_{n≥0} a_{n}^{(0)}ϖ^{n} ∈ O_{ℂ_{K}} (see Def. 3.8 and Prop. 3.9 of [CD15]).

Lemma 2.1.1. There is a unique set-theoretic section {·}_{f} : R → W (R)_{F} to the reduction modulo ϖ map which satisfies φ({x}_{f}) = f({x}_{f}) for all x ∈ R.

Proof. This is^{3} [Col02, Lemme 9.3]. Using the fact that f(u) ≡ u^{p} mod ϖ, one checks that the endomorphism f ∘ φ^{−1} of W (R)_{F} is a ϖ-adic contraction, so that for any lift x̃ ∈ W (R) of x ∈ R, the limit

{x}_{f} := lim_{n→∞}(f ∘ φ^{−1})^{(n)}(x̃)

exists in W (R)_{F} and is the unique fixed point of f ∘ φ^{−1}, which uniquely characterizes it independent of our choice of x̃. □

From Lemma 2.1.1 we obtain a unique continuous embedding ι : 𝔖_{F} ↪ W (R)_{F} of O_{F} -algebras with ι(u) := \{\underline{π}\}_{f}. Via ι, we henceforth identify 𝔖_{F} with a φstable O_{F} -subalgebra of W (R)_{F} on which we have φ(u) = f(u).

Example 2.1.2 (Cyclotomic case). Let {ζ_{p^{n}}}_{n≥0} be a compatible system of primitive p^{n}-th roots of unity. Let K = ℚ_{p}(ζ_{p}) with π = ζ_{p}−1 and take F = ℚ_{p} with f(u) = (u + 1)^{p} − 1. Choosing π_{n} = ζ_{p^{n+1}} − 1, we obtain K\underline{π} := ℚ_{p}(μ_{p^{∞}}). It is obvious that \underline{ϵ}_{1} := (ζ_{p^{n}})_{n≥1} ∈ R. In this case, ι(u) = [\underline{ϵ}_{1}] − 1 ∈ W (R).

Recall that R has the structure of a valuation ring via v_{R}(x) := v(x^{(0)}), where v is the normalized p-adic valuation of ℂ_{K} with v(ℤ_{p}) = ℤ.

Lemma 2.1.3. We have θ_{F} (u) = π and E generates Ker(θ_{F}) = Fil^{1}W (R)_{F}.

3In the version of Colmez’s article available from his website, it is Lemme 8.3.

Proof. The first assertion is [Col02, Lemme 9.3]. To compute θ_{F} (\{\underline{π}\}_{f}), we first choose [\underline{π}] as our lift of \underline{π} to W (R), and compute

θ_{F} (\{\underline{π}\}_{f}) = θ_{F} ( lim_{n→∞} f^{(n)}φ^{(−n)}([\underline{π}])) = lim_{n→∞} f^{(n)}θ_{F} ([\underline{π}^{p^{−n}}]) = lim_{n→∞} f^{(n)}(\underline{π}^{(n)})

But \underline{π}^{(n)} ≡ π_{n} mod ϖ, so

f^{(n)}(\underline{π}^{(n)}) ≡ f^{(n)}(π_{n}) ≡ π mod ϖ^{n+1},

which gives the claim. Now certainly θ_{F} (E(u)) = E(π) = 0, so E = E(u) lies in Fil^{1}W (R)_{F}. Since E ≡ \underline{π}^{e_{0}} mod ϖ, we conclude that

v_{R}(E mod ϖ) = e_{0}v_{R}(\underline{π}) = e_{0}v(π) = v(ϖ),

whence E generates Ker(θ_{F}) = Fil^{1}W (R)_{F} thanks to [Col02, Prop. 8.3]. □

Now let us recall the construction of B_{max}^{+} and B̃_{rig}^{+} from Berger’s paper [Ber02]. Let ξ be a generator of Fil^{1}W (R). By definition,

B_{max}^{+} := \{∑_{n≥0} a_{n} \frac{ξ^{n}}{p^{n}} ∈ B_{dR}^{+} | a_{n} ∈ W (R)[1/p] and lim_{n→∞} a_{n} = 0\} .

and B̃_{rig}^{+} := ⋂_{n≥1} φ^{n}(B_{max}^{+}). Write 𝔲 := [\underline{π}]. The discussion before Proposition 8.14 in [Col02] shows:

Lemma 2.1.4.

B_{max,F}^{+} =  \{∑_{n≥0} a_{n} \frac{E^{n}}{ϖ^{n}} ∈ B_{dR}^{+} | a_{n} ∈ W (R)_{F} [1/p] and lim_{n→∞} a_{n} = 0\}

= \{∑_{n≥0} a_{n} \frac{𝔲^{e_{0}n}}{ϖ^{n}} ∈ B_{dR}^{+} | a_{n} ∈ W (R)_{F} [1/p] and lim_{n→∞} a_{n} = 0\} .

We can now prove the following result, which will be important in §4.4:

Lemma 2.1.5. Let x ∈ B_{max,F}^{+} , and suppose that xE^{r} = φ^{m}(y) holds for some y ∈ B_{max,F}^{+} . Then x = φ^{m}(y′) with y′ ∈ B_{max,F}^{+} .

Proof. By Lemma 2.1.4, we may write y = ∑_{n} b_{n} \frac{𝔲^{e_{0}n}}{ϖ^{n}} with b_{n} ∈ W (R)_{F} [1/p] converging to 0. Write E = E(u) = 𝔲^{e_{0}} + ϖz with z ∈ W (R)_{F}. We then have

φ^{m}(y) = ∑_{n=0}^{∞} φ^{m}(b_{n})\frac{𝔲^{e_{0}p^{m}n}}{ϖ^{n}} = ∑_{n=0}^{∞} φ^{m}(b_{n})\frac{(E − ϖz)^{p^{m}n}}{ϖ^{n}} = ∑_{n=0}^{∞} c_{n} \frac{E^{p^{m}n}}{ϖ^{n}}

with c_{n} ∈ W (R)_{F} [1/p] converging to 0. By Lemma 2.1.3, E is a generator of

Fil^{1}W (R)_{F}, so definining s := 1+max\{n | p^{m}n < r\}, it follows that ∑_{n=0}^{s−1} c_{n} \frac{E^{p^{m}n}}{ϖ^{n}}

On F -Crystalline Representations 231

is divisible by E^{r} in W (R)_{F} [1/p] so we may write ∑_{n=0}^{s−1} c_{n} \frac{E^{p^{m}n}}{ϖ^{n}} = E^{r}x_{0} for some x_{0} ∈ W (R)[1/p]. Without loss of generality, replacing x by x − x_{0}, gives

x = ∑_{n=s}^{∞} c_{n} \frac{E^{p^{m}n−r}}{ϖ^{n}} = ∑_{n=s}^{∞} d_{n−s} \frac{E^{p^{m}(n−s)}}{ϖ^{n−s}} = ∑_{n=0}^{∞} d_{n} \frac{E^{p^{m}n}}{ϖ^{n}}

with d_{n−s} = c_{n} \frac{E^{p^{m}s−r}}{ϖ^{s}} . Using again the equality E = 𝔲^{e_{0}} + ϖz, we then obtain x = ∑_{n=0}^{∞} e_{n} \frac{𝔲^{e_{0}p^{m}n}}{ϖ^{n}} with e_{n} ∈ W (R)_{F} [1/p] converging to 0. We now have

x = φ^{m}(y′) for y′ := ∑_{n=0}^{∞} f_{n} \frac{𝔲^{e_{0}n}}{ϖ^{n}} with f_{n} = φ^{−m}(e_{n}). As f_{n} ∈ W (R)_{F} [1/p]

converges to 0, we conclude that y′ ∈ B_{max,F}^{+} , as desired. □

2.2. Some subrings of B_{cris,F}^{+} . For a subinterval I ⊂ [0, 1), we write 𝔒_{I} for the subring of F_{0}((u)) consisting of those Laurent series which converge for those x ∈ ℂ_{K} with |x| ∈ I, and we will simply write 𝔒 = 𝔒_{[0,1)}. Let B̃_{α} := W (R)_{F} [[\frac{E^{p}}{ϖ}]][1/p] ⊂ B_{cris,F}^{+} . We claim that Fil^{n}B̃_{α} = E^{n}B̃_{α}. To see this, set c = ⌈\frac{n}{p}⌉ and n = pc − s with 0 ≤ s < p. For any x ∈ Fil^{n}B̃_{α}, we write x = ∑_{i=0}^{∞} a_{i} \frac{E^{pi}}{ϖ^{i}} with a_{i} ∈ W (R)_{F} [1/p] converging to 0 in W (R)_{F} [1/p]. Since

x ∈ Fil^{n}B_{dR}^{+}, ∑_{i=0}^{c−1} a_{i} \frac{E^{pi}}{ϖ^{i}} = E^{n}x_{0} with x_{0} ∈ W (R)_{F} [1/p]. It suffices to show

that x − x_{0} = E^{n}y with y ∈ B̃_{α}. Now

y = ∑_{i≥c}^{∞} a_{i} \frac{E^{pi−n}}{ϖ^{i}} = ∑_{i≥c}^{∞} a_{i}E^{s}ϖ^{−c} (\frac{E^{p(i−c)}}{ϖ^{i−c}} ) ∈ B_{dR}^{+}.

As a_{i} converges to 0 in W (R)_{F} [1/p], so does a_{i}E^{s}ϖ^{−c}, whence y lies in B̃_{α}. Lemma 2.2.1. There are canonical inclusions of rings 𝔒 ⊂ B̃_{rig,F}^{+} ⊂ B̃_{α}.

Proof. We first show that 𝔒 ⊂ B̃_{rig,F}^{+} . For any h(u) = ∑_{n=0}^{∞} a_{n}u^{n} ∈ 𝔒, we have to show that h_{m}(u) = ∑_{n=0}^{∞} φ^{−m}(a_{n}u^{n}) is in B_{max,F}^{+} for all m ≥ 0. Writing u = 𝔲 + ϖz with 𝔲 = [\underline{π}] and z ∈ W (R)_{F}, we have φ^{−m}(u) = 𝔲^{p^{−m}} + ϖz^{(m)} with z^{(m)} = φ^{−m}(z) ∈ W (R)_{F}. Setting a_{n}^{(m)} := φ^{−m}(a_{n}) ∈ F_{0}, we then have

h_{m}(u) = h(𝔲^{\frac{1}{p^{m}}} + ϖz^{(m)}) = ∑_{k=0}^{∞} \frac{h^{(k)}(𝔲^{\frac{1}{p^{m}}})}{k!}(ϖz^{(m)})^{k},

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for h^{(k)} the k-th derivative of h̃(X) := ∑_{n=0}^{∞} a_{n}^{(m)}X^{n}. Therefore,

h_{m}(u) = ∑_{n=0}^{∞} (∑_{k=0}^{∞} \binom{k + n}{k}a_{n+k}^{(m)}(ϖz^{(m)})^{k}) 𝔲^{\frac{n}{p^{m}}} .

Since h(u) ∈ 𝔒_{[0,1)}, we have lim_{n→∞} |a_{n}^{(m)}|r^{n} = 0 for any r < 1. It follows that

the inner sum ∑_{k=0}^{∞} \binom{k+n}{k}a_{n+k}^{(m)}(ϖz^{(m)})^{k} converges to b_{n} ∈ W (R)_{F} [1/p]. Since

lim_{n→∞} |a_{n}^{(m)}|r^{n} = 0 for r = |ϖ^{\frac{1}{e_{0}p^{m}}} | ≥ |ϖ|, for any ϵ > 0, there exists N so that

|a_{n+k}^{(m)}ϖ^{\frac{n}{e_{0}p^{m}}} ϖ^{k}| < ϵ for any n > N and k ≥ 0. This implies that b_{n}ϖ^{\frac{n}{e_{0}p^{m}}} converges to 0 in W (R)_{F}. We may therefore write

h_{m}(u) = ∑_{n=0}^{∞} b_{n}𝔲^{\frac{n}{p^{m}}} = ∑_{n=0}^{∞} b_{n}ϖ^{\frac{n}{e_{0}p^{m}}} \frac{(𝔲^{e_{0}})^{\frac{n}{e_{0}p^{m}}}}{ϖ^{\frac{n}{e_{0}p^{m}}}} ,

and Lemma 2.1.4 implies that h_{m}(u) ∈ B_{max,F}^{+} , so 𝔒 ⊂ B̃_{rig,F}^{+} as desired.

To show that B̃_{rig,F}^{+} ⊂ B̃_{α}, we first observe that

(2.2.1)  B̃_{α} = W (R)_{F} [[\frac{u^{e_{0}p}}{ϖ}]][1/p] = W (R)_{F} [[\frac{𝔲^{e_{0}p}}{ϖ}]][1/p].

For any x ∈ B̃_{rig,F}^{+} , we may write x = φ(y) with y = ∑_{n=0}^{∞} a_{n} \frac{𝔲^{e_{0}n}}{ϖ^{n}} ∈ B_{max,F}^{+} , and

we see that x = ∑_{n=0}^{∞} φ(a_{n})\frac{𝔲^{e_{0}pn}}{ϖ^{n}} indeed lies in B̃_{α} by (2.2.1). □

Finally let us record the following technical lemma: recall that our Frobenius lift on 𝔖_{F} is determined by φ(u) := f(u), with f(u) = u^{p}+a_{p−1}u^{p−1}+⋯+a_{1}u. We define 𝔒_{α} := 𝔖_{F} [[\frac{u^{e_{0}p}}{ϖ}]][1/p] ⊂ B̃_{α}.

Lemma 2.2.2. Suppose that ϖ^{r+1}|a_{1} in O_{F}. Then there exists h_{i}^{(n)}(u) ∈ O_{F} [u] such that

f^{(n)}(u) = ∑_{i=0}^{n} h_{n−i}^{(n)}(u)u^{2^{n−i}}ϖ^{(r+1)i}.

In particular, φ^{n}(u)/ϖ^{rn} converges to 0 in 𝔒_{α}.

Proof. We proceed by induction on m = n. When m = 1, we may write

(2.2.2) f(u) = u^{p} +a_{p−1}u^{p_{1}} +⋯+a_{1}u = u^{2}h(u)+b_{0}ϖ^{r+1}u with b_{0} ∈ O_{F} .

Supposing that the assertion holds for m = n and using (2.2.2) we compute

f^{(n+1)}(u) = ∑_{i=0}^{n} h_{n−i}^{(n)}(f(u))f(u)^{2^{n−i}}ϖ^{(r+1)i}

= ∑_{i=0}^{n} h_{n−i}^{(n)}(f(u))(u^{2}h(u) + b_{0}ϖ^{r+1}u)^{2^{n−i}}ϖ^{(r+1)i}

= ∑_{i=0}^{n} h_{n−i}^{(n)}(f(u)) (∑_{k=0}^{2^{n−i}} \binom{2^{n−i}}{k}(u^{2}h(u))^{2^{n−i}−k}(b_{0}ϖ^{r+1}u)^{k}) ϖ^{(r+1)i}

= ∑_{i=0}^{n} ∑_{k=0}^{2^{n−i}} (h_{n−i}^{(n)}(f(u))\binom{2^{n−i}}{k}h(u)^{2^{n−i}−k}b_{0}^{k}) u^{2^{n+1−i}−k}ϖ^{(r+1)(i+k)}

To complete the inductive step, it therefore suffices to show that whenever i + k ≤ n + 1, we have 2^{n+1−i} − k ≥ 2^{n+1−i−k}. Equivalently, and writing j := n + 1 − i − k, we must show that 2^{j+k} − k ≥ 2^{j} for all j ≥ 0, which holds as 2^{k} ≥ k + 1 for all k ≥ 0. □

2.3. The action of G on u. In this subsection, we study the action of G on the element u ∈ W (R)_{F} corresponding to our choice of f-compatible sequence \{π_{n}\}_{n} in \overline{K} and our Frobenius lift determined by f. From the very construction of the embedding 𝔖_{F} ↪ W (R)_{F} in Lemma 2.1.1, the action of G\underline{π} on u is trivial. However, for arbitrary g ∈ G \ G\underline{π}, in contrast to the classical case, we know almost nothing about the shape of g(u); cf. the discussion in §3.1. Fortunately, we are nonetheless able to prove the following facts, which are sufficient for our applications. Define

I_{F}^{[1]} := {x ∈ W (R)_{F} | φ^{n}(x) ∈ Fil^{1}W (R)_{F} , ∀n ≥ 0}.

Recall that e_{F} := [F : ℚ_{p}], and for x ∈ W (R)_{F} write x̄ := x mod ϖ ∈ R. Thanks to Example 3.3.2, there exists 𝔱_{F} ∈ W (R)_{F} satisfying φ(𝔱_{F}) = E𝔱_{F}. As E ∈ Fil^{1}W (R)_{F}, it is easy to see that φ(𝔱_{F}) ∈ I^{[1]}W (R)_{F}, and since 𝔱̄_{F}^{p} = u^{e_{0}}𝔱̄_{F}, we have v_{R}(\overline{φ(𝔱_{F})}) = \frac{p}{e_{F} (p−1)}.

Lemma 2.3.1. The ideal I_{F}^{[1]} is principal. Moreover, x ∈ I_{F}^{[1]} is a generator of I_{F}^{[1]} if and only if v_{R}(x̄) = \frac{p}{e_{F} (p−1)}.

Proof. When F = ℚ_{p}, this follows immediately from [Fon94a, Proposition 5.1.3] with r = 1. The general case follows from a slight modification of this argument, as follows: For y ∈ I_{F}^{[1]}, we first claim that v_{R}(ȳ) ≥ \frac{p}{e_{F} (p−1)}. To

see this, we write y = ∑_{n=0}^{∞} ϖ^{n}[y_{i}] with y_{i} ∈ R given by the p-power compatible sequence y_{i} = (α_{i}^{(n)})_{n≥0} for α_{i}^{(n)} ∈ O_{ℂ_{K}}. Then

0 = θ_{F} (φ^{m}(y)) = ∑_{n=0}^{∞} ϖ^{n}(α_{i}^{(0)})^{p^{m}}.

By induction on n and m, it is not difficult to show that

v(α_{i}^{(0)}) ≥ \frac{1}{e_{F}} p^{−i}(1 + p^{−1} ⋯ + p^{−j})

for all j ≥ 0. In particular, v_{R}(ȳ) = v(α_{0}^{(0)}) ≥ \frac{p}{e_{F} (p−1)}.

Now pick a x ∈ I_{F}^{[1]} with v_{R}(x̄) = \frac{p}{e_{F} (p−1)} (take, for example, x = φ(𝔱_{F})). Since v_{R}(y) ≥ v_{R}(x), we may write y = ax + ϖz with a, z ∈ W (R)_{F}. One checks that z ∈ I_{F}^{[1]} and hence that z ∈ (ϖ, x). An easy induction argument then

shows that y = ∑_{n=0}^{∞} ϖ^{n}a_{n}x, and it follows that I_{F}^{[1]} is generated by x. □

It follows at once from Lemma 2.3.1 that φ(𝔱_{F}) is a generator of I_{F}^{[1]}. Write I^{+} for the kernel of the canonical projection ρ : W (R)_{F} → W (k̄)_{F} induced by the projection R → k̄. Using the very construction of u, one checks that u ∈ I^{+}: Indeed, writing 𝔲 = [\underline{π}] as before, we obviously have 𝔲 ∈ I^{+}, and it follows from the proof of Lemma 2.1.1 that u = lim_{n→∞} f^{(n)} ∘ φ^{−n}(𝔲) lies in I^{+} as well.

Lemma 2.3.2. Let g ∈ G be arbitrary. Then g(u) − u lies in I^{[1]}W (R)_{F}. Moreover, if ϖ^{2}|a_{1} in O_{F} then \frac{g(u)−u}{φ(𝔱_{F})} lies in I^{+}.

Proof. As before, writing f^{(n)} = f ∘ ⋯ ∘ f for the n-fold composition of f with itself, we have θ_{F} (φ^{n}(u)) = f^{(n)}(π) ∈ K, from which it follows that g(u) − u is in I_{F}^{[1]}. By Lemma 2.3.1, we conclude that z := \frac{g(u)−u}{φ(𝔱_{F})} lies in W (R)_{F}. It remains to show that z ∈ I^{+} when ϖ^{2}|a_{1}. We first observe that

φ(z) = \frac{f((g(u)) − f(u)}{φ^{2}(𝔱_{F})} = \frac{∑_{i=1}^{p} a_{i} ((g(u))^{i} − u^{i})}{φ(E)φ(𝔱_{F})}.

For each i, we may write (g(u))^{i}−u^{i} = (g(u)−u)h_{i}(g(u), u) = φ(𝔱_{F})zh_{i}(g(u), u) for some bivariate degree i − 1 homogeneous polynomials h_{i} with coefficients in W (R)_{F}. We therefore have

(2.3.1)  φ(E)φ(z) = ∑_{i=1}^{p} a_{i} (zh_{i}(g(u), u)) .

Reducing modulo I^{+} and noting that both u and g(u) lie in I^{+}, we conclude from (2.3.1) that ϖφ(ρ(z)) = a_{1}ρ(z), where ρ : W (R)_{F} → W (k̄)_{F} is the natural projection as above. Using the fact that v(φ(ρ(z))) = v(ρ(z)), our assumption that v(a_{1}) > v(ϖ) then implies that ρ(z) = 0. That is, z ∈ I^{+} as desired. □

Example 2.3.3. The following example shows that the condition ϖ^{2}|a_{1} in O_{F} is genuinely necessary for the conclusion of Lemma 2.3.2 to hold. Recall the situation of Example 2.1.2, with K = ℚ_{p}(ζ_{p}), π = ζ_{p} − 1, f(u) = (u + 1)^{p} − 1 and u = [\underline{ϵ}_{1}] − 1, where \underline{ϵ}_{1} = (ζ_{p^{n}})_{n≥1} ∈ R. We may choose g ∈ G with g(\underline{ϵ}_{1}) = \underline{ϵ}_{1}^{1+p}. We then have g(u) − u = [\underline{ϵ}_{1}]([\underline{ϵ}_{1}]^{p} − 1). Now it is well-known that [\underline{ϵ}_{1}]^{p} − 1 is a generator of I_{ℚ_{p}}^{[1]} (or one can appeal to Lemma 2.3.1). Then z = (g(u) − u)/φ(𝔱_{F}) is a unit in W (R) and does not lie in I^{+}.

We conclude this discussion with the following lemma, needed in §5.1:

Lemma 2.3.4. The ideal 𝔱_{F} I^{+} ⊂ W (R)_{F} is stable under the canonical action of G: that is, g(𝔱_{F} I^{+}) ⊂ 𝔱_{F} I^{+} for all g ∈ G.

Proof. It is clear that I^{+} is G-stable, so it suffices to show that g(𝔱_{F}) = x𝔱_{F} for some x ∈ W (R)_{F}. Since φ(𝔱_{F}) is a generator of I^{[1]}, which is obviously G-stable from the definition, we see that g(φ(𝔱_{F})) = yφ(𝔱_{F}) with y ∈ W (R)_{F}. Hence g(𝔱_{F}) = φ^{−1}(y)𝔱_{F}. □

3. Étale φ-modules and Kisin modules

In this section, following Fontaine, we establish a classification of G\underline{π}- representations by étale φ-modules and Kisin modules. To do this, we must first show that K\underline{π}/K is strictly Arithmetically Profinite, or APF, in the sense of Fontaine–Wintenberger [Win83], so that the theory of norm fields applies.

3.1. Arithmetic of f-iterate extensions. We keep the notation and conventions of §2. Recall that our choice of an f-compatible sequence {π_{n}}_{n} (in the sense that f(π_{n}) = π_{n−1} with π_{0} = π a uniformizer of K) determines an element \underline{π} := \{π_{n} mod ϖ\}_{n} of R. It also determines an infinite, totally wildly ramified extension K\underline{π} := ∪_{n≥1}K(π_{n}) of K, and we write G\underline{π} = Gal(\overline{K}/K\underline{π}).

Lemma 3.1.1. The extension K\underline{π}/K is strictly APF in the sense of [Win83]; in particular, the associated norm field E_{K\underline{π}/K} is canonically identified with the subfield k((\underline{π})) of Fr(R).

Proof. That K\underline{π}/K is strictly APF follows immediately from [CD15], which handles a more general situation. In the present setting with f(u) ≡ u^{p} mod ϖ, we can give a short proof as follows. As before, let us write

f(u) = a_{1}u + a_{2}u^{2} + ⋯ + a_{p−1}u^{p−1} + a_{p}u^{p},

with a_{i} ∈ ϖO_{F} for 1 ≤ i ≤ p−1 and a_{p} := 1. For each n ≥ 1, set f_{n} := f −π_{n−1} and put K_{n} := K(π_{n−1}). We compute the “ramification polynomial”

g_{n} := \frac{f_{n}(π_{n}u + π_{n})}{u} = ∑_{i=0}^{p−1} b_{i}u^{i},

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with coefficients b_{i} given by

b_{i} = ∑_{j=i+1}^{p} a_{j}π_{n}^{j}\binom{j}{i + 1}  for  0 ≤ i ≤ p − 1.

For ease of notation, put v_{n} := v_{K_{n+1}}, and denote by e_{n} := v_{n}(ϖ) the ramification index of K_{n+1}/F and by e := v_{F} (p) the absolute ramification index of F. Since K_{n+1}/K_{n} is totally ramified of degree p, we have e_{n} = p^{n}e_{0}; in particular, v_{n}(a_{j}\binom{j}{i+1}π_{n}^{j}) ≡ j mod p^{n}. It follows that v_{n}(b_{p−1}) = p, and for 0 ≤ i ≤ p − 2 we have

v_{n}(b_{i}) = min{e_{n}e + p, e_{n}v_{F} (a_{j}) + j : i + 1 ≤ j ≤ p − 1}.

It is easy to see that for n ≥ 1 the lower convex hull of these points is the straight line with endpoints (0, v_{n}(b_{0})) and (p − 1, p). In other words, defining

(3.1.1)  i_{min} := min_{i}{i : ord_{ϖ}(a_{i}) ≤ e, 1 ≤ i ≤ p}.

the Newton polygon of g_{n} is a single line segment with slope the negative of

(3.1.2)  i_{n} := \frac{e_{n} (v_{F} (a_{i_{min}}) + ⌊i_{min}/p⌋e) + i_{min} − p}{p − 1}.

In particular, for n ≥ 1 the extension K_{n+1}/K_{n} is elementary of level i_{n} in the sense of [Win83, 1.3.1]; concretely, this condition means that

(3.1.3)  v_{n}(π_{n} − σπ_{n}) = i_{n} + 1

for every K_{n}-embedding σ : K_{n+1} ↪ \overline{K}. It follows from this and [Win83, 1.4.2] that K\underline{π}/K is APF. Now let c(K\underline{π}/K) be the constant defined in [Win83, 1.2.1]. Then by [Win83, §1.4]

c(K\underline{π}/K) = inf_{n>0} \frac{i_{n}}{[K_{n+1} : K]},

so from (3.1.1) we deduce

c(K\underline{π}/K) = inf_{n>0} \frac{e_{n} (v_{F} (a_{i_{min}}) + ⌊i_{min}/p⌋e) + i_{min} − p}{p^{n}(p − 1)}

= \frac{e_{0}}{p − 1} (v_{F} (a_{i_{min}}) + ⌊i_{min}/p⌋e) − \frac{p − i_{min}}{p(p − 1)}

since p − i_{min} ≥ 0, so the above infimum occurs when n = 1. As i_{min} ≥ 1, the above constant is visibly positive, so by the very definition [Win83, 1.2.1], K\underline{π}/K is strictly APF. The canonical embedding of the norm field of K\underline{π}/K into Fr(R) is described in [Win83, §4.2]; that the image of this embedding coincides with k((\underline{π})) is a consequence of [Win83, 2.2.4, 2.3.1]. □

Remark 3.1.2. Observe that if the coefficient a_{1} of the linear term of f(u) has v(a_{1}) ≤ 1, then we have i_{min} = 1 and

c(K\underline{π}/K) = \frac{e_{0}}{p − 1}v_{F} (a_{1}) − \frac{1}{p}.

In this situation, v_{F} (a_{1})—which plays an important role in our theory—is encoded in the ramification structure of K\underline{π}/K.

It is natural to ask when two given polynomials f and f′ with corresponding compatible choices \underline{π} and \underline{π}′ give rise to the same iterate extension. Let us write f(x) = x^{p} + a_{p−1}x^{p−1} + ⋯ + a_{1}x and f′(x) = x^{p} + a′_{p−1}x^{p−1} + ⋯ + a′_{1}x, with a_{i}, a′_{i} ∈ O_{F} and a_{i} ≡ a′_{i} ≡ 0 mod ϖ for 1 ≤ i < p. Let {π_{n}} (respectively {π′_{n}}) be an f (resp. f′) compatible sequence of elements in \overline{K}. Set K_{n} := K(π_{n−1}) (resp. K′_{n} = K(π′_{n−1})) and let a_{s}u^{s} and a′_{s′}u^{s′} be the lowest degree terms of f(u) and f′(u) respectively.

Proposition 3.1.3. If K\underline{π} = K\underline{π}′ as subfields of \overline{K}, then K_{n} = K′_{n} for all n ≥ 1 and there exists an invertible power series ξ(x) ∈ O_{F} [[x]] with ξ(x) = μ_{0}x + ⋯ and μ_{0} ∈ O_{F}^{×} such that

f(ξ(x)) = ξ(f′(x)).

In particular, s = s′ and v(a_{s}) = v(a′_{s}) are numerical invariants of K\underline{π} = K\underline{π}′. Conversely, if f and f′ are given with s = s′ and v(a_{s}) = v(a′_{s}), then we have a_{s} = μ_{0}^{1−s}a′_{s} for a unique μ_{0} ∈ O_{F}^{×} and there is a unique power series ξ(x) ∈ F [[x]] with ξ(x) ≡ μ_{0}x mod x^{2} satisfying f(ξ(x)) = ξ(f′(x)) as formal power series in F [[x]]. If ξ(x) lies in O_{F} [[x]], then for any choice {π′_{n}}_{n} of f′-compatible sequence with π′_{0} a uniformizer of K, the sequence defined by π_{n} := ξ(\underline{π}′_{n}) is f-compatible with π_{0} = ξ(π′_{0}) a uniformizer of K and K\underline{π} = K\underline{π}′. Furthermore, if v(a_{s}) = v(a′_{s}) = v(ϖ), then ξ(x) always lies in O_{F} [[x]].

Proof. Suppose first that K\underline{π} = K\underline{π}′, and write simply K_{∞} for this common, strictly APF extension of K in \overline{K}. It follows from the proof of Lemma 3.1.1 that K_{n+1} and K′_{n+1} are both the n-th elementary subextension of K_{∞}; i.e. the fixed field of G_{K}^{b_{n}}G_{K_{∞}}, where b_{n} is the n-th break in the ramification filtration G_{K}^{u}G_{K_{∞}}; see [Win83, 1.4]. In particular, K_{n+1} = K′_{n+1} for all n ≥ 0. Now let W_{ϖ}(•) be the functor of ϖ-Witt vectors; it is the unique functor from O_{F} -algebras to O_{F} -algebras satisfying

(1) For any O_{F} -algebra A, we have W_{ϖ}(•) = ∏_{n≥0} • =: •^{N} as functors from O_{F} -algebras to sets.

(2) The ghost map W_{ϖ}(•) → •^{N} given by

(a_{0}, a_{1}, a_{2}, …) ↦ (a_{0}, a_{0}^{p} + ϖa_{1}, a_{0}^{p^{2}} + ϖa_{1}^{p} + ϖ^{2}a_{2}, …)

is a natural transformation of functors from O_{F} -algebras to O_{F} - algebras.

We remark that W_{ϖ}(•) exists and depends only on ϖ, and is equipped with a unique natural transformation φ : W_{ϖ}(•) → W_{ϖ}(•) which on ghost components has the effect (a_{0}, a_{1}, …) ↦ (a_{1}, a_{2}, …); see [CD15, §2]. Define the ring

A_{K_{∞}/K}^{+} := \{(\underline{x}_{i})_{i} ∈ \underleftarrow{lim}_{φ} W_{ϖ}(OK̂_{∞}) : \underline{x}_{n} ∈ W_{ϖ}(O_{K_{n+1}}) for all n\},

which depends only on F, ϖ, and K_{∞}/K. The main theorem of [CD15], implies that A_{K_{∞}/K}^{+} is a ϖ-adically complete and separated O_{F} -algebra equipped with a Frobenius endomorphism φ, which is canonically a Frobenius-stable subring of W (R)_{F} that is closed under the weak topology on W (R)_{F}. Giving A_{K_{∞}/K}^{+} the subspace topology, the proof of [CD15, Prop. 7.13] then shows that the f (respectively f′)-compatible sequence \underline{π} (respectively \underline{π}′) determine isomorphisms of topological O_{F} -algebras

\xymatrix{η, η′ : O_{F} [[x]] \ar[r] & A_{K_{∞}/K}^{+}}

characterized by the requirement that the ghost components of (η)_{n} (resp. (η′)_{n}) are (π_{n}, f(π_{n}), f^{(2)}(π_{n}), …) (resp. (π′_{n}, f′(π′_{n}), f^{′(2)}(π′_{n}), …)); here we give O_{F} [[x]] the (ϖ, x)-adic topology. These isomorphisms moreover satisfy

η(f(x)) = φ(η(x))  and  η′(f′(x)) = φ(η′(x)).

We therefore obtain a continuous automorphism ξ : O_{F} [[x]] → O_{F} [[x]] satisfying

(3.1.4)  f(ξ(x)) = ξ(f′(x)).

Since ξ is a continuous automorphism of O_{F} [[x]], we have that ξ preserves the maximal ideal (ϖ, x). This implies that ξ(x) ≡ μ_{0}x mod x^{2} with μ_{0} ∈ O_{F}^{×}. Then (3.1.4) forces a_{s}μ_{0}^{s}x^{s} = a′_{s′}μ_{0}x^{s′} which implies s = s′ and v(a_{s}) = v(a′_{s}). Conversely, suppose given f and f′ with s = s′ and v(a_{s}) = v(a′_{s}) and let μ_{0} ∈ O_{F}^{×} be the unique unit with a_{s} = μ_{0}^{1−s}a′_{s}; note that this exists because s − 1 < p. We inductively construct degree i polynomials ξ_{i}(x) = ∑_{j=1}^{i} μ_{j}x^{j} so that f(ξ_{i}(x)) ≡ ξ_{i}(f′(x)) mod x^{i+s}. As μ_{0}^{s}a_{s} = μ_{0}a′_{s}, we may clearly take ξ_{1}(x) = μ_{0}x. If ξ_{i}(x) has been constructed, we write ξ_{i+1}(x) = ξ_{i}(x)+μ_{i+1}x^{i+1} and f(ξ_{i}(x)) − ξ_{i}(f′(x)) ≡ λx^{i+s} mod x^{i+s+1} and seek to solve

(3.1.5)  f(ξ_{i+1}(x)) ≡ ξ_{i+1}(f′(x)) mod x^{i+s+1}.

As f(ξ_{i+1}(x)) = f(ξ_{i}(x)) + \frac{df}{dx}(ξ_{i}(x))(μ_{i+1}x^{i+1}) + ⋯, we see that (3.1.5) is equivalent to

(3.1.6) λ = μ_{i+1}(a_{1} − a′_{1}^{i+1}) if s = 1, and λ = μ_{i+1}sa_{s}μ_{0}^{s−1} if s > 1

which admits a unique solution μ_{i+1} ∈ F. We set ξ(x) = lim_{i} ξ_{i}(x) ∈ F [[x]], which by construction satisfies the desired intertwining relation (3.1.4). If ξ lies in O_{F} [[x]], it is clear that any f′-compatible sequence π′_{n} with π′_{0} a uniformizer of K yields an f-compatible sequence π_{n} := ξ(π′_{n}) with π_{0} a uniformizer of K and K_{n} := K(π_{n−1}) = K(π′_{n−1}) = K′_{n} for all n ≥ 1. Finally, since we have f(x) = f′(x) ≡ x^{p} mod ϖ, it follows that f(ξ_{i}(x)) − ξ_{i}(f′(x)) ≡ 0 mod ϖ, i.e. λ ≡ 0 mod ϖ in the above construction. When v(a_{s}) = v(a′_{s}) = v(ϖ), it then follows from (3.1.6) that μ_{i+1} ∈ O_{F}, and ξ(x) ∈ O_{F} [[x]] as claimed. □

As an immediate consequence of Proposition (3.1.3), one sees that there are infinitely many distinct f-iterate extensions K\underline{π} inside of \overline{K}.

3.2. Étale φ-modules. Let O_{E} be the p-adic completion of 𝔖_{F} [1/u], equipped with the unique continuous extension of φ. Our fixed embedding 𝔖_{F} ↪ W (R) determined by f and \underline{π} uniquely extends to a φ-equivariant embedding ι : O_{E} ↪ W (Fr R)_{F}, and we identify O_{E} with its image in W (Fr R)_{F}. We note that O_{E} is a complete discrete valuation ring with uniformizer ϖ and residue field k((\underline{π})), which, as a subfield of Fr R, coincides with the norm field of K\underline{π}/K thanks to Lemma 3.1.1. As Fr R is algebraically closed, there is a unique separable closure k((\underline{π}))^{sep} of k((\underline{π})) in Fr R, and the maximal unramified extension (i.e. strict Henselization) O_{E^{ur}} of O_{E} with residue field k((\underline{π}))^{sep} is uniquely determined up to unique isomorphism. The universal property of strict Henselization guarantees that ι uniquely extends to an embedding O_{E^{ur}} ↪ W (Fr R)_{F}, which moreover realizes O_{E^{ur}} as a φ-stable subring. We write OÊ^{ur} for the p-adic completion of O_{E^{ur}}, which is again a φ-stable subring of W (Fr R)_{F}. Again using the universal property of strict Henselization, one sees that each of O_{E}, O_{E^{ur}} and OÊ^{ur} are G\underline{π}-stable subrings of W (Fr R)_{F}, with G\underline{π} acting trivially on O_{E}. As suggested by the notation, we write E, E^{ur}, and Ê^{ur} for the fraction fields of O_{E}, O_{E^{ur}} and OÊ^{ur}, respectively. Finally, we define 𝔖_{F}^{ur} := W (R)_{F} ∩ OÊ^{ur}.

Lemma 3.2.1. With notation as above:

(1) The natural action of G\underline{π} on OÊ^{ur} induces an isomorphism of profinite groups

G\underline{π} := Gal(\overline{K}/K\underline{π}) ≃ Aut(OÊ^{ur}/O_{E}) = Gal(Ê^{ur}/E).

(2) The inclusions O_{F} ↪ (OÊ^{ur})^{φ=1} and O_{E} ↪ (OÊ^{ur})^{G\underline{π}} are isomorphisms.

Proof. By the very construction of OÊ^{ur} and the fact that the residue field of O_{E} is identified with the norm field E_{K\underline{π}/K} by Lemma 3.1.1, we have an isomorphism of topological groups Gal(E_{K\underline{π}/K}^{sep}/E_{K\underline{π}/K}) ≃ Aut(OÊ^{ur}/O_{E}) by the theory of unramified extensions of local fields. On the other hand, the theory of norm fields [Win83, 3.2.2] provides a natural isomorphism of topological groups G\underline{π} ≃ Gal(E_{K\underline{π}/K}^{sep}/E_{K\underline{π}/K}), giving (1).

To prove (2), note that the maps in question are local maps of ϖ-adically separated and complete local rings, so by a standard successive approximation argument it suffices to prove that these maps are surjective modulo ϖ. Now left-exactness of φ-invariants (respectively G\underline{π}-invariants) gives an F_{p}-linear (respectively E_{K_{\underline{π}/K}} -linear) injection

(OÊ^{ur})^{φ=1}/(ϖ) ↪ (E_{K_{\underline{π}/K}}^{sep} )^{φ=1} = F_{p} = O_{F} /(ϖ),

respectively

(OÊ^{ur})^{G\underline{π}}/(ϖ) ↪ (E_{K_{\underline{π}/K}}^{sep} )^{G\underline{π}} = E_{K_{\underline{π}/K}} = O_{E}/(ϖ)

which must be an isomorphism of vector spaces over F_{p} (respectively E_{K_{\underline{π}/K}}) as the source is nonzero and the target is 1-dimensional. We conclude that O_{F} ↪ (OÊ^{ur})^{φ=1} (respectively O_{E} ↪ (OÊ^{ur})^{G\underline{π}}) is surjective modulo ϖ, and therefore an isomorphism as desired. □

Denote by Mod_{O_{E}}^{φ} (resp. Mod_{O_{E}}^{φ,tor}) the category of pairs (M, φ_{M}), where M is a finite free O_{E}-module (resp. a finite O_{E}-module killed by a power of ϖ) and φ_{M} : M → M is a φ-semilinear and additive map whose linearization 1 ⊗ φ_{M} : φ^{∗}M → M is an isomorphism. In each case, morphisms are φequivarant O_{E}-module homomorphisms. Let Rep_{O_{F}} (G\underline{π}) (resp. Rep_{O_{F}}^{tor} (G\underline{π})) be the category of finite, free O_{F} -modules (resp. finite O_{F} -modules killed by a power of ϖ) that are equipped with a continuous and O_{F} -linear action of G\underline{π}. For M in Mod_{O_{E}}^{φ} or in Mod_{O_{E}}^{φ,tor}, we define

\underline{V} (M) := (OÊ^{ur} ⊗_{O_{E}} M)^{φ=1},

which is an O_{F} -module with a continuous action of G\underline{π}. For V in Rep_{O_{F}} (G\underline{π}) or in Rep_{O_{F}}^{tor} (G\underline{π}), we define

\underline{M}(V) = (OÊ^{ur} ⊗_{O_{F}} V)^{G\underline{π}},

which is an O_{E}-module with a φ-semilinear endomorphism φ\underline{M} := φ_{OÊ^{ur}} ⊗ 1.

Theorem 3.2.2. The functors \underline{V} and \underline{M} are quasi-inverse equivalences between the exact tensor categories Mod_{O_{E}}^{φ} (resp. Mod_{O_{E}}^{φ,tor}) and Rep_{O_{F}} (G\underline{π}) (resp. Rep_{O_{F}}^{tor} (G\underline{π})).

Proof. As in the proof of [KR09, Theorem 1.6], the original arguments of Fontaine [Fon90, A1.2.6] carry over to the present situation. Indeed, by standard arguments with inverse limits, it is enough to prove the Theorem for ϖ-power torsion objects. To do so, one first proves that \underline{M} is exact, which by (faithful) flatness of the inclusion O_{E} ↪ O_{E^{ur}} amounts to the vanishing of H^{1}(G\underline{π}, ·) on the category of finite length O_{E^{ur}}-modules with a continuous semilinear G\underline{π}-action. By a standard dévissage, such vanishing is reduced to the case of modules killed by ϖ, where it follows from Hilbert’s Theorem 90 and Lemma 3.2.1. One then checks that for any torsion V, the natural comparison map \underline{M}(V) ⊗_{O_{E}} O_{E^{ur}} → V ⊗_{O_{F}} O_{E^{ur}} induced by multiplication in O_{E^{ur}} is an O_{E^{ur}}-linear, φ, and G\underline{π}-compatible isomorphism by dévissage (using the settled exactness of \underline{M}) to the case that V is ϖ-torsion, where it again follows from Hilbert Theorem 90. Passing to submodules on which φ acts as the identity and using Lemma 3.2.1(2) then gives a natural isomorphism \underline{V} ∘ \underline{M} ≃ id. In a similar fashion, the exactness of \underline{V} and the fact that the comparison map

(3.2.1)  \xymatrix{\underline{V} (M) ⊗_{O_{F}} O_{E^{ur}} \ar[r] & M ⊗_{O_{E}} O_{E^{ur}}}

induced by multiplication in O_{E^{ur}} is an isomorphism for general ϖ-power torsion modules M follows by dévissage from the the truth of these claims in the case of M killed by ϖ. In this situation, the comparison map (3.2.1) is shown to be injective by checking that any F_{p}-linearly independent set of vectors in \underline{V} (M) remains E_{K_{∞/K}}^{sep} -linearly independent in E_{K_{∞/K}}^{sep} ⊗_{F_{p}} \underline{V} (M), which is accomplished by a standard argument using the Frobenius endomorphism and Lemma 3.2.1(2). To check surjectivity is then a matter of showing that both sides of (3.2.1) have the same E_{K_{∞/K}}^{sep} -dimension, i.e. that the F_{p}-vector space \underline{V} (M) has dimension d := dim_{E_{K_{∞/K}}} M. Equivalently, we must prove that \underline{V} (M) has p^{d} elements. Identifying M with E_{K_{∞/K}}^{d} by a choice of E_{K_{∞/K}} -basis and writing (c_{ij}) for the resulting matrix of φ, one (noncanonically) realizes \underline{V} (M) as the set of E_{K_{∞/K}}^{sep} -solutions to the system of d-equations x_{i}^{p} = ∑ c_{ij}x_{j} in d-unknowns, which has exactly p^{d} solutions as φ is étale, so the matrix (c_{ij}) is invertible. □

In what follows, we will need a contravariant version of Theorem 3.2.2, which follows from it by a standard duality argument (e.g. [Fon90, §1.2.7]). For any M ∈ Mod_{O_{E}}^{φ} (respectively M ∈ Mod_{O_{E}}^{φ,tor}), we define

\underline{T} (M) := Hom_{O_{E},φ}(M, OÊ^{ur}), respectively \underline{T} (M) := Hom_{O_{E},φ}(M, Ê^{ur}/OÊ^{ur}),

which is naturally an O_{F} -module with a continuous action of G\underline{π}.

Corollary 3.2.3. The contravariant functor \underline{T} induces an anti-equivalence between Mod_{O_{E}}^{φ} (resp. Mod_{O_{E}}^{φ,tor}) and Rep_{O_{F}} (G\underline{π}) ( resp. Rep_{O_{F}}^{tor} (G\underline{π})).

3.3. Kisin modules and Representations of finite E-height. For an integer r ≥ 0, we write ′Mod_{𝔖_{F}}^{φ,r} for the category of finite-type 𝔖_{F} -modules 𝔐 equipped with a φ_{𝔖_{F}} -semilinear endomorphism φ_{𝔐} : 𝔐 → 𝔐 satisfying

• the cokernel of the linearization 1 ⊗ φ : φ^{∗}𝔐 → 𝔐 is killed by E^{r};

• the natural map 𝔐 → O_{E} ⊗_{𝔖_{F}} 𝔐 is injective.

One checks that together these conditions guarantee that the scalar extension O_{E} ⊗_{𝔖_{F}} 𝔐 is an object of Mod_{O_{E}}^{φ} when 𝔐 is torsion free, and an object of Mod_{O_{E}}^{φ,tor} if 𝔐 is killed by a power of ϖ. Morphisms in ′Mod_{𝔖_{F}}^{φ,r} are φcompatible 𝔖_{F} -module homomorphisms. By definition, the category of Kisin modules of E-height r, denoted Mod_{𝔖_{F}}^{φ,r}, is the full subcategory of ′Mod_{𝔖_{F}}^{φ,r} consisting of those objects which are finite and free over 𝔖_{F}. For any such Kisin module 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r}, we define

T_{𝔖}(𝔐) := Hom_{𝔖_{F} ,φ}(𝔐, 𝔖_{F}^{ur}),

with 𝔖_{F}^{ur} := W (R)_{F} ∩ OÊ^{ur} as above Lemma 3.2.1; this is naturally an O_{F} - module with a linear action of G\underline{π}.

Proposition 3.3.1. Let 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r} and write M = O_{E} ⊗_{𝔖_{F}} 𝔐 for the corresponding object of Mod_{O_{E}}^{φ}.

(1) There is a canonical O_{F} -linear and G\underline{π}-equivariant isomorphism T_{𝔖}(𝔐) ≃ \underline{T} (M). In particular, T_{𝔖}(𝔐) is an object of Rep_{O_{F}} (G\underline{π}) and rank_{O_{F}} (T_{𝔖}(𝔐)) = rank_{𝔖_{F}} (𝔐).

(2) The inclusion 𝔖_{F}^{ur} ↪ W (R)_{F} induces a natural isomorphism of O_{F} [G\underline{π}]-modules T_{𝔖}(𝔐) ≃ Hom_{𝔖_{F} ,φ}(𝔐, W (R)_{F}).

Proof. As in the proofs of [Kis06, 2.1.2, 2.1.4] and [KR09, 3.2.1], the Lemma follows from B1.4.2 and B1.8.3 of [Fon90] (cf. B1.8.6), using [Fon90, A1.2] and noting that Fontaine’s arguments—which are strictly speaking only for F = ℚ_{p}—carry over mutatis mutandis to our more general situation. □

Example 3.3.2. Let 𝔐 be a Kisin module of rank 1 over 𝔖_{F}. Choosing a basis 𝔢 of 𝔐 and identifying 𝔐 = 𝔖_{F} ·𝔢, it follows from Weierstrass preparation that we must have φ(𝔢) = μE^{m}𝔢 for some μ ∈ 𝔖_{F}^{×}. Consider the particular case that φ(𝔢) = E𝔢, which is a rank-1 Kisin module of E-height 1. Proposition (3.3.1) then shows that T_{𝔖}(𝔐) gives an O_{F} -valued character of G\underline{π} and that there exists 𝔱 ∈ W (R)_{F} satisfying φ(𝔱) = E𝔱 and 𝔱 mod ϖ ≠ 0 inside R. We will see in §5 that the character of G\underline{π} furnished by T_{𝔖}(𝔐) can be extended to a Lubin-Tate character of G if we assume that ϖ^{2}|a_{1} in O_{F}, where a_{1} is the linear coefficient of f(x) ∈ O_{F} [x].

Let Rep_{F} (G\underline{π}) denote the category of continuous, F -linear representations of G\underline{π}. An object V of Rep_{F} (G\underline{π}) is of E-height r if there exists a Kisin module 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r} with V ≃ T_{𝔖_{F}} (𝔐)[1/p], and V is of finite E-height if there exists an integer r such that V is of E-height r. As E = E(u) is fixed throughout the paper, we will simply say that V is of (finite) height r. For 𝔐 an arbitrary object of Mod_{𝔖_{F}}^{φ,r}, we write V_{𝔖}(𝔐) := T_{𝔖}(𝔐)[1/p] for the associated height-r representation of G\underline{π}. We will need the following generalization of [Kis06, Lemma 2.1.15] (or [Liu07, Corollary 2.3.9]):

Proposition 3.3.3. If V ∈ Rep_{F} (G\underline{π}) is of height r then for any G\underline{π}-stable O_{F} - lattice L ⊂ V, there exists 𝔑 ∈ Mod_{𝔖_{F}}^{φ,r} such that T_{𝔖}(𝔑) ≃ L in Rep_{O_{F}} (G\underline{π}).

The proof of Proposition 3.3.3 will make use of the following key lemma:

Lemma 3.3.4. Let 𝔐 be an object of ′Mod_{𝔖_{F}}^{φ,r} that is torsion-free. Then the intersection 𝔐′ := 𝔐[1/p] ∩ (O_{E} ⊗_{𝔖_{F}} 𝔐), taken inside of E ⊗_{𝔖_{F}} 𝔐, is an object in Mod_{𝔖_{F}}^{φ,r} and there are canonical inclusions 𝔐 ⊂ 𝔐′ ⊂ O_{E} ⊗_{𝔖_{F}} 𝔐.

Proof. The proof of Lemma 2.3.7 in [Liu07] carries over mutatis mutandis to the present situation. □

Proof of Proposition 3.3.3. As the proof is a simple adaptation of that of Corollary 2.3.9 in [Liu07], we simply sketch the highlights. Let V ∈ Rep_{F} (G\underline{π}) be of height r, and select 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r} with V ≃ V_{𝔖}(𝔐). Put T := T_{𝔖}(𝔐), which is a G\underline{π}-stable O_{F} -lattice in V, and let L ⊂ V be an arbitrary G\underline{π}-stable O_{F} -lattice. Put M := O_{E} ⊗_{𝔖_{F}} 𝔐 and let N ∈ Mod_{O_{E}}^{φ} be the object of Mod_{O_{E}}^{φ} corresponding to L via Corollary 3.2.3, so \underline{T} (N) ≃ L in Rep_{O_{F}} (G\underline{π}). Without loss of generality, we may assume that N ⊂ M. Writing 𝔣 : M → M/N for the natural projection, it is easy to check that 𝔣(𝔐) is an object of ′Mod_{𝔖_{F}}^{φ,r}. It then follows from Proposition [Fon90, B 1.3.5] that 𝔑′ := ker(𝔣|_{𝔐}) ⊂ N is an object of ′Mod_{𝔖_{F}}^{φ,r}. Writing 𝔑 := 𝔑′[1/p] ∩ N, we have that 𝔑 is an object of Mod_{𝔖_{F}}^{φ,r} thanks to Lemma 3.3.4, and by construction we have O_{E} ⊗_{𝔖_{F}} 𝔑 ≃ N, so that T_{𝔖}(𝔑) ≃ L as O_{F} [G\underline{π}]-modules thanks to Proposition 3.3.1 and the choice of N. □

Proposition 3.3.5. Assume that φ^{n}(f(u)/u) is not power of E = E(u) for any n ≥ 0. Then the functor T_{𝔖} : Mod_{𝔖_{F}}^{φ,r} ⇝ Rep_{O_{F}} (G\underline{π}) is fully faithful.

Proof. We use an idea of Caruso [Car, Proposition 3.1]. Fix 𝔐, 𝔐′ ∈ Mod_{𝔖_{F}}^{φ,r}. Using Corollary 3.2.3 and Lemma 3.3.1, we easily reduce the proof of Proposition 3.3.5 to that of the following assertion: if 𝔣 : O_{E} ⊗_{𝔖_{F}} 𝔐 → O_{E} ⊗_{𝔖_{F}} 𝔐′ is a morphism in Mod_{O_{E}}^{φ} then 𝔣(𝔐) ⊂ 𝔐′. By applying Lemma 3.3.4 to 𝔣(𝔐)+𝔐′, we may further reduce the proof to that of the following statement: if 𝔐 ⊂ 𝔐′ ⊂ O_{E} ⊗_{𝔖_{F}} 𝔐 then 𝔐 = 𝔐′. Writing d := rk_{𝔖_{F}} (𝔐) = rk_{𝔖_{F}} (𝔐′) and applying ∧^{d}, we may reduce to the case d = 1, and now calculate with bases. Let e (resp. e′) be an 𝔖_{F} -basis of 𝔐 (resp. 𝔐′), and let a ∈ 𝔖_{F} be the unique element with e = ae′. Since O_{E} ⊗_{𝔖_{F}} 𝔐 = O_{E} ⊗_{𝔖_{F}} 𝔐′, by Weierstrass preparation, we may modify our choices of e and e′ to assume that a = A(u) = u^{s} + c_{s−1}u^{s−1} + ⋯ + c_{1}u + c_{0} with c_{i} ∈ ϖO_{F_{0}}. As in Example 3.3.2, we may suppose that φ(e′) = γ′E^{n}e′ and φ(e) = γE^{n′}e with γ, γ′ ∈ 𝔖_{F}^{×}. Then

γE^{n′}A(u)e′ = γE^{n′}e = φ(e) = φ(A(u))φ(e′) = φ(A(u))γ′E^{n}e′

which necessitates γA(u)E^{n′} = γ′φ(A(u))E^{n}. Reducing modulo ϖ and comparing u-degrees, we see easily that n′ ≥ n. We therefore have

(3.3.1)  γ_{0}A(u)E^{n′−n} = φ(A(u)) for γ_{0} = γ(γ′)^{−1} ∈ 𝔖_{F}^{×}.

As γ_{0} is a unit, it follows from (3.3.1) that A(u)E^{n′−n} and φ(A(u)) must have the same roots. Since A(u), φ(A(u)) and E are monic polynomials with roots either 0 or with positive valuation, we conclude that A(u)E^{n′−n} = φ(A(u)). Let us put A(u) = u^{l}A_{0}(u) with A_{0}(0) ≠ 0 and m = n′ − n. Then (3.3.1) simplifies to

(3.3.2)  A_{0}(u)E^{m} = (f(u)/u)^{l}φ(A_{0}(u)).

We first treat the case l = 0 (so A = A_{0}); we will then reduce the general case to this one. Put A^{φ}(u) := u^{s} + φ(c_{s−1})u^{s−1} + ⋯ + φ(c_{1})u + φ(c_{0}). There is then a bijection between the roots of A^{φ}(u) and the roots of A(u) which preserves valuation. Let x_{0} be a nonzero root of A(u) which achieves the maximal valuation. Then A(u)E^{m} = φ(A(u)) implies that x_{0} is root of φ(A(u)) = A^{φ}(f(u)). That is f(x_{0}) is a root of A^{φ}(u). If f(x_{0}) ≠ 0, then since f(u) ≡ u^{p} mod ϖ and x_{0} has positive valuation, we have v(f(x_{0})) > v(x_{0}), so there exists a root of A(u) with valuation strictly greater than v(x_{0}), which contradicts our choice of x_{0}. We must therefore have that f(x_{0}) = 0 is root of A^{φ}(u), which contradicts our assumption that A(0) ≠ 0 (l = 0). We conclude that A(u) = A_{0}(u) has degree zero, and hence that 𝔐 = 𝔐′ as desired. Now suppose that l > 0 and let r_{1} ≥ 0 be the unique integer satisfying (f(u)/u)^{l} = E^{r_{1}}h_{1}(u) for some unique monic h_{1} ∈ 𝔖_{F} with E ∤ h_{1}(u). Comparing u-degrees in (3.3.2) gives r_{1} ≤ m, so h_{1}(u)|A_{0}(u) and we have A_{0}(u) = h_{1}(u)A_{1}(u) for a unique monic polynomial A_{1} and m_{1} := m − r_{1} ≥ 0. Equation (3.3.2) then becomes

A_{1}(u)E^{m_{1}} = φ(h_{1}(u))φ(A_{1}(u)).

Now let r_{2} ≥ 0 be the unique integer with φ(h_{1}(u)) = E^{r_{2}}h_{2}(u) for h_{2} a monic polynomial with E ∤ h_{2}(u), and write A_{1}(u) = h_{2}(u)A_{2}(u) with A_{2} monic and m_{2} := m_{1} − r_{2} ≥ 0. We then have

A_{2}(u)E^{m_{2}} = φ(h_{2}(u))φ(A_{2}(u)).

We continue in this manner, constructing nonnegative integers r_{n}, m_{n} with m_{n+1} := m_{n} − r_{n} and monic A_{n}, h_{n} ∈ 𝔖_{F} with E ∤ h_{n}, h_{n}E^{r_{n}} = φ(h_{n−1}) and A_{n−1} = h_{n}A_{n} satisfying the equation

(3.3.3)  A_{n}(u)E^{m_{n}} = φ(h_{n}(u))φ(A_{n}(u)).

So long as h_{n} and A_{n} are non-constant, we have deg A_{n} < deg A_{n−1}, which can not continue indefinitely. We conclude that there is some n ≥ 1 with either h_{n} or A_{n} constant, which forces h_{n} = 1 or A_{n} = 1 by monicity. In the latter case, (3.3.3) implies that h_{n+1} = 1, so in any case there is some n > 0 with h_{n} = 1. By the construction of the h_{m}, we then have

(3.3.4)  φ^{n−1}((f(u)/u)^{l}) = ∏_{m=1,r_{m}≠0}^{n} φ^{n−m}(E^{r_{m}}).

We claim that in fact there is only one m with r_{m} ≠ 0. Indeed, if there exist m_{1} > m_{2} with r_{m_{i}} ≠ 0 for i = 1, 2, then writing f_{0}(u) = f(u)/u, we see that f_{0}(f^{(m_{i})}(π)) = 0 for i = 1, 2. Since f(u) = f_{0}(u)u, this implies that f^{(m_{2}+1)}(π) = 0. Then

0 = f_{0}(f^{(m_{1})}(π)) = f_{0}(f^{(m_{1}−m_{2}−1)}(f^{(m_{2}+1)}(π))) = f_{0}(f^{(m_{1}−m_{2}−1)}(0)) =f_{0}(0),

which implies that u|f_{0}(u). But this contradicts (3.3.2) because u ∤ A_{0}(u). We conclude that there is a unique m such that r_{m} ≠ 0, and it follows from (3.3.4) that there exists n ≥ 0 such that φ^{n}(f(u)/u) is a power of E(u), contradicting our hypothesis. We must therefore in fact have l = 0, whence 𝔐 = 𝔐′ as we showed above. □

Remark 3.3.6. The assumption that φ^{(n)}(f(u)/u) is not a power of E for any n ≥ 0 is satisfied in many cases of interest. For example, it is always satisfied when a_{1} = 0 (which includes the classical situation f(u) = u^{p}), as then f(u)/u has no constant term while any power of E = E(u) has nonzero constant term.

Example 3.3.7. The hypothesis of Proposition 3.3.5 that φ^{(n)}(f(u)/u) is not a power of E for any n ≥ 0 is necessary, as the following examples show:

(1) Fix r, let 0 ≤ l ≤ r be an integer and suppose that we have φ^{(n)}(f(u)/u) = E^{l}. Setting A(u) := f(u) · φ(f(u)/u) ⋯ φ^{n−1}(f(u)/u) if n > 0 and A(u) = u if n = 0, we have AE^{l} = φ(A). In particular, definining 𝔐 = A(u)𝔖_{F} and 𝔐′ := 𝔖_{F}, we have 𝔐 ⊆ 𝔐′ and both 𝔐 and 𝔐′ are objects of Mod_{𝔖_{F}}^{φ,r} with height l ≤ r. However, 𝔐 ≠ 𝔐′ and it follows that the conclusion of Proposition 3.3.5 does not hold.

(2) Concretely, recall the situation in Example 2.1.2 where K = ℚ_{p}(ζ_{p}), F = ℚ_{p}, π = ζ_{p} − 1 and φ(u) = (u + 1)^{p} − 1. In this case, E = φ(u)/u, and the Kisin modules 𝔐′ := 𝔖_{F} and 𝔐 := u𝔖_{F} are both of height 1 and are non-isomorphic, but T_{𝔖}(𝔐) ≃ T_{𝔖}(𝔐′).

(3) As a less familiar variant, we can take φ(u) = (u − p)^{p−1}u and let E = φ(u) − p. Then φ(f(u)/u) = E^{p−1}, and the construction of (1) provides a counterexample.

Corollary 3.3.8. Suppose that φ^{n}(f(u)/u) is not a power of E for any n ≥ 0 and ψ : V′ → V is a morphism of height-r representations. Then there are exact sequences

\xymatrix{0 \ar[r] & 𝔏 \ar[r] & 𝔐 \ar[r] & 𝔑 \ar[r] & 0, & and & 0 \ar[r] & 𝔑 \ar[r] & 𝔐′ \ar[r] & 𝔑′ \ar[r] & 0}

in Mod_{𝔖_{F}}^{φ,r} which correspond via V_{𝔖}(·) to the exact sequences in Rep_{F} (G\underline{π}):

\xymatrix{0 \ar[r] & ψ(V′) \ar[r] & V \ar[r] & V/ψ(V′) \ar[r] & 0},

and

\xymatrix{0 \ar[r] & ker(ψ) \ar[r] & V′ \ar[r]^{ψ} & ψ(V′) \ar[r] & 0.}

Proof. We may and do select G\underline{π}-stable O_{F} -lattices T ⊆ V and T′ ⊆ V′ with ψ(T′) ⊆ T and T/ψ(T′) torsion-free. Thanks to Proposition 3.3.3, there exist 𝔐 and 𝔐′ in Mod_{𝔖_{F}}^{φ,r} with T = T_{𝔖}(𝔐) and T′ = T_{𝔖}(𝔐′), and we define M := O_{E} ⊗_{𝔖_{F}} 𝔐 and M′ := O_{E} ⊗_{𝔖_{F}} 𝔐′ and write 𝔣 : M → M′ for the unique morphism in Mod_{O_{E}}^{φ} with \underline{T} (𝔣) = ψ|_{T′}. Let N′ := M′/𝔣(M) and write 𝔤 : M′ → N′ for the natural projection. Writing N := 𝔣(M) = ker 𝔤, we then have exact sequences in Mod_{O_{E}}^{φ}

\xymatrix{0 \ar[r] & ker(𝔣) \ar[r] & M \ar[r]^{𝔣} & N \ar[r] & 0}

and

\xymatrix{0 \ar[r] & N \ar[r] & M′ \ar[r]^{𝔤} & N′ \ar[r] & 0}

which correspond, via \underline{T} (·), to the exact sequences in Rep_{O_{F}} (G\underline{π})

\xymatrix{0 \ar[r] & ψ(T′) \ar[r] & T \ar[r] & T/ψ(T′) \ar[r] & 0},

and

\xymatrix{0 \ar[r] & ker(ψ) \ar[r] & T′ \ar[r]^{ψ} & ψ(T′) \ar[r] & 0.}

Since N′ corresponds to T/ψ(T′), which is torsion-free, it follows that N′ is also torsion free and hence finite and free as an O_{E}-module. Define 𝔑 := ker(𝔤)∩𝔐′, the intersection taken inside of M′. We claim that 𝔑 is an object in Mod_{𝔖_{F}}^{φ,r}. First note that by [Fon90, B 1.3.5], the fact that 𝔐′ has height r implies that both 𝔤(𝔐′) and 𝔑 have height r, and we need only show that 𝔑 is free over 𝔖_{F}. To do this, it suffices by Lemma 3.3.4 to prove that 𝔑 = 𝔑[1/p]∩N inside E ⊗_{O_{E}} N, or equivalently that 𝔑[1/p]∩N ⊆ 𝔑. For any x ∈ 𝔑[1/p]∩N, we have by the very definition of 𝔑 that x ∈ 𝔐′[1/p] ∩ M′ = 𝔐′. As x ∈ N = ker 𝔤, we then have x ∈ ker 𝔤 ∩ 𝔐′ = 𝔑′ as desired. A similar argument shows that 𝔏 := ker(𝔣) ∩ 𝔐 is a Kisin module in Mod_{𝔖_{F}}^{φ,r} as well.

Again using Lemma 3.3.4, both 𝔑̃ := 𝔣(𝔐)[1/p] ∩ N and 𝔑′ := 𝔤(𝔐′)[1/p] ∩ N′ are objects of Mod_{𝔖_{F}}^{φ,r}. As O_{E} ⊗_{𝔖_{F}} 𝔑̃ = N = O_{E} ⊗_{𝔖_{F}} 𝔑, it follows from Proposition 3.3.5 that 𝔑̃ = 𝔑. We therefore have exact sequences

\xymatrix{0 \ar[r] & 𝔏[1/p] \ar[r] & 𝔐[1/p] \ar[r]^{𝔣} & 𝔑[1/p] \ar[r] & 0,}

and

\xymatrix{0 \ar[r] & 𝔑[1/p] \ar[r] & 𝔐′[1/p] \ar[r]^{𝔤} & 𝔑′[1/p] \ar[r] & 0}.

Unfortunately, it need not be true in general that 𝔣(𝔐) = 𝔑 or 𝔤(𝔐′) = 𝔑′. To remedy this defect, we modify 𝔐 and 𝔐′ as follows. Using the inclusion 𝔑 ⊆ 𝔐′ and the above exact sequences, we may select an 𝔖_{F} [1/p]-basis 𝔢_{1}, …, 𝔢_{s}, 𝔢_{s+1}, …, 𝔢_{d} of 𝔐[1/p] with the property that 𝔢_{1}, …, 𝔢_{s} is an 𝔖_{F} -basis of 𝔏 and 𝔢′_{s+1} := 𝔣(𝔢_{s+1}), …, 𝔢′_{d} := 𝔣(𝔢_{d}) is an 𝔖_{F} -basis of 𝔑. We may further complete 𝔢′_{s+1}, …, 𝔢′_{d} to an 𝔖_{F} [1/p]-basis 𝔢′_{s+1}, …, 𝔢′_{d}, 𝔢′_{d+1}, …, 𝔢′_{d′} of 𝔐′ with the property that 𝔢′_{d+1}, …, 𝔢′_{d′} projects via 𝔤 to an 𝔖_{F} -basis of 𝔑′. We then have matrix equations

φ(𝔢_{1}, …, 𝔢_{d}) = (𝔢_{1}, …, 𝔢_{d}) (\begin{matrix}A & C \\ 0 & A′ \end{matrix})

and

φ(𝔢′_{s+1}, …, 𝔢′_{d}, 𝔢′_{d+1}, …, 𝔢′_{d′}) = (𝔢′_{s+1}, …, 𝔢′_{d}, 𝔢′_{d+1}, …, 𝔢′_{d′}) (\begin{matrix}B & D \\ 0 & B′ \end{matrix}) ,

where the entries of A, A′, B, B′ are in 𝔖_{F}, while the entries of C and D are in 𝔖_{F} [1/p]. Let m ≥ 0 be such that p^{m}C and p^{m}D have all entries in 𝔖_{F}. Replacing 𝔐 by the 𝔖_{F} -submodule of 𝔐[1/p] generated over 𝔖_{F} by p^{−m}𝔢_{1}, …, p^{−m}𝔢_{s}, 𝔢_{s+1}, …, 𝔢_{d}, and 𝔐′ by the 𝔖_{F} -submodule of 𝔐′[1/p] generated by (𝔢′_{s+1}, …, 𝔢′_{d}, p^{m}𝔢′_{d+1}, …, p^{m}𝔢′_{d′}) does the trick. □

4. Constructing Kisin modules from F -crystalline representations

In this section, we associate to any F -crystalline representation a Kisin module in the sense of §3.3 and employ our construction to prove Theorems 1.0.1 and 1.0.2. Throughout, and especially in §4.2, we make free use of many of the ideas of [Kis06] and [KR09]. To surmount the difficulty that we do not in general have a natural N_{∇}-structure (see the introduction), we will compare our modules over the Robba ring to those of Kisin’s classical setting in in §4.4, which will allow us to descend these modules to the desired Kisin modules. The proofs of our main results (Theorems 1.0.1 and 1.0.2) occupies §4.5.

4.1. Generalities on F -crystalline representations. Let V be an F - linear representation of G = G_{K} or of G\underline{π}. We write V^{∨} for the F -linear dual of V with its natural G or G\underline{π}-action. We warn the reader at the outset that our notational conventions regarding Fontaine’s functors are dual to the standard ones; we have chosen to depart from tradition here as it will be more convenient to deal with the integral theory. Let V be an object of Rep_{F} (G). Then D_{dR}(V) := (V^{∨} ⊗_{ℚ_{p}} B_{dR})^{G} is naturally a module over the semilocal ring K_{F} := K ⊗_{ℚ_{p}} F, so we have a decomposition

D_{dR}(V) = ∏_{𝔪} D_{dR}(V)_{𝔪}

with the product running over all maximal ideals of K_{F}. We give each D_{dR}(V)_{𝔪} the filtration induced from that of D_{dR}(V), and we denote by 𝔪_{0} the kernel of the natural map K ⊗_{ℚ_{p}} F → K coming from the given inclusion F ↪ K and multiplication. Following [KR09], we define:

Definition 4.1.1. We say that V ∈ Rep_{F} (G) is F -crystalline if it is crystalline (as a ℚ_{p}-linear G-representation) and the filtration on D_{dR}(V)_{𝔪} is trivial (Fil^{j}D_{dR}(V)_{𝔪} = 0 if j > 0 and Fil^{0}D_{dR}(V)_{𝔪} = D_{dR}(V)_{𝔪}) when 𝔪 ≠ 𝔪_{0}. We write Rep_{F}^{F -cris}(G) for the category of F -crystalline F -representations of G.

We now wish to describe the category of F -crystalline G-representations in terms of filtered φ-modules. To do this, we define:

Definition 4.1.2. Let MF_{F_{0},K}^{φ} be the category of triples (D, φ, Fil^{j}D_{F_{0},K}) where D is a finite dimensional F_{0}-vector space, φ : D → D is a semilinear (over the F -linear extension φ of the p-power Frobenius map K_{0} → K_{0}) endomorphism whose linearization is an F_{0}-linear isomorphism, and Fil^{j}D_{F_{0},K} is a separated and exhaustive descending filtration by K-subspaces on the scalar extension D_{F_{0},K} := D ⊗_{F_{0}} K. Morphisms in this category are φ-compatible F_{0}-linear maps D → D′ which are filtration-compatible after applying ⊗_{F_{0}}K.

Let V be an F -crystalline G-representation with F -dimension d. Then

D := D_{cris}(V) := (V^{∨} ⊗_{ℚ_{p}} B_{cris})^{G}

is naturally a module over F ⊗_{ℚ_{p}} K_{0}, equipped with a semilinear (over 1 ⊗ σ for σ the p-power Frobenius automorphism of K_{0}) Frobenius endomorphism φ : D → D which linearizes to an isomorphism. By our assumption that K_{0} ∩ F = ℚ_{p}, the natural multiplication map F ⊗_{ℚ_{p}} K_{0} → F K_{0} =: F_{0} is an isomorphism, so D is an F_{0}-vector space which, as V is crystalline as a ℚ_{p}-representation, has K_{0}-dimension d[F : ℚ_{p}], so must have F_{0}-dimension d. The natural injective map

D ⊗_{K_{0}} K = D_{cris}(V) ⊗_{K_{0}} K ↪ D_{dR}(V) := (V^{∨} ⊗_{ℚ_{p}} B_{dR})^{G}

is necessarily an isomorphism of F_{K} := F ⊗_{ℚ_{p}} K-modules, so since V is F - crystalline we have a direct sum decomposition of filtered K-vector spaces D ⊗_{K_{0}} K = ⨁_{𝔪} D_{K,𝔪}, with D_{K,𝔪} having trivial filtration unless 𝔪 = 𝔪_{0}. Noting the canonical identification

D_{K,𝔪_{0}} = D ⊗_{F_{0}} K =: D_{F_{0},K},

we therefore obtain a filtration on D_{F_{0},K}. In this way we obtain an object

D_{cris,F} (V) := (D, φ, Fil^{j}D_{F_{0},K})

of MF_{F_{0},K}^{φ}. Conversely, if D is any object of MF_{F_{0},K}^{φ}, we define

V_{cris,F} (D) := Hom_{F_{0},φ}(D, B_{cris,F}^{+} ) ∩ Hom_{K,Fil^{•}}(D_{F_{0},K}, B_{cris}^{+} ⊗_{K_{0}} K)

with the intersection taken inside of Hom_{K}(D_{F_{0},K}, B_{cris}^{+} ⊗_{K_{0}} K), via the map

\xymatrix{Hom_{F_{0},φ}(D, B_{cris,F}^{+} ) \ar@{^{(}->}[r] & Hom_{K}(D_{F_{0},K}, B_{cris}^{+} ⊗_{K_{0}} K)}

that sends an F_{0}-linear h : D → B_{cris,F}^{+} to its linear extension along F_{0} → K.

Proposition 4.1.3. Let V ∈ Rep_{F}^{F -cris}(G). Then V_{cris,F} (D_{cris,F} (V)) ≃ V.

Proof. Set D = D_{cris}(V) := (V^{∨} ⊗_{ℚ_{p}} B_{cris}^{+})^{G} and put D_{K} := K ⊗_{K_{0}} D. As it is well-known (e.g. [Fon94b, §5.3.7]) that V ≃ V_{cris}(D) as F [G]-modules, for

V_{cris}(D) = Hom_{K_{0},φ}(D, B_{cris}^{+}) ∩ Hom_{K,Fil}(D_{K}, B_{cris}^{+} ⊗_{K_{0}} K),

it is enough to prove that V_{cris}(D) ≃ V_{cris,F} (D) as F [G]-modules. We will first construct an F -linear isomorphism

(4.1.1)  \xymatrix{ι : Hom_{K_{0}}(D, B_{cris}^{+}) \ar[r]^{≃} & Hom_{F_{0}}(D, B_{cris,F}^{+} ).}

Writing D_{F_{0}} = D ⊗_{K_{0}} F_{0}, which is an F_{0} ⊗_{K_{0}} F_{0}-module, we note that F_{0} ≃ F_{0} ⊗_{F_{0}} F_{0} is a subfield of F_{0} ⊗_{K_{0}} F_{0}, so we may and do regard D ≃ D ⊗_{F_{0}} F_{0} as an F_{0}-subspace of D_{F_{0}}. Thus, restricting homomorphisms from D_{F_{0}} to the subspace D gives a natural map of F -vector spaces ι′ : Hom_{F_{0}}(D_{F_{0}}, B_{cris,F}^{+} ) → Hom_{F_{0}}(D, B_{cris,F}^{+} ). As Hom_{K_{0}}(D, B_{cris}^{+}) is easily checked to be an F -subspace of Hom_{F_{0}}(D_{F_{0}}, B_{cris,F}^{+} ), restriction of ι′ to Hom_{K_{0}}(D, B_{cris}^{+}) then gives the desired map (4.1.1). To check that (4.1.1) is an isomorphism, we explicitly compute with bases: Let e_{1}, …, e_{d} be an F_{0}-basis of D and β_{1}, …, β_{e_{F}} a K_{0}-basis of F_{0}. Any x ∈ D can then be uniquely expressed as a linear combination x = ∑_{ij} a_{ij}β_{j}e_{i} for some a_{ij} ∈ K_{0}, while any y ∈ D_{F_{0}} admits a unique representation of the form y = ∑_{i,j,l} a_{ijl}β_{j}e_{i} ⊗ β_{l} with a_{ijl} ∈ K_{0}. The natural F -linear inclusion D ↪ D_{F_{0}} induced by F_{0} ⊗_{F_{0}} F_{0} ⊂ F_{0} ⊗_{K_{0}} F_{0} carries x ∈ D above to

x = ∑_{ij} a_{ij}β_{j}e_{i} ⊗ β_{j} ∈ D_{F_{0}}.

In particular, if h ∈ Hom_{K_{0}}(D, B_{cris}^{+}), then h is uniquely determined by the matrix {c_{ij}} with c_{ij} := h(β_{j}e_{i}) ∈ B_{cris}^{+}, and it follows from definitions that ι(f)(x) = ∑_{ij} a_{ij}c_{ij} ⊗ β_{j} as an element of B_{cris}^{+} ⊗_{K_{0}} F_{0}. From this explicit description of ι, one checks easily that ι is indeed an isomorphism. From the very definition of (4.1.1), one checks that ι induces an isomorphism

Hom_{K_{0},φ}(D, B_{cris}^{+}) ≃ Hom_{F_{0},φ}(D, B_{cris,F}^{+} ),

so to complete the proof it remains to show that for any h ∈ Hom_{K_{0}}(D, B_{cris}^{+}), the scalar extension h⊗1 : D⊗_{K_{0}}K → B_{cris}^{+}⊗_{K_{0}}K is compatible with filtrations if and only if this is true of ι(h) ⊗ 1 : D ⊗_{F_{0}} K → B_{cris,F}^{+} ⊗_{F_{0}} K. Observe that the construction of the map (4.1.1) gives the following commutative diagram,

[figure: D_{K}; h⊗_{K_{0}} 1; B_{cris}^{+} ⊗_{K_{0}} K; D_{F_{0},K}; ι(h)⊗_{F_{0}} 1; B_{cris,F}^{+} ⊗_{F_{0}} K]

(4.1.2)

where we make the identification B_{cris}^{+} ⊗_{K_{0}} K = B_{cris,F}^{+} ⊗_{F_{0}} K. As V is F - crystalline, we have Fil^{i}D_{K} = Fil^{i}D_{F_{0},K} for i ≥ 1 by definition, and it follows from this and (4.1.2) that

(h ⊗_{K_{0}}1)(Fil^{i}D_{K}) ⊂ Fil^{i}B_{cris}^{+} ⊗_{K_{0}}K ⟺ (ι(h)⊗_{F_{0}}1)(Fil^{i}D_{F_{0},K}) ⊂ Fil^{i}B_{cris,F}^{+} ⊗_{F_{0}}K,

which completes that proof of V_{cris,F} (D) ≃ V_{cris}(D) ≃ V as F [G]-modules. □

Let V be an F -linear representation of G. For each field embedding τ : F → \overline{K}, we define the set τ-Hodge-Tate weights of V:

HT_{τ} (V) := {i ∈ ℤ | (V ⊗_{F,τ} ℂ_{K}(−i))^{G} ≠ {0}},

where ℂ_{K} is the p-adic completion of \overline{K}. It is easy to see that V is F -crystalline if and only if V is crystalline and HT_{τ} (V) = {0} unless τ is the trivial embedding τ_{0} : F ⊂ K ⊂ \overline{K}. For the remainder of this paper, we will fix a nonnegative integer r with the property that HT_{τ_{0}}(V) ⊂ {0, …, r}, or equivalently, Fil^{r+1}D_{F_{0},K} = {0}. We denote by Rep_{F}^{F -cris,r}(G) the category of F -crystalline representations V of G with HT_{τ_{0}}(V) ⊂ {0, …, r}.

4.2. φ-modules over 𝔒. Recall that we equip F_{0}((u)) with the Frobenius endomorphism φ : F_{0}((u)) → F_{0}((u)) which acts as the canonical Frobenius on K_{0}, acts trivially on F, and sends u to f(u). For any sub-interval I ⊂ [0, 1), we write 𝔒_{I} for the subring of F_{0}((u)) consisting of those elements which converge for all x ∈ \overline{K} with |x| ∈ I. For ease of notation, we put 𝔒 = 𝔒_{[0,1)} and as before we set K_{n} = K(π_{n−1}). We denote by 𝔖̂_{n} the completion of K_{n+1}⊗_{F_{0}} 𝔖_{F} at the maximal ideal (u−π_{n}). Equip 𝔖̂_{n} with its (u−π_{n})-adic filtration, which extends to a filtration on the quotient field Fr 𝔖̂_{n} = 𝔖̂_{n}[1/(u − π_{n})]. Note that for any n we have natural maps of F_{0}-algebras 𝔖_{F} [1/p] ↪ 𝔒 ↪ 𝔖̂_{n} given by sending u to u, where the first map has dense image. We will write φ_{W} : 𝔖_{F} → 𝔖_{F} for the O_{F} [[u]]-linear map which acts on W (k) via the canonical lift of Frobenius, and by φ_{𝔖/W} : 𝔖_{F} → 𝔖_{F} the O_{F_{0}}-linear map which sends u to f(u). Let c_{0} = E(0) ∈ F_{0} and set

λ := ∏_{n=0}^{∞} φ^{n}(E(u)/c_{0}) ∈ 𝔒.

A φ-module over 𝔒 is a finite free 𝔒-module M equipped with a semilinear endomorphism φ_{M} : M → M. We say that M is of finite E-height if the cokernel of the 𝔒-linear map 1 ⊗ φ_{M} : φ^{∗}M → M is killed by E^{r} for some r, and we write Mod_{𝔒}^{φ,r} for the category of φ-modules over 𝔒 of E-height r. Scalar extension along the inclusion 𝔖_{F} ↪ 𝔒 gives a functor Mod_{𝔖_{F}}^{φ,r} → Mod_{𝔒}^{φ,r} from height-r Kisin modules to φ-modules over 𝔒 of E-height r. Now let V ∈ Rep_{F}^{F -cris,r}(G) be any F -crystalline representation of G, and write D := D_{cris,F} (V) for the corresponding filtered φ-module. We functorially associate to D an 𝔒-module M(D) as follows: For each nonnegative integer n, let ι_{n} be the composite map:

(4.2.1) \xymatrix{𝔒 ⊗_{F_{0}} D \ar[r]^{φ_{W}^{−n}⊗φ_{D}^{−n}} & 𝔒 ⊗_{F_{0}} D \ar[r] & 𝔖̂_{n} ⊗_{F_{0}} D = 𝔖̂_{n} ⊗_{K} D_{F_{0},K},}

where the second morphism is induced by the canonical inclusion 𝔒 → 𝔖̂_{n}. We again write ι_{n} for the canonical extension

ι_{n} : 𝔒[1/λ] ⊗_{F_{0}} D ⟶ Fr 𝔖̂_{n} ⊗_{K} D_{F_{0},K},

and we define

M(D) := {x ∈ 𝔒[1/λ] ⊗_{F_{0}} D | ι_{n}(x) ∈ Fil^{0} (Fr 𝔖̂_{n} ⊗_{K} D_{F_{0},K}) , ∀n ≥ 0} .

Proposition 4.2.1. M(D) is a φ-module over 𝔒 of E-height r.

Proof. This is Lemma 1.2.2 in [Kis06] (also see Lemma (2.2.1) in [KR09]) with the following minor modifications: first note that we only discuss crystalline representation here, so we do not need the “logarithm element” ℓ_{n} which occurs in Kisin’s classical setting (strictly speaking, we do not know how to construct ℓ_{n} in our general setting). Likewise, we may replace D_{0} := (𝔒[ℓ_{n}] ⊗_{K_{0}} D)^{N=0} in the proof of [Kis06, 1.2.2] with D_{0} = 𝔒 ⊗_{F_{0}} D throughout. In the classical setting, Kisin showed that M(D) also has an N_{∇}-structure, which we entirely ignore here (once again, we do not know how to construct N_{∇} in general). This is of no harm, as the proof of Lemma 1.2.2 does not use the N_{∇}-structure of 𝔒 in any way. Finally, we note that Lemma 1.1.4 of [Kis06], which plays an important role in the proof of [Kis06, 1.2.2], is well-known for 𝔒-modules in our more general context.^{4} □ As above, let us write D_{0} := 𝔒 ⊗_{F_{0}} D. We record here the following useful facts, which arise out of (our adaptation of) Kisin’s proof of Proposition 4.2.1:

4Indeed, Kisin’s proof of [Kis06, 1.1.4] relies on §4 of Berger’s paper [Ber02] as well as results of Lazard [Laz62, §7–8] and Lemma 2.4.1 of [Ked04], while the required facts in [Ber02] build on Lazard’s work in a natural way. But [Laz62] already deals in the generality we need, as does Kedlaya [Ked04]. Thus, one checks that all the proofs of the results needed to establish [Kis06, 1.1.4] (as well as Kisin’s argument itself) carry over mutatis mutandis to our more general situation.

(1) D_{0} ⊂ M ⊂ λ^{−r}D_{0}.

(2) ι_{0} induces an isomorphism of 𝔖̂_{0}-modules

𝔖̂_{0} ⊗_{𝔒} M(D) ≃ ∑_{j≥0}(u − π)^{−j}𝔖̂_{0} ⊗_{K} Fil^{j}D_{F_{0},K} = ∑_{j≥0} E^{−j}𝔖̂_{0} ⊗_{K} Fil^{j}D_{F_{0},K}.

Consider now the obvious inclusions D ↪ D_{0} ⊂ M(D). As Frobenius induces a linear isomorphism φ^{∗}D ≃ D, we obtain a linear isomorphism φ^{∗}D_{0} ≃ D_{0} and hence an injection ξ : D_{0} → φ^{∗}(M(D)). Defining 𝔒_{α} := 𝔖_{F} [[\frac{E^{p}}{ϖ}]][1/p], one checks that 𝔒_{α} = 𝔖_{F} [[\frac{u^{e_{0}p}}{ϖ}]][1/p] and that 𝔒 ⊂ 𝔒_{α} ⊂ 𝔒_{[0,|π|^{1/p})}.

Lemma 4.2.2. The map ξ_{α} := 𝔒_{α} ⊗_{𝔒} ξ : 𝔒_{α} ⊗_{F_{0}} D → 𝔒_{α} ⊗_{𝔒} φ^{∗}M(D) is an isomorphism.

Proof. Using the containments D_{0} ⊂ M(D) ⊂ λ^{−r}D_{0} and D_{0} ≃ φ^{∗}D_{0}, we see that D_{0} ⊂ φ^{∗}(M(D)) ⊂ φ(λ)^{−r}D_{0}. It is easy to check that φ(λ) is a unit in 𝔒_{α}, and it follows that ξ_{α} is an isomorphism. □

For an object M ∈ Mod_{𝔒}^{φ,r}, we define a decreasing filtration on φ^{∗}M by:

(4.2.2)  Fil^{i}(φ^{∗}M) := {x ∈ φ^{∗}M | (1 ⊗ φ)(x) ∈ E^{i}M}.

On the other hand, using the evident inclusions 𝔒_{α} ⊂ 𝔒_{[0,|π|^{1/p})} ⊂ 𝔖̂_{0} we obtain a canonical injection 𝔒_{α} ⊗_{F_{0}} D ↪ 𝔖̂_{0} ⊗_{K} D_{F_{0},K}, which allows us to equip 𝔒_{α} ⊗_{𝔒} D_{0} with the natural subspace filtration, using the tensor product filtration on 𝔖̂_{0} ⊗_{K} D_{F_{0},K}.

Lemma 4.2.3. The inverse isomorphism

\xymatrix{ξ′_{α} : 𝔒_{α} ⊗_{𝔒} φ^{∗}M(D) \ar[r]^{≃}_{_{(ξ_{α})_{−1}}} & 𝔒_{α} ⊗_{F_{0}} D}

of Lemma 4.2.2 is compatible with filtrations and Frobenius.

Proof. Clearly, ξ′_{α} is compatible with Frobenius. To prove that ξ′_{α} is filtration compatible, we use the two facts recorded after Proposition 4.2.1. As noted above, φ(λ) is a unit in 𝔖̂_{0}, so the first fact implies that the injective map ξ : D_{0} ≃ φ^{∗}D_{0} ↪ φ^{∗}M(D) is an isomorphism after tensoring with 𝔖̂_{0}. Put D̂_{0} := 𝔖̂_{0} ⊗_{𝔒} D_{0} and define an auxillary filtration on D̂_{0} by

F̃il^{i}D̂_{0} := D̂_{0} ∩ E^{i}(𝔖̂_{0} ⊗_{𝔒} M(D)).

From the very definition (4.2.2), it is clear that 1 ⊗ ξ : D̂_{0} ≃ 𝔖̂_{0} ⊗_{𝔒} φ^{∗}M(D)

carries F̃il^{i}D̂_{0} isomorphically onto Fil^{i}(𝔖̂_{0} ⊗_{𝔒} φ^{∗}M(D)). On the other hand, the second fact above implies that an element d ∈ D̂_{0} lies in E^{i}(𝔖̂_{0} ⊗_{𝔒} M(D))

if and only if ι_{0}(d) ∈ Fil^{i}(𝔖̂_{0} ⊗_{K} D_{F_{0},K}), from which F̃il^{i}D̂_{0} = Fil^{i}D̂_{0} follows. Hence ξ′_{α} is indeed compatible with filtrations. □

For simplicity, let us put M := M(D). It follows from Lemma 4.2.3 that the isomorphism ξ′_{α} specializes to give a natural identification of φ-modules D ≃ φ^{∗}M/uφ^{∗}M as well as a natural identification of filtered K-vector spaces D_{F_{0},K} ≃ φ^{∗}M/Eφ^{∗}M. Writing ψ_{π} for the composite mapping

ψ_{π} : φ^{∗}M ↠ φ^{∗}M/Eφ^{∗}M ≃ D_{F_{0},K},

we therefore obtain:

Corollary 4.2.4. The map ψ_{π} : φ^{∗}M(D) → D_{F_{0},K} is filtration compatible.

Remark 4.2.5. In the classical situation where F = ℚ_{p} and f(u) = u^{p}, to any object M of Mod_{𝔒}^{φ,r}, Kisin functorially associates a filtered φ-module via D(M) := φ^{∗}M/uφ^{∗}M with Fil^{i}(D(M)_{K}) := ψ_{π}(Fil^{i}φ^{∗}M). That this is possible rests crucially on the existence of a unique φ-equivariant isomorphism

ξ_{α} : 𝔒_{α} ⊗_{F_{0}} D(M) ≃ 𝔒_{α} ⊗_{𝔒} φ^{∗}M

reducing modulo u to the given identification D(M) = φ^{∗}M/uφ^{∗}M, which is Lemma 1.2.6 of [Kis06]. For more general F and f(u) = u^{p} + ⋯ + a_{1}u, we are only able to construct such a map ξ_{α} under the restriction ϖ^{r+1}|a_{1} in O_{F}; see Lemma 4.5.6.

To conclude this section, we record the following further consequence of Lemma 4.2.3: Setting B̃_{α} := W (R)[[\frac{E^{p}}{ϖ}]][1/p] ⊂ B_{cris,F}^{+} , one checks that the subspace filtration {Fil^{n}B̃_{α}}_{n} coincides with the filtration {E^{n}B̃_{α}}_{n}. As 𝔖_{F} ⊂ W (R)_{F}, we have a canonical inclusion 𝔒_{α} ⊂ B̃_{α}, and the map ξ_{α} of Lemma 4.2.2 induces a natural isomorphism

(4.2.3)  ξ′ ; B̃_{α} : B̃_{α} ⊗_{𝔒} φ^{∗}M ≃ B̃_{α} ⊗_{F_{0}} D.

As the inclusion 𝔖̂_{0} ⊂ B_{dR}^{+} is compatible with filtrations, we deduce:

Corollary 4.2.6. The map (4.2.3) is compatible with Frobenius and filtrations.

4.3. The classical setting. For future reference, we now recall the main results in Kisin’s classical situation, where F = ℚ_{p} and f(u) = u^{p}. In this subsection only, we fix a choice \underline{π} := \{π_{n}\}_{n} of p-power compatible roots of a fixed uniformizer π = π_{0} in K, and set K_{∞} := K\underline{π} and G_{∞} := G\underline{π}. The following summarizes the main results in this setting:

Theorem 4.3.1 ([Kis06]). Let V be a ℚ_{p}-valued crystalline representation of G with Hodge-Tate weights in {0, …, r} and T ⊂ V a G_{∞}-stable ℤ_{p}-lattice. Then:

(1) There exists a unique Kisin module 𝔐 so that T_{𝔖}(𝔐) ≃ T as ℤ_{p}[G_{∞}]- modules.

(2) If D = D_{cris}(V) is the associated filtered φ-module, then one has M(D) ≃ 𝔒 ⊗_{𝔖} 𝔐 as φ-modules.

Proof. These are the main results of [Kis06] restricted to crystalline representations. □

Now let F be an arbitrary extension of ℚ_{p} contained in K and let V be an F -crystalline representation and T a G_{∞}-stable O_{F} -lattice in V. Viewing V as a crystalline ℚ_{p}-valued representation and T as a G_{∞}-stable ℤ_{p}-module, by Theorem 4.3.1 there is a unique (classical) Kisin module 𝔐 attached to T, which is of finite Ẽ-height, for Ẽ := Ẽ(u) the minimal polynomial of π over K_{0} (we write Ẽ to distinguish this polynomial from our fixed E, which by definition is the minimal polynomial of π over F_{0} = F K_{0}). The additional O_{F} -structure on T is reflected on the classical Kisin module in the following way:

Corollary 4.3.2. The classical Kisin module 𝔐 is naturally a finite and free 𝔖_{F} -module and as such has E-height r.

Proof. Proposition 3.4 of [GLS14] shows that 𝔐 is naturally a finite and free 𝔖_{F} -module (see also the proof of [Kis08, Prop. 1.6.4]). Factor Ẽ in O_{F_{0}}[u] as Ẽ = E_{1} ⋯ E_{e_{F}} with E_{1} = E, and for each i write 𝔖̂_{E_{i}} for the completion of the localization of 𝔖_{F} at the ideal (E_{i}). We must prove that the injective map 1 ⊗ φ : φ^{∗}𝔐 → 𝔐 has cokernel killed by a power of E = E_{1}. To do this, it suffices to prove that the scalar extension

(4.3.1)  \xymatrix{φ_{i}^{♯} : 𝔖̂_{E_{i}} ⊗_{𝔖_{F}} φ^{∗}𝔐 \ar[r]^{1⊗(1⊗φ)} & 𝔖̂_{E_{i}} ⊗_{𝔖_{F}} 𝔐}

of 1⊗φ along 𝔖_{F} → 𝔖̂_{E_{i}} is an isomorphism when i > 1. Writing M := M(D), we recall that the map ψ_{π} : φ^{∗}M → D_{K} is compatible with filtrations thanks to Corollary 4.2.4, from which it follows that the map

\xymatrix{ψ_{π}|_{φ^{∗}𝔐} : φ^{∗}𝔐 \ar[r] & φ^{∗}𝔐/Ẽφ^{∗}𝔐 \ar@{^{(}->}[r] & D_{K}}

is also filtration-compatible. As V is F -crystalline, for any i > 1 we have Fil^{j}D_{K,𝔪_{i}} = 0 for all j ≥ 1, where 𝔪_{i} is the maximal ideal of F ⊗_{ℚ_{p}} K corresponding to E_{i}, and it follows that Fil^{1}φ^{∗}𝔐 ⊂ E_{i}φ^{∗}𝔐 for all i > 1. We then claim that for i > 1 the map \overline{1 ⊗ φ} : φ^{∗}𝔐/E_{i}φ^{∗}𝔐 → 𝔐/E_{i}𝔐 induced from 1 ⊗ φ by reduction modulo E_{i} is injective. To see this, observe that if x ∈ φ^{∗}𝔐 has (1 ⊗ φ)(x) = E_{i}m for m ∈ 𝔐, then writing y := ∏_{j≠i} E_{j}x, we have (1 ⊗ φ)(y) = Ẽm so that y ∈ Fil^{1}φ^{∗}𝔐 by the very definition of the filtration on φ^{∗}𝔐. By what we have seen above, we then have y ∈ E_{i}φ^{∗}𝔐, so since E_{i} is coprime to ∏_{j≠i} E_{j}, we obtain x ∈ E_{i}φ^{∗}𝔐 as claimed. Now both φ^{∗}𝔐 and 𝔐 are 𝔖_{F} -free of the same rank, so as \overline{1 ⊗ φ} is injective, we see that ℚ_{p} ⊗_{ℤ_{p}} \overline{1 ⊗ φ} is an isomorphism for i > 1. But this map coincides with the map \overline{φ}_{i}^{♯} obtained from (4.3.1) by reduction modulo E_{i}, so it follows that φ_{i}^{♯} is an isomorphism as well, as desired. □

4.4. Comparing constructions. Let us first recall some standard facts about the Robba ring as in [Kis06]. For finer details of the Robba ring R and its subring R^{b}, we refer to §2 (in particular §2.3) of [Ked04], noting that several different notations are commonly used (in particular, we advise the reader that R^{b} = E^{†} = Γ_{con}^{k((t))}[1/p]). The Robba ring is defined as

R := lim_{s→1^{−}} 𝔒_{(s,1)}

and comes equipped with a Frobenius endomorphism, which is induced by the canonical maps φ : 𝔒_{(s,1)} → 𝔒_{(s^{1/p},1)}. Writing 𝔒_{(s,1)}^{b} ⊂ 𝔒_{(s,1)} for the subring of functions which are bounded, we also define the bounded Robba ring:

R^{b} := lim_{s→1^{−}} 𝔒_{(s,1)}^{b},

which is naturally a Frobenius-stable subring of R. Finally, we put

O_{R^{b}} := {∑_{n∈ℤ} a_{n}u^{n} ∈ R^{b} | a_{n} ∈ O_{F_{0}} ∀n ∈ ℤ};

this is a Henselian discrete valuation ring with uniformizer ϖ and residue field k((u)). One checks that the fraction field of O_{R^{b}} is R^{b}, justifying our notation; in particular, R^{b} is a field. Note that 𝔒 is canonically a Frobenius-stable subring of R. By definition, a φ-module over R is a finite free R-module M equipped with a φ-semilinear map φ_{M} : M → M whose linearization 1 ⊗ φ : φ^{∗}M → M is an isomorphism. One checks that E(u) ∈ 𝔒 is a unit in R, so that scalar extension along 𝔒 ↪ R gives a functor from φ-modules over 𝔒 to φ-modules over R. A φ-module M over R is étale if M admits a basis with the property that the corresponding matrix of φ_{M} lies in GL_{d}(O_{R^{b}}); by a slight abuse of terminology, we will say that a φ-module over 𝔒 is étale if its scalar extension to R is. Our main result of this subsection is the following:

Theorem 4.4.1. Let V ∈ Rep_{F}^{F -cris,r}(G) and write D := D_{cris,F} (V) for the corresponding filtered φ-module. If M(D) is the φ-module over 𝔒 attached to D as in §4.2, we have:

(1) M(D) is étale;

(2) There exists a Kisin module 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r} such that 𝔒⊗_{𝔖_{F}} 𝔐 ≃ M(D).

First note that there is a canonical inclusion 𝔖_{F} ↪ O_{R^{b}}, so that (2) implies (1). It follows that the above theorem is true in the classical setting of Kisin by Theorem 4.3.1. In what follows, we will reduce the general case of Theorem 4.4.1 to the known instance of it in the classical setting. To ease notation, we will adorn various objects with a superscript of “c” to signify that they are objects in the classical setting. We likewise abbreviate M := M(D) and M^{c} := M^{c}(D). We note that 𝔒_{α}^{c} ⊂ B̃_{α}, as E(u^{c}) is another generator of Fil^{1}W (R)_{F} so E(u^{c}) = μE(u) for some μ ∈ W (R)_{F}^{×} thanks to Lemma 2.1.3.

By Corollary 4.2.6, the B̃_{α}-linear isomorphism ξ′ ; B̃_{α} : B̃_{α} ⊗_{𝔒} φ^{∗}M ≃ B̃_{α} ⊗_{F_{0}} D is Frobenius and filtration compatible. The key point is that the Frobenius and filtration on B̃_{α} ⊗_{F_{0}} D are canonical (recall that the filtration on B̃_{α} ⊗_{F_{0}} D is induced by the inclusion B̃_{α} ⊗_{F_{0}} D ↪ B_{dR}^{+} ⊗_{K} D_{F_{0},K}) and are independent of the choice of φ(u) = f(u). We therefore have a natural isomorphism

(4.4.1)  ξ̃: B̃_{α} ⊗_{𝔒} φ^{∗}M ≃ B̃_{α} ⊗_{𝔒^{c}} φ^{∗}M^{c},

that is Frobenius and filtration compatible.

Lemma 4.4.2. There is a B̃_{α}-linear and Frobenius-compatible isomorphism

η : B̃_{α} ⊗_{𝔒} M ≃ B̃_{α} ⊗_{𝔒^{c}} M^{c}.

Proof. Choose an 𝔒-basis e_{1}, …, e_{d} of M, and let A ∈ M_{d}(𝔒) be the corresponding matrix of Frobenius, so (φ(e_{1}), …, φ(e_{d})) = (e_{1}, …, e_{d})A. We write 𝔢_{i} := 1 ⊗ e_{i} ∈ φ^{∗}M for the induced basis of φ^{∗}M. Using the definition of Fil^{i}φ^{∗}M, it is not difficult to see that there is a matrix B ∈ M_{d}(𝔒) satisfying AB = BA = E^{r}I_{d} and with the property that Fil^{r}φ^{∗}M is generated by (α_{1}, …, α_{d}) := (𝔢_{1}, …, 𝔢_{d})B. As promised, we denote by 𝔢_{i}^{c}, A^{c}, etc. the objects in the classical setting corresponding to a choice e_{1}^{c}, …, e_{d}^{c} of 𝔒^{c}-basis of M^{c}. Let X ∈ GL_{d}(B̃_{α}) be the matrix determined by the requirement ξ̃(𝔢_{1}, …, 𝔢_{d}) = (𝔢_{1}^{c}, …, 𝔢_{d}^{c})X. As ξ̃ is compatible with both Frobenius and filtrations, we find that

ξ̃∘φ(𝔢_{1}, …, 𝔢_{d}) = (𝔢_{1}^{c}, …, 𝔢_{d}^{c})Xφ(A) = φ∘ξ̃(𝔢_{1}, …, 𝔢_{d}) = (𝔢_{1}^{c} …, 𝔢_{d}^{c})φ(A^{c})φ(X)

and there exists a matrix Y ∈ GL_{d}(B̃_{α}) with

ξ̃(α_{1}, …, α_{d}) = (α_{1}^{c}, …, α_{d}^{c})Y.

We conclude that Xφ(A) = φ(A^{c})φ(X) and XB = B^{c}Y. Since B̃_{α} is an integral domain, the facts that B = E(u)^{r}A^{−1} and B^{c} = E(u^{c})^{r}(A^{c})^{−1} imply that A^{c}XE(u)^{r} = E(u^{c})^{r}Y A. Due to Lemma 2.1.3, both E(u) and E(u^{c}) are generators of Fil^{1}W (R)_{F}, so μ := E(u^{c})/E(u) is a unit in W (R)_{F}. We therefore have the relation A^{c}X = μ^{r}Y A. Combining this with the equality Xφ(A) = φ(A^{c})φ(X) yields X = φ(μ^{r}Y), and we deduce A^{c}φ(μ^{r}Y) = μ^{r}Y A. Defining a B̃_{α}-linear map

\xymatrix{η : B̃_{α} ⊗_{𝔒} M \ar[r] & B̃_{α} ⊗_{𝔒^{c}} M^{c}}

by the requirement (η(e_{1}), …, η(e_{d})) = (e_{1}^{c}, …, e_{d}^{c})μ^{r}Y, one then checks that η provides the claimed Frobenius-compatible isomorphism. □

Recall that Lemma 2.2.1 gives inclusions 𝔒 ⊂ B̃_{rig,F}^{+} ⊂ B̃_{α}.

Corollary 4.4.3. The isomorphism η of Lemma 4.4.2 descends to a B̃_{rig,F}^{+} - linear and Frobenius-compatible isomorphism

η_{rig} : B̃_{rig,F}^{+} ⊗_{𝔒} M ≃ B̃_{rig,F}^{+} ⊗_{𝔒^{c}} M^{c}.

Proof. We will use the notation of the proof of Lemma 4.4.2. Let us put Z := μ^{r}Y ∈ GL_{d}(B̃_{α}), so that η(e_{1}, …, e_{d}) = (e_{1}, …, e_{d})Z, and note that A^{c}φ(Z) = ZA as η is compatible with Frobenius. To prove the corollary, it suffices to show that both Z and Z^{−1} have entries in B̃_{rig,F}^{+} . We will show that Z ∈ M_{d}(B̃_{rig,F}^{+} ); the proof of the corresponding fact for Z^{−1} is similar and is left to the reader. By the definition of B̃_{rig,F}^{+} , it suffices to show that for any m, there exists Z_{m} ∈ M_{d}(B_{max,F}^{+} ) with φ^{m}(Z_{m}) = Z, which we prove by induction on m. The base case m = 0 is obvious, as B̃_{α} ⊂ B_{max,F}^{+} . Now suppose that Z_{m} exists, and note that from the equality A^{c}φ(Z) = ZA we obtain E(u)^{r}Z = A^{c}φ(Z)B. We may write A^{c} = φ^{m+1}(A_{m+1}) and B = φ^{m+1}(B_{m+1}) thanks to Lemma 2.2.1, and we then have E(u)^{r}Z = φ^{m+1}(A_{m+1}Z_{m}B_{m+1}). Finally, Lemma 2.1.5 implies the existence of Z_{m+1}. □

We can now prove Theorem 4.4.1:

Proof of Theorem 4.4.1. We first prove that M is étale, and to do so we will freely use the results and notation of [Ked04]. By the main theorem of [Ked04], M is étale if and only if M is pure of slope 0. Hence M^{c} is pure of slope 0 thanks Theorem 4.3.1 and our remarks immediately following Theorem 4.4.1. Since the slope filtration of M does not change after tensoring with the ring Γ_{an,con}^{alg} constructed in [Ked04], it is enough to show that

Γ_{an,con}^{alg} ⊗_{𝔒} M ≃ Γ_{an,con}^{alg} ⊗_{𝔒^{c}} M^{c}.

as φ-modules over Γ_{an,con}^{alg}, and to do this it is enough thanks to Lemma 4.4.2 to prove that B̃_{rig,F}^{+} ⊂ Γ_{an,con}^{alg}. But this follows from Berger’s construction [Ber02, §2.3] (strictly speaking, [Ber02, §2.3] deals only with the case F = ℚ_{p}, but see the last paragraph of [Ber14, §3] for the general case. We also warn the readers that Berger use B̃_{rig}^{†} to denote B̃_{rig,F}^{†} in this paper, while his B̃_{rig,F}^{†} means a different ring from ours), as he proves that B̃_{rig,F}^{+} ⊂ B_{rig,F}^{†} = B̃_{rig}^{†} = Γ_{an,con}^{alg} (see the table over [Ber08, Prop. 1.1.12] for a helpful comparison of the various notations used by different authors). It follows that M is étale. Now the proof that M := M(D) admits a descent to a Kisin module 𝔐 is exactly the same as the proof of Lemma 1.3.13 in [Kis06], so we just sketch the highlights. As M is étale, there exists a finite free O_{R^{b}}-module N with Frobenius endomorphism φ_{N} satisfying

(4.4.2)  R ⊗_{O_{R^{b}}} N ≃ R ⊗_{𝔒} M =: M_{R}

Proposition 6.5 in [Ked04] shows that it is possible to select an R-basis of M_{R} whose R^{b}-span is exactly N [1/p] and whose 𝔒-span is M via the identifications (4.4.2). Define M^{b} ⊆ M to be the 𝔖_{F} [1/p]-span of this basis. The equality 𝔖_{F} [1/p] = R^{b} ∩ 𝔒 provides the intrinsic description M^{b} = M ∩ N [1/p]; in particular, M^{b} is φ-stable and of E-height r. Let 𝔐′ := M^{b} ∩ N and put 𝔐 := (O_{R^{b}} ⊗_{𝔖_{F}} 𝔐′) ∩ 𝔐′[1/p] ⊂ N [1/p]. Then 𝔐 is a finite and φ-stable 𝔖_{F} -submodule of N [1/p]. It follows from the structure theorem of finite 𝔖_{F} - modules [Fon90, Proposition 1.2.4] that 𝔐 is in fact finite and free over 𝔖_{F}. To see that 𝔐 has E-height r, it suffices to check that det(φ_{𝔐}) = E^{s}w for some w ∈ 𝔖_{F}^{×}. But 𝔐[1/p] = M^{b} and M^{b} is of finite E-height, so det(φ_{𝔐}) = p^{m}E^{s}w for some w ∈ 𝔖_{F}^{×}; as M is pure of slope 0 (equivalently, det(φ_{𝔐}) ∈ O_{R^{b}}^{×}), we must in fact have m = 0. □

4.5. Full-faithfulness of restriction. Fix an object V of Rep_{F}^{F -cris,r}(G), and let 𝔐 be a Kisin module associated to V via Theorem 4.4.1 (2).

Proposition 4.5.1. There exists a natural O_{F} -linear injection

ι : T_{𝔖_{F}} (𝔐) ↪ V ≃ V_{cris,F} (D)

that is moreover G\underline{π}-equivariant. In particular, V_{𝔖}(𝔐) ≃ V as F [G\underline{π}]-modules.

Proof. Set M := M(D). As M ≃ 𝔒 ⊗_{𝔖_{F}} 𝔐, we have a natural injection

ι′ : T_{𝔖_{F}} (𝔐) = Hom_{𝔖_{F} ,φ}(𝔐, 𝔖_{F}^{ur}) ↪ Hom_{𝔒,φ,Fil}(φ^{∗}M, B_{cris,F}^{+} ),

uniquely determined by the requirement that for any h : 𝔐 → 𝔖_{F}^{ur}, the value of ι′(h) on any simple tensor s ⊗ m ∈ 𝔒 ⊗_{φ,𝔖_{F}} 𝔐 ≃ φ^{∗}M is given by

ι′(h)(s ⊗ m) = sφ(h(m)).

Using the fact that E ∈ Fil^{1}W (R)_{F}, one checks that this really does define a filtration-compatible 𝔒-linear homomorphism ι′(h) : φ^{∗}M → B_{cris,F}^{+} .

On the other hand, the isomorphism ξ_{α} : 𝔒_{α} ⊗_{F_{0}} D ⟶^{∼} 𝔒_{α} ⊗_{𝔒} φ^{∗}M of Lemma 4.2.2 induces, thanks to Lemma 4.2.3, a natural injection

\xymatrix{Hom_{𝔒,φ,Fil}(φ^{∗}M, B_{cris,F}^{+} ) \ar@{^{(}->}[r]^{h↦1⊗h} & Hom_{𝔒_{α},φ,Fil}(𝔒_{α} ⊗_{𝔒} φ^{∗}M, B_{cris,F}^{+} ) \ar[d]_{≃}^{h↦h∘ξ_{α}} \\  & Hom_{𝔒_{α},φ,Fil}(𝔒_{α} ⊗_{F_{0}} D, B_{cris,F}^{+} )}

and we claim that Hom_{𝔒_{α},φ,Fil}(𝔒_{α} ⊗_{F_{0}} D, B_{cris,F}^{+} ) = V_{cris,F} (D). By definition,

V_{cris,F} (D) = Hom_{F_{0},φ}(D, B_{cris,F}^{+} ) ∩ Hom_{K,Fil}(D_{F_{0},K}, B_{dR}^{+}),

and it is clear that Hom_{𝔒_{α},φ}(𝔒_{α} ⊗_{F_{0}} D, B_{cris,F}^{+} ) = Hom_{F_{0},φ}(D, B_{cris,F}^{+} ). Since the injection 𝔒_{α} ⊗_{𝔒} D ↪ 𝔖̂_{0} ⊗ D_{F_{0},K} is compatible with filtrations by the very construction of the filtration on 𝔒_{α} ⊗_{F_{0}} D, we conclude that

Hom_{F_{0}}(D, B_{cris,F}^{+} ) ∩ Hom_{𝔒_{α},Fil}(𝔒_{α} ⊗_{F_{0}} D, B_{cris,F}^{+} )
≃ Hom_{F_{0}}(D, B_{cris,F}^{+} ) ∩ Hom_{K,Fil}(D_{F_{0},K}, B_{dR}^{+}),

which gives our claim. We thus obtain a natural injection ι : T_{𝔖_{F}} (𝔐) ↪ V_{cris,F} (D) which is visibly compatible with the given G\underline{π}-actions. □

Combining Theorem 4.4.1 and Proposition 3.3.3, we deduce Theorem 1.0.1:

Corollary 4.5.2. Let V be an object of Rep_{F}^{F -cris,r}(G) and T ⊂ V a G\underline{π}-stable O_{F} -lattice. Then there is a Kisin module 𝔐 of E-height r with T_{𝔖_{F}} (𝔐) ≃ T.

Remark 4.5.3. It is an important point that in our general setup, the Kisin module 𝔐 may not be unique for a given T, contrary to the classical situation. Indeed, the “cyclotomic case” of Example 2.1.2 is a prototypical instance of such non-uniqueness: let T be the trivial character, 𝔐 = 𝔖_{F} the trivial rank-1 Kisin module, and 𝔐′ = u𝔖_{F} ⊂ 𝔐. Since φ(u) = E(u)u, one sees that 𝔐′ is also a Kisin module and T_{𝔖_{F}} (𝔐) = T_{𝔖_{F}} (𝔐′) = T.

We now prove Theorem 1.0.2:

Theorem 4.5.4. Assume that φ^{n}(f(u)/u) is not a power of E(u) for any n ≥ 0 and that v_{F} (a_{1}) > r. Then the restriction functor Rep_{F}^{F -cris,r}(G) → Rep_{F} (G\underline{π}) given by V ⇝ V |_{G\underline{π}} is fully faithful.

Remark 4.5.5. We suspect that the theorem remains valid if we drop the assumption that φ^{n}(f(u)/u) is not a power of E for any n ≥ 0. When r = 1, we will show that this is indeed the case in the next section.

In order to prove Theorem 4.5.4, we prepare several preliminaries. In what follows, we keep our running notation with f(u) = u^{p} + a_{p−1}u^{p−1} + ⋯ + a_{1}u, and we assume throughout that ϖ^{r+1}|a_{1} in O_{F}. Let 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r} and set M := φ^{∗}𝔐/uφ^{∗}𝔐.

Lemma 4.5.6. There exists a unique φ-equivariant isomorphism

\xymatrix{ξ_{α} : 𝔒_{α} ⊗_{O_{F_{0}}} M \ar[r]^{≃} & 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐}

whose reduction modulo u is the identity on M.

Proof. The proof is similar to that of Proposition 2.4.1 in [Liu11], and is motivated by the proof of Lemma 1.2.6 in [Kis06]. Choose an 𝔖_{F} - basis 𝔢_{1}, …, 𝔢_{d} of 𝔐 and let A ∈ M_{d}(𝔖_{F}) be the resulting matrix of φ; i.e. (φ(𝔢_{1}), …, φ(𝔢_{d})) = (𝔢_{1}, …, 𝔢_{d})A. Then e_{i} := 1 ⊗ 𝔢_{i} forms a basis of φ^{∗}𝔐 and we have (φ(e_{1}), …, φ(e_{d})) = (e_{1}, …, e_{d})φ(A). Put A_{0} := A mod u and ē_{i} := e_{i} mod u. Then we have (φ(ē_{1}), …, φ(ē_{d})) = (ē_{1}, …, ē_{d})φ(A_{0}). If the map ξ_{α} of the Lemma exists, then writing f_{i} := ξ_{α}(ē_{i}) ∈ 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐 and denoting by Y ∈ GL_{d}(𝔒_{α}) the matrix with (f_{1}, …, f_{d}) = (e_{1}, …, e_{d})Y, we necessarily have Y ≡ I_{d} mod u and

(4.5.1)  Y φ(A_{0}) = φ(A)φ(Y).

Conversely, if (4.5.1) has a solution Y ∈ GL_{d}(𝔒_{α}) satisfying Y ≡ I_{d} mod u, then we may define ξ_{α} by ξ_{α}(ē_{1}, …, ē_{d}) = (e_{1}, …, e_{d})Y. Thus, it remains to solve Equation (4.5.1). Put

(4.5.2)  Y_{n} := φ(A) ⋯ φ^{n}(A)φ^{n}(A_{0}^{−1}) ⋯ φ(A_{0}^{−1}).

We claim that the sequence {Y_{n}}_{n} converges to a matrix Y ∈ M_{d}(𝔒_{α}). To see this, note that there exists B_{0} ∈ GL_{d}(𝔒_{α}) with A_{0}B_{0} = ϖ^{r}I_{d} since 𝔐 has height r. It follows that AA_{0}^{−1} = I_{d} + \frac{u}{ϖ^{r}} Z for Z ∈ M_{d}(𝔖_{F}). Thus,

Y_{n} = Y_{n−1} + φ(A) ⋯ φ^{n−1}(A)\frac{φ^{n}(u)}{ϖ^{rn}}φ^{n}(Z)φ^{n−1}(B_{0}) … φ(B_{0}),

so to prove our claim it suffices to show that φ^{n}(u)/ϖ^{rn} converges to 0 in 𝔒_{α}, which is the content of Lemma 2.2.2. To prove that Y is invertible, we compute its determinant. Put d := rank_{𝔖_{F}} 𝔐, and observe that since ∧^{d}𝔐 has finite E-height, we have det(A) = γE^{m} for some γ ∈ 𝔖_{F}^{×}. It follows that det(φ(A)φ(A_{0}^{−1})) = γ′(\frac{φ(E)}{ϖ})^{m} for some γ′ ∈ 𝔖_{F}^{×}. One then checks that φ(E)/ϖ is a unit in 𝔒_{α}, and hence that det(Y) is a unit in 𝔒_{α} so Y is invertible as desired. Finally, we prove that the solution Y to (4.5.1) that we have constructed is unique. Suppose that equation (4.5.1) admits two solutions Y, Y′ satisfying Y, Y′ ≡ I_{d} mod u. Then their difference is also a solution Y − Y′ = uZ for Z ∈ M_{d}(𝔒_{α}). Equation (4.5.1) then implies that for all n we have

Y − Y′ = φ(A)φ(Y)φ(A_{0}^{−1}) = φ(A) ⋯ φ^{n}(A)φ^{n}(Y)φ^{n}(A_{0}^{−1}) ⋯ φ(A_{0}^{−1})
= φ(A) ⋯ φ^{n}(A)\frac{φ^{n}(u)}{ϖ^{rn}}φ^{n}(Z)φ^{n}(B_{0}) ⋯ φ(B_{0})

As φ^{n}(u)/ϖ^{rn} converges to 0 in 𝔒_{α}, we conclude that Y = Y′ as desired. □

For 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r}, let us write D(𝔐) = ξ_{α}(M[1/p]) ⊂ 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐 for the image of M[1/p] under the map of Lemma 4.5.6. If 𝔐 is a Kisin module associated to some F -crystalline G-representation V with D := D_{cris,F} (V) (i.e. 𝔒 ⊗_{𝔖_{F}} 𝔐 ≃ M(D)), then by the very construction of M(D) there is a natural φ-compatible inclusion D ≃ φ^{∗}D ↪ φ^{∗}M(D) which, thanks to Lemma 4.2.2, becomes an isomorphism after tensoring over 𝔒 with 𝔒_{α}. Recalling that 𝔒_{α} ⊗_{𝔖_{F}} 𝔐 ≃ M(D), we therefore have a φ-equivariant inclusion

(4.5.3)  [figure: D; d↦1⊗d; 𝔒_{α} ⊗_{F_{0}} D; ≃; 4.2.2; 𝔒_{α} ⊗_{𝔒} φ^{∗}M(D) ≃ 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐]

via which we view D as a φ-stable F_{0}-subspace of 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐.

Corollary 4.5.7. Let V ∈ Rep_{F}^{F -cris,r}(G). If 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r} is a Kisin module attached to D := D_{cris,F} (V), then D(𝔐) = D inside 𝔒_{α} ⊗_{𝔖} φ^{∗}𝔐.

Proof. The reduction of (4.5.3) modulo u is the φ-compatible isomorphism

D ≃ (𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐) mod u ≃ M[1/p].

Since the map ξ_{α} of Lemma 4.5.6 reduces to the identity modulo u, we conclude that both D and D(𝔐) inside 𝔒_{α}⊗_{𝔖_{F}} φ^{∗}𝔐 are φ-equivariant liftings of M[1/p]; the uniqueness aspect of Lemma 4.5.6 then forces D = D(𝔐) as claimed. □

It follows from Corollary 4.5.7 that the map ξ_{α} of Lemma 4.5.6 coincides with that of Lemma 4.2.2, which justifies our notation. Recall that V_{𝔖}(𝔐) = T_{𝔖}(𝔐)[1/p] for 𝔐 ∈ Mod_{𝔖_{F}}^{φ,r}.

Lemma 4.5.8. Let 𝔣 : 𝔐 → 𝔐′ be any morphism of height-r Kisin modules, and let 𝔣_{α} be the scalar extension 𝔣_{α} : 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐 → 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐′ of φ^{∗}𝔣 along 𝔖_{F} → 𝔒_{α}. Then 𝔣_{α}(D(𝔐)) ⊂ D(𝔐′).

Proof. Put V = V_{𝔖}(𝔐) and V′ = V_{𝔖}(𝔐′) and denote by ψ = V_{𝔖}(𝔣) : V′ → V the induced map. By Proposition 3.3.3, we can modify 𝔐 and 𝔐′ (inside 𝔐[1/p] and 𝔐′[1/p], respectively) so that 𝔣 may be decomposed by two exact sequences inside Mod_{𝔖_{F}}^{φ,r}:

\xymatrix{0 \ar[r] & 𝔏 \ar[r] & 𝔐 \ar[r]^{𝔣} & 𝔑 \ar[r] & 0} and \xymatrix{0 \ar[r] & 𝔑 \ar[r] & 𝔐′ \ar[r] & 𝔑′ \ar[r] & 0},

where 𝔑 = 𝔣(𝔐). From the construction of ξ_{α} in Lemma 4.5.6 (in particular, from the explicit construction of Y in (4.5.2)), we obtain the exact sequences

\xymatrix{0 \ar[r] & D(𝔏) \ar[r] & D(𝔐) \ar[r]^{𝔣_{α}} & D(𝔑) \ar[r] & 0}

and

\xymatrix{0 \ar[r] & D(𝔑) \ar[r] & D(𝔐′) \ar[r] & D(𝔑′) \ar[r] & 0},

which shows that 𝔣_{α}(D(𝔐)) ⊂ D(𝔐′) as claimed. □

Proof of Theorem 4.5.4. Let V′, V be two objects of Rep_{F}^{F -cris,r}(G), and set D = D_{cris,F} (V) and D′ = D_{cris,F} (V′). Suppose that there exists an F -linear map h : V_{cris,F} (D′) → V_{cris,F} (D) that is G\underline{π}-equivariant. By Corollary 4.5.2, there exist G\underline{π}-stable O_{F} -lattices T and T′ inside V_{cris,F} (D) and V_{cris,F} (D′), respectively, with h(T′) ⊆ T, and objects 𝔐 and 𝔐′ of Mod_{𝔖_{F}}^{φ,r} such that T_{𝔖}(𝔐) ≃ T and T_{𝔖}(𝔐′) ≃ T′ via the map ι of Proposition 4.5.1. By Proposition 3.3.5, there exists a map 𝔣 : 𝔐 → 𝔐′ in Mod_{𝔖_{F}}^{φ,r} with V_{𝔖_{F}} (𝔣) ≃ h. We may therefore realize h as the composite

\xymatrix{V_{cris,F} (D′) \ar[r]^{ι^{−1}}_{∼} & V_{𝔖}(𝔐′)[1/p] \ar[r]^{V_{𝔖}(𝔣)} & V_{𝔖}(𝔐)[\frac{1}{p}] \ar[r]^{ι}_{∼} & V_{cris,F} (D)} ,

where ι is constructed using the isomorphism ξ′_{α} : 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐 ≃ 𝔒_{α} ⊗ D of Lemma 4.2.3. Due to Lemma 4.5.8 and Corollary 4.5.7, we know that 𝔣 induces a map 𝔣_{α} : 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐 → 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐′ carrying D to D′, so for a ∈ V_{cris,F} (D′) ⊂ Hom_{F_{0},φ}(D, B_{cris,F}^{+} ) we have h(a) = a ∘ 𝔣̄ ∈ V_{cris,F} (D′) ⊂ Hom_{F_{0},φ}(D′, B_{cris,F}^{+} ) where we write 𝔣̄ : D → D′ for the map 𝔣_{α}|_{D}. It follows at once that h is compatible with the action of G = G_{K}, as desired. □

We note that Theorem 4.5.4 is false if we replace “Rep_{F}^{F -cris}(G)” with “Rep_{F}^{ℚ_{p}-cris}(G),” as the following example shows:

Example 4.5.9. Consider again the setting of Example 2.1.2, with K = ℚ_{p}(ζ_{p}), π = ζ_{p} − 1 and φ(u) = f(u) = (1 + u)^{p} − 1, and K\underline{π} = ⋃_{n≥1} ℚ_{p}(ζ_{p^{n}}). Let F = ℚ_{p}. Then the assumption of Theorem 4.5.4 is not satisfied as a_{1} = p, and the restriction functor Rep_{F}^{ℚ_{p}-cris}(G) → Rep_{F} (G_{∞}) induced by V ⇝ V |_{G\underline{π}} is visibly not fully faithful: letting χ denote the p-adic cyclotomic character, we have χ|_{G\underline{π}} = 1|_{G\underline{π}}, but χ ≄ 1 as G-representations. On the other hand, if F = K then we easily check that the assumptions of Theorem 4.5.4 are satisfied. Of course, there is no contradiction here as χ is not an F -crystalline representation because HT_{τ} (χ) = 1 for all τ.

5. F -Barsotti-Tate groups

Recall that by an F -Barsotti–Tate group over O_{K}, we mean a Barsotti–Tate group over O_{K} whose p-adic Tate module is an F -crystalline representation of G := G_{K}. In this section, we prove that the category of F -Barsotti–Tate groups over O_{K} is (anti)equivalent to the category of height-1 Kisin modules:

Theorem 5.0.10. Assume v_{F} (a_{1}) > 1. Then there is an (anti)equivalence of categories between the category of Kisin modules of height 1 and the category of F -Barsotti-Tate groups.

Using well-known results of Breuil, Kisin, Raynaud, and Tate, one shows as in [Liu13b, Theorem 2.2.1] that the p-adic Tate module gives an equivalence between thet category of F -Barsotti–Tate groups over O_{K} and the category Rep_{O_{F}}^{F -cris,1}(G) of G-stable O_{F} -lattices inside F -crystalline representations with Hodge-Tate weights in {0, 1}. Thus, to prove Theorem 5.0.10 we must construct an (anti)equivalence between Mod_{𝔖_{F}}^{φ,1} and Rep_{O_{F}}^{F -cris,1}(G). In what follows, we show that for each 𝔐 ∈ Mod_{𝔖_{F}}^{φ,1} the natural G\underline{π}-action on T_{𝔖}(𝔐) can be functorially extended to to a G-action such that T_{𝔖}(𝔐) ∈ Rep_{O_{F}}^{F -cris,1}(G). This construction will provide a contravariant functor T_{𝔖} : Mod_{𝔖_{F}}^{φ,1} → Rep_{O_{F}}^{F -cris,1}(G) that we will then prove is an (anti)equivalence.

5.1. A natural G-action on T_{𝔖}(𝔐). Fix a Kisin module 𝔐 of height 1. In this subsection, we will construct a natural G-action on T_{𝔖}(𝔐) which extends the given action of G\underline{π}. The key input to this construction is:

Lemma 5.1.1. There exists a unique W (R)_{F} -semilinear G-action on W (R)_{F} ⊗_{𝔖_{F}} 𝔐 that commutes with φ and satisfies

(1) If g ∈ G\underline{π} and m ∈ 𝔐 then g(1 ⊗ m) = 1 ⊗ m;
(2) If m ∈ 𝔐 then 1 ⊗ (g(m) − m) ∈ 𝔱_{F} I^{+}(W (R)_{F} ⊗_{𝔖_{F}} 𝔐).

Here, we remind the reader that 𝔱_{F} ∈ W (R)_{F}, constructed in Example 3.3.2, satisfies φ(𝔱_{F}) = E𝔱_{F} and 𝔱_{F} ≢ 0 mod ϖ.

Proof. Fix an 𝔖_{F} -basis 𝔢_{1}, …, 𝔢_{d} of 𝔐 and let A be the resulting matrix of Frobenius, so (φ(𝔢_{1}), …, φ(𝔢_{d})) = (𝔢_{1}, …, 𝔢_{d})A. Supposing that the required G-action exists, for any g ∈ G we have a matrix X_{g} ∈ M_{d}(W (R)_{F}) with (g𝔢_{1}, …, g𝔢_{d}) = (𝔢_{1}, …, 𝔢_{d})X_{g}, and the requirement that g and φ commute is equivalent the matrix equation

(5.1.1)  X_{g}g(A) = Aφ(X_{g}).

We claim that for each g ∈ G, equation (5.1.1) has a unique solution X_{g} satisfying the condition X_{g} − I_{d} ∈ M_{d}(𝔱_{F} I^{+}). Granting this for a moment, it is easy to see that the Lemma follows once we check that g ↦ X_{g} really defines an action of G, which is equivalent to the cocycle condition X_{σ}σ(X_{τ}) = X_{στ} for all σ, τ ∈ G. But it is clear that X_{σ}σ(X_{τ}) and X_{στ} are both solutions to Xστ(A) = Aφ(X), and the condition X−I_{d} ∈ M_{d}(𝔱_{F} I^{+}) holds for X = X_{στ} by our claim and for X = X_{σ}σX_{τ} thanks to Lemma 2.3.4. Thus, the uniqueness aspect of our claim gives X_{σ}σ(X_{τ}) = X_{στ}, as desired. It remains to prove our claim. Let us first dispense with the uniqueness aspect. Suppose that for some g ∈ G, equation (5.1.1) has two solutions X_{1}, X_{2} satisfying X_{i} − I_{d} ∈ M_{d}(𝔱_{F} I^{+}) for i = 1, 2. Then their difference is a solution as well, and has the form X_{1} − X_{2} = 𝔱_{F} Z for some Z ∈ M_{d}(I^{+}). Equation (5.1.1) then takes the shape

(5.1.2)  𝔱_{F} Zg(A) = Aφ(𝔱_{F} Z),

and we will show that this forces Z = 0, giving uniqueness. First, writing Z̄ := Z mod ϖ ∈ M_{d}(R), we note that it suffices to prove that Z̄ = 0: indeed, if Z = ϖZ_{1} for some Z_{1} ∈ M_{d}(W (R)_{F}), then Z_{1} ∈ M_{d}(I^{+}) is another solution to (5.1.2), so boot-strapping the argument gives Z ∈ ∩_{n≥1}ϖ^{n}W (R)_{F} = {0}. Now since 𝔐 has height 1, there exists a matrix B ∈ M_{d}(𝔖_{F}) with AB = EI_{d}. On the other hand, we have φ(𝔱_{F}) = E𝔱_{F} as noted above, so it follows from (5.1.2) that there exists a matrix C ∈ M_{d}(W (R)_{F}) with Z = Aφ(Z)C. Reducing modulo ϖ gives a matrix equation Z̄ = Āφ(Z̄)C̄ in M_{d}(R). If Z̄ ≠ 0, then there exists an entry z, say, of Z̄ which has minimal valuation. On the other hand, as Z ∈ M_{d}(I^{+}), we must have v_{R}(z) > 0. But the minimal possible valuation of entries in Āφ(Z̄)C̄ is pv_{R}(z) > v_{R}(z), which is a contradiction. Thus Z̄ = 0, settling uniqueness. Finally, let us prove the existence of X_{g} solving (5.1.1) for each g ∈ G. For ease of notation, put

P_{n} := Aφ(A) ⋯ φ^{n}(A) and Q_{n} := φ^{n}(g(A^{−1})) ⋯ φ(g(A^{−1}))g(A^{−1}) (5.1.3)

and define X_{n} := P_{n}Q_{n}. It suffices to prove the following:

(1) X_{n} ∈ M_{d}(W (R)_{F}) for all n;
(2) X_{n} − I_{d} ∈ M_{d}(𝔱_{F} I^{+}) for all n;
(3) X_{n} converges as n → ∞.

For (1) and (2), we argue by induction on n. When n = 0, by definition we have X_{0} = P_{0}Q_{0} = Ag(A^{−1}) = g(g^{−1}(A)A^{−1}). On the other hand, by Lemma 2.3.2, we may write g^{−1}A = A + φ(𝔱_{F})C for some C ∈ M_{d}(I^{+}), which gives

g^{−1}(A)A^{−1} = I_{d} + φ(𝔱_{F})CA^{−1} = I_{d} + 𝔱_{F} CEA^{−1} = I_{d} + 𝔱_{F} CB

thus proving (1) and (2) in the base case n = 0. Now suppose we have proved X_{n} = I_{d} + 𝔱_{F} C_{n} with C_{n} ∈ M_{d}(I^{+}), and let us show that X_{n+1} satisfies the same equation for some C_{n+1} ∈ M_{d}(I^{+}). Writing Ag(A^{−1}) = I_{d} + 𝔱_{F} C_{0} with C_{0} ∈ M_{d}(I^{+}), we have

X_{n+1} = X_{n} + P_{n}φ^{n+1}(𝔱_{F})φ^{n+1}(C_{0})Q_{n}

Now Eg(A^{−1}) ∈ M_{d}(W (R)_{F}) as g(E) = μ_{g}E for some unit μ_{g} ∈ W (R)_{F}, and we have φ^{n+1}(𝔱_{F}) = φ^{n}(E) ⋯ φ(E)E𝔱_{F}. We conclude that the matrix Q̃_{n} := φ^{n+1}(𝔱_{F})Q_{n} lies in M_{d}(𝔱_{F} W (R)_{F}), which gives X_{n+1} ∈ M_{d}(W (R)_{F}) and X_{n+1} − I_{d} ∈ M_{d}(𝔱_{F} I^{+}) as desired. By construction, we then have

X_{n+1} − X_{n} = P_{n} · φ^{n+1}(C_{0}) · Q̃_{n},

with P_{n}, Q̃_{n} ∈ M_{d}(W (R)_{F}) and since φ^{n+1}(C_{0}) converges to 0 in W (R)_{F}, we conclude that X_{n} converges, which gives (3) and completes the proof. □

Corollary 5.1.2. The natural G\underline{π}-action on T_{𝔖}(𝔐) can be functorially extended to an action of G. In particular, T_{𝔖} extends to a contravariant functor from Mod_{𝔖_{F}}^{φ,1} to Rep_{O_{F}} (G).

Proof. By Lemma 3.3.1 (2), we have isomorphisms of O_{F} [G\underline{π}]-modules

\begin{aligned}T_{𝔖}(𝔐) & ≃ Hom_{𝔖_{F} ,φ}(𝔐, W (R)_{F}) \\ & ≃ Hom_{W (R)_{F} ,φ}(W (R)_{F} ⊗_{𝔖_{F}} 𝔐, W (R)_{F}). \end{aligned} (5.1.4)

Thanks to Lemma 5.1.1, we have an action of G on W (R)_{F} ⊗_{𝔖_{F}} 𝔐 that extends the given action of G\underline{π}, so the final term in (5.1.4) has an action of G given by

(g∘h)(x) = g(h(g^{−1}(x))), ∀g ∈ G, ∀h ∈ Hom_{W (R)_{F} ,φ}(W (R)_{F} ⊗_{𝔖_{F}} 𝔐, W (R)_{F})

and one checks easily that this action extends the given action of G\underline{π} on T_{𝔖}(𝔐). It remains to prove that T_{𝔖} is a functor. So suppose that h : 𝔐 → 𝔐′ is a map in Mod_{𝔖_{F}}^{φ,1} and let us check that the induced map T_{𝔖}(h) : T_{𝔖}(𝔐′) → T_{𝔖}(𝔐) is indeed a map of O_{F} [G]-modules. To do this, using (5.1.4), it suffices to show that the map

1 ⊗ h : W (R)_{F} ⊗_{𝔖_{F}} 𝔐 → W (R)_{F} ⊗_{𝔖_{F}} 𝔐′

is G-equivariant, i.e. that (1 ⊗ h) ∘ g = g ∘ (1 ⊗ h) for all g ∈ G. Choose 𝔖_{F} - bases 𝔢_{1}, …, 𝔢_{d} and 𝔢′_{1}, …, 𝔢′_{d′} of 𝔐 and 𝔐′, respectively, and let A and A′ be the corresponding matrices of Frobenius, so (φ(𝔢_{1}), …, φ(𝔢_{d})) = (𝔢_{1}, …, 𝔢_{d})A and (φ(𝔢′_{1}), …, φ(𝔢′_{d′})) = (𝔢′_{1}, …, 𝔢′_{d′})A′. Letting Z be the d′ × d-matrix with entries in 𝔖_{F} determined by the relation h(𝔢_{1}, …, 𝔢_{d}) = (𝔢′_{1}, …, 𝔢′_{d′})Z, we seek to prove that g ∘ (1 ⊗ h)(𝔢_{1}, …, 𝔢_{d}) = (1 ⊗ h) ∘ g(𝔢_{1}, …, 𝔢_{d}), which is equivalent to the matrix equation

X′_{g}g(Z) = ZX_{g},

where X_{g} (resp. X′_{g}) is the matrix constructed in the proof of Lemma 5.1.1 giving the action of g on 𝔐 (resp. 𝔐′). By construction, X_{g} = lim_{n→∞} X_{n}, and similarly for X′_{g}, so it suffices to check that X′_{n}g(Z) = ZX_{n} for all n. From the very definition of X_{n} = P_{n}Q_{n} and X′_{n} = P′_{n}Q′_{n} via (5.1.3), this amounts to

P′_{n}Q′_{n}g(Z) = ZAP_{n}Q_{n}. (5.1.5)

But as φ∘h = h∘φ, we have A′φ(Z) = ZA, or equivalently, A′^{−1}Z = φ(Z)A^{−1}, and the truth of equation (5.1.5) follows easily from the definition (5.1.3). □

5.2. An equivalence of categories. In this subsection, we prove Theorem 5.0.10. Let us first recall the setup and some notation. For 𝔐 ∈ Mod_{𝔖_{F}}^{φ,1}, put M := 𝔒 ⊗_{𝔖_{F}} 𝔐 and define a decreasing filtration on φ^{∗}M as in (4.2.2). Since 𝔐 has height 1, we have Fil^{i}φ^{∗}M = E^{i−1}Fil^{1}φ^{∗}M for i ≥ 2. Recall that we set M := φ^{∗}𝔐/uφ^{∗}𝔐 and let us put D := D(𝔐) := ξ_{α}(M[1/p]), which is naturally a φ-stable F_{0}-subspace of 𝔒_{α} ⊗_{𝔖_{F}} φ^{∗}𝔐 via the unique isomorphism ξ_{α} constructed in Lemma 4.5.6. Since ξ_{α} : 𝔒_{α} ⊗_{F_{0}} D → 𝔒_{α} ⊗ φ^{∗}M is an isomorphism, we may identify φ^{∗}M/Eφ^{∗}M with D_{F_{0},K} = K ⊗_{F_{0}} D, and we write ψ_{π} : φ^{∗}M → D_{F_{0},K} for the natural projection. We then define Fil^{i}D_{F_{0},K} := ψ_{π}(Fil^{i}φ^{∗}M), and note that since Fil^{2}φ^{∗}M ⊂ Eφ^{∗}M, we have Fil^{2}D_{F_{0},K} = 0. In this way we obtain from 𝔐 an object D = D(𝔐) of MF_{F_{0},K}^{φ}. Suppose that 𝔐′ = 𝔖_{F} · 𝔢 is a rank-1 Kisin module with 𝔖_{F} -basis 𝔢. Then we have φ(𝔢) = γE^{m}𝔢 with γ ∈ 𝔖_{F}^{×} a unit thanks to Example 3.3.2, and we call m the minimal height of 𝔐′.

Lemma 5.2.1. With notation as above,

(1) The natural injection

\xymatrix{𝔒_{α} ⊗_{𝔒} φ^{∗}M \ar[r]^{ξ′_{α}} & 𝔒_{α} ⊗_{F_{0}} D \ar@{^{(}->}[r] & 𝔖̂_{0} ⊗ D_{F_{0},K}}

is compatible with filtrations, where ξ′_{α} = (ξ_{α})^{−1}.

(2) Suppose 𝔐 has rank d. Then the minimal height of ∧^{d}𝔐 is dim_{K_{0}} Fil^{1}D_{F_{0},K}.

Proof. Since Fil^{i}φ^{∗}M = E^{i−1}Fil^{1}φ^{∗}M for i ≥ 2, to prove (1) it suffices to check the given injection is compatible with Fil^{1}. As E is a generator of Fil^{1}𝔖̂_{0}, such compatibility is equivalent to the condition that x ∈ Fil^{1}φ^{∗}M if and only if ψ_{π}(x) ∈ Fil^{1}D_{F_{0},K}. But this is clear as Eφ^{∗}M ⊂ Fil^{1}φ^{∗}M. We now prove (2). Fix an 𝔖_{F} -basis 𝔢_{1}, …, 𝔢_{d} of 𝔐 and let A ∈ M_{d}(𝔖_{F}) be the corresponding matrix of Frobenius. Since 𝔐 has height 1, there exists a matrix B ∈ M_{d}(𝔖_{F}) with AB = EI_{d}. Defining e_{i} = 1 ⊗ 𝔢_{i} ∈ φ^{∗}M, we easily check that {e_{i}} is an 𝔒-basis of φ^{∗}M with (α_{1}, …, α_{d}) := (e_{1}, …, e_{d})B an 𝔒-basis of Fil^{1}φ^{∗}M. Now the inclusion φ^{∗}M/Fil^{1}φ^{∗}M ⊂ φ^{∗}M/Eφ^{∗}M = D_{F_{0},K} realizes φ^{∗}M/Fil^{1}φ^{∗}M as a K-subspace of D_{F_{0},K}, so there exists a basis f_{1}, …, f_{d} of φ^{∗}M with the property that f_{1}, … f_{s}, Ef_{s+1}, …, Ef_{d} generates Fil^{1}φ^{∗}M. Since Fil^{1}D_{F_{0},K} = ψ_{π}(Fil^{1}φ^{∗}M) we have dim_{K} Fil^{1}D_{F_{0},K} = s. On the other hand, since α_{1}, …, α_{d} also generates Fil^{1}φ^{∗}M, there exist invertible matrices X, Y ∈ GL_{d}(𝔒) with

B = XΛY for Λ = diag(1, …, 1, E, …, E)

the diagonal matrix with s many 1’s and d − s many E’s along the diagonal. Thus, det B = E^{d−s}γ for γ ∈ 𝔒^{×} a unit and since AB = EI_{d} we then have det(A) = E^{s}γ^{−1}. It follows that the minimal height of ∧^{d}𝔐 is s = dim_{K} Fil^{1}D_{F_{0},K}, as desired. □

Recall that we have defined V_{𝔖}(𝔐) := F ⊗_{O_{F}} T_{𝔖}(𝔐).

Proposition 5.2.2. With notation as above, we have V_{𝔖}(𝔐) ≃ V_{cris,F} (D(𝔐)) as F [G]-modules. In particular, V_{𝔖}(𝔐) is crystalline with Hodge-Tate weights in {0, 1}.

Proof. The proof of Proposition 4.5.1 carries over mutatis mutandis to show that there exists a natural injection of O_{F} [G\underline{π}]-modules

ι : T_{𝔖}(𝔐) ↪ Hom_{𝔒,φ,Fil}(φ^{∗}M, B_{cris,F}^{+} ) ↪ Hom_{𝔒_{α},φ,Fil}(𝔒_{α} ⊗_{𝔒} φ^{∗}M, B_{cris,F}^{+} )

≃ Hom_{𝔒_{α},φ,Fil}(𝔒_{α} ⊗_{F_{0}} D, B_{cris,F}^{+} ) ≃ V_{cris,F} (D),

where instead of using Lemma 4.2.3, we must appeal to Lemma 4.5.6 and Lemma 5.2.1 (note that a priori we know neither that M(D) ≃ M nor that D is admissible). Since dim_{F_{0}}(D) = rank_{𝔖_{F}} 𝔐 and ι is injective, we conclude that D is admissible. In particular, V_{cris,F} (D) is crystalline with Hodge-Tate weights in {0, 1}. It remains to show that ι is compatible with the given actions of G = G_{K}. By construction, the G-action on T_{𝔖}(𝔐) is induced from the identification

(5.2.1)  T_{𝔖}(𝔐) ≃ Hom_{W (R)_{F} ,φ}(W (R)_{F} ⊗_{𝔖_{F}} 𝔐, W (R)_{F})

of (5.1.4) with G-action on the right side that of Lemma 5.1.1. Now the right side of (5.2.1) is clearly naturally isomorphic as an O_{F} [G]-module to

Hom_{W (R)_{F} ,φ,Fil}(W (R)_{F} ⊗_{𝔖_{F}} φ^{∗}𝔐, W (R)_{F}),

which is an O_{F} -lattice in Hom_{B_{cris,F}^{+} ,φ,Fil}(B_{cris,F}^{+} ⊗_{F_{0}} D, B_{cris,F}^{+} ). Thus, to prove

that ι is G-equivariant, we must show that the G-action on B_{cris,F}^{+} ⊗_{𝔖_{F}} φ^{∗}𝔐 deduced from Lemma 5.1.1 agrees with the G-action on B_{cris,F}^{+} ⊗_{F_{0}} D via the map

\xymatrix{B_{cris,F}^{+} ⊗_{F_{0}} D \ar[r]^{≃} & B_{cris,F}^{+} ⊗_{𝔖_{F}} φ^{∗}𝔐}

deduced from (4.5.3) (which is an isomorphism thanks to Lemma 4.5.6); here, G acts trivially on D. Equivalently, we must show that the G-action on B_{cris,F}^{+} ⊗_{𝔖_{F}} φ^{∗}𝔐 provided by Lemma 5.1.1 restricts to the trivial action on D(𝔐), viewed as a subspace of this tensor product again via (4.5.3). As in the proofs of Lemma 4.5.6 and Lemma 5.1.1, let 𝔢_{1}, …, 𝔢_{d} be an 𝔖_{F} -basis of 𝔐 and put {e_{i} := 1 ⊗ 𝔢_{i}}, which is then an 𝔖_{F} -basis of φ^{∗}𝔐. The proof of Lemma 4.5.6 shows that (f_{1}, …, f_{d}) := (e_{1}, …, e_{d})Y is a basis of D(𝔐) for

Y = lim_{n→∞} φ(A) ⋯ φ^{n}(A)φ^{n}(A_{0}^{−1}) ⋯ φ(A_{0}^{−1}).

Now for any g ∈ G, by the proof of Lemma 5.1.1 we have the equality g(e_{1}, …, e_{d}) = (e_{1}, …, e_{d})φ(X_{g}) with

φ(X_{g}) = lim_{n→∞} φ(A) ⋯ φ^{n}(A)φ^{n}(g(A^{−1})) ⋯ φ(g(A^{−1})).

Thus, g(f_{1}, …, f_{d}) = (e_{1}, …, e_{d})φ(X_{g})g(Y) = (e_{1}, …, e_{d}) lim_{n→∞} φ(X_{n})g(Y_{n})

with

φ(X_{n})g(Y_{n}) = (φ(A) ⋯ φ^{n}(A)φ^{n}(g(A^{−1})) ⋯ φ(g(A^{−1})))
× (φ(g(A)) ⋯ φ^{n}(g(A))φ^{n}(g(A_{0}^{−1})) ⋯ φ(g(A_{0}^{−1})))
= Y_{n}

In other words, g(f_{1}, …, f_{d}) = (f_{1}, …, f_{d}), which completes the proof. □

Proof of Theorem 5.0.10. Thanks to Proposition 5.2.2 and Corollary 5.1.2, we have a contravariant functor T_{𝔖} : Mod_{𝔖_{F}}^{φ,1} → Rep_{O_{F}}^{F -cris,1}(G), which it remains to prove is fully faithful and essentially surjective. For full-faithfulness, suppose given a map h : T_{𝔖}(𝔐) → T_{𝔖}(𝔐′) of O_{F} [G]- modules. Restricting to G\underline{π} gives a map h|_{G\underline{π}} : T_{𝔖}(𝔐)|_{G\underline{π}} → T_{𝔖_{F}} (𝔐′)|_{G\underline{π}}, and by Corollary 3.2.3 we obtain a morphism 𝔣 : O_{E} ⊗_{𝔖_{F}} 𝔐′ → O_{E} ⊗_{𝔖_{F}} 𝔐 with T_{𝔖}(𝔣) = h|_{G\underline{π}}. It then suffices to show that 𝔣(𝔐′) ⊂ 𝔐. Arguing as in the proof of Proposition 3.3.5, it suffices to check that if 𝔐 ⊂ 𝔐′ ⊂ O_{E} ⊗_{𝔖_{F}} 𝔐 then 𝔐 = 𝔐′. Applying ∧^{d}, we then easily reduce to proving that ∧^{d}𝔐 and ∧^{d}𝔐′ have the same minimal height. By our reductions are now in the situation that T_{𝔖}(𝔐) ≃ T_{𝔖}(𝔐′) as O_{F} [G]-modules thanks to Corollary 5.1.2, so by Proposition 5.2.2 we have D(𝔐) ≃ D(𝔐′) as filtered φ-modules. In particular, Fil^{1}D(𝔐)_{F_{0},K} ≃ Fil^{1}D(𝔐′)_{F_{0},K} and the minimal heights of ∧^{d}𝔐 and ∧^{d}𝔐′ are the same by Lemma 5.2.1 (2). Thus, T_{𝔖} is fully faithful. We now show that T_{𝔖} is essentially surjective. Fix T ∈ Rep_{O_{F}}^{F -cris,1}(G), put V := F ⊗_{O_{F}} T and let D := D_{cris,F} (V) be the corresponding filtered φ-module. By Corollary 4.5.2, there exists 𝔐 ∈ Mod_{𝔖_{F}}^{φ,1} with M(D) ≃ 𝔒 ⊗_{𝔖_{F}} 𝔐 and ι : T_{𝔖}(𝔐) ⟶^{≃} T |_{G\underline{π}}. It suffices to show that ι is compatible with the actions of G on source and target, with the G-action on the source provided by Corollary 5.1.2. Using Proposition 5.2.2, we obtain an isomorphism of F [G]-modules ι′ : V_{𝔖}(𝔐) ≃ V_{cris,F} (D(𝔐)), which one verifies is compatible with the identification ι. It therefore remains to check that D(𝔐) ≃ D as filtered φ-modules. Thanks to Lemma 4.2.2 and Lemma 4.5.6, we can identify each of D and D(𝔐) as the image of the unique φ-equivariant section to projection φ^{∗}M(D) ↠ φ^{∗}M(D)/uφ^{∗}M(D), which gives D ≃ D(𝔐) as φ-modules. Thus, it remains to prove that Fil^{i}D_{K} = Fil^{i}D(𝔐)_{K} for all i > 0, or equivalently that Fil^{1}D_{F_{0},K} = Fil^{1}D(𝔐)_{F_{0},K}. Thanks to Corollary 4.2.4, the projection ψ_{π} : φ^{∗}M(D) ↠ φ^{∗}M(D)/Eφ^{∗}M(D) ≃ D_{F_{0},K} is compatible with filtrations, and one checks using the very definition of Fil^{1}φ^{∗}M(D) that x ∈ Fil^{1}φ^{∗}M(D) if and only if ψ_{π}(x) ∈ Fil^{1}D_{F_{0},K}. Thus, Fil^{1}D_{F_{0},K} = ψ_{π}(Fil^{1}φ^{∗}M(D)) = Fil^{1}D(𝔐)_{F_{0},K}, as desired. □

Remark 5.2.3. In the classical situation, let S be the p-adic completion of the divided-power envelope of the surjection W (k)[[u]] ↠ O_{K} sending u to π. If 𝔐 is the Kisin module attached to a Barsotti–Tate group H over O_{K}, then one can show ([Kis06, §2.2.3]) that there is a functorial isomorphism of Breuil modules φ^{∗}𝔐 ⊗_{𝔖} S ≃ 𝔻(H)_{S}, where 𝔻(H) is the Dieudonné crystal attached to H, which gives a geometric interpretation of 𝔐 in terms of the crystalline cohomology of H. It is natural to ask for such an interpretation in the general case, for arbitrary F and f(u) as in the introduction of this paper. If F/ℚ_{p} is unramified, then this interpretation is provided by [CL14]. However, for F ramified over ℚ_{p}, things are more subtle as it is necessary to use the O-divided powers of Faltings [Fal02]. For general F, A. Henniges has obtained the analogous relation with the Dieudonné crystal under the restriction v_{F} (a_{1}) = 1 (the so-called Lubin–Tate setting) in his Ph. D thesis. The general case remains open, but we nonetheless conjecture that one has a natural isomorphism A_{cris,F} ⊗_{𝔖_{F}} φ^{∗}𝔐 ≃ 𝔻(H)_{A_{cris}}, and expect to be able to prove this conjecture using the ideas of §6.3.

6. Further Questions

As Theorem 1.0.1 and Theorem 1.0.2 provide the foundations of the theory of Kisin modules and its variants (e.g. the theory of (φ, Ĝ)-modules as in [Liu10]), it is natural to ask to what extent we can extend these theories to accommodate general F and f(u). In this section, we list some questions that are natural next steps to consider in furthering the general theory we have laid out in this paper.

6.1. The case q = p^{s}. Recall the setup of the introduction: F/ℚ_{p} is an arbitrary finite extension with uniformizer ϖ and residue field k_{F} of cardinality q = p^{s}, and f(u) ∈ O_{F} [[u]] is any power series f(u) = a_{1}u + ⋯ satisfying f(u) ≡ u^{q} mod ϖ. We allow K to be any finite extension of F with uniformizer π = π_{0} and residue field k ⊇ k_{F}, and consider the Frobenius-iterate extension K\underline{π} formed by adjoing to K a choice of f-compatible system \{π_{n}\}_{n} in \overline{K} with f(π_{n}) = π_{n−1}. Such extensions and their associated norm fields are considered in [CD15] and [CDL]. In this paper, we have restricted ourselves to q = p, or what is the same, that F/ℚ_{p} is totally ramified. Certainly this restriction is unnecessary, and we are confident that the results of this paper can be adapted to the general case of arbitrary F with minor modifications. In particular, in this general case, for any W (k)-algebra A we set A_{F} := A ⊗_{W (k_{F})} O_{F}, and we equip 𝔖_{F} with the “q-power Frobenius” φ_{q} which acts on F -trivially, acts on W (k) via φ_{W (k)}^{s} and sends u to f(u). We write F_{0} := K_{0}F and again denote by E ∈ O_{F_{0}}[u] the minimal polynomial of π over F_{0}. Then our theory should be able to be adapted to functorially associate Kisin modules of finite E-height to O_{F} -lattices in F -crystalline G-representations. We note that such a theory is already known in the “Lubin–Tate” case that v_{F} (a_{1}) = 1 and K ⊆ F\underline{ϖ} thanks to the work of Kisin and Ren [KR09], but that there are many details in our general setup that still need to be checked.

6.2. Semi-stable representations and Breuil theory. In the classical situation, Theorem 4.4.1 includes semi-stable representations. This fact is one of the key inputs for Breuil’s classification of lattices in semistable representations via strongly divisible lattices over S (see [Liu08]). It is therefore natural to ask if Theorem 4.4.1 remains valid for semi-stable representations and general f(u). This appears to be a rather nontrivial question, as the case of semi-stable representations requires a monodromy operator. But for general F and f, we do not even know how to define a reasonable monodromy operator over 𝔖_{F} (i.e., one satisfying Nφ = pφN as in the classical situation). New ideas are needed for this direction.

6.3. Comparison between different choices of f(u). For a fixed F - crystalline representation V of G and a fixed uniformizer π ∈ K, we may select different f(u). It is then natural to ask for the relationship between the associated Kisin modules attached to V and f(u), as f varies. Motivated by [Liu13a], we conjecture that all such Kisin modules become isomorphic after base change to W (R)_{F}. Note that if true, this result provides a proof of the conjecture mentioned in Remark 5.2.3, because we know that A_{cris,F} ⊗_{𝔖_{F}} φ^{∗}𝔐 ≃ 𝔻(H)(A_{cris}) in the classical situation. To prove such comparison results, the key point is to generalize [Liu07, Theorem 3.2.2] to allow general f(u). This is likely relatively straightforward, as we have recovered many results of [Liu07] in §3 already.

6.4. Torsion theory. A major advantage of the theory of Kisin modules is that it provides a powerful set of tools for dealing with torsion representations. It is therefore natural to try and rebuild the torsion theory in our general situation, and we hope that such a theory will have some striking applications, for example, to the computation of the reduction of potentially crystalline representations as discussed in the introduction. One obvious initial goal is to establish the equivalence between torsion Kisin modules of height 1 and finite flat group schemes over O_{K}; this would be achievable quickly once we know the truth of the conjecture formulated in Remark 5.2.3.

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Bryden Cais Department of Mathematics University of Arizona Tucson, Arizona 85721 USA cais@math.arizona.edu Tong Liu Department of Mathematics Purdue University West Lafayette IN 47907 USA tongliu@math.purdue.edu