Galois Representations and Lubin-Tate Groups

Mark Kisin and Wei Ren

Received: May 3, 2009 Revised: June 6, 2009

Communicated by Takeshi Saito

Abstract. Using Lubin-Tate groups, we develop a variant of Fontaine’s theory of (φ, Γ)-modules, and we use it to give a description of the Galois stable lattices inside certain crystalline representations.

Introduction

In his Grothendieck Festschrift paper [Fo 1], Fontaine introduced a new way to classify local Galois representation, using the theory of so called (φ, Γ)- modules. To recall this, let k be a perfect field of characteristic p, K_{0} = Fr W (k) and K/K_{0} a finite, totally ramified extension. Fix an algebraic closure K̄ of K. Fontaine’s theory starts with an infinite extension K_{∞}/K which is required to have certain ramification properties. Miraculously, these properties ensure that G_{K_{∞}} = Gal(K̄/K_{∞}) can be identified with the absolute Galois group of a local field of equal characteristic p, X(K). It is well known that representations of such a Galois group on finite dimensional 𝔽_{p}-vector spaces can be classified rather concretely in terms of finite dimensional vector spaces over X(K) equipped with an étale Frobenius. If K_{∞}/K is Galois, then Γ = Gal(K_{∞}/K) acts naturally on X(K), and one obtains a classification of G_{K}- representations on finite dimensional 𝔽_{p}-vector spaces by adding a semi-linear action of Γ to the étale φ-modules over X(K). To obtain a classification of G_{K}-representations on finite ℤ_{p}-modules, one needs to lift the action of φ and Γ on X(K) to commuting operators on a Cohen ring for X(K). This is probably not always possible, but can be done when K_{∞} is the p-cyclotomic extension of K. Much of the work on Fontaine’s theory by Berger, Colmez, Wach and others has focused on this case. In this paper we focus on the case when K_{∞} is generated by the p-power torsion points of a Lubin-Tate group for a finite extension L/ℚ_{p} contained in K. As an application we obtain a description of the G_{K}-stable lattices in a certain class of crystalline G_{K}-representations. This is possible using the p-cyclotomic theory only when K is an unramified extension of some ℚ_{p}(μ_{p^{n}}).

More precisely, let G be a Lubin-Tate group over O_{L}, write k_{L} for the residue field of L, fix a uniformizer π_{L} of L, and write R = \underleftarrow{lim}OK̄/p, where the transition maps in the inverse limit are given by Frobenius. The action of G_{K} on the Tate module T G of G gives rise to a character χ : Γ → O_{L}^{×}. It turns out that, using the periods of T G one can construct a subring O_{E} ⊂ W (Fr R) ⊗_{W (k_{L})} L which is naturally a Cohen ring for X(K). The action of O_{L}^{×} ⊂ O_{L} on G gives rise to a natural lifting of the action of Γ to O_{E} (via χ), while the action of π_{L} on G allows one to lift the q-Frobenius φ_{q} = φ^{r} to O_{E}, where q = |k_{L}|. This allows one to classify G_{K}-representations on finite O_{L}-modules in terms of étale (φ_{q}, Γ)-modules (see Theorem 1.6 below), and is explained in §1 of the paper. At least some part of this construction was certainly known to experts. The construction of the periods involved is in Colmez’s paper [Col 1], and some of the ideas go back to Coleman [Co]. This material is also closely related to the subject of Fourquaux’s thesis [Fou, §1.4]. In §2,3 we use this classification to give a classification of Galois stable lattices in certain crystalline G_{K}-representations, assuming that K ⊂ K_{0} · L_{∞} where L_{∞}/L is the field generated by the torsion points of G. To explain the classification, assume for simplicity that K = K_{0} · L, and let 𝔖_{L} = O_{K}[[u]]. Fix a co-ordinate X on G, and for a ∈ O_{L} denote by [a] ∈ O_{L}[[X]] the power series giving the action of a on G. Then γ ∈ Γ acts on 𝔖_{L} by u ↦ [χ(γ)](u), while φ_{q} acts on 𝔖_{L} by u ↦ [π_{L}](u). Let Q = [π_{L}](u)/u. We denote by Mod_{/𝔖_{L}}^{φ_{q},Γ} the category of finite free 𝔖_{L}-modules equipped with a continuous semi-linear action of Γ which induces the trivial action of 𝔐/u𝔐, and an isomorphism φ_{q}^{∗}𝔐[1/Q] ⟶^{∼} 𝔐[1/Q] such that the map 1 ⊗ φ_{q} : 𝔐 → 𝔐[1/Q] commutes with the action of Γ. Inside this category is a subcategory Mod_{/𝔖_{L}}^{φ_{q},Γ,an} consisting of objects 𝔐 on which the Γ-action is O_{L}-analytic. this means that there is O_{L}-linear map of Lie algebras dΓ : Lie Γ → End_{K}(𝔐 ⊗_{𝔖_{L}} K[[u]]), such that the action of an open subgroup of Γ is obtained by exponentiating dΓ. To describe the crystalline representations we allow, consider any crystalline G_{K}-representation on an L-vector space V. Then

D_{dR}(V) = (B_{dR} ⊗_{ℚ_{p}} V)^{G_{K}} = ⊕_{𝔪}D_{dR}(V)_{𝔪}

where 𝔪 runs over the maximal ideals of K⊗_{ℚ_{p}} L. We say that V is L-crystalline if the filtration on D_{dR}(V)_{𝔪} is trivial, unless 𝔪 is the kernel of the natural map K ⊗_{ℚ_{p}} L ↪ K corresponding to the inclusion L → K. One of our main results is then the following

Theorem (0.1). There is an exact equivalence of ⊗-categories between Mod_{/𝔖_{L}}^{φ_{q},Γ,an} and the category of G_{K}-stable O_{L}-lattices in L-crystalline G_{K}- representations.

The theorem is a generalization of the classification of G_{K}-stable lattices in crystalline representations in terms of Wach lattices due to Wach [Wa], Colmez [Col 2] and Berger [Be 3], when K_{∞} is the p-cyclotomic extension and K is unramified.

It is also analogous to the classification G_{K_{∞}}-stable lattices, obtained in [Ki] in the case when K_{∞} is obtained from K by adjoining the p-power roots of a uniformizer. The advantage of Theorem (0.1) is that it applies without restriction on the ramification of K, and gives a precise description of G_{K}-stable lattices. Unfortunately, it applies only to a rather special kind of crystalline G_{K}-representation. It seems likely that in order to obtain a classification valid for any crystalline G_{K}-representation one needs to consider higher dimensional subrings of W (Fr R), constructed using the periods of all the conjugates of G.

Acknowledgment: The results presented here were to be the subject of the Ph. D thesis of Wei Ren. They were written up by the first author after Ren’s premature and tragic death. We would like to thank Barry Mazur and Jean-Marc Fontaine for useful conversations on some of the material presented here. Finally we thank the referee for a careful reading of the paper and some useful remarks.

§1 Étale (φ_{q}, Γ)-modules

(1.1) Throughout the paper we fix a perfect field k, of characteristic p > 0. Let W = W (k), K_{0} = W [1/p] and K/K_{0} a finite totally ramified extension with ring of integers O_{K}, and uniformizer π. We also fix an algebraic closure K̄ of K with ring of integers OK̄, and set G_{K} = Gal(K̄/K). Let L/ℚ_{p} be a finite extension of ℚ_{p} contained in K. Let O_{L} denote the ring of integers of L, and k_{L} ⊂ k its residue field. Write O_{L_{0}} = W (k_{L}), L_{0} = O_{L_{0}}[1/p], and q = p^{r} = |k_{L}|. For an O_{L_{0}}-algebra A, it will be convenient to write A_{L} = A ⊗_{O_{L_{0}}} O_{L}. Let G be a Lubin-Tate group over L corresponding to a uniformizer π_{L} ∈ L. Fix a local co-ordinate X on G so that the formal Hopf algebra O_{G} may be identified with O_{L}[[X]]. For a ∈ O_{L} we denote by [a] ∈ O_{L}[[X]] = O_{G} the power series giving the endomorphism a of G. For n ≥ 1, let K_{n} ⊂ K̄ denote the subfield generated by the π_{L}^{n}-torsion points of G. We set K_{∞} = ∪_{n}K_{n} and we write Γ = Gal(K_{∞}/K) and G_{K_{∞}} = Gal(K̄/K_{∞}). Let T G denote the p-adic Tate module of G. Then T G is a free O_{L}-module of rank 1, and the action of Γ induces a faithful character χ : Γ → O_{L}^{×}. We let R = \underleftarrow{lim}OK̄/p with the transition maps being given by Frobenius. We may also identify R with \underleftarrow{lim}OK̄/π_{L} with the transition map being given by the q-Frobenius φ^{r}. Evaluation of X at π_{L}-torsion points then induces a map ι : T G → R. Namely if v = (v_{n})_{n≥0} ∈ T G with v_{n} ∈ G[π_{L}^{n}](OK̄) and π_{L} · v_{n+1} = v_{n}, then ι(v) = (v_{n}^{∗}(X))_{n≥0}.

Lemma (1.2). There is a unique map { } : R → W (R)_{L} such that {x} is a lifting of x, and φ^{r}({x}) = [π_{L}](x). Moreover { } respects the action of G_{K}, and for v ∈ T G we have

(1) If a ∈ O_{L} then {ι(av)} = [a]({ι(v)}).

(2) The action of G_{K} on {ι(T G)} factors through Γ and for γ ∈ Γ

[χ(γ)]({ι(v)}) = {ι(γv)} = {γ · ι(v)} = γ · {ι(v)}

In particular, if v ∈ T G is an O_{L}-generator, there is an embedding W_{L}[[u]] ↪ W (R)_{L} sending u to {ι(v)} which identifies W_{L}[[u]] with a G_{K}-stable, φ^{r}-stable subring of W (R)_{L} such that {ι(T G)} lies in the image of W_{L}[[u]].

Proof. The existence and uniqueness of { } is [Col 1, Lem. 9.3]. The map {x} is given by

{x} = lim_{n}[π_{L}^{n}](φ^{−rn}(x̃))

where x̃ ∈ W (R)_{L} is any lifting of x. That { } respects the action of G_{K} follows by functoriality. In particular, the action of G_{K} on {ι(T G)} factors through Γ. For (1) note that

[π_{L}][a]{ι(v)} = [a][π_{L}]{ι(v)} = [a]φ^{r}{ι(v)} = φ^{r}([a]{ι(v)}).

Since [a]{ι(v)} and {ι(av)} both have image [a](ι(v)) in R, this proves (1). (Here R is viewed as a O_{K} algebra via O_{K} → k.) Now the first equality in (2) follows from (1), while the other two equalities follows from the compatibility of ι and { } with the action of G_{K}. Finally, since ι(v) has positive valuation with respect to the canonical valuation on R, u ↦ {ι(v)} induces a well defined map W_{L}[[u]] → W (R)_{L}. Its image is φ-stable by definition of { } and Γ-stable by (2). If this map had a non-trivial kernel, then so would its reduction modulo π_{L}. The latter map k[[u]] → R, sending u to ι(v) is easily seen to be injective, as ι(v) has positive valuation. □

(1.3) Write 𝔖_{L} = W_{L}[[u]]. We fix an O_{L}-generator v ∈ T_{p}G, and we identify 𝔖_{L} with a subring of W (R)_{L} by sending u to {ι(v)}. Let O_{E} denote the p-adic completion of 𝔖_{L}[1/u]. Then O_{E} is a complete discrete valuation ring with uniformizer π_{L} and residue field k((u)). We may view O_{E} as a subring of W (Fr R)_{L}. Let O_{E^{ur}} ⊂ W (Fr R)_{L} denote the maximal integral, unramified extension of O_{E}. We denote by OÊ^{ur} the p-adic completion of O_{E^{ur}}, which is again naturally a subring of W (Fr R)_{L}. We write E, E^{ur} and Ê^{ur} for the fields of fractions of O_{E}, O_{E^{ur}} and OÊ^{ur} respectively. These rings are all stable by φ^{r}, and by the action of G_{K}. Moreover the G_{K}-action on O_{E} factors through Γ.

Lemma (1.4). The residue field of OÊ^{ur} is a separable closure of k((u)). There is a natural isomorphism

Gal(E^{ur}/E) ⟶^{∼} Gal(K̄/K_{∞}).

Proof. This is a consequence of the theory of norm fields [Wi]. Since Γ is a p-adic Lie group the theory of loc. cit applies [Wi, 1.2.2]. For any finite extension F/K write X_{K}(F) = \underleftarrow{lim}(F ·K_{n}) where the maps in the inverse limit are given by the norm. We set X_{K}(K̄) = ∪_{F} X_{K}(F) where the limit runs over finite extensions F/K in K̄. Then X_{K}(F) has the structure of a local field of characteristic p, which is a finite separable extension of X_{K}(K), X_{K}(K̄) is a separable closure of X_{K}(K), and the functor X_{K} induces an isomorphism [Wi, 3.2.2]

Gal(X_{K}(K̄)/X_{K}(K)) ⟶^{∼} Gal(K̄/K_{∞}).

On the other hand, there is a natural embedding X_{K}(K) ↪ Fr R [Wi ,§4]. To see this explicitly note that one has well defined maps of rings

(1.4.1)  \underleftarrow{lim}O_{K_{n}} → \underleftarrow{lim}O_{K_{n}}/(v_{1}) ↪ \underleftarrow{lim}OK̄/π_{L} = R,

where the transition maps in the first two inverse limits are given by the norm, and the final inverse limit by x ↦ x^{q}. The image of (1.4.1) is easily seen to be k[[u]] ⊂ R. Hence we may identify O_{E}/π_{L}O_{E} with X_{K}(K). It follows that O_{E^{ur}}/π_{L}O_{E^{ur}} ⊂ Fr R may be identified with X_{K}(K̄). The lemma follows. □

(1.5) Note that the above proof shows that the map ι induces a map

T G → \underleftarrow{lim}K_{∞},

where the transition maps are given by the norm. This is Coleman’s map [Co, Thm. A]. We will write φ_{q} for the q-Frobenius φ^{r} (for example on the ring W (Fr R)). Now denote by Mod_{/O_{E}}^{φ_{q}} (resp. Mod_{/O_{E}}^{φ_{q},tor}) the category of finite free (resp. finite torsion) O_{E}-modules M, equipped with an isomorphism (φ_{q})^{∗}M ⟶^{∼} M. We denote by Mod_{/O_{E}}^{φ_{q},Γ} (resp. Mod_{/O_{E}}^{φ_{q},Γ,tor}) the category of consisting of a module

M in Mod_{/O_{E}}^{φ_{q}} (resp. Mod_{/O_{E}}^{φ_{q},tor}) equipped with a continuous semi-linear action of Γ which commutes with the action of φ_{q}. We denote by Rep_{G_{K∞}} (resp. Rep_{G_{K∞}}^{tor} ) the category of finite free (resp. finite torsion) O_{L}-modules V, equipped with a linear action of G_{K_{∞}} Similarly, we denote by Rep_{G_{K}} (resp. Rep_{G_{K}}^{tor} ) the category of finite free (resp. finite torsion) O_{K}-modules V, equipped with a linear action of G_{K}. For M in Mod_{/O_{E}}^{φ_{q}} (resp. Mod_{/O_{E}}^{φ_{q},tor}, resp. Mod_{/O_{E}}^{φ_{q},Γ}, resp. Mod_{/O_{E}}^{φ_{q},Γ,tor}) we set

V (M) = (OÊ^{ur} ⊗_{O_{E}} M)^{φ_{q}=1}.

For V in Rep_{G_{K∞}} (resp. Rep_{G_{K∞}}^{tor} resp. Rep_{G_{K}} resp. Rep_{G_{K}}^{tor} )) we set

M_{O_{E}} (V) = (V ⊗_{O_{K}} OÊ^{ur})^{G_{K∞}} .

Theorem (1.6). V and M_{O_{E}} are quasi-inverse equivalences between the exact tensor categories Mod_{/O_{E}}^{φ_{q}} (resp. Mod_{/O_{E}}^{φ_{q},tor}, resp. Mod_{/O_{E}}^{φ_{q},Γ}, resp. Mod_{/O_{E}}^{φ_{q},Γ,tor}) and Rep_{G_{K∞}} (resp. Rep_{G_{K∞}}^{tor} , resp. Rep_{G_{K}} , resp. Rep_{G_{K}}^{tor} )

Proof. The argument for this is identical to that in [Fo 1, 1.2.6, 3.4.3]. For the convenience of the reader we sketch it: It suffices to prove that V and M induce quasi-inverse, exact tensor equivalences between Mod_{/O_{E}}^{φ_{q},tor} and Rep_{G_{K∞}}^{tor} . We first remark that both functors are exact. It suffices to prove this for objects killed by p. For M_{O_{E}} this follows from the fact that for M in Mod_{/O_{E}}^{φ_{q},tor}, 1−φ_{q} is étale locally (on Spec k((u))) surjective. For V this is a consequence of Hilbert’s theorem 90, and (1.4). For M in Mod_{/O_{E}}^{φ_{q},tor}, we have a natural map

(1.6.1)  (M ⊗_{O_{E}} O_{E^{ur}})^{φ_{q}=1} ⊗_{O_{K}} O_{E^{ur}} → M ⊗_{O_{E}} O_{E^{ur}}

and taking G_{K_{∞}} invariants of both sides induces a map M_{O_{E}} (V (M)) → M. To show that this map is an isomorphism one reduces to the case of objects killed by p, using the exactness proved above. In this case, (1.6.1) is an isomorphism, because étale locally M is spanned by its φ_{q}-invariants. Similarly, one obtains an isomorphism V (M_{O_{E}} (V)) → M for V in Rep_{G_{K∞}}^{tor} , using dévissage, Hilbert theorem 90, and (1.4). □

§2 (φ_{q}, Γ)-modules and weakly admissible modules

(2.1) We keep the notation of the previous section, so in particular we write K_{0,L} for the field K_{0} ⊗_{L_{0}} L = Fr W_{L} ⊂ K. In order not to overload notation we will write v_{n} for (v_{n})^{∗}(X) ∈ K̄. We now also assume that K ⊂ K_{0,L}(v_{n})_{n≥0}. Fix an integer m ≥ 1 such that K ⊂ K_{0,L}(v_{m}). As in [Ki, 1.1.1], denote by D[0, 1) the rigid analytic disk of radius 1, over K_{0,L}, and denote by u the co-ordinate on D[0, 1). For I ⊂ [0, 1) an interval, denote by D(I) ⊂ D[0, 1) the open subspace whose K̄ points consist of x ∈ K̄ with |x| ∈ I. We denote by O_{I} the ring of rigid analytic functions on D(I), and we write O = O_{[0,1)}. We will often use the fact that D[0, 1) is a p-adic Stein space, so that a coherent sheaf on D[0, 1) can be recovered from its global sections. In particular, we may regard a finite free O-module as a coherent sheaf on D[0, 1). We regard 𝔖_{L} ⊂ O by u ↦ u. The action of φ_{q} and Γ on 𝔖_{L} have a unique continuous extension to O, regarded with its canonical Frechet topology.^{1} Let Q = [π_{L}^{m}](u)/[π_{L}^{m−1}](u). Denote by Mod_{/O}^{φ_{q}} the category of finite free Omodules M equipped with an isomorphism φ_{q}^{∗}(M)[1/Q] ⟶^{∼} M[1/Q]. We denote by Mod_{/O}^{φ_{q},Γ} the category whose objects consist of an object of Mod_{/O}^{φ_{q}} equipped with a continuous semi-linear action of Γ such that Γ acts trivially on M/uM and the φ_{q}-semi-linear map 1 ⊗ φ_{q} : M → M[1/Q] commutes with Γ. We now explain how to differentiate the action of Γ on an object in Mod_{/O}^{φ_{q},Γ} following [Be 1, §IV,V].

1Contrary to our usual conventions, the symbol O_{L} will continue to denote the ring of integers of L, rather than O ⊗_{L_{0}} L.

Lemma (2.1.1). The action of Γ on O, defined above, is continuous. In particular, O with its action of Γ and φ_{q} is an object of Mod_{/O}^{φ_{q},Γ}.

Proof. For r ∈ (0, 1), denote by | · |_{r} the sup norm on O_{[0,r]}. If f(u) = ∑_{i≥0} a_{i}u^{i} ∈ O_{[0,r]} then |f|_{r} = sup_{i} |a_{i}|r^{i}. We have to show that, for any r, as γ → 1, |γ(f) − f|_{r} → 0, uniformly in f with |f|_{r} ⩽ 1. Any γ ∈ Γ acts on O by composition with [χ(γ)]. Write

[χ(γ)] = ∑_{i=1}^{∞} b_{i}X^{i} = exp_{G}(χ(γ)log_{G}X)

where b_{i} ∈ O_{L}, log_{G} denotes the logarithm of G and exp_{G} denotes its inverse.^{2} Then b_{1} = χ(γ) and for i > 1, b_{i} is a polynomial in χ(γ), which vanishes at χ(γ) = 1. Given ϵ > 0, choose i_{0} so that r^{i_{0}} < ϵ. Then for γ sufficiently close to 1, |b_{i}| < ϵ for 1 < i < i_{0}, and |b_{1} − 1| < ϵ so |[χ(γ)](u) − u|_{r} < ϵ. Hence

|γ(f) − f|_{r} = |f([χ(γ)](u)) − f(u)|_{r} ⩽ ∑_{i=1}^{∞} |a_{i}([χ(γ)](u)^{i} − u^{i})|_{r} ⩽ ϵ|f(u)|_{r}.

The lemma follows. □

Lemma (2.1.2). Let M be in Mod_{/O}^{φ_{q},Γ}. For each r ∈ (0, 1) and γ ∈ Γ sufficiently close to 1 (depending on r) the series

logγ = ∑_{i=1}^{∞}(γ − 1)^{i}(−1)^{i−1}/i

induces a well defined operator on M|_{D[0,r)}. This induces a well defined ℤ_{p}- linear map of Lie algebras

dΓ_{M} : Lie Γ → End_{K_{0}}M; β ↦ log(exp β).

such that for β ∈ Lie Γ, dΓ_{O}(β) is a derivation and dΓ_{M}(β) is a differential operator over dΓ_{O}(β). That is, for m ∈ M, f ∈ O and β ∈ Lie Γ,

dΓ_{M}(β)(fm) = dΓ_{O}(β)(f)m + fdΓ_{M}(β)(m).

Proof. Let M_{0} ⊂ M be finite free W_{L}-submodule of rank equal to d = rk_{O}M, which spans M. Choosing a basis for M_{0}, we may identify M with O^{d}. As in (2.1.1), choose r ∈ (0, 1) and denote by | |_{r} the norm on M_{D[0,r]} = O_{D[0,r]}^{d} induced by the sup norm on O_{D[0,r]}. For any ϵ > 0, and γ sufficiently small we have |γ(m) − m|_{r} ⩽ ϵ|m|_{r} for m ∈ M_{0} and |γ(f) − f|_{r} ⩽ ϵ|f|_{r} for f ∈ O_{D[0,r]} by (2.1.1). Hence

2So if G = 𝔾_{m}, then log_{G}(X) = log(1 + X).

|γ(mf) − mf|_{r} ⩽ |γ(m) − m|_{r}|γ(f)|_{r} + |γ(f) − f|_{r}|m|_{r} ⩽ 2ϵ|m|_{r}|f|_{r} = 2ϵ|fm|_{r}.

This shows that logγ is well defined. It follows that the map

dΓ_{M} =: Lie Γ → End_{K_{0}}M; β ↦ log(exp β)

is well defined for β sufficiently small, and we extend it to all of Lie Γ by ℤ_{p}- linearity. That dΓ_{O}(β) is a derivation and dΓ_{M}(β) is a differential operator over dΓ_{O}(β) follows from a simple computation, as does the fact that dΓ_{M} is a map of Lie algebras. Note that the latter statement just means that the differential operators dΓ_{M}(β) for β ∈ Lie Γ commute. □

(2.1.3) We say that M in Mod_{/O}^{φ_{q},Γ} is O_{L}-analytic if the map dΓ_{M} is O_{L}-

linear, not just ℤ_{p}-linear. We denote by Mod_{/O}^{φ_{q},Γ,an} the full subcategory of

Mod_{/O}^{φ_{q},Γ} consisting of O_{L}-analytic objects. One checks easily that this is a ⊗-subcategory, which is stable under taking subobjects and quotients.

Lemma (2.1.4). Let M be in Mod_{/O}^{φ_{q},Γ,an}.

(1) For each r ∈ (0, 1) the operator N_{∇} =: logγ/logχ(γ) is well defined for γ ≠ 1 sufficiently close to 1, and is independent of γ.

(2) The operators in (1) induce a K_{0,L}-linear map N_{∇} : M → M, which is a differential operator over the derivation N_{∇} : O → O, and which commutes with φ_{q} on M.

(3) There is a singular connection ∇ on M with simple poles at the zeroes of [π_{L}^{n}]/u for n ≥ 1 (that is at the non-trivial π_{L}-power torsion points of G) such that N_{∇} = ⟨∇, \frac{∂F}{∂Y} (u, 0)log_{G}u · d/du⟩, where F (X, Y) denotes the formal group law of G with respect to X, and \frac{∂F}{∂Y} (u, 0) ∈ O_{L}[[u]]^{×}.

Proof. For γ sufficiently close to 1, we may write γ = exp β with β ∈ Lie Γ. Since β ↦ log(exp β) is O_{L}-linear by assumption. and β ↦ log(χ(exp(β))) is obviously O_{L}-linear, logγ/logχ(γ) is independent of γ. This proves (1) and (2) follows by viewing M as a coherent module on D[0, 1). The fact that N_{∇} commutes with φ_{q} follows from the fact that φ_{q} commutes with the action of Γ. To see (3), we first compute the derivation N_{∇} on O. For γ ∈ Γ write a_{γ} = χ(γ) − 1. Then

N_{∇}(u) = lim_{γ↦1} \frac{[χ(γ)](u) − u}{logχ(γ)} = lim_{𝔞_{γ}→0} \frac{exp_{G}((1 + a_{γ})log_{G}u) − u}{logχ(γ)}

= lim_{𝔞_{γ}→0} \frac{F (u, exp_{G}(a_{γ}log_{G}u)) − u}{logχ(γ)} = \frac{∂F}{∂Y} (u, 0)log_{G}(u).

Documenta Mathematica 14 (2009) 441–461

Hence N_{∇} is given on O by N_{∇}(f) = \frac{∂F}{∂Y} (u, 0)(log_{G}u)\frac{df}{du}. As \frac{∂F}{∂Y} (u, 0) has constant term 1, and coefficients in O_{L}[[u]], it is a unit O. Now for any M, define ∇(m) for m ∈ M by ∇(m) = (\frac{∂F}{∂Y} (u, 0)log_{G}u)^{−1}N_{∇}(m). Since N_{∇} on M is a differential operator over the derivation \frac{∂F}{∂Y} (u, 0)(log_{G}u)\frac{df}{du} on O, ∇ is a (singular) connection. A priori ∇(m) has a simple pole at each [π_{L}]-torsion point of G, however since the action of Γ on M is trivial mod u, the operator N_{∇} vanishes mod u, and ∇(m) has no pole at u = 0. This proves (3). □

(2.2) Denote by Mod_{/K_{0,L}}^{F,φ_{q}} the category of finite dimensional K_{0,L}-vector spaces

D equipped with an isomorphism φ_{q}^{∗}D ⟶^{∼} D and a decreasing, separated filtration on D_{K} = D ⊗_{K_{0,L}} K, indexed by ℤ, by K-subspaces. Our next task to to show that there is an exact ⊗-equivalence between Mod_{/K_{0,L}}^{F,φ_{q}} and Mod_{/O}^{φ_{q},Γ,an}. The construction is analogous to that in [Be 2] and [Ki, 1.2]. Since many of the proofs from [Ki] go over verbatim, we often only sketch the argument.^{3} For n ≥ 0, denote by 𝔖̂_{n} the complete local ring at the point x_{n} of D[0, 1), corresponding to u = v_{m+n}. That is, 𝔖̂_{n} is the completion of the localization of O at the maximal ideal generated by [π_{L}^{m+n}](u)/[π_{L}^{m+n−1}](u). Then 𝔖̂_{n} is a discrete valuation ring with residue field K_{m+n} = K(v_{m+n}) ⊃ K, which is canonically a subfield of 𝔖̂_{n}. In particular, u − v_{m+n} is a uniformizer for 𝔖̂_{n}. Let

λ = ∏_{n≥0} φ_{q}^{n}(Q(u)/Q(0))) = ∏_{n≥0}[π_{L}^{m+n}](u)/[π_{L}^{m+n−1}](u)π_{L},

and write φ_{q,W_{L}} : O → O for the O_{L}-linear automorphism given by applying φ^{r} to the coefficients of a series in O. Given D in Mod_{/K_{0,L}}^{F,φ_{q}} and n ≥ 0, we denote by ι_{n} the composite

ι_{n} : D ⊗_{K_{0,L}} O[1/λ] ↪^{φ_{q}^{−n}⊗φ_{q,W_{L}}^{−n}} D ⊗_{K_{0,L}} 𝔖̂_{n}[1/λ] ⟶^{∼} D_{K} ⊗_{K} 𝔖̂_{n}[1/u − v_{m+n}].

We set

M(D) = {d ∈ D⊗_{K_{0,L}} O[1/λ] : ∀n ≥ 0, ι_{n}(d) ∈ Fil^{0}(D_{K} ⊗_{K} 𝔖̂_{n}[1/u−v_{m+n}])}.

Lemma (2.2.1). For D in Mod_{/K_{0,L}}^{F,φ_{q}}, M(D) is naturally an object of

Mod_{/O}^{φ_{q},Γ,an}.

Proof. That M(D) is a finite free O-module, and the fact that φ_{q} on D ⊗_{K_{0,L}} O[1/λ] induces an isomorphism φ_{q}^{∗}(M(D))[1/Q] ⟶^{∼} M(D)[1/Q] is proved exactly as in [Ki, 1.2.2].

3In fact they often simplify since one only has to consider the case when N = 0 in [Ki].

Note that for γ ∈ Γ,

γ([π_{L}](u)) = [π_{L}] ∘ [χ(γ)](u) = [π_{L}χ(γ)](u).

Hence γ(λ) = λ ∘ [χ(γ)] has a simple zero at each [π_{L}^{m}]-torsion point which is not a [π_{L}^{m−1}]-torsion point. It follows that λ/γ(λ) ∈ 𝔖_{L}[1/p]^{×}. In particular, if γ ∈ Γ acts on D ⊗_{K_{0,L}} O by 1 ⊗ γ, then this induces an action of Γ on D ⊗_{K_{0,L}} O[1/λ]. The same argument shows that γ induces an automorphism of 𝔖̂_{n} for n ≥ 0. As φ_{q,W_{L}}[χ(γ)] = [χ(γ)], one sees that M(D) is stable by the action of Γ. Finally, this action is O_{L}-analytic, as the action of Γ on O is O_{L}-analytic. □

Lemma (2.2.2). Let M be in Mod_{/O}^{φ_{q},Γ,an}. There exists a unique K_{0,L}-linear section ξ : M/uM → M[1/λ] such that the elements of ξ(M/uM) are Γinvariant. Moreover, we have

(1) ξ is φ_{q}-equivariant.

(2) ξ induces an isomorphism

M/uM ⊗_{K_{0,L}} O[1/λ] ⟶^{∼} M[1/λ].

(3) The image of ξ ⊗ 1 : M/uM ⊗_{K_{0,L}} O → M[1/λ] coincides with (1 ⊗ φ_{q})(φ_{q}^{∗}M) over an admissible open neighborhood of u = v_{m}.

Proof. Consider the connection ∇ on M defined in (2.1.4)(3). For r ∈ (0, 1) sufficiently small there exists a unique ∇-parallel section ξ_{r} : M/uM → M|_{D[0,r)}. Since N_{∇} commutes with φ_{q}, the section φ_{q} ∘ ξ_{r} ∘ φ_{q}^{−1} is also ∇-parallel, and hence equal to ξ_{r}. Hence ξ_{r} is φ_{q}-invariant. Similarly γ ∘ ξ_{r} ∘ γ^{−1} is ∇-parallel for γ ∈ Γ, so ξ_{r} is Γ-invariant. Now ξ may be constructed from ξ_{r} exactly as in [Ki, 1.2.6], by repeatedly pulling ξ_{r} back by φ_{q}^{∗} and using the isomorphism φ_{q}^{∗}M[1/Q] ⟶^{∼} M[1/Q]. The claims (2) and (3) also follow exactly as in loc. cit. □

(2.2.3) Suppose that M is in Mod_{/O}^{φ_{q},Γ,an}. For i an integer denote by Fil^{i}φ_{q}^{∗}M the preimage of Q^{i}M under φ_{q}^{∗}M[1/Q] ⟶^{∼} M[1/Q]. Note that this filtration is Γ-stable. Let D(M) = M/uM. By (2.2.2), ξ induces an isomorphism D(M) ⊗_{K_{0,L}} O ⟶^{∼} φ^{∗}(M) near the point u = v_{m}. Hence we obtain an isomorphism

(2.2.4)  D(M) ⊗_{K_{0,L}} K(v_{m}) ⟶^{∼} φ_{q}^{∗}(M)/Qφ_{q}^{∗}(M).

Give the right hand side of (2.2.4) the filtration induced by that on φ_{q}^{∗}M, and pull this filtration back to D(M) ⊗_{K_{0,L}} K(v_{m}). This gives rise to a Γ-stable filtration on D(M) ⊗_{K_{0,L}} K(v_{m}), which necessarily descends to a filtration on D(M)_{K}. This gives D(M) the structure of an object in Mod_{/K_{0,L}}^{F,φ_{q}}.

Lemma (2.2.5). Let M be in Mod_{/O}^{φ_{q},Γ,an} and D = D(M). Then for all i ∈ ℤ the map ξ induces an isomorphism

∑_{j≥0} Q^{j}𝔖̂_{0} ⊗_{K} Fil^{i−j}D(M)_{K} ⟶^{∼} 𝔖̂_{0} ⊗_{O} Fil^{i}φ^{∗}(M).

Proof. This is the analogue of [Ki, 1.2.12(4)] in our situation, and the proof is identical, so we only sketch it here. Since D(M)_{K} ⊗ 𝔖̂_{0} and 𝔖̂_{0} ⊗_{O} φ_{q}^{∗}(M) induce the same filtration on their common quotient D(M)_{K}, one sees easily that it suffices to check that for all i ∈ ℤ,

ξ(Fil^{i}D(M)_{K}) ⊂ 𝔖̂_{0} ⊗_{O} (1 ⊗ φ_{q})(Fil^{i}φ^{∗}(M)).

We will identify φ_{q}^{∗}M with its image (1 ⊗ φ_{q})(φ_{q}^{∗}M) in M[1/Q]. An element d ∈ ξ(Fil^{i}D(M)_{K}) can be written as d = d_{0} + d_{1} with d_{0} ∈ 𝔖̂_{0} ⊗_{O} Fil^{i}φ_{q}^{∗}(M) and d_{1} ∈ Q𝔖̂_{0} ⊗_{O} φ_{q}^{∗}(M). As N_{∇}(d) = 0, we have

N_{∇}(d_{1}) = −N_{∇}(d_{0}) ∈ 𝔖̂_{0} ⊗_{O} (Fil^{i}φ_{q}^{∗}(M) ∩ Qφ_{q}^{∗}(M)) =: M_{i}.

Thus it suffices to show that for all i ∈ ℤ, N_{∇} induces a bijection on M_{i}, for then d_{1} ∈ M_{i} ⊂ 𝔖̂_{0} ⊗_{O} Fil^{i}φ_{q}^{∗}(M). For i sufficiently small this follows from the isomorphism D(M) ⊗_{O} 𝔖̂_{0} ⟶^{∼} φ_{q}^{∗}M ⊗_{O} 𝔖̂_{0}. The general case follows by descending induction on i and an application of the snake lemma. □

Proposition (2.2.6). The functors M and D between Mod_{/K_{0,L}}^{F,φ_{q}} and

Mod_{/O}^{φ_{q},Γ,an} are quasi-inverse, exact, ⊗-equivalences.

Proof. Let D be in Mod_{/K_{0,L}}^{F,φ_{q}}. From the definition of M(D), there is a natural Γ-equivariant inclusion D ⊂ M(D)[1/Q], which induces an isomorphism of D with D(M(D)) = M(D)/uM(D). Hence the image of this inclusion coincides with ξ(D(M(D))), and one sees from the definitions that the filtration on D_{K} coincides with the one on D(M(D))_{K}. This produces a natural isomorphism

D ⟶^{∼} D(M(D)).

Conversely, let M be in Mod_{/O}^{φ_{q},Γ,an}. Then both M and M(D(M)) may be identified with O-submodules of D(M) ⊗_{K_{0,L}} O[1/λ]. At any point of D[0, 1) other u = v_{n}, n ≥ m, both submodules coincide with D(M) ⊗_{K_{0,L}} O[1/λ]. Since both M and M(D(M)) are in Mod_{/O}^{φ_{q},Γ}, to show these submodules are equal it suffices to check that these two submodules coincide at u = v_{m}. This follows from (2.2.5). (cf. [Ki, 1.2.13]). Hence we have a natural isomorphism

M(D(M)) ⟶^{∼} M.

That M and D are exact follows from (2.2.5). One checks easily that M and D respect ⊗-products (cf. [Ki, 1.2.15]). □

(2.3) We now apply Kedlaya’s slope filtration as in [Be 2] and [Ki, §1.3] to show that an object M of Mod_{/O}^{φ_{q},Γ,an} can be descended to 𝔖_{L} if and only if D(M) is weakly admissible.^{4} Again, as many of the arguments are identical to those of [Ki] we sometimes only sketch the proofs. Let R = lim_{r→1^{−}} O_{(r,1)} denote the Robba ring, and R^{b} = lim_{r→1^{−}} O_{(r,1)}^{b}, where O_{(r,1)}^{b} ⊂ O_{(r,1)} denotes the subring of bounded functions. Then R^{b} is a discrete valuation field, with valuation

v_{R^{b}}(f) = −log_{π_{L}} lim_{r→1^{−}} sup_{x∈D(r,1)} |f(x)|

and uniformizer π_{L}. The endomorphism φ_{q} and the derivation N_{∇} of O induce an automorphism and a derivation respectively of R and R^{b}, which we will again denote by φ_{q} and N_{∇}. Denote by Mod_{/R}^{φ_{q}} (resp. Mod_{/R^{b}}^{φ_{q}}) the category of finite free R-modules (resp.

R^{b}-modules) M equipped with an isomorphism φ_{q}^{∗}M ⟶^{∼} M. For an M in Mod_{/R}^{φ_{q}}, we denote by

0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{r} = M

Kedlaya’s slope filtration [Ke 1], [Ke 2]. We write s_{i} for the slope of the pure slope quotient M_{i}/M_{i−1}, which is finite free over R. The filtration is functorial for maps in Mod_{/R}^{φ_{q}}. One of Kedlaya’s results about the filtration says that a module of pure slope s has a canonical descent to a module M^{b} in Mod_{/R^{b}}^{φ_{q}} which has slope s in the sense of Dieudonné-Manin theory (and the valuation on R^{b} normalized so that v(π_{L}) = 1). For M in Mod_{/O}^{φ,Γ,an} we write M_{R} = M ⊗_{O} R. The operators φ_{q} and N_{∇} on M induce operators φ_{q} and N_{∇} on M_{R}.

Lemma (2.3.1). Let M be in Mod_{O}^{φ_{q},Γ,an}. The slope filtration on M_{R} is induced by a unique filtration on M by saturated, finite free O-submodules. This filtration on M is stable by φ_{q} and the action of Γ.

Proof. It is clear that a such filtration on M, if it exists is unique and stable by φ_{q}. The functoriality of the slope filtration on M_{R} implies that it is stable by Γ, and hence so is the filtration on M. It remains to show the existence of such a filtration. As φ_{q}(λ) = π_{L}[π_{L}^{m−1}]/[π_{L}^{m}]λ, for any integer s, the slope filtration on λ^{−s}M_{R} is given by λ^{−s}M_{R,i}, and the slopes of λ^{−s}M_{R} are those of M_{R} shifted by −s. Since [π_{L}^{m}]/[π_{L}^{m−1}]π_{L} has a unique, simple zero on D[0, 1) at x_{0}, we may replace M by λ^{−s}M for s sufficiently large, and assume that φ_{q} induces a map φ_{q}^{∗}M → M. We first show that the slope filtration is induced by a filtration on M|_{D(0,1)} by saturated O_{(0,1)}-submodules. For some r_{0} sufficiently close to 1, the slope filtration on M_{R} is induced by a filtration on M|_{D(r_{0},1)} by saturated O_{(r_{0},1)}- submodules. Let r_{0} > r_{1} > … be a sequence approaching 0, and such that φ_{q}^{−1}(D(r_{i}, 1)) ⊂ D(r_{i−1}, 1) for i ≥ 1. The same argument as in [Ki, 1.3.4] shows that for j ≥ 0, M_{R,i} is induced by a filtration on M|_{D(r_{j},1)} by closed, saturated O_{(r_{j},1)}-submodules, and hence by a filtration on M|_{D(0,1)} by closed, saturated O_{(0,1)}-submodules. The filtration on M_{D(0,1)} is stable by Γ, by uniqueness, and hence it is stable by N_{∇}. Consider the operator ∂ = ⟨∇, −u\frac{d}{du}⟩ on M. This is well defined in a neighbourhood of 0, and preserves the filtration on M|_{D(0,1)} over this neighbourhood as N_{∇} does. Hence the filtration on M|_{D(0,1)} is induced by a filtration on M by closed, saturated O-submodules by [Ki, 1.3.5]. □

4The idea of relating Kedlaya’s slope filtration to the condition of weak admissibility comes from Berger’s beautiful paper [Be 2], however our treatment here is closer to that of [Ki].

(2.3.2) Let D in Mod_{/K_{0,L}}^{F,φ_{q}} be 1-dimensional over K_{0,L}. Choose a basis vector v for D, and set t_{N,L}(D) = v_{π_{L}}(α) where α ∈ K_{0,L} satisfies φ_{q}(v) = αv. We write t_{H,L}(D) for the unique integer i such that gr^{i}D_{K} is non-zero. For D of arbitrary dimension d, we set t_{N,L}(D) = t_{N,L}(∧^{d}D) and t_{H,L}(D) = t_{H,L}(∧^{d}D). We will say that D is weakly admissible if the usual conditions of weak admissibility are satisfied with these invariants in place of the usual ones. That is if t_{H,L}(D) = t_{N,L}(D) and t_{H,L}(D′) ⩽ t_{N,L}(D′) for all φ_{q}-stable submodules D′ ⊂ D.

Proposition (2.3.3). Let D be in Mod_{/K_{0,L}}^{F,φ_{q}} and M = M(D). Then D is weakly admissible if and only if M_{R} is pure of slope 0.

Proof. Suppose first that dim_{K_{0,L}}D = 1. Let v be a basis vector for D, and write φ(v) = αv for some α ∈ K_{0,L}^{×}. From the definition of M(D) one finds that M(D) = λ^{−t_{H,L}(D)}(D ⊗_{K_{0,L}} O), so

φ_{q}(λ^{−t_{H,L}(D)}e) = ([π_{L}^{m−1}]π_{L}/[π_{L}^{m}])^{−t_{H,L}(D)}αλ^{−t_{H,L}(D)}e

As, [π_{L}^{m}] = [π_{L}] ∘ [π_{L}^{m−1}], we have that [π_{L}^{m}]/[π_{L}^{m−1}] ∈ 𝔖_{L}, is an element whose reduction modulo π_{L} is u^{q^{m}−q^{m−1}}. Hence [π_{L}^{m}]/[π_{L}^{m−1}] is a unit in R^{b}. It follows that M(D) has slope

v_{π_{L}}(α) − t_{H,L}(D) = t_{N,L}(D) − t_{H,L}(D).

This proves the proposition when D has dimension 1. The general case follows from exactly the same argument as in [Ki, 1.3.8], using the equivalence (2.2.6) and (2.3.1). □

(2.4) Denote by Mod_{/𝔖_{L}}^{φ_{q}} the category consisting of finite free 𝔖_{L}-modules 𝔐 equipped with an isomorphisms 1 ⊗ φ_{q} : φ_{q}^{∗}𝔐[1/Q] ⟶^{∼} 𝔐[1/Q]. We denote by Mod_{/𝔖_{L}}^{φ_{q},Γ} the category whose objects consist of an object of Mod_{/𝔖_{L}}^{φ_{q}} equipped a semi-linear action of Γ on 𝔐 which commutes with the action of φ_{q}, and such that Γ acts trivially on 𝔐/u𝔐. We denote by Mod_{/O}^{φ_{q},0} (resp. Mod_{/O}^{φ_{q}.Γ,0}) the full subcategory of Mod_{/O}^{φ_{q}} (resp.

Mod_{/O}^{φ_{q},Γ}) consisting of objects M such that M_{R} = M ⊗_{O} R is pure of slope 0.

Lemma (2.4.1). There is an equivalence of ⊗-categories

Mod_{/𝔖_{L}}^{φ_{q}} ⊗_{ℤ_{p}} ℚ_{p} → Mod_{/O}^{φ_{q},0}; 𝔐 ↦ 𝔐 ⊗_{W_{L}[[u]]} O

where the left hand side means the category obtained from Mod_{/𝔖_{L}}^{φ_{q},Γ} by applying ⊗_{ℤ_{p}}ℚ_{p} to the Hom groups.

Proof. This is identical to [Ki, 1.3.13]. □

Corollary (2.4.2). There is an equivalence of exact ⊗-categories

Mod_{/𝔖_{L}}^{φ_{q},Γ} ⊗_{ℤ_{p}} ℚ_{p} → Mod_{/O}^{φ_{q},Γ,0}; 𝔐 ↦ 𝔐/u𝔐.

Proof. This follows from (2.4.1), since the action of γ ∈ Γ can be thought of as an isomorphism γ^{∗}(𝔐) ⟶^{∼} 𝔐 for 𝔐 in Mod_{/𝔖_{L}}^{φ_{q}} or Mod_{/O}^{φ_{q},Γ,0}. □

(2.4.3) We denote by Mod_{/𝔖_{L}}^{φ_{q},Γ,an} the full subcategory of Mod_{/𝔖_{L}}^{φ_{q},Γ} such that

the corresponding object in Mod_{/O}^{φ_{q},Γ} is in Mod_{/O}^{φ_{q},Γ,an}

Corollary (2.4.4). There is an exact, fully faithful ⊗-functor

Mod_{/𝔖_{L}}^{φ_{q},Γ,an} ⊗_{ℤ_{p}} ℚ_{p} → Mod_{/K_{0,L}}^{F,φ_{q}}

whose essential image consists of the weakly admissible modules in Mod_{/K_{0,L}}^{F,φ_{q}}.

Proof. This follows by combining (2.4.2), (2.2.6) and (2.3.3). □

§3 (φ_{q}, Γ)-modules and crystalline representations

(3.1) Recall the ring R = \underleftarrow{lim}OK̄/p introduced in (1.1). Denote by B_{dR}, B_{cris} ⊃ W (R) the usual rings introduced by Fontaine [Fo 2]. Write B_{cris,L} = B_{cris}⊗_{L_{0}}L. We write φ_{q} for the L-linear extension of the operator φ^{r} on B_{cris}. Note that we have an embedding

(3.1.1)  B_{cris,L} ⊗_{K_{0,L}} K ⟶^{∼} B_{cris} ⊗_{K_{0}} K ↪ B_{dR}.

For D in Mod_{/K_{0,L}}^{F,φ_{q}} this gives rise to an embedding

(3.1.2)  B_{cris,L} ⊗_{K_{0,L}} D_{K} ⟶^{∼} B_{cris} ⊗_{K_{0}} D_{K} ↪ B_{dR} ⊗_{K} D_{K}.

We say an element in B_{cris,L} ⊗_{K_{0,L}} D is in Fil^{0} if its image in B_{dR} ⊗_{K} D_{K} under (3.1.2) is. Dually, we say K_{0,L}-linear map D → B_{cris,L} is compatible with filtrations if the map D_{K} → B_{dR} induced by (3.1.1) is compatible with filtrations.

For D in Mod_{K_{0,L}}^{F,φ_{q}} let

V_{L}(D) = Fil^{0}(B_{cris,L} ⊗_{K_{0,L}} D)^{φ_{q}=1}

and

V_{L}^{∗}(D) = Hom_{φ_{q},Fil}(D, B_{cris,L}).

There is a canonical isomorphism V_{L}^{∗}(D) ⟶^{∼} V_{L}(D^{∗}), where D^{∗} denotes the dual of D in Mod_{/K_{0,L}}^{F,φ_{q}}.

We will prove an analogue for the category Mod_{/K_{0,L}}^{F,φ_{q}} of the result that weakly admissible modules are admissible.

Lemma (3.1.3). Let D in Mod_{/K_{0,L}}^{F,φ_{q}} be of K_{0,L}-dimension 1 such that t_{H,L}(D) ⩽ t_{N,L}(D). Then V_{L}(D) = 0 unless t_{H,L}(D) = t_{N,L}(D), in which case dim_{L}V_{L}(D) = 1.

Proof. Give D(G) = K_{0,L} the structure of an object of Mod_{/K_{0,L}}^{F,φ_{q}} by equipping it with a φ_{q} semi-linear automorphism given by sending 1 to π_{L}^{−1} and defining Fil^{i}D(G)_{K} = D(G)_{K} if i ⩽ −1 and 0 otherwise. Then by [Col 1, Prop. 9.19]

V_{L}(D(G)) = Fil^{0}(D(G) ⊗_{K_{0,L}} B_{cris,L})^{φ_{q}=1} = Fil^{1}B_{cris,L}^{φ_{q}=π_{L}} = t_{L} · L,

where t_{L} = log_{G}u, is a unit in B_{cris,L} [Col 1, Prop. 9.10, 9.17]. Let D in Mod_{/K_{0,L}}^{F,φ_{q}} be of K_{0,L}-dimension 1 such that t_{H,L}(D) ⩽ t_{N,L}(D). Let

d ∈ D be a K_{0,L}-basis vector. Since multiplication by t_{L}^{j} induces a bijection of V_{L}(D) with V_{L}(D ⊗ D(G)^{j}), we may assume that φ_{q}(d) = αd with α ∈ W_{L}^{×}. Furthermore, if k̄ denotes the residue field of K̄, then there exists β ∈ W (k̄)_{L}^{×} such that φ_{q}(β) = αβ, so we may assume that α = 1. Then t_{N,L}(D) = 0 ≥ t_{H,L}(D),

V_{L}(D) = Fil^{−t_{H,L}(D)}B_{cris,L}^{φ_{q}=1}

and the lemma follows from [Col 1, Lem. 9.14]. □

Lemma (3.1.4). Suppose that D in Mod_{/K_{0,L}}^{F,φ_{q}} is weakly admissible. Then

dim_{L}V_{L}(D) ⩽ dim_{K_{0,L}}D.

Proof. This is very similar to [CF, Prop. 4.5]. Denote by C_{cris,L} the field of fractions of B_{cris,L}. Let V denote the C_{cris,L}-span of V_{L}(D) ⊂ C_{cris} ⊗_{K_{0,L}} D. Then V is invariant under the action of G_{K}, and [CF, Lem. 4.6] implies that there exists a unique K_{0,L}-subspace D′ ⊂ D such that C_{cris,L} ⊗_{K_{0,L}} D′ is equal to V. Let s = dim_{K_{0,L}}D′ and d_{1}, …, d_{s} ∈ D′ a K_{0,L}-basis. We also choose v_{1}, …, v_{s} ∈ V_{L}(D) which span V. Then

w := v_{1} ∧ ⋯ ∧ v_{s} = bd_{1} ∧ ⋯ ∧ d_{s}

for some b ∈ C_{cris,L}. As 0 ≠ w ∈ V_{L}(∧^{s}D′), t_{H,L}(D′) = t_{N,L}(D′) by (3.1.3) and dim_{L}V_{L}(∧^{s}D′) = 1. Moreover, (3.1.3) implies that there is a perfect pairing

V_{L}(∧^{s}D′) ⊗_{L} V_{L}((∧^{s}D′)^{∗}) → V_{L}(K_{0,L}) ⟶^{∼} L,

so V_{L}((∧^{s}D′)^{∗}) = b^{−1}d_{1} ∧ ⋯ ∧ d_{s} and b is a unit in B_{cris,L}. Finally, if v ∈ V_{L}(D), write v = ∑_{i=1}^{s} b_{i}v_{i} with b_{i} ∈ C_{cris}. Then for 1 ⩽ i ⩽ s

v_{1} ∧ ⋯ ∧ v_{i−1} ∧ v ∧ v_{i+1} ⋯ ∧ v_{s} = b_{i}w ∈ V_{L}(∧^{s}D′).

Hence b_{i} ∈ L, which shows that dim_{L}V_{L}(D) = s ⩽ dim_{K_{0,L}}D. □

(3.2) Write 𝔖_{L}^{ur} = OÊ^{ur} ∩ W (R)_{L} ⊂ W (Fr R)_{L}. We set P = [π_{L}^{m−1}](u). For γ ∈ Γ, we have γ(P) = [χ(γ)] ∘ [π_{L}^{m−1}] = [χ(γ)π_{L}^{m−1}], so γ(P)/P is a unit in 𝔖_{L}. In particular 𝔖_{L}^{ur}[1/P] is G_{K}-stable. Note that the embedding 𝔖_{L} ↪ W (R)_{L}, extends uniquely to a continuous embedding O ↪ B_{cris,L}^{+}, where B_{cris,L}^{+} = B_{cris}^{+} ⊗_{O_{L_{0}}} O_{L}, as usual.

Lemma (3.2.1). Let 𝔐 be in Mod_{/𝔖_{L}}^{φ_{q}}. The natural map

(3.2.2) V_{𝔖_{L}}(𝔐) := Hom_{𝔖_{L},φ_{q}}(𝔐, 𝔖_{L}^{ur}[1/P]) → Hom_{O_{E},φ_{q}}(O_{E} ⊗_{𝔖_{L}} 𝔐, Ô_{E^{ur}})

is an isomorphism, and both sides are free O_{L}-modules of rank rk_{𝔖_{L}}𝔐. Moreover, if φ_{q} on 𝔐 induces a map φ_{q}^{∗}𝔐 → 𝔐 then the left hand side of (3.2.2) is equal to Hom_{𝔖_{L},φ_{q}}(𝔐, 𝔖_{L}^{ur}).

Proof. Suppose first that φ_{q} induces a map φ_{q}^{∗}𝔐 → 𝔐. In this case the proof of the lemma is identical to the proof of [Ki, 2.1.4], using [Fo 1, §A.1.2]. Next let t_{L} = log_{G}u ∈ O as in the proof of (3.1.3). Then φ_{q}(λ^{−1}t_{L}) = Q(u)λ^{−1}t_{L}, and the zeroes of λ^{−1}t_{L} on D[0, 1) coincide with those of P. Hence λ^{−1}t_{L}P^{−1} ∈ 𝔖_{L}[1/p]^{×}, and there exists w ∈ 𝔖_{L}[1/P]^{×} such that φ_{q}(w) = Q(u)w. Let 𝔐(G) = 𝔖_{L} equipped with a semi-linear action of φ_{q} which takes 1 to Q(u). Then multiplication by w^{i} induces a bijection

V_{𝔖_{L}}(𝔐) → V_{𝔖_{L}}(𝔐 ⊗_{𝔖_{L}} 𝔐(G)^{⊗i}).

Hence the lemma for general 𝔐 in Mod_{/𝔖_{L}}^{φ_{q}} follows from the case considered above. □

Proposition (3.2.3). Let 𝔐 be in Mod_{𝔖_{L}}^{φ_{q},Γ,an} and D in Mod_{/K_{0,L}}^{F,φ_{q}} the weakly admissible module associated to 𝔐 by (2.4.4). Then there is a a canonical G_{K}- equivariant bijection

Hom_{𝔖_{L},φ_{q}}(𝔐, 𝔖_{L}^{ur}[1/P]) ⟶^{∼} Hom_{Fil,φ_{q}}(D, B_{cris,L})

where the right hand side means maps compatible with filtrations in the sense explained in (3.1). In particular, both sides of the above isomorphism have dimension dim_{K_{0,L}}D over L.

Proof. The argument is similar to [Ki, 2.1.5]. Let M = 𝔐 ⊗_{𝔖_{L}} O. Note that P^{−1} ∈ λt_{L}^{−1}𝔖_{L}[1/p]^{×} ⊂ B_{cris,L}. Similarly, λ^{−1} ∈ P t_{L}^{−1}𝔖_{L}[1/p]^{×} ⊂ B_{cris,L}.

Consider the composite (3.2.4) Hom_{𝔖_{L},φ_{q}}(𝔐, 𝔖_{L}^{ur}[1/P]) → Hom_{O,φ_{q}}(M, B_{cris,L}) → Hom_{O,φ_{q}}(φ_{q}^{∗}M, B_{cris,L}).

We claim the image of composite map consists of morphisms respecting filtrations. To see this suppose first that φ_{q} on M induces a map φ_{q}^{∗}M → M. Then by (3.2.1), the left hand side of (3.2.4) is equal to Hom_{𝔖_{L},φ_{q}}(𝔐, 𝔖_{L}^{ur}). A 𝔖_{L}-linear map f : 𝔐 → 𝔖_{L}^{ur} induces an O-linear map f : M → B_{cris,L}^{+}. If m ∈ φ_{q}^{∗}M satisfies (1⊗φ_{q})(m) ∈ Q^{i}M for some integer i, then f ∘(1⊗φ_{q})(m) ∈ Q^{i}B_{cris,L}^{+} ⊂ Fil^{i}B_{dR}, as Q(u) ∈ Fil^{1}B_{dR} [Col 1, Lem. 9.3]. This proves the claim when φ_{q}^{∗}M maps to M. To prove the claim for general M we use the notation of the proof of (3.2.1). Let 𝔐(i) = 𝔐 ⊗_{𝔖} 𝔐(G)^{⊗i}, and M(i) = 𝔐(i) ⊗_{𝔖_{L}} O, where i is an integer which is large enough that φ_{q} induces a map φ_{q}^{∗}𝔐(i) → 𝔐(i). The underlying O-module of M(i) may be identified with M, and the induced identification φ_{q}^{∗}M = φ_{q}^{∗}M(i) identifies Fil^{j}φ_{q}^{∗}M with Fil^{i+j}φ_{q}^{∗}M for all j. As we have a commutative diagram

\xymatrix{Hom_{𝔖_{L},φ_{q}}(𝔐(i), 𝔖_{L}^{ur}[1/P]) \ar[r] \ar[d]^{w^{−i}}_{∼} & Hom_{O,Fil,φ_{q}}(φ_{q}^{∗}M(i), B_{cris,L}) \ar[d]^{w^{−i}}_{∼} \\ Hom_{𝔖_{L},φ_{q}}(𝔐, 𝔖_{L}^{ur}[1/P]) \ar[r] & Hom_{O,φ_{q}}(φ_{q}^{∗}M, B_{cris,L})}

the claim follows for general 𝔐. We now compose (3.2.4) with the map

(3.2.5)  Hom_{O,Fil,φ_{q}}(φ_{q}^{∗}M, B_{cris,L}) → Hom_{K_{0,L},Fil,φ_{q}}(D, B_{cris,L}).

induced by extending a map in the left hand side to φ_{q}^{∗}M[1/λ] and composing with φ_{q}^{∗}(ξ) : D → φ_{q}^{∗}M[1/λ], where ξ is the map of (2.2.2). Note that (3.2.5) respects filtrations as φ_{q}^{∗}(ξ) is compatible with filtrations by (2.2.5), and λ vanishes to order 1 at u = v_{m}. Combining (3.2.4) and (3.2.5) we obtain a canonical G_{K}-equivariant map

(3.2.6)  Hom_{𝔖_{L},φ_{q}}(𝔐, 𝔖_{L}^{ur}[1/P]) → Hom_{K_{0,L},Fil,φ_{q}}(D, B_{cris,L}).

It is easy to see that (3.2.6) is injective. By (3.2.1) the left hand side of (3.2.6) has L-dimension d = rk_{𝔖_{L}}𝔐, while by (3.1.4) the right hand side has dimension ⩽ d. Hence (3.2.6) is an isomorphism. □

(3.3) We now explain how to pass from φ_{q}-modules to φ-modules, using an induction procedure which was explained to us by Fontaine (cf. [FY, §7.3]). Suppose that D is a finite dimensional K_{0,L}-vector space. We set

D̃ := ⊕_{i=0}^{r−1}φ^{i∗}(D) = ⊕_{σ:L_{0}↪L}σ^{∗}(D), so that D̃ is a finite free K_{0} ⊗_{ℚ_{p}} L-module. Here we have denoted by φ^{i} the

map K_{0} ⊗_{L_{0}} L →^{φ^{i}⊗1} K_{0} ⊗_{φ^{i},L_{0}} L. We put D̃_{K} = D̃ ⊗_{K_{0}} K.

We denote by Mod_{/K_{0}⊗_{ℚp} L}^{F,φ} the category of finite free K ⊗_{ℚ_{p}} L-modules D̃ equipped with an isomorphism φ^{∗}(D̃) ⟶^{∼} D̃ and a decreasing separated filtration on D̃_{K} = D̃ ⊗_{K_{0}} K, indexed by ℤ, by K ⊗_{ℚ_{p}} L-submodules. For any D̃ in Mod_{K_{O}⊗_{ℚp} L}^{F,φ} we denote by t_{N}(D̃) and t_{H}(D̃) the usual invariants when D̃ is considered as a filtered φ-module over K_{0}. For any D̃ in Mod_{/K_{0}⊗_{ℚp} L}^{F,φ} we may write D̃_{K} = ⊕_{𝔪}(D̃_{K})_{𝔪} where 𝔪 runs over the maximal ideals of K ⊗_{ℚ_{p}} L. Denote by 𝔪_{0} the kernel of the natural map K ⊗_{ℚ_{p}} L → K. Given D in Mod_{/K_{0,L}}^{F,φ_{q}} we give D̃ the structure of an object in Mod_{/K_{0}⊗_{ℚp} L}^{F,φ} by noting that (D̃_{K})_{𝔪_{0}} may be identified with D_{K}, and giving D̃_{K} the direct sum of the filtration on D_{K} and the trivial filtration on ⊕_{𝔪≠𝔪_{0}}(D̃_{K})_{𝔪}.

Lemma (3.3.1). The functor D ↦ D̃ induces an fully faithful ⊗-functor

Mod_{/K_{0,L}}^{F,φ_{q}} ⟶^{∼} Mod_{K_{0}⊗_{ℚp} L}^{F,φ}

The essential image of this functor consists of those objects such that the filtration on ⊕_{𝔪≠𝔪_{0}} D̃_{K} is trivial.

Proof. Given D in Mod_{K_{0,L}}^{F,φ_{q}} there is a natural isomorphism

φ^{∗}(D̃) = ⊕_{i=1}^{r}φ^{i∗}(D) ⟶^{∼} ⊕_{i=0}^{r−1}φ^{i∗}(D) = D̃,

which sends φ^{i∗}(D) identically to φ^{i∗}(D) for i ≠ r and maps φ^{r∗}(D) = φ_{q}^{∗}(D) to D using the map φ_{q} on D. This defines the functor of the lemma. To define a quasi-inverse on the essential image of the functor, let D̃ be in Mod_{/K_{0}⊗_{ℚp} L}^{F,φ} be such that the filtration on ⊕_{𝔪≠𝔪_{0}} D̃_{K} is trivial, and set D′ = D̃ ⊗_{K_{0}⊗_{ℚp} L} (K_{0} ⊗_{L_{0}} L). There is an isomorphism φ^{∗r}(D′) ⟶^{∼} D′ induced by φ^{∗r}(D̃) ⟶^{∼} D̃. Using the decomposition

K_{0} ⊗_{ℚ_{p}} L ⟶^{∼} ⊕_{i=0}^{r−1}K_{0} ⊗_{φ^{i},L_{0}} L

one sees that there is a canonical isomorphism D̃′ ⟶^{∼} D̃. In particular this makes D′ into an object of Mod_{/K_{0,L}}^{F,φ_{q}}. One checks immediately that these two functors are quasi-inverse. □

Lemma (3.3.2). Let D be in Mod_{/K_{0,L}}^{F,φ_{q}}. Then

(3.3.3)  t_{N,L}(D) = t_{N}(D̃) and t_{H,L}(D) = t_{H}(D̃)

and D is weakly admissible if and only if D̃ is weakly admissible.

Proof. Since the functor of (3.3.1) respects ⊗-products, it suffices to prove (3.3.3) when D is 1-dimensional over K_{0,L}. Moreover, as the essential image of the functor in (3.3.1) is stable under subobjects, (3.3.3) implies the claim regarding weak admissibility. Let D be 1-dimensional over K_{0,L} with basis vector v, and φ_{q}(v) = αv for some α ∈ K_{0,L}. Then for i = 0, …, r − 1 the K_{0}-vector space ∧_{K_{0}}^{[L:L_{0}]}φ^{i∗}(D) has a basis vector e_{i} such that φ(e_{i}) = e_{i+1} if i < r−1, and φ(e_{r−1}) = N_{K_{0,L}/K_{0}}(α)e_{0}. Hence φ takes

e_{0} ∧ ⋯ ∧ e_{r−1} ∈ ∧_{K_{0}}^{[L:ℚ_{p}]}D̃ ⟶^{∼} ⊗_{i=0}^{r−1} ∧_{K_{0}}^{[L:L_{0}]} φ^{i∗}(D)

to (−1)^{r}N_{K_{0,L}/K_{0}}(α)e_{0} ∧ ⋯ ∧ e_{r−1}, and

t_{N}(D̃) = v_{p}(N_{K_{0,L}/K_{0}}(α)) = [L : L_{0}]v_{p}(α) = v_{π_{L}}(α) = t_{N,L}(D).

On the other hand, t_{H,L}(D) = t_{H}(D̃), from the definition of the filtration on D̃. □

Proposition (3.3.4). Let D be a weakly admissible object in Mod_{/K_{0,L}}^{F,φ_{q}}. Then there is a canonical G_{K}-equivariant isomorphism

V_{L}^{∗}(D) ⟶^{∼} V^{∗}(D̃) := Hom_{Fil,φ}(D̃, B_{cris}).

Proof. Suppose f̃: D̃ → B_{cris} ⊗_{ℚ_{p}} L is a K_{0} ⊗_{ℚ_{p}} L-linear, φ-compatible map, such that

f_{K} : D̃_{K} → K ⊗_{K_{0}} B_{cris} ⊗_{ℚ_{p}} L ↪ B_{dR} ⊗_{ℚ_{p}} L

is compatible with filtrations. Consider the composite

θ(f) : D ↪ D̃ → B_{cris} ⊗_{ℚ_{p}} L → B_{cris,L}.

This is a φ_{q}-compatible map, such that the composite

θ(f)_{K} : D_{K} → K ⊗_{K_{0,L}} B_{cris,L} ↪ B_{dR}.

is obtained from f_{K} by localizing at 𝔪_{0}. In particular, θ(f)_{K} is compatible with filtrations. Note also that f can be recovered from θ(f) : The decomposition

K_{0} ⊗_{ℚ_{p}} L ⟶^{∼} ⊕_{i=0}^{r−1}K_{0} ⊗_{φ^{i},L_{0}} L

allows us to view B_{cris,L} as a direct summand in B_{cris} ⊗_{ℚ_{p}} L. Then f is the unique φ-linear extension of the φ_{q}-linear map

D →^{θ(f)} B_{cris,L} ↪ B_{cris} ⊗_{ℚ_{p}} L.

Hence we have an injective map

(3.3.5)  Hom_{Fil,φ,K_{0}⊗_{ℚp} L}(D̃, B_{cris} ⊗_{ℚ_{p}} L) ↪ Hom_{Fil,φ_{q},K_{0,L}}(D, B_{cris,L})

On the other hand, the trace map L → ℚ_{p} induces an isomorphism

(3.3.6)  Hom_{Fil,φ,K_{0}⊗_{ℚp} L}(D̃, B_{cris} ⊗_{ℚ_{p}} L) ⟶^{∼} Hom_{Fil,φ,K_{0}}(D̃, B_{cris}).

Composing the inverse of (3.3.6) with (3.3.5) gives an injective map V^{∗}(D̃) → V_{L}^{∗}(D). As D̃ is admissible by (3.3.2), dim_{L}V^{∗}(D̃) = dim_{K_{0,L}}D, and this is equal to dim_{L}V_{L}^{∗}(D) by (3.3.2). Hence we have V^{∗}(D̃) ⟶^{∼} V_{L}^{∗}(D). □

(3.3.7) Denote by Rep_{G_{K}}^{L-cris} the full subcategory of Rep_{G_{K}} consisting of those objects V such that V ⊗_{O_{L}} L is a crystalline representation and, if D_{dR}(V) = (V ⊗_{ℚ_{p}} B_{dR})^{G_{K}} , then the filtration on D_{dR}(V)_{𝔪} is trivial for 𝔪 ≠ 𝔪_{0} a maximal ideal of K ⊗_{ℚ_{p}} L.

Corollary (3.3.8). There is an exact equivalence of ⊗-categories

Mod_{/𝔖_{L}}^{φ_{q},Γ,an} ⟶^{∼} Rep_{G_{K}}^{L-cris}; 𝔐 ↦ V (O_{E} ⊗_{𝔖_{L}} 𝔐)

where V is the functor introduced in (1.5).

Proof. Using (3.2.3) and (3.3.4) one sees that the functor of (1.6) induces a fully faithful, exact ⊗-functor as in the corollary. To show that this functor is essentially surjective let V be in Rep_{G_{K}}^{L-cris}, and let M = M_{O_{E}} (V). By (2.4.4) and (3.3.1), there exists an 𝔐′ in Mod_{/𝔖_{L}}^{φ_{q},Γ,an} such that V (O_{E} ⊗_{𝔖_{L}} 𝔐′) is isomorphic to a G_{K}-stable O_{L}-lattice V′ ⊂ V ⊗_{O_{L}} L. Thus, by the equivalence of (1.6) there is an isomorphism E ⊗_{O_{E}} M ⟶^{∼} E ⊗_{𝔖_{L}} 𝔐′. Then 𝔐 = M ∩ 𝔐′[1/p] ⊂ M[1/p] is in Mod_{/𝔖_{L}}^{φ_{q},Γ,an} and satisfies O_{E} ⊗_{𝔖_{L}} 𝔐 ⟶^{∼} M. Hence V (O_{E} ⊗_{𝔖_{L}} 𝔐) ⟶^{∼}

V (M) ⟶^{∼} V. □

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Mark Kisin Department of Mathematics University of Chicago USA kisin@ math.uchicago.edu Current address: Department of Mathematics Harvard University USA kisin@math.harvard.edu Wei Ren Department of Mathematics University of Chicago USA 462

Documenta Mathematica 14 (2009)