On Reductions of Families

of Crystalline Galois Representations

Gerasimos Dousmanis

Received: September 3, 2010

Communicated by Peter Schneider

Abstract. Let K_{f} be the finite unramified extension of ℚ_{p} of degree f and E any finite large enough coefficient field containing K_{f}. We construct analytic families of étale (φ, Γ)-modules which give rise to families of crystalline E-representations of the absolute Galois group G_{K_{f}} of K_{f}. For any irreducible effective two-dimensional crystalline E-representation of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}} induced from a crystalline character of G_{K_{2f}} , we construct an infinite family of crystalline E-representations of G_{K_{f}} of the same Hodge-Tate type which contains it. As an application, we compute the semisimplified mod p reductions of the members of each such family.

2010 Mathematics Subject Classification: 11F80, 11F85. Keywords and Phrases: Wach modules, (φ, Γ)-modules, reductions of crystalline Galois representations.

Contents

1 Introduction 874 1.1 Preliminaries and statement of results . . . . . . . . . . . . . . 876

2 Overview of the theory 884 2.1 Étale (φ, Γ)-modules and Wach modules . . . . . . . . . . . . 884 2.2 Wach modules of restricted representations . . . . . . . . . . . 887

3 Effective Wach modules of rank one 889

4 Families of effective Wach modules of arbitrary weight and rank 895 5 Families of two-dimensional crystalline representations 902

5.1 Families of rank two Wach modules . . . . . . . . . . . . . . . . 903 5.2 Corresponding families of rank two filtered φ-modules . . . . . 912

6 Reductions of crystalline representations 914

6.1 Reductions of reducible two-dimensional crystalline representa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 6.2 Proof of theorem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . 919 6.3 Proof of theorem 1.7 . . . . . . . . . . . . . . . . . . . . . . . . 928

1 Introduction

Let p be a prime number and ℚ̄_{p} a fixed algebraic closure of ℚ_{p}. Let N be a positive integer and g = ∑_{n≥1} a_{n}q^{n} a newform of weight k ≥ 2 over Γ_{1}(N)

with character ψ. The complex coefficients a_{n} are algebraic over ℚ and may be viewed as elements of ℚ̄_{p} after fixing embeddings ℚ̄ → ℂ and ℚ̄ → ℚ̄_{p}. By work of Eichler-Shimura when k = 2 and Deligne when k > 2, there exists a continuous irreducible two-dimensional p-adic representation ρ_{g} : G_{ℚ} ⟶ GL_{2}(ℚ̄_{p}) attached to g. If l ∤ pN, then ρ_{g} is unramified at l and det(X − ρ_{g}(Frob_{l})) = X^{2} − a_{l}X + ψ (l) l^{k−1}, where Frob_{l} is any choice of an arithmetic Frobenius at l. The contraction of the maximal ideal of the ring of integers of ℚ̄_{p} via an embedding ℚ̄ → ℚ̄_{p} gives rise to the choice of a place of ℚ̄ above p, and the decomposition group D_{p} at this place is isomorphic to the local Galois group G_{ℚ_{p}} via the same embedding. The local representation

ρ_{g,p} : G_{ℚ_{p}} ⟶ GL_{2}(ℚ̄_{p}),

Reductions of Families of Crystalline Representations 875

obtained by restricting ρ_{g} to D_{p}, is de Rham with Hodge-Tate weights {0, k−1} ([Tsu99]). If p ∤ N the representation ρ_{g,p} is crystalline and the characteristic polynomial of Frobenius of the weakly admissible filtered φ-module 𝔻_{k,a_{p}} := 𝔻_{cris} (ρ_{g,p}) attached to ρ_{g,p} by Fontaine is X^{2} − a_{p}X + ψ (p) p^{k−1} ([Fal89] and [Sc90]). The roots of Frobenius are distinct if k = 2 and conjecturally distinct if k ≥ 3 (see [CE98]). In this case, weak admissibility imposes a unique up to isomorphism choice of the filtration of 𝔻_{k,a_{p}}, and the isomorphism class of the crystalline representation ρ_{g,p} is completely determined by the characteristic polynomial of Frobenius of 𝔻_{k,a_{p}}. The mod p reduction ρ̄_{g,p} : G_{ℚ_{p}} ⟶ GL_{2}(𝔽̄_{p}) of the local representation ρ_{g,p} is well defined up to semisimplification and plays a role in the proof of Serre’s modularity conjecture, now a theorem of Khare and Wintenberger [KW09a], [KW09b], which states that any irreducible continuous odd Galois representation ρ : G_{ℚ} ⟶ GL_{2}(𝔽̄_{p}) is similar to a representation of the form ρ̄_{g} for a certain newform g which should occur in level N(ρ), an integer prime-to-p, and weight κ(ρ) ≥ 2, which Serre explicitly defined in [Ser87]. If ρ_{g,p} is crystalline, the semisimplified mod p reduction ρ̄_{g,p} has been given concrete descriptions in certain cases by work of Berger-Li-Zhu [BLZ04] combined with work of Breuil [Bre03], which extended previous results of Deligne, Fontaine, Serre and Edixhoven, and more recently by Buzzard-Gee [BG09] using the padic Langlands correspondence for GL_{2} (ℚ_{p}) . For a more detailed account and the shape of these reductions see [Ber10, §5.2].

Recall that (up to unramified twist) all irreducible two-dimensional crystalline representations of G_{ℚ_{p}} with fixed Hodge-Tate weights in the range [0; p] have the same irreducible mod p reduction. Reductions of crystalline representations of G_{ℚ_{p^{f}}} := Gal(ℚ̄_{p}/ℚ_{p^{f}}) with f ≠ 1, where ℚ_{p^{f}} is the unramified extension of ℚ_{p} of degree f, are more complicated. For example, in the simpler case where f = 2, there exist irreducible two-dimensional crystalline representation of G_{ℚ_{p^{2}}} with Hodge-Tate weights in the range [0; p − 1] sharing the same characteristic polynomial and filtration, with distinct irreducible or reducible reductions (cf. Proposition 6.22).

The purpose of this paper is to extend the constructions of [BLZ04] to twodimensional crystalline representations of G_{ℚ_{p^{f}}} , and to compute the semisimplified mod p reductions of the crystalline representations constructed. The strategy for computing reductions is to fit irreducible representations of G_{K_{f}} which are not induced from crystalline characters of G_{K_{2f}} into families of representations of the same Hodge-Tate type and with the same mod p reduction, which contain some member which is either reducible or irreducible induced.

Serre’s conjecture has been recently generalized by Buzzard, Diamond and Jarvis [BDJ] for irreducible totally odd two-dimensional 𝔽̄_{p}-representations of the absolute Galois group of any totally real field unramified at p, and has subsequently been extended by Schein [Sch08] to cases where p is odd and tamely ramified in F. Crystalline representations of the absolute Galois group of finite unramified extensions of ℚ_{p} arise naturally in this context of the conjecture of Buzzard, Diamond and Jarvis, and their modulo p reductions are crucial for the weight part of this conjecture (see [BDJ, §3]).

Let F be a totally real number field of degree d > 1, and let I = {τ_{1}, ..., τ_{d}} be the set of real embeddings of F. Let k = (k_{τ_{1}}, k_{τ_{2}}, ..., k_{τ_{d}}, w) ∈ ℕ_{≥1}^{d+1} with k_{τ_{i}} ≡ wmod 2. We denote by O the ring of integers of F and we let 𝔫 ≠ 0 be an ideal of O. The space S_{k}(U_{1}(𝔫)) of Hilbert modular cusp forms of level 𝔫 and weight k is a finite dimensional complex vector space endowed with actions of Hecke operators T_{𝔮} indexed by the nonzero ideals 𝔮 of O (for the precise definitions see [Tay89]). Let 0 ≠ g ∈ S_{k}(U_{1}(𝔫)) be an eigenform for all the T_{𝔮}, and fix embeddings ℚ̄ → ℂ and ℚ̄ → ℚ̄_{p}. By constructions of Rogawski-Tunnell [RT83], Ohta [Oht84], Carayol [Car86], Blasius-Rogawski [BR89], Taylor [Tay89], and Jarvis [Jar97], one can attach to g a continuous Galois representation ρ_{g} : G_{F} → GL_{2}(ℚ̄_{p}), where G_{F} is the absolute Galois group of the totally real field F. Fixing an isomorphism between the residue field of ℚ̄_{p} with 𝔽̄_{p}, the mod p reduction ρ̄_{g} : G_{F} → GL_{2}(𝔽̄_{p}) is well defined up to semisimplification. A continuous representation ρ : G_{F} ⟶ GL_{2}(𝔽̄_{p}) is called modular if ρ ∼ ρ̄_{g} for some Hilbert modular eigenform g. Conjecturally, every irreducible totally odd continuous Galois representation ρ : G_{F} ⟶ GL_{2}(𝔽̄_{p}) is modular ([BDJ]). We now assume that k_{τ_{i}} ≥ 2 for all i. We fix an isomorphism ℚ̄_{p} ≃^{i} ℂ and an algebraic closure F̄ of F. For each prime ideal 𝔭 of O lying above p we denote by F_{𝔭} the completion of F at 𝔭, and we fix an algebraic closure F̄_{𝔭} of F_{𝔭} and an F -embedding F̄ ↪ F̄_{𝔭}. These determine a choice of a decomposition group D_{𝔭} ⊂ G_{F} an isomorphism D_{𝔭} ≃ G_{F_{𝔭}}. For each embedding τ : F_{𝔭} → ℚ̄_{p},

let k_{τ} be the weight of g corresponding to the embedding τ_{|F} : F → ℚ̄_{p} ≃^{i} ℂ. By works of Blasius-Rogawski [BR93], Saito [Sai09], Skinner [Ski09], and T. Liu [Liu09], the local representation ρ_{g,F_{𝔭}} : G_{F_{𝔭}} ⟶ GL_{2}(ℚ̄_{p}), obtained by restricting ρ_{g} to the decomposition subgroup G_{F_{𝔭}}, is de Rham with labeled Hodge-Tate weights \{\frac{k−k_{τ}}{2} , \frac{k+k_{τ} −2}{2}\}_{τ:F_{𝔭}→ℚ̄_{p}}, where k = max\{k_{τ_{i}}\}. This has also been proved by Kisin [Kis08, Theorem 4.3] under the assumption that ρ_{g,F_{𝔭}} is residually irreducible. If p is odd, unramified in F and prime to 𝔫, then ρ_{g,F_{𝔭}} is crystalline by works of Breuil [Bre99, Théorème 1(1)] and Berger [Ber04a, Théorème IV.2.1]. In the newform case, assuming that ρ_{g,p} is crystalline, the weight of g and the eigenvalue of the Hecke operator T_{p} on g completely determine the structure of the filtered φ-module 𝔻_{cris}(ρ_{g,p}). In the Hilbert modular newform case, assuming that ρ_{g,F_{𝔭}} is crystalline, the structure of 𝔻_{cris}(ρ_{g,F_{𝔭}}) is more complicated and the characteristic polynomial of Frobenius and the labeled Hodge-Tate weights do not suffice to completely determine its structure. The filtration of 𝔻_{cris}(ρ_{g,F_{𝔭}}) is generally unknown, and, even worse, the characteristic polynomial of Frobenius and the filtration are not enough to determine the structure of the filtered φ-module 𝔻_{cris}(ρ_{g,F_{𝔭}}). In this case, the isomorphism class is (roughly) determined by an extra parameter in (ℚ̄_{p}^{×})^{f_{𝔭}−1} (for a precise statement see [Dou10, §§6, 7]). As a consequence, if f_{𝔭} ≥ 2 there exist infinite families of non-isomorphic, irreducible two-dimensional crystalline representations of G_{ℚ_{p^{f𝔭}}} sharing the same characteristic polynomial and filtration. For higher-dimensional crystalline E-representations of G_{ℚ_{p^{f}}} , we mention that even in the simpler case of three-dimensional crystalline representations of G_{ℚ_{p}}, there exist non-isomorphic Frobenius-semisimple crystalline representations sharing the same characteristic polynomial and filtration, with the same mod p reductions with respect to appropriately chosen Galois-stable O_{E}-lattices. This follows by applying the constructions of §4 to the higher-dimensional case, and a proof is not included in this paper.

Acknowledgements. I thank Fred Diamond for suggesting this problem and for his feedback, Laurent Berger for useful suggestions, and Seunghwan Chang for detailed comments on drafts. The last parts of the paper were written during visits at the I.H.P. and the I.H.É.S. in Spring 2010. I thank both institutions for their hospitality and the C.N.R.S. and the S.F.B. 478 “Geometrische Strukturen in der Mathematik” of Münster University for financial support.

1.1 Preliminaries and statement of results

Throughout this paper p will be a fixed prime number, K_{f} = ℚ_{p^{f}} the finite unramified extension of ℚ_{p} of degree f, and E a finite large enough extension of K_{f} with maximal ideal 𝔪_{E} and residue field k_{E}. We simply write K whenever the degree over ℚ_{p} plays no role. We denote by σ_{K} the absolute Frobenius of K. We fix once and for all an embedding K↪^{τ_{0}} E and we let τ_{j} = τ_{0} ∘ σ_{K}^{j} for all j = 0, 1, ..., f − 1. We fix the f-tuple of embeddings | τ |:= (τ_{0}, τ_{1}, ..., τ_{f−1}) and

we denote E^{|τ|} := ∏_{τ:K↪E} E. The map ξ : E ⊗ K → E^{|τ|} with ξ_{K}(x ⊗ y) =

(xτ(y))_{τ} and the embeddings ordered as above is a ring isomorphism. The ring automorphism 1_{E} ⊗σ_{K} : E⊗K → E⊗K transforms via ξ to the automorphism φ : E^{|τ|} → E^{|τ|} with φ(x_{0}, x_{1}, ..., x_{f−1}) = (x_{1}, ..., x_{f−1}, x_{0}). We denote by e_{j} = (0, ..., 1, ..., 0) the idempotent of E^{|τ|} where the 1 occurs in the τ_{j}-th coordinate for each j ∈ {0, 1, ..., f − 1}. It is well-known (see for instance [BM02, Lemme 2.2.1.1]) that every continuous representation ρ : G_{K} → GL_{n}(ℚ̄_{p}) is defined over some finite extension of ℚ_{p}. Let ρ : G_{K} → GL_{E}(V) be a continuous E-linear representation. Recall that 𝔻_{cris}(V) = (𝔹_{cris} ⊗_{ℚ_{p}} V)^{G_{K}} , where 𝔹_{cris} is the ring constructed by Fontaine in [Fon88], is a filtered φ-module over K with E-coefficients, and V is crystalline if and only if 𝔻_{cris}(V) is free over E ⊗ K of rank dim_{E} V. One can easily prove that V is crystalline as an E-linear representation of G_{K} if and only if it is crystalline as a ℚ_{p}-linear representation of G_{K} (cf. [CDT99] appendix B). We may therefore extend E whenever appropriate without affecting crystallinity. By a variant of the fundamental theorem of Colmez and Fontaine ([CF00], Théorème A) for nontrivial coefficients, the functor V ↦ 𝔻_{cris}(V) is an equivalence of categories from the category of crystalline E-linear representations of G_{K} to the category of weakly admissible filtered φ-modules (𝔻, φ) over K with E-coefficients (see [BM02], §3). Such a filtered module 𝔻 is a module over E ⊗ K and may be viewed as a module over E^{|τ|} via the ring isomorphism ξ defined above. Its Frobenius endomorphism is bijective and semilinear with respect to the automorphism φ of E^{|τ|}. For each embedding τ_{i} of K into E we define 𝔻_{i} := e_{i}𝔻. We

have the decomposition 𝔻 = ⨁_{i=0}^{f−1} 𝔻_{i}, and we filter each component 𝔻_{i} by set-

ting Fil^{j}𝔻_{i} := e_{i}Fil^{j}𝔻. An integer j is called a labeled Hodge-Tate weight with respect to the embedding τ_{i} of K in E if and only if e_{i}Fil^{−j}𝔻 ≠ e_{i}Fil^{−j+1}𝔻 and is counted with multiplicity dim_{E} (e_{i}Fil^{−j}𝔻/e_{i}Fil^{−j+1}𝔻) . Since the Frobenius endomorphism of 𝔻 restricts to an E-linear isomorphism from 𝔻_{i} to 𝔻_{i−1} for all i, the components 𝔻_{i} are equidimensional over E. As a consequence, there are n = rank_{E⊗K}(𝔻) labeled Hodge-Tate weights for each embedding, counting multiplicities. The labeled Hodge-Tate weights of 𝔻 are by definition the f-tuple of multisets (W_{i})_{τ_{i}}, where each such multiset W_{i} contains n integers, the opposites of the jumps of the filtration of 𝔻_{i}. For crystalline characters we usually write (−k_{0}, −k_{1}, ..., −k_{f−1}) instead of {−k_{i}}_{τ_{i}}. The characteristic polynomial of a crystalline E-linear representation of G_{K} is the characteristic polynomial of the E^{|τ|}-linear map φ^{f}, where (𝔻, φ) is the weakly admissible filtered φ-module corresponding to it by Fontaine’s functor.

Definition 1.1. A filtered φ-module (𝔻, φ) is called F-semisimple, non-F-semisimple, or F-scalar if the E^{|τ|}-linear map φ^{f} has the corresponding property.

We may twist 𝔻 by some appropriate rank one weakly admissible filtered φmodule (see Proposition 3.5) and assume that W_{i} = {−w_{in−1} ≤ ... ≤ −w_{i2} ≤ −w_{i1} ≤ 0} for all i = 0, 1, ..., f − 1, for some non-negative integers w_{ij}. The Hodge-Tate weights of a crystalline representation V are the opposites of the jumps of the filtration of 𝔻_{cris}(V). If they are all non-positive, the crystalline representation is called effective or positive. To avoid trivialities, throughout the paper we assume that at least one labeled Hodge-Tate weight is strictly negative.

Notation 1.2. Let k_{i} be nonnegative integers which we call weights. Assume that after ordering them and omitting possibly repeated weights we get w_{0} < w_{1} < ... < w_{t−1}, where w_{0} is the smallest weight, w_{1} the second smallest weight,..., and w_{t−1}, for some 1 ≤ t ≤ f, is the largest weight. The largest weight w_{t−1} will be usually denoted by k. For convenience we define w_{−1} = 0. Let I_{0} = {0, 1, ..., f − 1} and I_{0}^{+} = {i ∈ I_{0} : k_{i} > 0}. For j = 1, 2, ..., t − 1 we let I_{j} = {i ∈ I_{0} : k_{i} > w_{j−1}} and for j = t we define I_{t} = ∅. Let f^{+} = |I_{0}^{+}| be the number of strictly positive weights. For each subset J ⊂ I_{0} we write f_{J} = ∑_{i∈J} e_{i} and E^{|τ_{J} |} = f_{J} · E^{|τ|}. We may

visualize the sets E^{|τ_{I_{j}} |} as follows: E^{|τ_{I_{0}} |} is the Cartesian product E^{f}. Starting with E^{|τ_{I_{0}} |}, we obtain E^{|τ_{I_{1}} |} by killing the coordinates where the smallest weight occurs i.e. by killing the i-th coordinate for all i with k_{i} = w_{0}. We obtain E^{|τ_{I_{2}} |} by further killing the coordinates where the second smallest weight w_{1} occurs and so on. For any vector x ∈ E^{|τ|} we denote by x_{i} its i-th coordinate and by J_{x} its support {i ∈ I_{0} : x_{i} ≠ 0}. We define as norm of x with respect to φ the

vector Nm_{φ}(x) := ∏_{i=0}^{f−1} φ^{i}(x) and we write v_{p}(Nm_{φ}(x)) := v_{p} ( ∏_{i=0}^{f−1} x_{i}) , where

v_{p} is the normalized p-adic valuation of ℚ̄_{p}. If ℓ is an integer we write ℓ = (ℓ, ℓ, ..., ℓ) and v_{p}(x) > ℓ (resp. if v_{p}(x) ≥ ℓ) if and only if v_{p}(x_{i}) > ℓ (resp. v_{p}(x_{i}) ≥ ℓ) for all i. Finally, for any matrix A ∈ M_{n}(E^{|τ|}) we define as its φ-norm the matrix Nm_{φ}(A) := Aφ(A)···φ^{f−1}(A), with φ acting on each entry of A.

In §3 we construct the effective crystalline characters of G_{K_{f}} . More precisely, for i = 0, 1, ..., f − 1 we construct E-characters χ_{i} of G_{K_{f}} with labeled Hodge-Tate weights −e_{i+1} = (0, ..., −1, ...0) with the −1 appearing in the (i + 1)- place for all i, and we show that any crystalline E-character of G_{K_{f}} with labeled Hodge-Tate weights {−k_{i}}_{τ_{i}} can be written uniquely in the form χ = η · χ_{0}^{k_{1}} · χ_{1}^{k_{2}} · · · · · χ_{f−2}^{k_{f−1}} · χ_{f−1}^{k_{0}} for some unramified character η of G_{K_{f}} . In the same section we prove the following.

Theorem 1.3. Let {ℓ_{i}, ℓ_{i+f}} = {0, k_{i}}, where the k_{i}, i = 0, 1, ..., f − 1 are nonnegative integers. Let f^{+} be the number of strictly positive k_{i} and assume that f^{+} ≥ 1.

(i) The crystalline character χ_{ℓ} = χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{2f−2}^{ℓ_{2f−1}} · χ_{2f−1}^{ℓ_{0}} of G_{K_{2f}} has labeled Hodge-Tate weights (−ℓ_{0}, −ℓ_{1}, ..., −ℓ_{2f−1}) and does not extend to G_{K_{f}} . The induced representation Ind_{K_{2f}}^{K_{f}} (χ_{ℓ}) is irreducible and crystalline with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}.

(ii) Let V be an irreducible two-dimensional crystalline E-representation of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, whose restriction to G_{K_{2f}} is reducible. There exist an unramified character η of G_{K_{f}} and nonnegative integers m_{i}, i = 0, 1, ..., 2f − 1, with {m_{i}, m_{i+f}} = {0, k_{i}} for all i = 0, 1, ..., f − 1, such that

V ≃ η ⊗ Ind_{K_{2f}}^{K_{f}} (χ_{0}^{m_{1}} · χ_{1}^{m_{2}} · · · · · χ_{2f−2}^{m_{2f−1}} · χ_{2f−1}^{m_{0}}) .

(iii) Ind_{K_{2f}}^{K_{f}} (χ_{ℓ}) ≃ Ind_{K_{2f}}^{K_{f}} (χ_{m}) if and only if χ_{ℓ} = χ_{m} or χ_{ℓ}^{σ} = χ_{m}, where

χ_{ℓ}^{σ} = χ_{0}^{ℓ′_{1}} ·χ_{1}^{ℓ′_{2}} ·· · ··χ_{2f−2}^{ℓ′_{2f−1}} ·χ_{2f−1}^{ℓ′_{f}}, with ℓ′_{i} = ℓ_{i+f} and indices viewed modulo 2f.

(iv) Up to twist by some unramified character, there exist precisely 2^{f^{+} −1} distinct isomorphism classes of irreducible two-dimensional crystalline Erepresentations of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, induced from crystalline characters of G_{K_{2f}} .

Next, we turn our attention to generically irreducible families of twodimensional crystalline E-representations of G_{K_{f}} . For any irreducible effective two-dimensional crystalline E-representation of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}} which is induced from a crystalline character of G_{K_{2f}} , we construct an infinite family of crystalline E-representations of G_{K_{f}} of the same Hodge-Tate type which contains it. The members of each of these families have the same semisimplified mod p reductions which we explicitly compute. Let V_{ℓ} = Ind_{K_{2f}}^{K_{f}} (χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{2f−2}^{ℓ_{2f−1}} · χ_{2f−1}^{ℓ_{0}}) , where {ℓ_{i}, ℓ_{i+f}} = {0, k_{i}} for all i = 0, 1, ..., f − 1, and assume that at least one k_{i} is strictly positive. Theorem 1.3 asserts that V_{ℓ} is irreducible and crystalline with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}. We describe the members of the family containing V_{ℓ} in terms of their corresponding by the Colmez-Fontaine theorem weakly admissible filtered φ-modules.

Definition 1.4. We define the following four types of matrices

t_{1}: ( \begin{matrix}p^{k_{i}} & 0 \\ X_{i} & 1 \end{matrix} ) , t_{2}: ( \begin{matrix}X_{i} & 1 \\ p^{k_{i}} & 0 \end{matrix} ) , t_{3}: ( \begin{matrix}1 & X_{i} \\ 0 & p^{k_{i}} \end{matrix} ) , t_{4} : ( \begin{matrix}0 & p^{k_{i}} \\ 1 & X_{i} \end{matrix} ) ,

where the X_{i} are indeterminates. Let k = max{k_{i}, i = 0, 1, ..., f − 1} and let

m := \{ \begin{matrix}⌊\frac{k−1}{p−1}⌋ if k ≥ p and k_{i} ≠ p for some i, \\ 0 if k ≤ p − 1 or k_{i} = p for all i. \end{matrix}

Let P (\overrightarrow{X}) = (P_{1} (X_{1}) , P_{2} (X_{2}) , ..., P_{f} (X_{f})) be a matrix whose coordinates P_{j} (X_{j}) are matrices of type 1, 2, 3 or 4. To each such f-tuple we attach a type-vector i ∈ {1, 2, 3, 4}^{f}, where for any j = 1, 2, ..., f, the j-th coordinate of i is defined to be the type of the matrix P_{j}. We write P (\overrightarrow{X}) = P^{i}(\overrightarrow{X}). The set of all f-tuples of matrices of type 1, 2, 3, 4 will be denoted by P. There is no loss to assume that the first f − 1 coordinates of P (\overrightarrow{X}) are of type 1 or 2 (see Remark 6.13) and unless otherwise stated we always assume so. Matrices of type t_{1} or t_{3} are called of odd type while matrices of type t_{2} or t_{4} are called of even type.

For any vector a = (α_{1}, α_{2}, ..., α_{f}) ∈ (p^{m}𝔪_{E})^{f} we obtain a matrix

P^{i} (α) = (P_{1} (α_{1}) , P_{2} (α_{2}) , ..., P_{f} (α_{f}))

by evaluating each indeterminate X_{i} at α_{i}. We view indices of f-tuples mod f, so P_{f} = P_{0}. To construct the family containing V_{ℓ} we choose the types of the matrices P_{i} as follows:

(1) If ℓ_{1} = 0, P_{1} = t_{2};

(2) If ℓ_{1} = k_{1} > 0, P_{1} = t_{1}. For i = 2, 3, ..., f − 1 we choose the type of the matrix P_{i} as follows:

(1) If ℓ_{i} = 0, then:

• If an even number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, P_{i} = t_{2};

• If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, P_{i} = t_{1}.

(2) If ℓ_{i} = k_{i} > 0, then:

• If an even number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, P_{i} = t_{1};

• If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, P_{i} = t_{2}.

Finally, we choose the type of the matrix P_{0} as follows:

(1) If ℓ_{0} = 0, then:

• If an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, P_{0} = t_{4}; • If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type,

P_{0} = t_{3}.

(2) If ℓ_{0} = k_{i} > 0, then:

• If an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, P_{0} = t_{2};

• If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, P_{0} = t_{1}.

We define families of rank two filtered φ-modules (𝔻_{k}^{i} (α) , φ) over E^{|τ|} by

equipping 𝔻_{k}^{i} (α) = E^{|τ|}η_{1} ⨁ E^{|τ|}η_{2} with the Frobenius endomorphism defined

by (φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2}) P^{i} (α) and the filtration

Fil^{j}(𝔻_{k}^{i} (α)) = \{ \begin{matrix}E^{|τ|}η_{1} ⨁ E^{|τ|}η_{2} if j ≤ 0, \\ E^{|τ_{I_{0}} |} (xη_{1} + yη_{2}) if 1 ≤ j ≤ w_{0}, \\ E^{|τ_{I_{1}} |} (xη_{1} + yη_{2}) if 1 + w_{0} ≤ j ≤ w_{1}, \\ ⋯ & ⋯ \\ E^{|τ_{I_{t−1}} |} (xη_{1} + yη_{2}) if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0  if j ≥ 1 + w_{t−1}, \end{matrix} (1.1)

where x = (x_{0}, x_{1}, ..., x_{f−1}) and y = (y_{0}, y_{1}, ..., y_{f−1}), with

(x_{i}, y_{i}) = \{ \begin{matrix}(1, −α_{i}) if P_{i} has type 1 or 2, \\ (−α_{i}, 1) if P_{i} has type 3 or 4. \end{matrix}  (1.2)

Theorem 1.5. Let i be the type-vector attached to the f-tuple (P_{1}, P_{2}, ..., P_{f}) defined above. For any α ∈ (p^{m}𝔪_{E})^{f} ,

(i) The filtered φ-module 𝔻_{k}^{i}(α) is weakly admissible and corresponds to a

two-dimensional crystalline E-representations V_{k}^{i}(α) of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}};

(ii) V_{k}^{i}(0) = Ind_{K_{2f}}^{K_{f}} (χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{2f−2}^{ℓ_{2f−1}} · χ_{2f−1}^{ℓ_{0}});

(iii) \overline{V}_{k}^{i} (α) = \overline{V}_{k}^{i}(0);

(iv) (\overline{V}_{k}^{i} (α)_{|I_{K_{f}}} )^{s.s.} = ω_{2f,τ̄_{0}}^{β} ⨁ ω_{2f,τ̄_{0}}^{p^{f} β}, where β = − ∑_{i=0}^{2f−1} p^{i}ℓ_{i};

(v) The residual representation \overline{V}_{k}^{i} (α) is irreducible if and only if 1 + p^{f} ∤ β;

(vi) Any irreducible member of the family {V_{k}^{i} (α) , α ∈ (p^{m}𝔪_{E})^{f}} , other

than V_{k}^{i}(0), is non-induced.

Notice that in the cases where 1 + p^{f} ∤ β, all the members of the family {V_{k}^{i} (α) , α ∈ (p^{m}𝔪_{E})^{f}} are forced to be irreducible. Next, we compute the semisimplified reduction of any reducible two-dimensional crystalline Erepresentation of G_{K_{f}} . After enlarging E if necessary, any reducible rank two weakly admissible filtered φ-module 𝔻 over E^{|τ|} with labeled Hodge-Tate weights \{0, −k_{i}\}_{τ_{i}} contains an ordered basis \underline{η} = (η_{1}, η_{2}) in which the matrix

of Frobenius takes the form Mat\underline{η}(φ) = ( \begin{matrix}α & 0 \\ ∗ & δ \end{matrix} ) such that 𝔻_{2} = (E^{|τ|}) η_{2} is

a φ-stable weakly admissible submodule (see Proposition 6.4). The filtration of 𝔻 in such a basis \underline{η} has the form

Fil^{j}(𝔻) = \{ \begin{matrix}E^{|τ|}η_{1} ⨁ E^{|τ|}η_{2} if j ≤ 0, \\ E^{|τ_{I_{0}} |} (xη_{1} + yη_{2}) if 1 ≤ j ≤ w_{0}, \\ E^{|τ_{I_{1}} |} (xη_{1} + yη_{2}) if 1 + w_{0} ≤ j ≤ w_{1}, \\ ⋯ & ⋯ \\ E^{|τ_{I_{t−1}} |} (xη_{1} + yη_{2}) if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0 if j ≥ 1 + w_{t−1}, \end{matrix}

for some vectors x, y ∈ E^{|τ|} with (x_{i}, y_{i}) ≠ (0, 0) for all i. For each i ∈ I_{0}, let

m_{i} = \{ \begin{matrix}0 if x_{i} ≠ 0 , \\ k_{i} if x_{i} = 0. \end{matrix}

Theorem 1.6. Let V be any reducible two-dimensional crystalline Erepresentation of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}} corresponding to the weakly admissible filtered φ-module 𝔻 as above.

(i) There exist unramified characters η_{i} of G_{K_{f}} such that

V ≃ ( \begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix} ) ,

where ψ_{1} = η_{1} · χ_{0}^{m_{1}} · · · · · χ_{f−2}^{m_{f−1}} · χ_{f−1}^{m_{0}} and ψ_{2} = η_{2} · χ_{0}^{k_{1}−m_{1}} · χ_{1}^{k_{2}−m_{2}} ·

⋯ χ_{f−2}^{k_{f−1}−m_{f−1}} · χ_{f−1}^{k_{0}−m_{0}};

(ii) (\overline{V}_{|I_{K}} )^{s.s.} = ω_{f,τ̄_{0}}^{β_{1}} ⨁ ω_{f,τ̄_{0}}^{β_{2}}, where β_{1} = − ∑_{i=0}^{f−1} m_{i}p^{i} and β_{2} =

∑_{i=0}^{f−1} (m_{i} − k_{i}) p^{i}.

The computation of the semisimplified mod p reduction of a reducible twodimensional crystalline representation is easy and does not require the construction of the Wach module (see §2.1 for the definition) corresponding to some G_{K_{f}} -stable lattice contained in it. Computing the non-semisimplified mod p reduction of a two-dimensional crystalline representations with reducible reduction is an interesting problem not pursued in this paper. For results of this flavour for K = ℚ_{p^{2}}, see [CD09]. Up to twist by some unramified character, any split-reducible two-dimensional crystalline E-representations of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}} is of the form

V_{ℓ,ℓ′} (η) = η · χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{f−2}^{ℓ_{f−1}} · χ_{f−1}^{ℓ_{0}} ⨁ χ_{0}^{ℓ′_{1}} · χ_{1}^{ℓ′_{2}} · · · · · χ_{f−2}^{ℓ′_{f−1}} · χ_{f−1}^{ℓ′_{0}},

for some unramified character η and some nonnegative integers ℓ_{i} and ℓ′_{i} such that {ℓ_{i}, ℓ′_{i}} = {0, k_{i}} for all i. In Theorem 1.5 we showed that each irreducible representation of G_{K_{f}} induced from some crystalline character of G_{K_{2f}} belongs to an infinite family of crystalline representations of the same Hodge-Tate types with the same mod p reductions. In the next theorem we prove the same for any split-reducible, non-ordinary two-dimensional crystalline E-representation of G_{K_{f}} . We list the weakly admissible filtered φ-modules corresponding to these families. In order to construct the infinite family containing V_{ℓ,ℓ′} (η) , we

define a matrix P^{i}(\overrightarrow{X}) ∈ P by choosing the (f − 1)-tuple (P_{1}, P_{2}, ..., P_{f−1}) as in Theorem 1.5. If η = η_{c} is the unramified character which maps the geometric Frobenius Frob_{K_{f}} of G_{K_{f}} to c, we replace the entry p^{k_{0}} in the definition of the matrix P_{0} by cp^{k_{0}}. The type of the matrix P_{0} is chosen as follows:

(1) If ℓ_{0} = 0, then:

• If an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, P_{0} = t_{3};

• If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, P_{0} = t_{4}.

(2) If ℓ_{0} = k_{0} > 0, then:

• If an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, P_{0} = t_{1};

• If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, P_{0} = t_{2}.

Using the matrices P^{i}(\overrightarrow{X}) we define families of two-dimensional crystalline Erepresentations {V_{k}^{i} (α) , α ∈ (p^{m}𝔪_{E})^{f}} of G_{K_{f}} as in Theorem 1.5 and prove the following.

Theorem 1.7. Let i be the type-vector attached to the f-tuple (P_{1}, P_{2}, ..., P_{f}) defined above.

(i) There exists some unramified character μ such that V_{k}^{i}(0) ≃ μ ⊗ V_{ℓ,ℓ′}(η);

(ii) Assume that ℓ ≠ 0 and ℓ′ ≠ 0. For any α ∈ (p^{m}𝔪_{E})^{f} , \overline{V}_{k}^{i}(α) ≃ \overline{V}_{k}^{i}(0);

(iii) \overline{V}_{ℓ,ℓ′}(η)_{|I_{K_{f}}} = ω_{f,τ̄_{0}}^{β} ⨁ ω_{f,τ̄_{0}}^{β′}, where β = − ∑_{i=0}^{f−1} ℓ_{i}p^{i} and β′ = − ∑_{i=0}^{f−1} ℓ′_{i}p^{i}.

A family as in Theorem 1.7 can contain simultaneously split and non-split reducible, as well as irreducible crystalline representations. For example, in the family {V_{k}^{(1,3)}(α), α ∈ (p^{m}𝔪_{E})^{2}} , the representation V_{k}^{(1,3)}(α) is split-

reducible if and only if α = 0, non-split-reducible if and only if precisely one of the coordinates α_{i} of α is zero, and irreducible if and only if α_{0}α_{1} ≠ 0 (cf. Proposition 6.21). The families of Wach modules which give rise to V_{k}^{(1,3)}(α) contain infinite sub-families of non-split reducible Wach modules which can be used to compute the non-semisimplified mod p reduction of the corresponding crystalline representations with respect to G_{K_{f}} -stable O_{E}-lattices. Some reducible two-dimensional crystalline representations with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}} are easily recognized by looking at their trace of Frobenius. More precisely, if Tr (φ^{f}) ∈ O_{E}^{×}, then the representation is reducible (cf. Proposition 6.5), with the converse being false.

2 Overview of the theory

2.1 Étale (φ, Γ)-modules and Wach modules

The general theory of (φ, Γ)-modules works for arbitrary finite extensions K of ℚ_{p}. However, a theory of Wach modules, which is our main tool and which we briefly recall in this section, currently exists only when K is unramified over ℚ_{p}. We temporarily allow K to be any finite extension of ℚ_{p}; we will go back to assume that K is unramified after Theorem 2.2. Let K_{n} = K(ζ_{p^{n}}), where ζ_{p^{n}} is a primitive p^{n}-th root of unity inside ℚ̄_{p}, and let K_{∞} = ∪_{n≥1}K_{n}. Let χ : G_{K} → ℤ_{p}^{×} be the cyclotomic character, and let H_{K} = ker χ = Gal(ℚ̄_{p}/K_{∞}) and Γ_{K} = G_{K}/H_{K} = Gal(K_{∞}/K). Fontaine ([Fon90]) has constructed topological rings 𝔸 and 𝔹 endowed with continuous commuting Frobenius φ and G_{ℚ_{p}}- actions. Unless otherwise stated and whenever applicable, continuity means continuity with respect to the topologies induced by the weak topologies of the rings 𝔸 and 𝔹. Let 𝔸_{K} = 𝔸^{H_{K}} and 𝔹_{K} = 𝔹^{H_{K}} , and define 𝔸_{K ,E} := O_{E} ⊗_{ℤ_{p}} 𝔸_{K} and 𝔹_{K ,E} := E ⊗_{ℚ_{p}} 𝔹_{K}. The actions of φ and Γ_{K} extend to 𝔸_{K,E} and 𝔹_{K ,E} by O_{E} (resp. E, k_{E})-linearity, and one easily sees that 𝔸_{K,E} = 𝔸_{E}^{H_{K}} and 𝔹_{K,E} = 𝔹_{E}^{H_{K}} .

Definition 2.1. A (φ, Γ)-module over 𝔸_{K,E} (resp. 𝔹_{K,E}) is an 𝔸_{K,E}-module of finite type (resp. a free 𝔹_{K,E}-module of finite type) endowed with a semilinear and continuous action of Γ_{K}, and with a semilinear map φ which commutes with the action of Γ_{K}. A (φ, Γ)-module M over 𝔸_{K,E} is called étale if φ^{∗}(M) = M, where φ^{∗}(M) is the 𝔸_{K,E}-module generated by the set φ(M). A (φ, Γ)- module M over 𝔹_{K,E} is called étale if it contains a basis (e_{1}, ..., e_{d}) over 𝔹_{K,E} such that (φ(e_{1}), ..., φ(e_{d})) = (e_{1}, ..., e_{d})A for some matrix A ∈ GL_{d} (𝔸_{K,E}) .

If V is a continuous E-linear representation of G_{K}, we equip the 𝔹_{K,E}-module 𝔻(V) := (𝔹_{E} ⊗_{E} V)^{H_{K}} with a Frobenius endomorphism φ defined by φ(b ⊗ v) := φ(b) ⊗ v, where φ on the right hand side is the Frobenius of 𝔹_{E}, and with an action of Γ_{K} given by ḡ(b ⊗ v) := gb ⊗ gv for any g ∈ G_{K}. This Γ_{K}-action commutes with φ and is continuous. Moreover, 𝔻(V) is an étale (φ, Γ)-module over 𝔹_{K,E}. Conversely, if D is an étale (φ, Γ)-module over 𝔹_{K,E}, let 𝕍(D) := (𝔹_{E} ⊗_{𝔹_{K,E}} D)^{φ=1} , where φ(b ⊗ d) := φ(b) ⊗ φ(d). The E-vector space 𝕍(D) is finite dimensional and is equipped with a continuous E-linear G_{K}-action given by g(b ⊗ d) := gb ⊗ ḡd. We have the following fundamental theorem of Fontaine.

Theorem 2.2. [Fon90]

(i) There is an equivalence of categories between continuous E-linear representations of G_{K} and étale (φ, Γ)-modules over 𝔹_{K,E} given by

𝔻 : Rep_{E} (G_{K}) → Mod_{(φ,Γ)}^{ét} (𝔹_{K,E}) : V ⟼ 𝔻(V) := (𝔹_{E} ⊗_{E} V)^{H_{K}} ,

with quasi-inverse functor

𝕍 : Mod_{(φ,Γ)}^{ét} (𝔹_{K,E}) → Rep_{E} (G_{K}) : D ⟼ 𝕍(D) := (𝔹_{E} ⊗_{𝔹_{K,E}} D)^{φ=1} .

(ii) There is an equivalence of categories between continuous O_{E}-linear representations of G_{K} and étale (φ, Γ)-modules over 𝔸_{K,E} given by

𝔻 : Rep_{O_{E}} (G_{K}) → Mod_{(φ,Γ)}^{ét} (𝔸_{K,E}) : T ⟼ 𝔻(T) := (𝔸_{E} ⊗_{O_{E}} T)^{H_{K}} ,

with quasi-inverse functor

𝕋 : Mod_{(φ,Γ)}^{ét} (𝔸_{K,E}) → Rep_{O_{E}} (G_{K}) : D ⟼ 𝕋(D) := (𝔸_{E} ⊗_{𝔸_{K,E}} D)^{φ=1}.

We return to assume that K is unramified over ℚ_{p}. In this case 𝔸_{K} has the form

𝔸_{K} = { ∑_{n=−∞}^{∞} α_{n}π_{K}^{n} : α_{n} ∈ O_{K} and lim_{n→−∞}α_{n} = 0} for some element π_{K} which

can be thought of as a formal variable. The Frobenius endomorphism φ of 𝔸_{K} extends the absolute Frobenius of O_{K} and is such that φ(π_{K}) = (1 + π_{K})^{p} − 1. The Γ_{K}-action of 𝔸_{K} is O_{K}-linear, commutes with Frobenius, and is such that γ(π_{K}) = (1 + π_{K})^{χ(γ)} − 1 for all γ ∈ Γ_{K}. For simplicity we write π instead of π_{K}. The ring 𝔸_{K} is local with maximal ideal (p), fraction field 𝔹_{K} = 𝔸_{K}[\frac{1}{p}], and residue field 𝔼_{K} := k_{K}((π)), where k_{K} is the residue field of K. The rings 𝔸_{K}, 𝔸_{K,E}, 𝔹_{K} and 𝔹_{K,E} contain the subrings 𝔸_{K}^{+} = O_{K}[[π]], 𝔸_{K,E}^{+} := O_{E} ⊗_{ℤ_{p}} 𝔸_{K}^{+}, 𝔹_{K}^{+} = 𝔸_{K}^{+}[\frac{1}{p}] and 𝔹_{K,E}^{+} := E ⊗_{ℚ_{p}} 𝔹_{K}^{+} respectively, and these subrings are equipped with the restrictions of the φ and the Γ_{K}-actions of the rings containing them. There is a ring isomorphism

ξ : 𝔸_{K,E}^{+} → ∏_{τ:K↪E} O_{E}[[π]]  (2.1)

given by ξ (a ⊗ b) = (aτ_{0} (b) , aτ_{1} (b) , ..., aτ_{f−1} (b)) , where τ_{i} ( ∑_{n=0}^{∞} β_{n}π^{n}) =

∑_{n=0}^{∞} τ_{i} (β_{n}) π^{n} for all b = ∑_{n=0}^{∞} β_{n}π^{n} ∈ 𝔸_{K}^{+}. The ring O_{E}[[π]]^{|τ|} := ∏_{τ:K↪E} O_{E}[[π]]

is equipped via ξ with commuting O_{E}-linear actions of φ and Γ_{K} given by

φ(α_{0}(π), α_{1}(π), ..., α_{f−1}(π)) = (α_{1}(φ(π)), ..., α_{f−1}(φ(π)), α_{0}(φ(π))) (2.2)
and γ(α_{0}(π), α_{1}(π), ..., α_{f−1}(π)) = (α_{0}(γπ), α_{1}(γπ), ..., α_{f−1}(γπ))  (2.3)

for all γ ∈ Γ_{K}.

Definition 2.3. Suppose k ≥ 0. A Wach module over 𝔸_{K,E}^{+} (resp. 𝔹_{K,E}^{+}) with weights in [−k; 0] is a free 𝔸_{K,E}^{+}-module (resp. 𝔹_{K,E}^{+}-module) N of finite rank, endowed with an action of Γ_{K} which becomes trivial modulo π, and also with a Frobenius map φ which commutes with the action of Γ_{K} and such that φ(N) ⊂ N and N/φ^{∗}(N) is killed by q^{k}, where q := φ(π)/π.

A natural question is to determine the types of étale (φ, Γ)-modules which correspond to crystalline representations via Fontaine’s functor. An answer is given by the following theorem of Berger who built on previous work of Wach [Wac96], [Wac97] and Colmez [Col99].

Theorem 2.4. [Ber04a]

(i) An E-linear representation V of G_{K} is crystalline with Hodge-Tate weights in [−k; 0] if and only if 𝔻(V) contains a unique Wach module ℕ(V) of rank dim_{E} V with weights in [−k; 0]. The functor V ↦ ℕ(V) defines an equivalence of categories between crystalline representations of G_{K} and Wach modules over 𝔹_{K,E}^{+}, compatible with tensor products, duality and exact sequences.

(ii) For a given crystalline E-representation V, the map T ↦ ℕ(T) := ℕ(V)∩ 𝔻(T) induces a bijection between G_{K}-stable, O_{E}-lattices of V and Wach modules over 𝔸_{K,E}^{+} which are 𝔸_{K,E}^{+}-lattices contained in ℕ(V). Moreover 𝔻(T) = 𝔸_{K,E} ⊗_{𝔸_{K,E}^{+}} ℕ(T).

(iii) If V is a crystalline E-representation of G_{K}, and if we endow ℕ(V) with the filtration Fil^{i}ℕ(V) = {x ∈ ℕ(V)|φ(x) ∈ q^{i}ℕ(V)}, then we have an isomorphism

𝔻_{cris}(V) → ℕ(V)/πℕ(V)

of filtered φ-modules over E^{|τ|} (with the induced filtration on ℕ(V)/πℕ(V)).

In view of Theorems 2.2 and 2.4, constructing the Wach module ℕ(T) of a G_{K}-stable O_{E}-lattice T in a crystalline representation V amounts to explicitly constructing the crystalline representation. Indeed, we have

V ≃ E ⊗_{O_{E}} (𝔸_{K,E} ⊗_{𝔸_{K,E}^{+}} ℕ(T))^{φ=1} .

An obvious advantage of using Wach modules is that instead of working with the more complicated rings 𝔸_{K,E} and 𝔹_{K,E}, one works with the simpler ones 𝔸_{K,E}^{+} and 𝔹_{K,E}^{+}.

2.2 Wach modules of restricted representations

In this section we relate the Wach module of an effective n-dimensional effective crystalline E-representation V_{K_{f}} of G_{K_{f}} to the Wach module of its restriction V_{K_{df}} to G_{K_{df}} .

Proposition 2.5. (i) The Wach module associated to the representation V_{K_{df}} is given by

ℕ(V_{K_{df}}) = 𝔹_{K_{df} ,E}^{+} ⊗_{𝔹_{K_{f} ,E}^{+}} ℕ(V_{K_{f}}),

where ℕ(V_{K_{f}}) is the Wach module associated to V_{K_{f}} .

(ii) If T_{K_{f}} is a G_{K_{f}} -stable O_{E}-lattice in V_{f} associated to the Wach-module ℕ(T_{K_{f}}), then V_{df} contains some G_{K_{df}} -stable O_{E}-lattice T_{K_{df}} whose associated Wach module is

ℕ(T_{K_{df}}) = 𝔸_{K_{df} ,E}^{+} ⊗_{𝔸_{K_{f} ,E}^{+}} ℕ(T_{K_{f}}).

Proof. (i) Since ℕ(V_{K_{f}}) is a free 𝔹_{K_{f} ,E}^{+}-module of rank dim_{E} V contained in 𝔻(V_{K_{f}}), N := 𝔹_{K_{df} ,E}^{+} ⊗_{𝔹_{K_{f} ,E}^{+}} ℕ(V_{K_{f}}) is a free 𝔹_{K_{df} ,E}^{+}-module of rank dim_{E} V

contained in 𝔻(V_{K_{df}}) ⊇ 𝔻(V_{K_{f}}). Moreover, N is endowed with an action of Γ_{K_{df}} which becomes trivial modulo π, and also with a Frobenius map φ which commutes with the action of Γ_{K_{df}} and such that φ(N) ⊂ N and N/φ^{∗}(N) is killed by q^{k}. Hence N= ℕ(V_{K_{df}}) by the uniqueness part of Theorem 2.4(i). Part (ii) follows immediately from Theorem 2.4(ii) since 𝔸_{K_{df} ,E}^{+} ⊗_{𝔸_{K_{f} ,E}^{+}} ℕ(T_{K_{f}})

is an 𝔸_{K_{df} ,E}^{+}-lattice in ℕ(V_{K_{df}}).

We fix once and for all an embedding τ_{K_{df}}^{0} : K_{df}↪ E and we let τ_{K_{df}}^{j} = τ_{K_{df}}^{0} ∘ σ_{K_{df}}^{j} for j = 0, 1, ..., df − 1, where σ_{K_{df}} is the absolute Frobenius of K_{df}.

We fix the df-tuple of embeddings | τ_{K_{df}} |:= (τ_{K_{df}}^{0} , τ_{K_{df}}^{1} , ..., τ_{K_{df}}^{df−1}). We adjust the notation of §1.1 for the embeddings of K_{f} into E to the relative situation considered in this section. Let ι be the natural inclusion of K_{f} into K_{df}, in the sense that ι ∘ σ_{K_{f}} = σ_{K_{df}} ∘ ι, where σ_{K_{f}} is the absolute Frobenius of K_{f}. This induces a natural inclusion of 𝔸_{K}^{+} to 𝔸_{K_{df}}^{+} which we also denote by ι. Let τ_{K_{f}}^{j} := τ_{K_{df}}^{0} ∘ ι ∘ σ_{K_{f}}^{j} for j = 0, 1, ..., f − 1. We fix the f-tuple of embeddings

| τ_{K_{f}} |:= (τ_{K_{f}}^{0} , τ_{K_{f}}^{1} , ..., τ_{K_{f}}^{f−1}). Since the restriction of σ_{K_{df}} to K_{f} is σ_{K_{f}} , we obtain the following commutative diagram

𝔸_{K_{f} ,E}^{+} ⟶^{ξ_{K_{f}}} O_{E}^{|τ_{K_{f}} |}[[π]]
1_{O_{E}} ⊗ι↓  ↓^{θ}
𝔸_{K_{df} ,E}^{+} ⟶^{ξ_{K_{df}}} O_{E}^{|τ_{K_{df}} |}[[π]]

where θ is the ring homomorphism defined by

θ(α_{0}, α_{1}, ..., α_{f−1}) = \underbrace{(α_{0}, α_{1}, ..., α_{f−1}, α_{0}, α_{1}, ..., α_{f−1}, ..., α_{0}, α_{1}, ..., α_{f−1})}_{d-times}

=: (α_{0}, α_{1}, ..., α_{f−1})^{⊗d}.

For any matrix A ∈ M_{n} (O_{E}^{|τ_{K_{f}} |}[[π]]) we denote by A^{⊗d} the matrix obtained

by replacing each entry α of A by α^{⊗d}. A similar commutative diagram is obtained by replacing 𝔸_{K}^{+} by 𝔹_{K}^{+} and O_{E}^{|τ_{K}|}[[π]] by O_{E}^{|τ_{K}|}[[π]][\frac{1}{p}]. The following proposition follows easily from the discussion above.

Proposition 2.6. Let V_{K_{f}} , V_{K_{df}} , T_{K_{f}} , and T_{K_{df}} be as in Proposition 2.5.

(i) If the Wach module ℕ(V_{K_{f}}) of V_{K_{f}} is defined by the actions of φ and Γ_{K_{f}} given by (φ(η_{1}), φ(η_{2}), ..., φ(η_{n})) = \underline{η} · Π_{K_{f}} and (γ(η_{1}), γ(η_{2}), ..., γ(η_{n})) = \underline{η} · G_{K_{f}}^{γ} for all γ ∈ Γ_{K_{f}} for some ordered basis \underline{η} = (η_{1}, η_{2}, ..., η_{n}), then the Wach module ℕ(V_{K_{df}}) of V_{K_{df}} is defined by (φ(η′_{1}), φ(η′_{2}), ..., φ(η′_{n})) = \underline{η}′ · Π_{K_{df}} and (γ(η′_{1}), γ(η′_{2}), ..., γ(η′_{n})) = \underline{η}′ · G_{K_{df}}^{γ} for all γ ∈ Γ_{K_{df}} , where

Π_{K_{df}} = (Π_{K_{f}} )^{⊗d} and G_{K_{df}}^{γ} = (G_{K_{f}}^{γ} )^{⊗d} for all γ ∈ Γ_{K_{df}} , for some ordered basis \underline{η}′ of ℕ(V_{K_{df}}).

(ii) If the Wach module ℕ(T_{K_{f}}) of T_{K_{f}} is defined by the actions of φ and Γ_{K_{f}} given by (φ(η_{1}), φ(η_{2}), ..., φ(η_{n})) = \underline{η} · Π_{K_{f}} and (γ(η_{1}), γ(η_{2}), ..., γ(η_{n})) = \underline{η}·G_{K_{f}}^{γ} for all γ ∈ Γ_{K_{f}} for some ordered basis \underline{η} = (η_{1}, η_{2}, ..., η_{n}), then the Wach module ℕ(T_{K_{df}}) of T_{K_{df}} is defined by (φ(η′_{1}), φ(η′_{2}), ..., φ(η′_{n})) = \underline{η}′ · Π_{K_{df}} and (γ(η′_{1}), γ(η′_{2}), ..., γ(η′_{n})) = \underline{η}′ · G_{K_{df}}^{γ} for all γ ∈ Γ_{K_{df}} , where

Π_{K_{df}} = (Π_{K_{f}} )^{⊗d} and G_{K_{df}}^{γ} = (G_{K_{f}}^{γ} )^{⊗d} for all γ ∈ Γ_{K_{df}} , for some ordered basis \underline{η}′ of ℕ(V_{K_{df}}).

Corollary 2.7. If V_{K_{f}} is a two-dimensional effective crystalline E-representation of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, i = 0, 1, ..., f − 1, then V_{K_{df}} is an effective crystalline E-representation of G_{K_{df}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, i = 0, 1, ..., df − 1, with k_{j} = k_{j} for all i, j = 0, 1, ..., df − 1 with i ≡ jmod f.

Proof. By Proposition 2.6 there exist ordered bases \underline{η} and \underline{η}′ of ℕ(V_{K_{f}}) and ℕ(V_{K_{df}}) respectively, such that φ (\underline{η}) = \underline{η} · Π_{K_{f}} , γ (\underline{η}) = \underline{η} · G_{K_{f}}^{γ} for all γ ∈

Γ_{K_{f}} and φ (\underline{η′}) = \underline{η}′ · (Π_{K_{f}} )^{⊗d} , γ (\underline{η}′) = \underline{η}′ · (G_{K_{f}}^{γ} )^{⊗d} for all γ ∈ Γ_{K_{df}} . By

Theorem 2.4, x ∈ Fil^{j} (ℕ(V_{K_{f}})) if and only if φ (x) ∈ q^{j}ℕ(V_{K_{f}}), from which

it follows that Fil^{j} (ℕ(V_{K_{df}})) = (Fil^{j} (ℕ(V_{K_{f}})))^{⊗d} for all j. By Theorem 2.4,

𝔻(V_{K_{f}}) ≃ ℕ(V_{K_{f}})/πℕ(V_{K_{f}}) as filtered φ-modules over E^{|τ_{K_{f}} |}. This implies

that Fil^{j} (𝔻(V_{K_{df}})) = (Fil^{j} (𝔻(V_{K_{f}})))^{⊗d} for all j and the corollary follows.

3 Effective Wach modules of rank one

In this section we construct the rank one Wach modules over O_{E}[[π]]^{|τ|} with labeled Hodge-Tate weights {−k_{i}}_{τ_{i}}.

Definition 3.1. Recall that q = \frac{φ(π)}{π} where φ (π) = (1 + π)^{p} − 1. We define

q_{1} = q and q_{n} = φ^{n−1} (q) for all n ≥ 1. Let λ_{f} = ∏_{n=0}^{∞} (\frac{q_{nf+1}}{p} ) . For each

γ ∈ Γ_{K}, we define λ_{f},_{γ} = \frac{λ_{f}}{γλ_{f}} .

Lemma 3.2. For each γ ∈ Γ_{K}, the functions λ_{f} and λ_{f,γ} ∈ ℚ_{p}[[π]] have the following properties:

(i) λ_{f}(0) = 1;

(ii) λ_{f,γ} ∈ 1 + πℤ_{p} [[π]] .

Proof. (i) This is clear since \frac{q_{n}(0)}{p} = 1 for all n ≥ 1. (ii) One can easily check that \frac{q}{γq} ∈ 1 + πℤ_{p} [[π]] . From this we deduce that λ_{f,γ} ∈ 1 + πℤ_{p} [[π]] .

Consider the rank one module ℕ_{k,c} = (O_{E}[[π]]^{|τ|}) η equipped with the semilinear φ and Γ_{K}-actions defined by φ(η) = (c · q^{k_{1}}, q^{k_{2}}, ..., q^{k_{f−1}}, q^{k_{0}})η and γ(η)= (g_{1}^{γ}(π), g_{2}^{γ}(π), ...g_{f−1}^{γ}(π), g_{0}^{γ}(π))η for all γ ∈ Γ_{K}, where c ∈ O_{E}^{×}. We want to define the functions g_{i}(π) = g_{i}^{γ}(π) ∈ O_{E}[[π]] appropriately to make ℕ_{k,c} a Wach module over O_{E}[[π]]^{|τ|}. The actions of φ and γ should commute and a short computation shows that g_{0} should satisfy the equation

φ^{f}(g_{0}) = g_{0} (\frac{γq}{q} )^{k_{0}} φ(\frac{γq}{q})^{k_{1}} ⋯ φ^{f−1}(\frac{γq}{q})^{k_{f−1}}.  (3.1)

Lemma 3.3. Equation 3.1 has a unique ≡ 1mod π solution in ℤ_{p}[[π]] given by

g_{0} = λ_{f,γ}^{k_{0}}φ(λ_{f,γ})^{k_{1}}φ^{2}(λ_{f,γ})^{k_{2}} ⋯ φ^{f−1}(λ_{f,γ})^{k_{f−1}}.

Proof. Notice that φ^{f}(λ_{f}) = \frac{λ_{f}}{( \frac{q}{p})} and φ^{f}(γλ_{f}) = \frac{γλ_{f}}{( \frac{γq}{p})}, hence λ_{f,γ} = \frac{λ_{f}}{γλ_{f}} solves

the equation φ^{f}(u) = u (\frac{γq}{q} ) . It is straightforward to check that

g_{0} = λ_{f,γ}^{k_{0}}φ(λ_{f,γ})^{k_{1}}φ^{2}(λ_{f,γ})^{k_{2}} ⋯ φ^{f−1}(λ_{f,γ})^{k_{f−1}}

is a solution of equation 3.1. By Lemma 3.2, g_{0} ≡ 1mod π. If g_{0} and g′_{0} are two solutions of equation 3.1 congruent to 1mod π, then (\frac{g′_{0}}{g_{0}} ) ∈ ℤ_{p}[[π]] is fixed by φ^{f} and is congruent to 1mod π, hence equals 1.

Commutativity of φ with the Γ_{K}-actions implies that

g_{1} = ( \frac{q}{γq})^{k_{1}}φ( \frac{q}{γq})^{k_{2}} ⋯ φ^{f−2}( \frac{q}{γq})^{k_{f−1}}φ^{f−1}(λ_{f,γ})^{k_{0}}φ^{f} (λ_{f,γ})^{k_{1}} ⋯ φ^{2f−2}(λ_{f,γ})^{k_{f−1}},

· · · · · ·

g_{f−2} = ( \frac{q}{γq})^{k_{f−2}}φ( \frac{q}{γq})^{k_{f−1}}φ^{2}(λ_{f,γ})^{k_{0}}φ^{3}(λ_{f,γ})^{k_{1}} ⋯ φ^{f+1}(λ_{f,γ})^{k_{f−1}},

g_{f−1} = ( \frac{q}{γq})^{k_{f−1}}φ(λ_{f,γ})^{k_{0}}φ^{2}(λ_{f,γ})^{k_{1}}φ^{3}(λ_{f,γ})^{k_{2}} ⋯ φ^{f} (λ_{f,γ})^{k_{f−1}},

and Lemma 3.2 implies that g_{i} ≡ 1mod π for all i.

Proposition 3.4. We equip ℕ_{k,c} = (O_{E}[[π]]^{|τ|}) η with semilinear φ and Γ_{K}-actions defined by φ(η) = (c · q^{k_{1}}, q^{k_{2}}, ..., q^{k_{f−1}}, q^{k_{0}})η and γ(η) = (g_{1}^{γ}(π), g_{2}^{γ}(π), ...g_{f−1}^{γ}(π), g_{0}^{γ}(π))η for the g_{i}(π) = g_{i}^{γ}(π) defined above, where c ∈ O_{E}^{×}. The module ℕ_{k,c} is a Wach module over O_{E}[[π]]^{|τ|} with labeled Hodge-

Tate weights {−k_{i}}_{τ_{i}}. Moreover, 𝔻_{k,c} ≃ E^{|τ|} ⨂_{O_{E}^{|τ|}} (ℕ_{k,c}/πℕ_{k,c}) as filtered φ-

modules over E^{|τ|}, where 𝔻_{k,c} = (E^{|τ|}) η is the filtered φ-module with Frobenius endomorphism φ(η) = (c · p^{k_{1}}, p^{k_{2}}, ..., p^{k_{f−1}}, p^{k_{0}})η and filtration

Fil^{j}(𝔻_{k,c}) = \{ \begin{matrix}E^{|τ_{I_{0}} |}η if j ≤ w_{0}, \\ E^{|τ_{I_{1}} |}η if 1 + w_{0} ≤ j ≤ w_{1}, \\ ⋯ & ⋯ \\ E^{|τ_{I_{t−1}} |}η if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0 if j ≥ 1 + w_{t−1}. \end{matrix}

Proof. (i) To prove that Γ_{K} acts on ℕ_{k,c}, it suffices to prove that g_{i}^{γ_{1}γ_{2}}(π) = g_{i}^{γ_{1}}γ_{1}(g_{i}^{γ_{2}}) for all γ_{1}, γ_{2} ∈ Γ_{K} and i ∈ I_{0}. This follows immediately from the cocycle relations

\frac{q}{γ_{1}γ_{2}(q)} = \frac{q}{γ_{1}(q)}γ_{1} (\frac{q}{γ_{2}(q)}) and λ_{f,γ_{1}γ_{2}} = λ_{f,γ_{1}}γ_{1}(λ_{f,γ_{2}}),

and the definition of the g_{i}^{γ}(π). Since g_{i}^{γ}(π) ≡ 1mod π for all i ∈ I_{0}, the action of Γ_{K} on ℕ_{k,c}/πℕ_{k,c} is trivial. (ii) Let k = max{k_{0}, k_{1}, ..., k_{f−1}} and let φ^{∗}(ℕ_{k,c}) be the O_{E}[[π]]^{|τ|}-linear span of the set φ(ℕ_{k,c}). Let c_{1} = c^{−1} and c_{i} = 1 if

i ≠ 1. Since q^{k}η = ∑_{i=0}^{f−1} (q^{k−k_{i}}c_{i}e_{i}) φ(η) ∈ φ^{∗}(ℕ_{k,c}), it follows that q^{k} kills

ℕ_{k,c}/φ^{∗}(ℕ_{k,c}). (iii) To compute the filtration of ℕ_{k,c}, we use the fact that q^{j} | φ(x) if and only if π^{j} | x for any x ∈ O_{E}[[π]]. Let x = (x_{0}, x_{1}, ..., x_{f−1})η ∈ ℕ_{k,c}. By Theorem 2.4, x ∈ Fil^{j}ℕ_{k,c} if and only if φ(x) ∈ q^{j}ℕ_{k,c} or equivalently q^{j} | φ(x_{i})q^{k_{i}} for all i ∈ I_{0}. If j ≤ k_{i} there are no restrictions on the x_{i}, whereas if j > k_{i} this is equivalent to x_{i} ≡ 0mod π^{j−k_{i}}. Therefore,

e_{i}Fil^{j}ℕ_{k,c} = \{ \begin{matrix}e_{i}ℕ_{k,c} if j ≤ k_{i}, \\ e_{i}π^{j−k_{i}}O_{E}[[π]]η if j ≥ 1 + k_{i}. \end{matrix}

This implies that

E^{|τ|} ⨂_{O_{E}^{|τ|}} e_{i}Fil^{j} (ℕ_{k,c}/πℕ_{k,c}) = \{ \begin{matrix}e_{i}E^{|τ|}\overline{η} if j ≤ k_{i}, \\ 0 if j ≥ 1 + k_{i}. \end{matrix}

For the filtration, we have

E^{|τ|} ⨂_{O_{E}^{|τ|}} Fil^{j} (ℕ_{k,c}/πℕ_{k,c}) = ⨁_{i=0}^{f−1} (E^{|τ|} ⨂_{O_{E}^{|τ|}} e_{i}Fil^{j}(ℕ_{k,c}/πℕ_{k,c})) .

Recall from Notation 1.2 that after ordering the weights k_{i} and omitting possibly repeated weights we get w_{0} < w_{1} < ... < w_{t−1}. By the formulas above,

Fil^{j}(𝔻_{k,c}) = \{ \begin{matrix}E^{|τ|} ( ∑_{i∈I_{0}} e_{i}) η  if j ≤ w_{0}, \\ E^{|τ|} ( ∑_{\{i∈I_{0}:k_{i}>w_{0}\}} e_{i}) η if 1 + w_{0} ≤ j ≤ w_{1}, \\ E^{|τ|} ( ∑_{\{i∈I_{0}:k_{i}>w_{1}\}} e_{i}) η if 1 + w_{1} ≤ j ≤ w_{2} \\ ⋯ & ⋯ \\ E^{|τ|} (  ∑_{\{i∈I_{0}:k_{i}>w_{t−2}\}} e_{i}) η if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0  if j ≥ 1 + w_{t−1}. \end{matrix}

The formula for the filtration follows immediately, recalling that I_{j} = {i ∈ I_{0} :

k_{i} > w_{j−1}} for each j = 1, 2, ..., t − 1, and E^{|τ_{Ir} |} := E^{f} ( ∑_{i∈I_{r}} e_{i}) for each

r = 0, 1, ..., t − 1. The isomorphism of filtered φ-modules is obvious.

Proposition 3.5. Let k_{0}, k_{1}, ..., k_{f−1} be arbitrary integers.

(i) The weakly admissible rank one filtered φ-modules over E^{|τ|} with labeled Hodge-Tate weights {−k_{i}}_{τ_{i}} are of the form 𝔻_{k,α} = (E^{|τ|}) η, with φ(η) = (α_{0}, α_{1}, ..., α_{f−1})η for some α = (α_{0}, α_{1}, ..., α_{f−1}) ∈ (E^{×})^{|τ|} such that

v_{p}(Nm_{φ}(α)) = ∑_{i∈I_{0}} k_{i} and

Fil^{j}(𝔻_{k,α}) = \{ \begin{matrix}E^{|τ_{I_{0}} |}η if j ≤ w_{0}, \\ E^{|τ_{I_{1}} |}η if 1 + w_{0} ≤ j ≤ w_{1}, \\ ⋯ & ⋯ \\ E^{|τ_{I_{t−1}} |}η if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0 if j ≥ 1 + w_{t−1}. \end{matrix}

(ii) The filtered φ-modules 𝔻_{k,α} and 𝔻_{v,β} are isomorphic if and only if k = v and Nm_{φ}(α) = Nm_{φ}(β).

Proof. Follows easily arguing as in [Dou10], §§4 and 6.

Corollary 3.6. All the effective crystalline E-characters of G_{K} are those constructed in Proposition 3.4.

Let c ∈ O_{E}^{×} and k = (−k_{1}, −k_{2}, ..., −k_{f−1}, −k_{0}) . We denote by χ_{c,k} the crystalline character of G_{K} corresponding to the Wach module ℕ_{k,c} = (O_{E}[[π]]^{|τ|}) η with φ action defined by φ(e) = (c·q^{k_{1}}, q^{k_{2}}, ..., q^{k_{f−1}}, q^{k_{0}})η and the unique commuting with it Γ_{K}-action defined in Proposition 3.4. When c = 1 we simply write χ_{k}. By Proposition 3.4 the crystalline character χ_{i} := χ_{e_{i}} has labeled Hodge-Tate weights −e_{i+1} for all i. By taking tensor products we see that χ_{c,k} = χ_{c,0} · χ_{0}^{k_{1}} · χ_{1}^{k_{2}} · · · · · χ_{f−2}^{k_{f−1}} · χ_{f−1}^{k_{0}}. As usual, we denote by Frob_{p} be the geometric Frobenius of G_{ℚ_{p}} and by Frob_{K} the geometric Frobenius of G_{K}. We have the following.

Lemma 3.7. (i) The unramified character of G_{K_{f}} which maps Frob_{K_{f}} to c equals χ_{c,0} for any c ∈ O_{E}^{×};

(ii) For any i = 0, 1, ..., f − 1, (χ_{i})_{|G_{K_{2f}}} = χ_{i} · χ_{i+f}, where the character on the left hand side is a character of G_{K_{f}} and the characters on the right hand side are characters of G_{K_{2f}};

(iii) If χ is a crystalline character of G_{K_{f}} with labeled Hodge-Tate weights {−k_{i}}_{τ_{i}}, i = 0, 1, ..., f − 1, its restriction to G_{K_{2f}} has labeled weights {−k_{i}}_{τ_{i}}, i = 0, 1, ..., 2f − 1, with k_{i+f} = k_{i} for all i = 0, 1, ..., f − 1;

(iv) If χ and ψ are crystalline characters of G_{K_{f}} , then χ_{|G_{K_{df}}} = ψ_{|G_{K_{df}}} if and only if χ = η · ψ, where η is an unramified character of G_{K_{f}} which maps Frob_{K_{f}} to a d-th root of unity.

Proof. (i) Let \sqrt[f]{c} be any choice of an f-th root of c in E. The filtered φ-module with trivial filtration and φ(e) = \sqrt[f]{c} · e corresponds to the unramified character η of G_{ℚ_{p}} which maps Frob_{p} to \sqrt[f]{c}. Since the Frob_{K_{f}} = (Frob_{p})_{|K_{f}}^{f} , the restriction of η_{c} of η to K_{f} maps Frob_{K_{f}} to c. By Proposition 2.6 the rank one filtered φ-module corresponding to the unramified character η_{c} has trivial filtration and Frobenius φ(e) = (\sqrt[f]{c}, \sqrt[f]{c}, ..., \sqrt[f]{c}) e, and by Proposition 3.5(ii) the latter is isomorphic to the rank one filtered φ-module with trivial filtration and φ(e) = (c, 1, ...1) e. Part (ii) follows from the definition of the characters χ_{i} and Proposition 2.6. Part (iii) follows immediately from part (ii). For part (iv) it suffices to prove that any crystalline character η of G_{K_{f}} with trivial restriction to G_{K_{df}} is an unramified character of G_{K_{f}} which maps Frob_{K_{f}} to a d-th root of unity. The restriction of η to G_{K_{df}} has all its labeled Hodge-Tate weights equal to zero, and by Corollary 2.7 so does η. By part (i) η is an unramified character of G_{K_{f}} which maps Frob_{K_{f}} to some constant, say c. The restriction of η to G_{K_{df}} is trivial and maps Frob_{K_{df}} = (Frob_{K_{f}} )_{|K_{df}}^{d} to c^{d}, therefore c is a d-th root of unity and part (iv) follows.

Let χ be any E-character of G_{K}, and let h ∈ G_{ℚ_{p}}. Since K is unramified over ℚ_{p}, it is h-stable and the character χ^{h} with χ^{h} (g) := χ (hgh^{−1}) is well defined. We have h_{|K} =: σ_{K}^{n(h)} for a unique integer n (h) modulo f. We denote by T (χ) the rank one O_{E}-representation of G_{K} defined by γe = χ (γ) e for any basis element e and any γ ∈ G_{K}.

Lemma 3.8. Let χ be the crystalline character corresponding to the Wach module defined in Proposition 3.4, and let h ∈ G_{ℚ_{p}}. Let η_{1} = (h̄_{|K}^{−1}) · η. The rank

one module ℕ^{h} := (O_{E}[[π]]^{|τ|}) η_{1} endowed with semilinear Frobenius and Γ_{K}- actions defined by

φ (η_{1}) = (c · q^{k_{f+1−n(h)}}, q^{k_{f+2−n(h)}}, ..., q^{k_{2f−n(h)}}) η_{1} and
γ (η_{1}) = (g_{f+1+n(h^{−1})}^{hγh^{−1}}, g_{f+2−n(h)}^{hγh^{−1}}, ..., g_{2f−1−n(h)}^{hγh^{−1}}, g_{2f−n(h)}^{hγh^{−1}}) η_{1},

where the indices are viewed modulo f, is a Wach module whose corresponding crystalline character is χ^{h}.

Proof. It is trivial to check that ℕ^{h} with the above defined actions is a Wach

module. By Theorems 2.2 and 2.4, T (χ) ≃ (𝔸_{K,E} ⊗_{𝔸_{K,E}^{+}} ℕ (T (χ)))^{φ=1} , hence there exists some α ∈ 𝔸_{K,E} such that φ (α ⊗ η) = α ⊗ η and γ (α ⊗ η) = χ (γ) (α ⊗ η) for all γ ∈ G_{K}. This is equivalent to

φ (α) · ξ^{−1} (c · q^{k_{1}}, q^{k_{2}}, ..., q^{k_{0}}) ⊗ η = α ⊗ η and  (3.2)
γ (α) · ξ^{−1} (g_{1}^{γ}, g_{2}^{γ}, ...g_{f−1}^{γ}, g_{0}^{γ}) ⊗ η = χ (γ) α ⊗ η  (3.3)

for all γ ∈ G_{K}, where ξ is the isomorphism defined in formula 2.1. A little computation shows that for any (x_{0}, x_{1}, ..., x_{f−1}) ∈ O_{E}[[π]]^{|τ|},

h^{−1} (ξ^{−1} (x_{0}, x_{1}, ..., x_{f−1})) = ξ^{−1} (x_{f−n(h)}, x_{f+1−n(h)}, ..., x_{2f−1−n(h)}) . (3.4)

Let α_{1} := h^{−1}α ∈ 𝔸_{K,E}. We show that φ (α_{1} ⊗ η_{1}) = α_{1} ⊗η_{1} and γ (α_{1} ⊗ η_{1}) = χ^{h} (γ) (α_{1} ⊗ η_{1}) for all γ ∈ G_{K}. Indeed,

φ (α_{1} ⊗ η_{1}) = φ (h^{−1}α) ⊗ φ (η_{1})

= h^{−1}φ (α) · ξ^{−1} (c · q^{k_{f+1−n(h)}}, q^{k_{f+2−n(h)}} , ..., q^{k_{2f−n(h)}}) ⊗ η_{1}

{3.4 \atop =} h^{−1}φ (α) · h^{−1}ξ^{−1} (c · q^{k_{1}}, q^{k_{2}}, ..., q^{k_{f−1}}, q^{k_{0}}) ⊗ h^{−1}η

{3.2 \atop =} h^{−1} (α ⊗ η) = α_{1} ⊗ η_{1}.

Also,

γ (α_{1} ⊗ η_{1}) = γ (h^{−1}α) · ξ^{−1} (g_{f+1−n(h)}^{hγh^{−1}}, g_{f+2−n(h)}^{hγh^{−1}}, ..., g_{2f−1−n(h)}^{hγh^{−1}}, g_{2f−n(h)}^{hγh^{−1}}) ⊗ η_{1}

{3.4 \atop =} h^{−1} (hγh^{−1}α · ξ^{−1} (g_{1}^{hγh^{−1}}, g_{2}^{hγh^{−1}}, ..., g_{f−1}^{hγh^{−1}}, g_{f}^{hγh^{−1}}) ⊗ η)

{3.3 \atop =} h^{−1} (χ (hγh^{−1}) α ⊗ η) = χ^{h} (γ) (α_{1} ⊗ η_{1})

for all γ ∈ G_{K}. By Theorems 2.2 and 2.4, it follows that the crystalline character which corresponds to ℕ^{h} is χ^{h}.

Corollary 3.9. If χ is a crystalline E-characters of G_{K} with labeled Hodge-Tate weights {−k_{i}}_{τ_{i}}, the character χ^{h} is crystalline with labeled Hodge-Tate weights {−ℓ_{i}}_{τ_{i}}, where ℓ_{i} = k_{f+i−n(h)} for all i, with the indices f + i − n (h) viewed modulo f.

Corollary 3.10. The representation

V_{K_{f}} ≃ Ind_{K_{2f}}^{K_{f}} (χ_{0}^{k_{1}} · χ_{1}^{k_{2}} · · · · · χ_{2f−2}^{k_{2f−1}} · χ_{2f−1}^{k_{0}})

is crystalline. Moreover, V_{K_{f}} is irreducible if and only if k_{i} ≠ k_{i+f} for some i ∈ {0, 1, ..., f − 1}.

Proof. Since V_{K_{2f}} is crystalline, V_{K_{f}} is crystalline. The corollary follows from Mackey’s irreducibility criterion and Corollary 3.9.

Proposition 3.11. Let V_{K} be an irreducible two-dimensional crystalline Erepresentation of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, whose restriction to G_{K_{2f}} is reducible. There exist some unramified character η of G_{K_{f}} and some nonnegative integers ℓ_{i}, i = 0, 1, ..., 2f − 1 with {ℓ_{i}, ℓ_{i+f}} = {0, k_{i}} for all i = 0, 1, ..., f − 1 and ℓ_{i} ≠ ℓ_{i+f} for some i ∈ {0, 1, ..., f − 1}, such that

V_{K_{f}} ≃ η ⊗ Ind_{K_{2f}}^{K_{f}} (χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{2f−2}^{ℓ_{2f−1}} · χ_{2f−1}^{ℓ_{0}}) .

Proof. Let χ be a constituent of V_{K_{2f}} . By Corollary 3.6, χ = χ_{c} · χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{2f−2}^{ℓ_{2f−1}} · χ_{2f−1}^{ℓ_{0}} for some c ∈ O_{E}^{×} and some integers ℓ_{i}. Let η be the unramified character of G_{K_{f}} which maps Frob_{K_{f}} to \sqrt[2]{c}. Arguing as in the proof of Lemma 3.7(i) we see that the restriction of η to G_{K_{2f}} is χ_{c}, hence χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{2f−2}^{ℓ_{2f−1}} · χ_{2f−2}^{ℓ_{0}} is a constituent of (η^{−1} ⊗ V_{K_{f}} )_{|K_{2f}} . Since η^{−1} ⊗ V_{K_{f}} is irreducible,

η^{−1} ⊗ V_{K_{f}} ≃ Ind_{K_{2f}}^{K_{f}} (χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{2f−2}^{ℓ_{2f−1}} · χ_{2f−1}^{ℓ_{0}})

by Frobenius reciprocity. By Mackey’s formula and Corollary 3.9,

V_{K_{2f}} ≃ (χ_{c} · χ_{0}^{ℓ_{1}} · χ_{1}^{ℓ_{2}} · · · · · χ_{2f−2}^{ℓ_{2f−1}} · χ_{2f−1}^{ℓ_{0}}) ⨁

⨁ (χ_{c} · χ_{0}^{ℓ_{1+f}} · χ_{1}^{ℓ_{2+f}} · · · · · χ_{2f−2}^{ℓ_{3f−1}} · χ_{2f−1}^{ℓ_{3f}}) ,

where the indices of the exponents of the second summand are viewed modulo 2f. By Corollary 2.7, the restricted representation V_{K_{2f}} has labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, i = 0, 1, 2, ..., 2f − 1, where k_{i+f} = k_{i} for all i = 0, 1, ..., f − 1. The labeled Hodge-Tate weights of the direct sum of characters in formula 3 with respect to the embedding τ_{i} of K_{2f} to E are {−ℓ_{i}, −ℓ_{i+f}} for all i = 0, 1, 2, ..., 2f − 1, with the indices i + f viewed modulo 2f. Therefore {ℓ_{i}, ℓ_{i+f}} = {0, k_{i}} for all i = 0, 1, ..., f − 1. The rest of the proposition follows from Corollary 3.10.

Proposition 3.12. Up to twist by some unramified character, there exist precisely 2^{f^{+} −1} distinct isomorphism classes of irreducible crystalline two-dimensional E-representations of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, whose restriction to G_{K_{2f}} is reducible.

Proof. In Proposition 3.11, notice that ℓ_{i+f} = k_{i}−ℓ_{l} for all i = 0, 1, ..., f−1. The corollary follows since Ind_{K_{2f}}^{K_{f}} (χ) ≃ Ind_{K_{2f}}^{K_{f}} (ψ) if and only if {χ, χ^{h}} = {ψ, ψ^{h}}, where h is any element in G_{ℚ_{p}} lifting a generator of Gal (K_{2f}/K_{f}) .

4 families of effective Wach modules of arbitrary weight and rank

We extend the method used by Berger-Li-Zhu in [BLZ04] for two-dimensional crystalline representations of G_{ℚ_{p}}, in order to construct families of Wach modules of effective crystalline representations of G_{K} of arbitrary rank. Fixing a basis, we need to exhibit matrices Π and G_{γ} such that Πφ(G_{γ}) = G_{γ}γ(Π) for all γ ∈ Γ_{K}, with some additional properties imposed by Theorem 2.4. In the two-dimensional case, for representations of G_{ℚ_{p}} and for a suitable basis, it is trivial to write down such a matrix Π assuming that the valuation of the trace of Frobenius of the corresponding filtered φ-module is suitably large, and the main difficulty is in constructing a Γ_{K}-action which commutes with Π. When K ≠ ℚ_{p}, finding a matrix Π which gives rise to a prescribed weakly admissible filtration seems to be already hard, even in the two-dimensional case. Assuming that such a matrix Π is available, it is usually very hard to explicitly write down the matrices G_{γ}. There are exceptions to this, for example some split-reducible two-dimensional crystalline representations. In the general case, instead of explicitly writing down the matrices G_{γ} we prove that such matrices exist using a successive approximation argument. Let S = {X_{i} ; i = 0, 1, ..., m − 1} be a set of indeterminates, were m ≥ 1 is any integer. We extend the actions of φ and Γ_{K} defined in equations 2.2 and 2.3 on the ring O_{E}[[π]]^{|τ|} := ∏_{τ:K↪E} O_{E}[[π]] to an action on O_{E}[[π, S]]^{|τ|} := ∏_{τ:K↪E}

O_{E}[[π, S]], by letting φ and Γ_{K} act trivially on each indeterminate X_{i}. We let φ and Γ_{K} act on the matrices of M_{n}^{S} := M_{n}(O_{E}[[π, S]]^{|τ|}) entry-wise for any integer n ≥ 2. For any integer s ≥ 0, we write π^{s} = (π^{s}, π^{s}, ..., π^{s}) , and for any α ∈ O_{E} [[π, S]] and any vector r = (r_{0}, r_{1}, ..., r_{f−1}) with nonnegative integer coordinates we write α^{r} = (α^{r_{0}}, α^{r_{1}}, ..., α^{r_{f−1}}) . As usual, we assume that k_{i} are nonnegative integers and we write k := w_{t−1} = max{k_{0}, k_{1},..., k_{f−1}}. Let ℓ ≥ k be any fixed integer. We start our constructions with the following lemma.

Lemma 4.1. Let Π_{i} = Π_{i}(S), i = 0, 1, ..., f − 1 be matrices in M_{n}(O_{E}[[π, S]]) such that det(Π_{i}) = c_{i}q^{k_{i}}, with c_{i} ∈ O_{E}[[π]]^{×}. We denote by Π(S) the matrix (Π_{1}, Π_{2}, ..., Π_{f−1}, Π_{0}) and view it as an element of M_{n}^{S} via the natural isomorphism M_{n}^{S} ≃ M_{n}(O_{E}[[π, S]])^{|τ|}. We denote by P_{i} = P_{i}(S) the reduction of Π_{i}mod π for all i. Assume that for each γ ∈ Γ_{K} there exists a matrix G_{γ}^{(ℓ)} = G_{γ}^{(ℓ)}(S) ∈ M_{n}^{S} such that:

1. G_{γ}^{(ℓ)}(S) ≡ \overrightarrow{Id}mod π^{ℓ};

2. G_{γ}^{(ℓ)}(S) − Π(S)φ(G_{γ}^{(ℓ)}(S))γ(Π(S)^{−1}) ∈ π^{ℓ}M_{n}^{S};

3. There is no nonzero matrix H ∈ M_{n}(O_{E}[[S]]^{|τ|}) such that HU = p^{ft}UH for some t > 0, where U = Nm_{φ}(P) and P = P (S) = (P_{1}, P_{2}, ..., P_{f−1}, P_{0}) ;

4. For each s ≥ ℓ + 1 the operator

H ⟼ H − Q_{f}H(p^{f(s−1)}Q_{f}^{−1}) : M_{n} (O_{E} [[S]]) ⟶ M_{n} (O_{E} [[S]]) , (4.1)

where Q_{f} = P_{1}P_{2} ⋯ P_{f−1}P_{0}, is surjective.

Then for each γ ∈ Γ_{K} there exists a unique matrix G_{γ}(S) ∈ M_{n}^{S} such that

(i) G_{γ}(S) ≡ \overrightarrow{Id}mod π and

(ii) Π(S)φ(G_{γ}(S)) = G_{γ}(S)γ(Π(S)).

Proof. Uniqueness: Suppose that both the matrices G_{γ}(S) and G′_{γ}(S) satisfy the conclusions of the lemma and let H = G′_{γ}(S)G_{γ}(S)^{−1}. We easily see that H ∈ Id + πM_{n}^{S} and HΠ(S) = Π(S)φ (H) . We’ll show that H = Id. We write H = Id + π^{t}H_{t} + ⋯, where H_{t} ∈ M_{n}(O_{E}[[S]]^{|τ|}) for some t ≥ 1 and Π(S) = P + πP^{(1)} + π^{2}P^{(2)} + ⋯, and we will show that H_{t} = 0. Since HΠ(S) = Π(S)φ (H), we have (H − Id)Π(S) = Π(S)φ(H − Id). We divide both sides of this equation by π^{t} using that φ(π) = qπ, and reduce mod π. Since q ≡ pmod π, this gives H_{t}P = p^{t}P φ (H_{t}) which implies that H_{t}U = p^{ft}Uφ^{f} (H_{t}), where U = Nm_{φ}(P). Since φ acts trivially on X_{i} and O_{E}, the map φ^{f} acts trivially on M_{n}(O_{E}[[S]]^{|τ|}). Therefore H_{t}U = p^{ft}UH_{t} and H_{t} = 0 by assumption (iii) of the lemma. Existence: Fix a γ ∈ Γ_{K}. By assumptions (i) and (ii) of the lemma, there exists a matrix G_{γ}^{(ℓ)} ∈ Id + π^{ℓ} M_{n}^{S} such that

G_{γ}^{(ℓ)} − Π(S)φ(G_{γ}^{(ℓ)})γ(Π(S)^{−1}) = π^{ℓ}R^{(ℓ)}

for some matrix R^{(ℓ)} = R^{(ℓ)}(γ) ∈ M_{n}^{S}. We shall prove that for each s ≥ ℓ + 1 there exist matrices R^{(s)} = R^{(s)}(γ) ∈ M_{n}^{S} and G_{γ}^{(s)} ∈ M_{n}^{S} such that G_{γ}^{(s)} ≡ G_{γ}^{(s−1)}mod π^{s−1}M_{n}^{S} and G_{γ}^{(s)} − Π(S)φ(G_{γ}^{(s)})γ(Π(S)^{−1}) = π^{s}R^{(s)}. Let

G_{γ}^{(s)} = G_{γ}^{(s−1)} + π^{s−1}H^{(s)}, where H^{(s)} ∈ M_{n}(O_{E}[[S]]^{|τ|}) and write R^{(s)} = R̄^{(s)} + π · C with C ∈ M_{n}^{S}. We need

(G_{γ}^{(s−1)} + π^{(s−1)}H^{(s)})−Π(S) (φ(G_{γ}^{(s−1)}) + φ (π)^{(s−1)}φ (H^{(s)})) γ(Π(S)^{−1}) ∈ π^{s}M_{n}^{S},

or equivalently

G_{γ}^{(s−1)} − Π(S)φ(G_{γ}^{(s−1)})γ(Π(S)^{−1}) + π^{(s−1)}H^{(s)}−
− (qπ)^{(s−1)}Π(S)φ (H^{(s)}) γ(Π(S)^{−1}) ∈ π^{s}M_{n}^{S}.

The latter is equivalent to

π^{(s−1)}R^{(s−1)} + π^{(s−1)}H^{(s)} − (qπ)^{(s−1)}Π(S)φ (H^{(s)}) γ(Π(S)^{−1}) ∈ π^{s}M_{n}^{S},

which is in turn equivalent to

H^{(s)} − q^{(s−1)}Π(S)φ (H^{(s)}) γ(Π(S)^{−1}) ≡ −R^{(s−1)}mod πM_{n}^{S}.

This holds if and only if

H^{(s)} − p^{(s−1)}P (S) φ (H^{(s)}) P (S)^{−1} = −R̄^{(s−1)}.  (4.2)

Notice that p^{(s−1)}P (S)^{−1} ∈ M_{n}(O_{E}[[S]]^{|τ|}) since s − 1 ≥ ℓ ≥ k = max{k_{0}, k_{1}, ..., k_{f−1}}. We write

H^{(s)} = (H_{1}^{(s)}, H_{2}^{(s)}, ..., H_{f−1}^{(s)}, H_{0}^{(s)}) and

−R̄^{(s−1)} = (R̄_{1}^{(s−1)}, R̄_{2}^{(s−1)}, ..., R̄_{f−1}^{(s−1)}, R̄_{0}^{(s−1)}) .

Equation 4.2 is equivalent to the system of equations in M_{n} (O_{E} [[S]])

H_{i}^{(s)} − P_{i} · H_{i+1}^{(s)} · (p^{s−1}P_{i}^{−1}) = R̄_{i}^{(s−1)},  (4.3)

where i = 1, 2, ..., f, with indices viewed mod f. These imply that

H_{1}^{(s)} − Q_{f}H_{1}^{(s)}(p^{f(s−1)}Q_{f}^{−1}) = R̄_{1}^{(s−1)} + Q_{1}R̄_{2}^{(s−1)}(p^{(s−1)}Q_{1}^{−1})+
+ Q_{2}R̄_{3}^{(s−1)}(p^{2(s−1)}Q_{2}^{−1}) + ⋯ + Q_{f−1}R̄_{0}^{(s−1)}(p^{(s−1)(f−1)}Q_{f−1}^{−1}),

where Q_{i} = P_{1} ⋯ P_{i} for all i = 1, 2, ..., f. From equations 4.3 we see that the matrices H_{i}^{(s)}, i = 2, 3, ..., f, are uniquely determined by the matrix H_{1}^{(s)}, so it suffices to prove that the operator defined in formula 4.1 contains

A = R̄_{1}^{(s−1)} + Q_{1}R̄_{2}^{(s−1)}(p^{(s−1)}Q_{1}^{−1}) + Q_{2}R̄_{3}^{(s−1)}(p^{2(s−1)}Q_{2}^{−1}) + ⋯

+Q_{f−1}R̄_{0}^{(s−1)}(p^{(s−1)(f−1)}Q_{f−1}^{−1})

in its image. Since p^{i(s−1)}Q_{i}^{−1} ∈ M_{n} (O_{E} [[S]]) for all i, this is true by assumption (iv) of the lemma. We define G_{γ}(S) = lim_{s→∞}G_{γ}^{(s)}(S) and the proof is complete.

Let M̃_{n} be the ring M_{n} (O_{E} [[S]]) /I where I is the ideal of M_{n} (O_{E} [[S]]) generated by the set {p · Id, X_{i} · Id : X_{i} ∈ S} . We use the notation of Lemma 4.1 and its proof, and we are interested in the image of the operator \overline{H} ↦ \overline{H − Q_{f}H(p^{fℓ}Q_{f}^{−1})} : M̃_{n} → M̃_{n} where bar denotes reduction modulo I.

Proposition 4.2. If the operator

\overline{H} ↦ \overline{H − Q_{f}H(p^{fℓ}Q_{f}^{−1})} : M̃_{n} → M̃_{n}  (4.4)

is surjective, then for each s ≥ ℓ + 1 the operator defined in formula 4.1 is surjective.

Proof. Case (i). s ≥ k + 2. In this case f(s − 1) − ∑_{i=0}^{f−1} k_{i} ≥ f (s − 1 − k) ≥

f ≥ 1. Since Q_{f}^{−1} = P_{0}^{−1}P_{f−1}^{−1}P_{f−2}^{−1}...P_{1}^{−1} and det(P_{i}) = c̄_{i}p^{k_{i}}, it follows that p^{f(s−1)}Q_{f}^{−1} ∈ pM_{n}(O_{E}[[S]]). Let B be any matrix in M_{n} (O_{E} [[S]]) . We write

B = B − Q_{f}B (p^{f(s−1)}Q_{f}^{−1}) + pB_{1}

for some matrix B_{1} ∈ M_{n} (O_{E} [[S]]) . Similarly,

B_{1} = B_{1} − Q_{f}B_{1} (p^{f(s−1)}Q_{f}^{−1}) + pB_{2} for some matrix B_{2} ∈ M_{n} (O_{E} [[S]]) and

B = (B + pB_{1}) − Q_{f} (B + pB_{1}) (p^{f(s−1)}Q_{f}^{−1}) + p^{2}B_{2}.

Continuing in the same fashion we get

B = (∑_{i=0}^{N} p^{i}B_{i}) − Q_{f} (∑_{i=0}^{N} p^{i}B_{i}) (p^{f(s−1)}Q_{f}^{−1}) + p^{N+1}B_{N+1}

for some matrix B_{N+1}∈ M_{n} (O_{E} [[S]]) with B_{0} = B. Let H = ∑_{i=0}^{∞} p^{i}B_{i}. Then

H ∈ M_{n} (O_{E} [[S]]) and B = H − Q_{f}H (p^{f(s−1)}Q_{f}^{−1}) . Case (ii). ℓ = k and s = k + 1. We reduce modulo the ideal I defined before Proposition 4.2. Let A be any element of M_{n} (O_{E} [[S]]) . The operator

\overline{H} ⟼ \overline{H − Q_{f}H (p^{fℓ}Q_{f}^{−1})} : M̃_{n} → M̃_{n}

contains Ā = Amod I in its image by the assumption of the lemma. Let A = A_{0} − Q_{f}A_{0} (p^{fℓ}Q_{f}^{−1}) mod I for some matrix A_{0} ∈ M_{n} (O_{E} [[S]]) . We write

A = A_{0} − Q_{f}A_{0} (p^{fℓ}Q_{f}^{−1}) + pB_{m} + X_{0}B_{0} + ⋯ + X_{m−1}B_{m−1}

for matrices B_{i} ∈ M_{n} (O_{E} [[S]]) . Similarly B_{i} = B_{i}^{0} − Q_{f}B_{i}^{0} (p^{fℓ}Q_{f}^{−1}) mod I

for matrices B_{i}^{0} ∈ M_{n} (O_{E} [[S]]) and for all i. Then

A = A_{0} − Q_{f}A_{0} (p^{fℓ}Q_{f}^{−1}) + pB_{m}^{0} − Q_{f} (pB_{m}^{0}) (p^{fℓ}Q_{f}^{−1}) +
+X_{0}B_{1}^{0} − Q_{f} (X_{0}B_{1}^{0}) (p^{fℓ}Q_{f}^{−1}) +
+ ⋯ + X_{m−1}B_{m−1}^{0} − Q_{f} (X_{m−1}B_{f−1}^{0}) (p^{fℓ}Q_{f}^{−1}) mod I^{2},

therefore

A = (A_{0} + pB_{m}^{0} + X_{0}B_{1}^{0} + ⋯ + X_{m−1}B_{m−1}^{0})−
− Q_{f}(A_{0} + pB_{m}^{0} + X_{0}B_{1}^{0} + ⋯ + X_{f−1}B_{m−1}^{0}) (p^{fℓ}Q_{f}^{−1}) mod I^{2}.

By induction, A = H − Q_{f}H (p^{fℓ}Q_{f}^{−1}) for some H ∈ M_{n} (O_{E} [[S]]) .

The surjectivity assumption of Proposition 4.2 is usually trivial to check thanks to the following proposition.

Proposition 4.3. Assume that ℓ > k or ℓ = k and the weights k_{i} are not all equal. Then the operator defined in formula 4.4 is surjective.

Proof. The proposition follows immediately because det Q_{f} = c̄p^{k_{1}+k_{2}+···+k_{f}} , where c̄ = c̄_{1}c̄_{2} ⋯ c̄_{f}, since fℓ > k_{1} + ⋯ + k_{f} and p ∈ I.

The following lemma summarizes the results of this section. We use the notation of Lemma 4.1.

Lemma 4.4. Let ℓ ≥ k be a fixed integer. We assume that for each γ ∈ Γ_{K} there exists a matrix G_{γ}^{(ℓ)} = G_{γ}^{(ℓ)}(S) ∈ M_{n}^{S} such that:

1. G_{γ}^{(ℓ)}(S) ≡ \overrightarrow{Id}mod π^{ℓ};

2. G_{γ}^{(ℓ)}(S) − Π(S)φ(G_{γ}^{(ℓ)}(S))γ(Π(S)^{−1}) ∈ π^{ℓ}M_{n}^{S};

3. There is no nonzero matrix H ∈ M_{n}(O_{E}[[S]]^{|τ|}) such that HU = p^{ft}UH for some t > 0;

4. If ℓ = k and k = k_{i} for all i, we additionally assume that the operator

\overline{H} ↦ \overline{H − Q_{f}H(p^{fℓ}Q_{f}^{−1})} : M̃_{n} → M̃_{n}

is surjective.

Then for each γ ∈ Γ_{K} there exists a unique matrix G_{γ}(S) ∈ M_{n}^{S} such that

(i) G_{γ}(S) ≡ \overrightarrow{Id}mod π, and

(ii) Π(S)φ(G_{γ}(S)) = G_{γ}(S)γ(Π(S)).

For any vector a = (a_{0}, a_{1}, ..., a_{f−1}) ∈ 𝔪_{E}^{|S|} we denote by Π(a) = (Π_{1}(a_{1}), Π_{2}(a_{2}), ..., Π_{f−1}(a_{f−1}), Π_{0}(a_{0})) the matrix obtained from Π(S) = (Π_{1}(X_{1}), Π_{2}(X_{2}), ..., Π_{f−1}(X_{f−1}), Π_{0}(X_{0})) by substituting a_{i} ∈ 𝔪_{E} in each indeterminate X_{i} of Π_{i}(X_{i}).

Proposition 4.5. For any a = (a_{0}, a_{1}, ..., a_{f−1}) ∈ 𝔪_{E}^{|S|} and any γ_{1}, γ_{2}, γ ∈ Γ_{K}, the following equations hold:

(i) G_{γ_{1}γ_{2}}(a) = G_{γ_{1}}(a)γ_{1}(G_{γ_{2}}(a)) and

(ii) Π(a)φ(G_{γ}(a)) = G_{γ}(a)γ(Π(a)).

Proof. Both matrices G_{γ_{1}γ_{2}}(S) and G_{γ_{1}}(S)γ_{1}(G_{γ_{2}}(S)) are ≡ \overrightarrow{Id}mod π and are solutions in A of the equation Π(S)φ(A) = Aγ(Π(S)). They are equal by the uniqueness part of Lemma 4.1. The second equation follows from part (ii) of the same lemma.

For any vector a ∈ 𝔪_{E}^{|S|} we equip the module ℕ(a) = ⨁_{i=1}^{n} (O_{E}[[π]]^{|τ|}) η_{i}

with semilinear φ and Γ_{K}-actions defined by (φ(η_{1}), φ(η_{2}), ..., φ(η_{n})) = (η_{1}, η_{2}, ..., η_{n})Π(a) and (γ(η_{1}), γ(η_{2}), ..., γ(η_{n})) = (η_{1}, η_{2}, ..., η_{n})G_{γ}(a) for any γ ∈ Γ_{K}. Proposition 4.5 implies that (γ_{1}γ_{2})x= γ_{1}(γ_{2}x) and φ(γx) = γ(φ(x)) for all x ∈ ℕ(a) and γ, γ_{1}, γ_{2} ∈ Γ_{K}. Since G_{γ}(a) ≡ \overrightarrow{Id}mod π, it follows that Γ_{K} acts trivially on ℕ(a)/πℕ(a).

Proposition 4.6. For any a ∈ 𝔪_{E}^{|S|}, the module ℕ(a) equipped with the φ and Γ_{K}-actions defined above is a Wach module over O_{E}[[π]]^{|τ|} corresponding (by Theorem 2.4) to some G_{K}-stable O_{E}-lattice inside some n-dimensional crystalline E-representation of G_{K} with Hodge-Tate weights in [−k; 0].

Proof. The only thing left to prove is that q^{k}ℕ(a) ⊂ φ^{∗}(ℕ(a)). Since det(Π_{i}) = c_{i}q^{k_{i}} we have det Π(a) = (c_{1}q^{k_{1}}, c_{2}q^{k_{2}}, ..., c_{0}q^{k_{0}}) and

(q^{k}η_{1}, q^{k}η_{2}, ..., q^{k}η_{n})
= (η_{1}, η_{2}, ..., η_{n}) det Π(a) (c_{1}^{−1}q^{k−k_{1}}, c_{2}^{−1}q^{k−k_{2}}, ..., c_{0}^{−1}q^{k−k_{0}})
= (η_{1}, η_{2}, ..., η_{n}) (Π(a) · adj (Π(a))) (c_{1}^{−1}q^{k−k_{1}}, c_{2}^{−1}q^{k−k_{2}}, ..., c_{0}^{−1}q^{k−k_{0}})
= (φ(η_{1}), φ(η_{2}), ..., φ(η_{n})) · (adjΠ(a)) (c_{1}^{−1}q^{k−k_{1}}, c_{2}^{−1}q^{k−k_{2}}, ..., c_{0}^{−1}q^{k−k_{0}}) .

Hence (q^{k}η_{1}, q^{k}η_{2}, ..., q^{k}η_{n}) ∈ φ^{∗}(ℕ(a)) and q^{k}ℕ(a) ⊂ φ^{∗}(ℕ(a)).

We proceed to prove the main theorem concerning the modulo p reductions of the crystalline representations corresponding to the families of Wach modules constructed in Proposition 4.6. By reduction modulo p we mean reduction modulo the maximal ideal 𝔪_{E} of the ring of integers of the coefficient field E. If T is a G_{K}-stable O_{E}-lattice in some E-linear representation V of G_{K}, we denote by \overline{V} = k_{E} ⨂_{O_{E}} T the reduction of V modulo p, where k_{E} is the residue

field of O_{E}. The reduction \overline{V} depends on the choice of the lattice T, and a theorem of Brauer and Nesbitt asserts that the semisimplification

\overline{V}^{s.s.} = (k_{E} ⨂_{O_{E}}T)^{s.s.}

is independent of T. Instead of the precise statement “there exist G_{K}-stable O_{E}-lattices T_{V} and T_{W} inside the E-linear representation V and W of G_{K} respectively, such that k_{E} ⨂_{O_{E}} T_{V} ≃ k_{E} ⨂_{O_{E}} T_{W} ”, we abuse notation and write

\overline{V} ≃ \overline{W} . For each a ∈ 𝔪_{E}^{|S|}, let V (a) = E⊗_{O_{E}} T(α), where T(α) = 𝕋(𝔻(a)), and 𝔻(a) = 𝔸_{K,E} ⨂_{𝔸_{K,E}^{+}} ℕ(a). The representations V (a) are n-dimensional crystalline

E-representations of G_{K} with Hodge-Tate weights in [−k; 0]. Concerning their mod p reductions, we have the following theorem.

Theorem 4.7. For any a ∈ 𝔪_{E}^{|S|}, the isomorphism \overline{V} (a) ≃ \overline{V} (0) holds.

Proof. We prove that the k_{E}-linear representations k_{E} ⨂_{O_{E}} T(a) and k_{E} ⨂_{O_{E}} T(0)

of G_{K} are isomorphic. Since Π(S) and G_{γ}(S) ∈ M_{n}^{S}, we have G_{γ}(a) ≡ G_{γ}(0)mod 𝔪_{E} and Π(a) ≡ Π(0)mod 𝔪_{E}. As (φ, Γ_{K})-modules over k_{E}((π))^{|τ|}, 𝔻(a)/𝔪_{E}𝔻(a) ≃ 𝔻(0)/𝔪_{E}𝔻(0). Hence 𝕋 (𝔻(a)/𝔪_{E}𝔻(a)) ≃ 𝕋 (𝔻(0)/𝔪_{E}𝔻(0)), where 𝕋 is Fontaine’s functor on representations mod 𝔪_{E}. Since Fontaine’s functor is exact, 𝕋 (𝔻(a)/𝔪_{E}𝔻(a)) ≃ T(a)/𝔪_{E}T(a) and T(a)/𝔪_{E}T(a) ≃ T(0)/𝔪_{E}T(0).

5 Families of two-dimensional crystalline representations

The main difficulty in applying Lemma 4.4 is in constructing the matrices G_{γ}^{(ℓ)}(S) which satisfy conditions (1) and (2). Conditions (3) and (4) are usually easy to check. Throughout this section we retain the notations of Lemma 4.4. We denote by E_{ij} the 2 × 2 matrix with 1 in the (i, j)-entry and 0 everywhere else. Recall that E_{ij} · E_{kl} = δ_{jk} · E_{il}, where δ is the Kronecker delta function. Also recall our assumption that at least one of the weights k_{i} is strictly positive.

Proposition 5.1. The operator \overline{H} ↦ \overline{H − Q_{f}H(p^{fℓ}Q_{f}^{−1})} : M̃_{2} → M̃_{2} is surjective, unless ℓ = k, k = k_{i} for all i and Q̄_{f} ∈ {E_{11}, E_{22}}.

Proof. It is straightforward to check that Q̄_{f} = E_{ij} for some i, j ∈ {1, 2} and

p^{kℓ}Q_{f}^{−1}mod I = \{ \begin{matrix}E_{22} if Q̄_{f} = E_{11}, \\ E_{11} if Q̄_{f} = E_{22}, \\ −E_{12} if Q̄_{f} = E_{12}, \\ −E_{21} if Q̄_{f} = E_{21}. \end{matrix}

If Q̄_{f} = E_{11} (respectively E_{22}), the image is the set of matrices with zero (1, 2) (respectively (2, 1)) entry, while if Q̄_{f} = E_{12} or Q̄_{f} = E_{21} the operator becomes

( \begin{matrix}h_{11} & h_{12} \\ h_{21} & h_{22} \end{matrix} ) ⟼ ( \begin{matrix}h_{11} & h_{12} + h_{21} \\ h_{21} & h_{22} \end{matrix} )

and

( \begin{matrix}h_{11} & h_{12} \\ h_{21} & h_{22} \end{matrix} ) ⟼ ( \begin{matrix}h_{11} & h_{12} \\ h_{21} + h_{12} & h_{22} \end{matrix} )

respectively and is clearly surjective. The proposition follows from Proposition 4.3.

Lemma 5.2. If the matrix Q_{f} = P_{1}P_{2} ⋯ P_{f−1}P_{f} (with P_{f} = P_{0}) does not have eigenvalues which are a scalar multiple of each other, then the matrix U = Nm_{φ}(P), where P = (P_{1}, P_{2}, ..., P_{f−1}, P_{0}) , satisfies condition (3) of Lemma 4.4.

Proof. Let H ∈ M_{n}(O_{E}[[S]]^{|τ|}) be a nonzero matrix such HU = p^{ft}UH for some t > 0. We write H = (H_{1}, H_{2}, ..., H_{f}) and U = (U_{1}, U_{2}, ..., U_{f}) . Since P · φ(U) · P^{−1} = U, we have P_{i}U_{i+1}P_{i}^{−1} = U_{i} for all i. Since Q_{f} = U_{1}, none of the U_{i} has eigenvalues which are a scalar multiple of each other. If H is invertible then U_{1} = Q_{f} has eigenvalues with quotient p^{ft} which contradicts the assumption of the lemma. If H is nonzero and not invertible, there exists an index i such that H_{i}U_{i} = p^{ft} U_{i}H_{i} and rank (H_{i}) = 1. There also exists an invertible matrix B such that

BH_{i}B^{−1} = ( \begin{matrix}α_{11} & 0 \\ α_{21} & 0 \end{matrix} )

with (α_{11}, α_{21}) ≠ (0, 0) . Let Γ = BU_{i}B^{−1} and write Γ = (γ_{ij}) . The equation H_{i}U_{i} = p^{ft}U_{i}H_{i} is equivalent to p^{ft}ΓBH_{i}B^{−1} = BH_{i}B^{−1}Γ which implies that γ_{12} = 0 and p^{ft}γ_{11}α_{11} = α_{11}γ_{11}. If α_{11} ≠ 0, then γ_{11} = 0 a contradiction since Γ is invertible. If α_{11} = 0, then p^{ft}α_{21}γ_{22} = α_{21}γ_{11} and p^{ft}γ_{22} = γ_{11} (since α_{21} ≠ 0). Since γ_{12} = 0, the latter implies that Γ has two eigenvalues with quotient p^{ft}. This in turn implies that U_{i} and its conjugate Q_{f} = U_{1} have eigenvalues with quotient p^{ft} and contradicts the assumption of the lemma. Hence H = 0.

In the two-dimensional case, instead of checking condition (3) of Lemma 4.4 it is often more convenient to use following corollary.

Corollary 5.3. If Tr (Q_{f}) ∉ ℚ̄_{p}, then the matrix U = Nm_{φ}(P) satisfies condition (3) of Lemma 4.4.

Proof. Since the determinant of Q_{f} is a nonzero scalar, the eigenvalues of Q_{f} are a scalar multiple of each other if and only if Tr (Q_{f}) is a scalar.

5.1 Families of rank two Wach modules

We now apply Lemma 4.4 for matrices Π_{i} as in the following definition.

Definition 5.4. For a fixed integer ℓ ≥ k = max{k_{0}, k_{1}, ..., k_{f−1}} we define matrices of the following four types

t_{1}: ( \begin{matrix}c_{i}q^{k_{i}} & 0 \\ X_{i}φ(z_{i}) & 1 \end{matrix} ) , t_{2}: ( \begin{matrix}X_{i}φ(z_{i}) & 1 \\ c_{i}q^{k_{i}} & 0 \end{matrix} ) ,

t_{3}: ( \begin{matrix}1 & X_{i}φ(z_{i}) \\ 0 & c_{i}q^{k_{i}} \end{matrix} ) , t_{4}: ( \begin{matrix}0 & c_{i}q^{k_{i}} \\ 1 & X_{i}φ(z_{i}) \end{matrix} ) ,

where X_{i} is an indeterminate, c_{i} ∈ O_{E}, and z_{i} is a polynomial of degree ≤ ℓ − 1 in ℤ_{p}[π] such that z_{i} ≡ p^{m_{ℓ}}mod π, where m_{ℓ} := ⌊\frac{ℓ−1}{p−1}⌋. Matrices of type t_{1} or t_{3} are called of odd type while matrices of type t_{2} or t_{4} are called of even type. We write Π^{i}(S) = (Π_{1}(X_{1}), Π_{2}(X_{2}), ..., Π_{f−1}(X_{f−1}), Π_{0}(X_{0})) with i = (i_{1}, i_{2}, ..., i_{f−1}, i_{0}) the vector in {1, 2, 3, 4}^{f} whose j-th coordinate i_{j} is the type of the matrix Π_{j} for all j ∈ I_{0}. We call i the type-vector attached to the matrix f-tuple Π^{i}(S).

The polynomials z_{i} appearing in the entries of the matrices Π_{i} will be defined shortly. We will also define functions x_{i}^{γ}, y_{i}^{γ} ∈ 1 + πℤ_{p}[[π]] such that

G_{γ}^{(ℓ)} − Π(S)φ(G_{γ}^{(ℓ)})γ(Π(S)^{−1}) ∈ π^{ℓ}M_{2}^{S}

for all γ ∈ Γ_{K}, where

G_{γ}^{(ℓ)} = diag ((x_{0}^{γ}, x_{1}^{γ}, ..., x_{f−1}^{γ}) , (y_{0}^{γ}, y_{1}^{γ}, ..., y_{f−1}^{γ})) .

904 Gerasimos Dousmanis Let

Π(S) = ( \begin{matrix}(α_{1}, α_{2}, ...α_{f−1}, α_{0}) & (β_{1}, β_{2}, ..., β_{f−1}, β_{0}) \\ (γ_{1}, γ_{2}, ..., γ_{f−1}, γ_{0}) & (δ_{1}, δ_{2}, ..., δ_{f−1}, δ_{0}) \end{matrix} ) with ( \begin{matrix}α_{i} & β_{i} \\ γ_{i} & δ_{i} \end{matrix} )

belonging to the set

\{( \begin{matrix}c_{i}q^{k_{i}} & 0 \\ X_{i}φ(z_{i}) & 1 \end{matrix} ) , ( \begin{matrix}X_{i}φ(z_{i}) & 1 \\ c_{i}q^{k_{i}} & 0 \end{matrix} ) , ( \begin{matrix}1 & X_{i}φ(z_{i}) \\ 0 & c_{i}q^{k_{i}} \end{matrix} ) , ( \begin{matrix}0 & c_{i}q^{k_{i}} \\ 1 & X_{i}φ(z_{i}) \end{matrix} )\} .

For each i = 1, 2, ..., f we demand that all of the elements

x_{i−1}^{γ} − \frac{α_{i}φ (x_{i}^{γ}) (γδ_{i}) − β_{i}φ (y_{i}^{γ}) (γγ_{i})}{ε_{i} (γq)^{k_{i}}}, \frac{β_{i}φ (y_{i}^{γ}) (γα_{i}) − α_{i}φ (x_{i}^{γ}) (γβ_{i})}{ε_{i} (γq)^{k_{i}}},

(5.1)

y_{i−1}^{γ} − \frac{δ_{i}φ (y_{i}^{γ}) (γα_{i}) − γ_{i}φ (x_{i}^{γ}) (γβ_{i})}{ε_{i} (γq)^{k_{i}}}, \frac{γ_{i}φ (x_{i}^{γ}) (γδ_{i}) − δ_{i}φ (y_{i}^{γ}) (γγ_{i})}{ε_{i} (γq)^{k_{i}}}

(5.2)

of O_{E}[[π, X_{0}, ..., X_{f−1}]][q^{−1}] which belong to O_{E}[[π]][q^{−1}] are zero, and those which contain an indeterminate belong to π^{ℓ}O_{E}[[π, X_{0}, ..., X_{f−1}]], where in the formulas above ε_{i} = 1 if Π_{i} has type 1 or 3 and ε_{i} = −1 if Π_{i} has type 2 or 4. As usual lower indices are viewed modulo f. Proposition 5.5. For each i, equations 5.1 and 5.2 imply that

x_{i−1}^{γ} = (\frac{q}{γq} )^{ℓ_{i}} φ (w_{i}^{γ}) and y_{i−1}^{γ} = (\frac{q}{γq} )^{ℓ′_{i}} φ ((w_{i}^{γ})′) ,  (5.3)

with ℓ_{i} ∈ \{0, k_{i}\}, w_{i}^{γ} ∈ \{x_{i}^{γ}, y_{i}^{γ}\}, ℓ′_{i} = k_{i} − ℓ_{i}, and (w_{i}^{γ})′ = \{ \begin{matrix}x_{i}^{γ} if w_{i}^{γ} = y_{i}^{γ}, \\ y_{i}^{γ} if w_{i}^{γ} = x_{i}^{γ}. \end{matrix}

Proof. If Π_{i} is of type 1, then β_{i} = 0, α_{i} = c_{i}q^{k_{i}} and δ_{i} = 1. We must have q^{k_{i}}φ (x_{i}^{γ}) = x_{i−1}^{γ} (γq)^{k_{i}} and φ (y_{i}^{γ}) = y_{i−1}^{γ}. The proposition holds with ℓ_{i} = k_{i}, w_{i}^{γ} = x_{i}^{γ}, ℓ′_{i} = 0, and (w_{i}^{γ})′ = y_{i}^{γ}. The cases where Π_{i} is of type 2, 3, or 4 are identical. From Proposition 5.5 it follows that

x_{0}^{γ} = (∏_{i=0}^{f−1} φ^{i} (\frac{q}{γq} )^{s_{i}}) φ^{f} (z_{f}^{γ}) and y_{0}^{γ} = (∏_{i=0}^{f−1} φ^{i} (\frac{q}{γq} )^{s′_{i}}) φ^{f} ((z_{f}^{γ})′) ,

(5.4)

with s′_{i}, s_{i} ∈ {ℓ_{i}, ℓ′_{i}}. If z_{f}^{γ} = x_{0}^{γ}, then (z_{f}^{γ})′ = y_{0}^{γ}, and by Lemma 3.3 equations 5.4 have unique ≡ 1mod π solutions given by

x_{0}^{γ} = ∏_{i=0}^{f−1} φ^{i} (λ_{f,γ})^{s_{i}} and y_{0}^{γ} = ∏_{i=0}^{f−1} φ^{i} (λ_{f,γ})^{s′_{i}} .  (5.5)

Documenta Mathematica 15 (2010) 873–938

If If z_{f}^{γ} = y_{0}^{γ}, then (z_{f}^{γ})′ = x_{0}^{γ} and equations 5.4 imply that

x_{0}^{γ} = ∏_{i=0}^{f−1} (φ^{i} (\frac{q}{γq} )^{s_{i}} · φ^{i+f} (\frac{q}{γq} )^{s′_{i}}) φ^{2f} (x_{0}^{γ}) ,  (5.6)

y_{0}^{γ} = ∏_{i=0}^{f−1} (φ^{i} (\frac{q}{γq} )^{s′_{i}} · φ^{i+f} (\frac{q}{γq} )^{s_{i}}) φ^{2f} (y_{0}^{γ}) ,  (5.7)

which by Lemma 3.3 have unique ≡ 1mod π solutions given by

x_{0}^{γ} = ∏_{i=0}^{f−1} (φ^{i} (λ_{2f,γ})^{s_{i}} · φ^{i+f} (λ_{2f,γ})^{s′_{i}}) ,  (5.8)

y_{0}^{γ} = ∏_{i=0}^{f−1} (φ^{i} (λ_{2f,γ})^{s′_{i}} · φ^{i+f} (λ_{2f,γ})^{s′_{i}}) .  (5.9)

Equations 5.3 for i = f give the unique ≡ 1mod π solutions for x_{f−1}^{γ} and y_{f−1}^{γ}, and continuing for i = f − 1, f − 2, ..., 2, we get the unique ≡ 1mod π solutions for x_{i}^{γ} and y_{i}^{γ}. We now define the polynomials z_{i} so that for each γ ∈ Γ_{K}, the matrix G_{γ}^{(ℓ)} ≡ \overrightarrow{Id}mod π satisfies the congruence G_{γ}^{(ℓ)}−Π(S)φ(G_{γ}^{(ℓ)})γ(Π(S)^{−1}) ∈ π^{ℓ}M_{2}^{S}.

Lemma 5.6. Let R = \{∑_{i≥0} a_{i}π^{i} ∈ ℚ_{p}[[π]] : v_{p}(a_{i})+ \frac{i}{p−1} ≥ 0 for all i ≥ 0\}. The

set R endowed with the addition and the multiplication of ℚ_{p}[[π]] is a subring of ℚ_{p}[[π]] which is stable under the φ and the Γ_{K}-actions. Moreover,

(i) (\frac{q_{n}}{p})^{±1} ∈ R for all n ≥ 1 and (λ_{f})^{±1} ∈ R for all f ≥ 1;

(ii) Let b = cp^{N}b^{∗}, where c ∈ O_{E}^{×}, n ∈ ℤ, and b^{∗} ∈ R \ {0} is such that \frac{b^{∗}}{γb^{∗}} ∈ 1 + πℤ_{p}[[π]] for all γ ∈ Γ_{K}. If ℓ ≥ 1 is a fixed integer, there exists some polynomial z = z(ℓ, b) ∈ ℤ_{p}[π] with deg_{π} z ≤ ℓ − 1 and z ≡ p^{m_{ℓ}}mod π, where m_{ℓ} = ⌊\frac{ℓ−1}{p−1}⌋, such that z − γz \frac{b}{γb} ∈ π^{ℓ}ℤ_{p}[[π]] for all γ ∈ Γ_{K}.

Proof. We notice that the coefficients a_{i} of π^{i} in \frac{q}{p} are such that v_{p}(a_{i})+\frac{i}{p−1} ≥ 0 for all i = 0, 1, .... Motivated by this we consider the set R of all functions of ℚ_{p}[[π]] with the same property. This is a ring with the obvious operations,

stable under φ and Γ_{K}. One easily checks that (\frac{p}{q})^{±1} ∈ R and therefore

(\frac{q_{n}}{p} )^{±1} ∈ R for all n ≥ 1 from which (i) follows easily. (ii) Since Γ_{K} acts

trivially on O_{E}^{×} we may replace b by c^{−1}b and assume that c = 1. We write b = p^{n}b^{∗}. Let p^{m}b = z + a, where a ∈ π^{ℓ}ℚ_{p}[[π]] and deg_{π} z ≤ ℓ − 1, for integer

m which will be chosen large enough so that z ∈ ℤ_{p}[π]. Let z = ∑_{j=0}^{ℓ−1} z_{j}π^{j}. Since p^{m+n}b^{∗} = z +a and b^{∗} ∈ R, we have v_{p}(z_{j})−m−n+ \frac{j}{p−1} ≥ 0 for all j ≥ 0. We need v_{p}(z_{j}) > −1 for all j = 0, 1, ..., ℓ − 1 and it suffices to have m + n − \frac{ℓ−1}{p−1} > −1. We choose m = ⌊\frac{ℓ−1}{p−1}⌋−n. Then z ∈ ℤ_{p} [π] , deg_{π} z ≤ ℓ−1 and z ≡ p^{m+n} = p^{m_{ℓ}}mod π, For any γ ∈ Γ_{K}, z − γz \frac{b}{γb} = p^{m}b − a − bγ(b^{−1}) (p^{m}(γb) − γa) = bγ(b^{−1})γa − a ∈ π^{ℓ}ℚ_{p}[[π]]. Since z ∈ ℤ_{p} [π] and bγ(b^{−1}) ∈ 1 + πℤ_{p}[[π]], we have z − γz \frac{b}{γb} ∈ π^{ℓ}ℤ_{p} [[π]] = ℤ_{p} [[π]] ∩ π^{ℓ}ℚ_{p}[[π]] for all γ ∈ Γ_{K}.

Lemma 5.7. For any γ ∈ Γ_{K} and i ∈ I_{0},

(i) x_{i}^{γ}, y_{i}^{γ} ∈ 1 + πℤ_{p} [[π]] ;

(ii) x_{i}^{γ} = \frac{a_{i}}{γa_{i}} and y_{i}^{γ} = \frac{b_{i}}{γb_{i}} for some a_{i} and b_{i} with (a_{i})^{±1} and (b_{i})^{±1} ∈ R.

Proof. (i) is clear by the definition of the x_{i}^{γ}, y_{i}^{γ} and Lemma 3.2. (ii) Let i = 0.

If z_{f}^{γ} = x_{0}^{γ}, by equation 5.5 we have x_{0}^{γ} = \frac{a_{0}}{γa_{0}} , where a_{0} = ∏_{i=0}^{f−1} φ^{i} (λ_{f})^{s_{i}} ∈ R.

Since (λ_{f})^{±1} ∈ R and R is φ-stable, (a_{0})^{±1} ∈ R. If z_{f}^{γ} = y_{0}^{γ}, by equation

5.8 we have x_{0}^{γ} = \frac{a_{0}}{γa_{0}} , where a_{0} = ∏_{i=0}^{f−1} (φ^{i} (λ_{f})^{s_{i}} φ^{i+f} (λ_{f})^{s′_{i}}) ∈ R, therefore

(a_{0})^{±1} ∈ R. The proof for y_{0}^{γ} and (b_{i})^{±1} is similar. For x_{f−1}^{γ}, notice that x_{f−1}^{γ} =

(\frac{q}{γq} )^{ℓ_{i}} φ (w_{i}^{γ}) = ^{\frac{γ(φ(c_{0})(\frac{q}{p})^{ℓ_{f}} )}{φ(c_{0})(\frac{q}{p})^{ℓ_{f}}}} with c_{0} ∈ \{a_{0}, b_{0}\}. Since (a_{0})^{±1} , (b_{0})^{±1} ∈ R, it

follows that x_{f−1}^{γ} ∈ R. Since (φ (c_{0}) (\frac{q}{p})^{ℓ_{f}} )^{±1} ∈ R, it follows that (a_{f−1})^{±1} ∈

R. Similarly y_{f−1}^{γ} and (b_{f−1})^{±1} ∈ R. The lemma follows by induction.

To define the polynomials z_{i} we will also need the following lemma.

Lemma 5.8. If α ∈ π^{ℓ}O_{E}[[π]] and 0 ≤ k ≤ ℓ is an integer, then \frac{φ(α)}{(γq)^{k}} ∈ π^{ℓ}O_{E}[[π]].

Proof. Since \frac{γq}{q} ≡ 1mod π, it suffices to prove that \frac{φ(α)}{q^{k}} ∈ π^{ℓ}O_{E}[[π]]. Let α = π^{ℓ}β for some β ∈ O_{E}[[π]]. We have φ (\frac{α}{π^{k}} ) = φ (π)^{ℓ−k} φ (β) = q^{ℓ−k}π^{ℓ−k}φ (β) . Hence \frac{φ(α)}{q^{k}} = π^{k}φ (\frac{α}{π^{k}} ) = π^{k}q^{ℓ−k}π^{ℓ−k}φ (β) = π^{ℓ}q^{ℓ−k}φ (β) ∈ π^{ℓ}O_{E}[[π]].

Proposition 5.9. Let k = max{k_{0}, k_{1}, ..., k_{f−1}}, let ℓ ≥ k be a fixed integer and let m_{ℓ} = ⌊\frac{ℓ−1}{p−1}⌋. There exist polynomials z_{i} ∈ ℤ_{p} [π] with deg_{π} z_{i} ≤ ℓ − 1 such that z_{i} ≡ p^{m_{ℓ}}mod π with the following properties:

(i) G_{γ}^{(ℓ)} ≡ \overrightarrow{Id}mod π;

(ii) G_{γ}^{(ℓ)} − Π(S)φ(G_{γ}^{(ℓ)})γ(Π(S)^{−1}) ∈ π^{ℓ}M_{2}^{S} for each γ ∈ Γ_{K}.

Proof. Suppose that P_{i} is of type 2 and α_{i} = X_{i}φ(z_{i}) for some polynomial z_{i} to be defined. Then β_{i} = 1 and β_{i}φ (y_{i}^{γ}) = x_{i−1}^{γ} (γβ_{i}) implies that x_{i−1}^{γ} = φ (y_{i}^{γ}) . We need

X_{i} (φ(z_{i})φ (x_{i}^{γ}) − x_{i−1}^{γ}φ(γz_{i})) \frac{1}{(γq)^{k_{i}}} ∈ π^{ℓ}O_{E}[[π, X_{0}, ..., X_{f−1}]]

for all γ ∈ Γ_{K}. By Lemma 5.8 it suffices to define z_{i} so that z_{i}x_{i}^{γ} − y_{i}^{γ}γz_{i} ∈ π^{ℓ}O_{E}[[π]] for all γ ∈ Γ_{K}. Since x_{i}^{γ} ∈ 1 + πℤ_{p} [[π]] for all γ ∈ Γ_{K}, this is equivalent to z_{i} − \frac{y_{i}^{γ}}{x_{i}^{γ}} γz_{i} ∈ π^{ℓ}O_{E}[[π]]. By Lemma 5.7 we have \frac{y_{i}^{γ}}{x_{i}^{γ}} = \frac{b}{γb}, where

b = a_{i} (b_{i})^{−1} ∈ R. Since \frac{y_{i}^{γ}}{x_{i}^{γ}} ∈ 1 + πℤ_{p} [[π]] , the existence of the z_{i} follows from Lemmata 5.6 and 5.7. The proof for P_{i} of type 1, 3 and 4 is identical.

Proposition 5.10. If α (π) = ∑_{n=0}^{∞} α_{n}π^{n} ∈ ℚ_{p}[[π]] is such that v_{p} (α_{i}) ≥ 0 for

all i = 0, 1, 2, ..., p − 2 and v_{p} (α_{p−1}) > −1, then the first p − 1 coefficients of α (π)^{p} are in ℤ_{p}. In particular, the first p − 1 coefficients of the p-th power of any element of R are in ℤ_{p}.

Proof. Follows easily using the binomial expansion.

Proposition 5.11. If k_{i} = p for all i, then there exist polynomials z_{i} ∈ ℤ_{p} [π] with deg_{π} z_{i} ≤ p − 1 such that z_{i} ≡ 1mod π, and such that G_{γ}^{(p)} − Π(S)φ(G_{γ}^{(p)})γ(Π(S)^{−1}) ∈ π^{p}M_{2}^{S} for any γ ∈ Γ_{K}.

Proof. Assume that P_{i} is of type 2 and let x_{0}^{γ} and y_{0}^{γ} be as in the proof of Proposition 5.9. First we notice that the exponents s_{i} and s′_{i} in formulas 5.5 or 5.8 and 5.9 for the x_{0}^{γ} and y_{0}^{γ} are either 0 or p. With the notation of Lemma 5.7 we have \frac{y_{0}^{γ}}{x_{0}^{γ}} = c_{0} (γc_{0}^{−1}) , where c_{0} = a_{0}^{−1}b_{0}. The formulas for a_{0}^{−1} and b_{0} in the proof of Lemma 5.7 imply that they are both p-th powers of elements of R. From the same formulas and Lemma 3.2 it follows that a_{0}^{−1} (0) = b_{0} (0) = 1. By Lemma 5.10, c_{0} = z_{0} + a for some polynomial z_{0} ∈ ℤ_{p}[π] of degree ≤ p − 1 and constant term 1 and some a ∈ π^{p}ℚ_{p}[[π]]. For any γ ∈ Γ_{K}, z_{0} − \frac{y_{0}^{γ}}{x_{0}^{γ}} γz_{0} = c_{0} −a−c_{0} (γc_{0}^{−1}) (γc_{0} − γa) = c_{0} (γc_{0}^{−1}) γa−a ∈ π^{p}ℚ_{p}[[π]]. Since

\frac{y_{0}^{γ}}{x_{0}^{γ}} ∈ 1 + πℤ_{p}[[π]] and z_{0} ∈ ℤ_{p}[π], z_{0} − \frac{y_{0}^{γ}}{x_{0}^{γ}} γz_{0} ∈ ℤ_{p}[[π]] ∩ π^{p}ℚ_{p}[[π]] = π^{p}ℤ_{p}[[π]]. The proof for the other z_{i} is similar, using formulas 5.3 and noticing that

(\frac{q}{γq} )^{±1} ∈ 1 + πℤ_{p}[[π]]. The proof for P_{i} of type 1, 3 or 4 is identical.

Remark 5.12. If k_{i} = p for all i, then there exist polynomials z_{i} ∈ ℤ_{p} [π] with deg_{π} z_{i} ≤ p − 1 and z_{i} ≡ 1mod π which satisfy properties (i) and (ii) of Proposition 5.9. This follows immediately from Proposition 5.11.

Next, we explicitly determine when Tr (Q_{f}) ∉ ℚ̄_{p}. We first need some definitions.

Definition 5.13. (i) We define C_{1} to be the set of f-tuples (P_{1}, P_{2}, ..., P_{f}) where the types of the matrices P_{i} are chosen as follows: P_{1} ∈ {t_{2}, t_{3}}. For i = 2, 3, ..., f − 1, P_{i} ∈ {t_{2}, t_{3}} if an even number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, and P_{i} ∈ {t_{1}, t_{4}} if an odd number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type. Finally, P_{0} = t_{3} if an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, and P_{0} = t_{4} otherwise.

(ii) We define C_{2} to be the set of f-tuples (P_{1}, P_{2}, ..., P_{f}) where the types of the matrices P_{i} are chosen as follows: P_{1} ∈ {t_{1}, t_{4}}. For i = 2, 3, ..., f−1, P_{i} ∈ {t_{1}, t_{4}} if an even number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, and P_{i} ∈ {t_{2}, t_{3}} if an odd number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type. Finally, P_{0} = t_{1} if an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, and P_{0} = t_{2} otherwise.

In Definition 5.13 the type of the matrix P_{0} has been chosen so that an even number of coordinates of the f-tuple (P_{1}, P_{2}, ..., P_{f−1}, P_{0}) is of even type.

Definition 5.14. (i) We define C_{1}^{∗} to be the set of f-tuples (P_{1}, P_{2}, ..., P_{f}) where the types of the matrices P_{i} are chosen as follows: P_{1} ∈ {t_{2}, t_{3}}. For i = 2, 3, ..., f − 1, P_{i} ∈ {t_{2}, t_{3}} if an even number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, and P_{i} ∈ {t_{1}, t_{4}} if an odd number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type. Finally, P_{0} = t_{2} if an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, and P_{0} = t_{1} otherwise.

(ii) We define C_{2}^{∗} to be the set of f-tuples (P_{1}, P_{2}, ..., P_{f}) where the types of the matrices P_{i} are chosen as follows: P_{1} ∈ {t_{1}, t_{4}}. For i = 2, 3, ..., f −1, P_{i} ∈ {t_{1}, t_{4}} if an even number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, and P_{i} ∈ {t_{2}, t_{3}} if an odd number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type. Finally, P_{0} = t_{4} if an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, and P_{0} = t_{3} otherwise.

In Definition 5.14 the type of the matrix P_{0} has been chosen so that an odd number of coordinates of the f-tuple (P_{1}, P_{2}, ..., P_{f−1}, P_{0}) is of even type.

Lemma 5.15. Assume that f ≥ 2 and, as before, let Q_{f} = P_{1}P_{2} ⋯ P_{f}.

(i) If (P_{1}, P_{2}, ..., P_{f}) ∈ C_{1}^{∗}, then Q_{f} = ( \begin{matrix}α & β \\ γ & 0 \end{matrix} ) with β, γ nonconstant

polynomials in X_{1}, X_{2}, ..., X_{f} (with X_{f} = X_{0}), linearly independent over ℚ̄_{p}, and α nonscalar.

(ii) If (P_{1}, P_{2}, ..., P_{f}) ∈ C_{2}^{∗}, then Q_{f} = ( \begin{matrix}0 & β \\ γ & δ \end{matrix} ) with β, γ nonconstant poly-

nomials in X_{1}, X_{2}, ..., X_{f}, linearly independent over ℚ̄_{p}, and δ nonscalar.

(iii) If (P_{1}, P_{2}, ..., P_{f}) ∈ C_{1}, then Q_{f} = ( \begin{matrix}α & β \\ 0 & δ \end{matrix} ) with β a nonzero polyno-

mial in X_{1}, X_{2}, ..., X_{f}, and α, δ nonzero scalars.

(iv) If (P_{1}, P_{2}, ..., P_{f}) ∈ C_{2}, then Q_{f} = ( \begin{matrix}α & 0 \\ γ & δ \end{matrix} ) with γ a nonzero polyno-

mial in X_{1}, X_{2}, ..., X_{f}, and α, δ nonzero scalars.

Proof. Follows easily by induction on f.

Lemma 5.16. Assume that f ≥ 2.

(i) If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f}) is of even type, then Q_{f} has one of the following forms:

(a) Q_{f} = ( \begin{matrix}0 & β \\ γ & δ \end{matrix} ) with β, γ nonconstant polynomials in

X_{1}, X_{2}, ..., X_{f}, linearly independent over ℚ̄_{p}, and δ nonscalar. This case occurs if and only if (P_{1}, P_{2}, ..., P_{f}) ∈ C_{2}^{∗}.

(b) Q_{f} = ( \begin{matrix}α & β \\ γ & 0 \end{matrix} ) with β, γ nonconstant polynomials in

X_{1}, X_{2}, ..., X_{f}, linearly independent over ℚ̄_{p}, and α nonscalar. This case occurs if and only if (P_{1}, P_{2}, ..., P_{f}) ∈ C_{1}^{∗}.

(c) In any other case, Q_{f} = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) with β, γ nonconstant polyno-

mials in X_{1}, X_{2}, ..., X_{f}, linearly independent over ℚ̄_{p}, αδ ≠ 0, and Tr (Q_{f}) nonscalar.

(ii) If an even number of coordinates of (P_{1}, P_{2}, ..., P_{f}) is of even type, then Q_{f} has one of the following forms:

(d) Q_{f} = ( \begin{matrix}α & β \\ 0 & δ \end{matrix} ) with β a nonzero polynomial in X_{1}, X_{2}, ..., X_{f}, and

α, δ nonzero scalars. This case occurs if and only if (P_{1}, P_{2}, ..., P_{f}) ∈ C_{1}.

(e) Q_{f} = ( \begin{matrix}α & 0 \\ γ & δ \end{matrix} ) with γ a nonzero polynomial in X_{1}, X_{2}, ..., X_{f}, and

α, δ nonzero scalars. This case occurs if and only if (P_{1}, P_{2}, ..., P_{f}) ∈ C_{2}.

(f) In any other case, Q_{f} = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) with β, γ nonconstant polyno-

mials in X_{1}, X_{2}, ..., X_{f}, linearly independent over ℚ̄_{p}, αγ ≠ 0 and Tr (Q_{f}) is nonscalar.

Proof. By induction on f. If f = 2 the proof of the lemma is by a direct computation. Suppose f ≥ 3. Case (i). An odd number of coordinates of (P_{1}, P_{2}, ..., P_{f}) is of even type.

(a) If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, then P_{0} ∈ {t_{1}, t_{3}}. We have the following three subcases:

(α) Q_{f−1} = ( \begin{matrix}0 & β \\ γ & δ \end{matrix} ) with β, γ nonconstant polynomials in X_{1}, X_{2}, , , .X_{f−1},

linearly independent over ℚ̄_{p}, and δ nonscalar. If P_{0} = t_{1}, then Q_{f} is as in case (c), and by Lemma 5.15 (P_{1}, P_{2}, ..., P_{f}) ∉ C_{1}^{∗} ∪ C_{2}^{∗}. If P_{0} = t_{3}, then Q_{f} is as in case (a). By the inductive hypothesis (P_{1}, P_{2}, ..., P_{f−1}) ∈ C_{2}^{∗}, and since P_{0} = t_{3}, (P_{1}, P_{2}, ..., P_{f}) ∈ C_{2}^{∗}.

(β) Q_{f−1} = ( \begin{matrix}α & β \\ γ & 0 \end{matrix} ) with β, γ nonconstant polynomials in X_{1}, X_{2}, , , .X_{f−1},

linearly independent over ℚ̄_{p}, and α nonscalar. If P_{0} = t_{1}, then Q_{f} is as in case (b). By the inductive hypothesis (P_{1}, P_{2}, ..., P_{f−1}) ∈ C_{1}^{∗}, and since P_{0} = t_{1}, (P_{1}, P_{2}, ..., P_{f}) ∈ C_{1}^{∗}. If P_{0} = t_{3}, then Q_{f} is as in case (c), and by Lemma 5.15 (P_{1}, P_{2}, ..., P_{f}) ∉ C_{1}^{∗} ∪ C_{2}^{∗}.

(γ) Q_{f−1} = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) with β, γ nonconstant polynomials in X_{1}, X_{2}, , , .X_{f−1},

linearly independent over ℚ̄_{p}, αδ ≠ 0, and Tr (Q_{f}) nonscalar. If P_{0} ∈ {t_{1}, t_{3}} then Q_{f} is as in case (c), and by Lemma 5.15 (P_{1}, P_{2}, ..., P_{f}) ∉ C_{1}^{∗} ∪ C_{2}^{∗}.

(b) If an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, then P_{0} ∈ {t_{2}, t_{4}}. We have the following three subcases:

(α) Q_{f−1} = ( \begin{matrix}α & β \\ 0 & δ \end{matrix} ) with β a nonzero polynomial in X_{1}, X_{2}, , , .X_{f−1},

and α, δ nonzero scalars. If P_{0} = t_{2}, then Q_{f} is as in case (b). Since (P_{1}, P_{2}, ..., P_{f−1}) ∈ C_{1} and P_{0} = t_{2}, (P_{1}, P_{2}, ..., P_{f}) ∈ C_{1}^{∗}. If P_{0} = t_{4}, then Q_{f} is as in case (c), and by Lemma 5.15 (P_{1}, P_{2}, ..., P_{f}) ∉ C_{1}^{∗} ∪ C_{2}^{∗}.

(β) Q_{f−1} = ( \begin{matrix}α & 0 \\ γ & δ \end{matrix} ) with γ a nonzero polynomial in X_{1}, X_{2}, , , .X_{f−1}, and

α, δ nonzero scalars. If P_{0} = t_{2}, then Q_{f} is as in case (c), and by Lemma 5.15 (P_{1}, P_{2}, ..., P_{f}) ∉ C_{1}^{∗} ∪ C_{2}^{∗}. If P_{0} = t_{4}, then Q_{f} is as in case (a). Since (P_{1}, P_{2}, ..., P_{f−1}) ∈ C_{2} and P_{0} = t_{4}, (P_{1}, P_{2}, ..., P_{f}) ∈ C_{2}^{∗}.

(γ) Q_{f−1} = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) with β, γ nonconstant polynomials in X_{1}, X_{2}, , , .X_{f},

linearly independent over ℚ̄_{p}, αγ ≠ 0 and Tr (Q_{f}) is nonscalar. Then (P_{1}, P_{2}, ..., P_{f−1}) ∉ C_{1} ∪ C_{2}. If P_{0} ∈ {t_{2}, t_{4}}, then Q_{f} is as in case (c), and by Lemma 5.15 (P_{1}, P_{2}, ..., P_{f}) ∉ C_{1}^{∗} ∪ C_{2}^{∗}. Case (ii). An even number of coordinates of (P_{1}, P_{2}, ..., P_{f}) is of even type. The rest of the lemma is proved by a case-by-case analysis similar to that of Case (i).

Corollary 5.17. Tr (Q_{f}) ∈ ℚ̄_{p} if and only if (P_{1}, P_{2}, ..., P_{f−1}, P_{0}) ∈ C_{1} ∪ C_{2}.

Remark 5.18. If (P_{1}, P_{2}, ..., P_{f}) ∈ C_{1} ∪ C_{2}, the filtered φ-modules 𝔻_{k}^{i} (0) are weakly admissible and the corresponding crystalline representation is splitreducible and ordinary (see §6.3). The filtered φ-modules 𝔻_{k}^{i} (α) make sense

for any α ∈ O_{E}^{f}. One can check by induction that Tr (φ^{f}) = 1+p^{∑_{i=0}^{f−1} k_{i}}, therefore whenever 𝔻_{k}^{i} (α) is weakly admissible the corresponding crystalline representation is reducible (see Proposition 6.5). Since we have not constructed the Wach modules which give rise to these filtered modules, weak admissibility is not automatic and has to be checked.

We now turn our attention to condition (iv) of Lemma 4.4. By Proposition 5.1 the problematic cases are those with ℓ = k, all the weights k_{i} equal and Q_{f} ∈ {E_{11}, E_{22}}. We have the following.

Lemma 5.19. If Q̄_{f} = E_{11} then (P_{1}, ..., P_{f}) ∈ C_{1}; (ii) If Q̄_{f} = E_{22}, then (P_{1}, ..., P_{f}) ∈ C_{2}.

Proof. By induction on f. First, we notice that

P mod (p · Id, X_{i} · Id) = \{ \begin{matrix}E_{22} if P = t_{1}, \\ E_{12} if P = t_{2}, \\ E_{11} if P = t_{3}, \\ E_{21} if P = t_{4}. \end{matrix}

Suppose that Q̄_{f} = E_{11} and f = 2. Then P_{1} ∈ {t_{2}, t_{3}}. If P_{1} = t_{2} then P_{0} = t_{4} and if P_{1} = t_{3} then P_{0} = t_{3}. Suppose Q̄_{f} = E_{11} and f > 2. Then Q̄_{f−1} = E_{11} and P_{f} = t_{3} or Q̄_{f−1} = E_{12} and P_{f} = t_{4}. In the first case the inductive hypothesis implies that (P_{1}, P_{2}, ..., P_{f−1}) ∈ C_{1} and P_{f} = t_{3}. If an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−2}) is of even type, then P_{f−1} = t_{3}. In this case an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type and P_{f} = t_{3}, hence (P_{1}, ..., P_{f}) ∈ C_{1}. If an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f−2}) is of even type, then P_{f−1} = t_{4}. In this case an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type and P_{f} = t_{3}, hence (P_{1}, ..., P_{f}) ∈ C_{1}. Now assume that Q̄_{f−1} = E_{12} and P_{f} = t_{4}. This implies that either Q̄_{f−2} = E_{12}, P_{f−1} = t_{4} and P_{f} = t_{4} which is absurd since in this case Q̄_{f} = 0, or Q̄_{f−2} = E_{11}, P_{f−1} = t_{2} and P_{f} = t_{4}. If f = 3, then P_{1} = t_{3}, P_{2} = t_{2}, P_{3} = t_{4} and the lemma holds. If f ≥ 4 and an even number of coordinates (P_{1}, P_{2}, ..., P_{f−3}) is of even type, then P_{f−2} = t_{3}, P_{f−1} = t_{2} and P_{f} = t_{4}. Then an odd number of coordinates (P_{1}, P_{2}, ..., P_{f−1}) is of even type and P_{f} = t_{4}, hence (P_{1}, ..., P_{f}) ∈ C_{1}. If an odd number of coordinates (P_{1}, P_{2}, ..., P_{f−3}) is of even type, then P_{f−2} = t_{4}, P_{f−1} = t_{2} and P_{f} = t_{4}. Then an odd number of coordinates (P_{1}, P_{2}, ..., P_{f−1}) is of even type and P_{f} = t_{4}, hence (P_{1}, ..., P_{f}) ∈ C_{1}. Part (ii) is proved similarly.

Corollary 5.20. If (P_{1}, P_{2}, ..., P_{f}) ∈ P and Tr (Q_{f}) ∉ ℚ̄_{p}, then the operator

\overline{H} ↦ \overline{H − Q_{f}H(p^{fℓ}Q_{f}^{−1})} : M̃_{2} → M̃_{2}

is surjective.

Documenta Mathematica 15 (2010) 873–938

5.2 Corresponding families of rank two filtered φ-modules

Let Π^{i}(S) = (Π_{1}(X_{1}), Π_{2}(X_{2}), ..., Π_{f−1}(X_{f−1}), Π_{0}(X_{0})) with i ∈ {1, 2, 3, 4}^{f} and matrices Π_{i}(X_{i}) as in Definition 5.4. The definition of the Π_{i} and P_{i} = Π_{i}mod π depends on the choice of the z_{i} in Proposition 5.9 and therefore on ℓ. For the rest of the paper we assume that (P_{1}, P_{2}, ..., P_{0}) ∉ C_{1} ∪ C_{2} and we choose ℓ = k = max{k_{0}, k_{1}, ..., k_{f−1}}.

Proposition 5.21. For any γ ∈ Γ_{K}, there exists a unique matrix G_{γ}(S) = G_{γ}(S) ∈ M_{2}^{S} such that:

(i) G_{γ}(S) ≡ \overrightarrow{Id}mod π;

(ii) Π^{i}(S)φ(G_{γ}(S)) = G_{γ}(S)γ(Π^{i}(S)).

Proof. Conditions (1) and (2) of Lemma 4.4 are satisfied by Proposition 5.9. Condition (3) of Lemma 4.4 is satisfied by the assumption that (P_{1}, P_{2}, ..., P_{0}) ∉ C_{1} ∪ C_{2} and Corollaries 5.3 and 5.17. Finally, condition (4) of Lemma 4.4 is satisfied by Proposition 5.1 and Lemma 5.19. The proposition follows from Lemma 4.4.

For any a ∈ 𝔪_{E}^{f} we equip the module ℕ_{k}^{i}(a) = (O_{E}[[π]]^{|τ|}) η_{1} ⨁ (O_{E}[[π]]^{|τ|}) η_{2} with semilinear φ and Γ_{K}-actions defined as in Proposition 4.6. For any a ∈ 𝔪_{E}^{f} we consider the matrices of GL_{2} (E^{|τ|}) obtained from the matrices P^{i}(a) = (P_{1}(X_{1}), P_{2}(X_{2}), ..., P_{f−1}(X_{f−1}), P_{0}(X_{0})) by substituting X_{j} = a_{j} in P_{j}(X_{j}). We define families of filtered φ-modules 𝔻_{k}^{i} (a) = (E^{|τ|}) η_{1} ⨁ (E^{|τ|}) η_{2}

with Frobenius endomorphisms given by (φ(η_{1}), φ(η_{2})) = (η_{1}, η_{2})P^{i}(a), and filtrations given by

Fil^{j}(𝔻_{k}^{i} (a)) = \{ \begin{matrix}E^{|τ|}η_{1} ⨁ E^{|τ|}η_{2} if j ≤ 0, \\ E^{|τ_{I_{0}} |} (xη_{1} + yη_{2}) if 1 ≤ j ≤ w_{0}, \\ E^{|τ_{I_{1}} |} (xη_{1} + yη_{2}) if 1 + w_{0} ≤ j ≤ w_{1}, \\ ⋯ & ⋯ \\ E^{|τ_{I_{t−1}} |} (xη_{1} + yη_{2}) if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0 if j ≥ 1 + w_{t−1}, \end{matrix}

(5.10)

where x = (x_{0}, x_{1}, ..., x_{f−1}) and y = (y_{0}, y_{1}, ..., y_{f−1}) with

(x_{i}, y_{i}) = \{ \begin{matrix}(1, −α_{i}) if P_{i} has type 1 or 2, \\ (−α_{i}, 1) if P_{i} has type 3 or 4, \end{matrix}  (5.11)

and α_{i} = a_{i}z_{i}(0) for all i. Since ℓ = k, Remark 5.12 implies that α_{i} ∈ p^{m}𝔪_{E} for all i, where

m := \{ \begin{matrix}⌊\frac{k−1}{p−1}⌋ if k ≥ p and k_{i} ≠ p for some i, \\ 0 if k ≤ p − 1 or k_{i} = p for all i. \end{matrix}

Proposition 5.22. For any a ∈ 𝔪_{E}^{f} the filtered φ-modules (𝔻_{k}^{i} (a) , φ) defined above are weakly admissible and 𝔻_{k}^{i} (a) ≃ E^{|τ|} ⨂_{O_{E}^{|τ|}}(ℕ_{k}^{i}(a)/πℕ_{k}^{i}(a)) as filtered

φ-modules over E^{|τ|}. Proof. By Theorem 2.4, xη_{1} + yη_{2} ∈ Fil^{j}(ℕ_{k}^{i}(a)) if and only if φ(x)φ(η_{1})+ φ(y)φ(η_{2}) ∈ q^{j}ℕ_{k}^{i}(a) or equivalently

e_{i}φ(x)φ(η_{1}) + e_{i}φ(y)φ(η_{2}) ∈ q^{j}e_{i}ℕ_{k}^{i}(a) for all i ∈ I_{0},  (5.12)

with the idempotents e_{i} as in §1.1. We fix some i ∈ I_{0} and calculate in the case where Π_{i} is of type 2. Then Π_{i}(a_{i}) = ( \begin{matrix}0 & c_{i}q^{k_{i}} \\ 1 & a_{i}φ(z_{i}) \end{matrix} ) and equation 5.12 is

equivalent to \{ \begin{matrix}q^{j} | φ(y_{i})q^{k_{i}}, \\ q^{j} | φ(x_{i} + y_{i}a_{i}z_{i}). \end{matrix} We use that q^{j} | φ(x) if and only if π^{j} | x

for any x ∈ O_{E}[[π]]. If j ≥ 1 + k_{i}, then x_{i}, y_{i} ≡ 0mod π. If 1 ≤ j ≤ k_{i}, the system above is equivalent to π^{j} | x_{i} + y_{i}a_{i}z_{i}. Since a_{i}z_{i} ≡ α_{i}mod π,

e_{i}xη_{1} +e_{i}yη_{2} +πℕ_{k}^{i}(a) = \{ \begin{matrix}α_{i}ȳ_{i}e_{i}η_{1} + ȳ_{i}e_{i}η_{2} + πℕ_{k}^{i}(a) if 1 ≤ j ≤ k_{i}, \\ 0 if j ≥ k_{i} \end{matrix}

where ȳ_{i} = y_{i}mod π can be any element of O_{E}. Since Fil^{0}(ℕ_{k}^{i}(a)/πℕ_{k}^{i}(a)) = (O_{E}^{|τ|})η_{1} ⨁(O_{E}^{|τ|})η_{2}, we get

e_{i}Fil^{j}(ℕ_{k}^{i}(a)/πℕ_{k}^{i}(a)) = \{ \begin{matrix}e_{i}(O_{E}^{|τ|})η_{1} ⨁ e_{i}(O_{E}^{|τ|})η_{2} if j ≤ 0, \\ e_{i}(O_{E}^{|τ|})(x^{i}η_{1} + y^{i}η_{2}) if 1 ≤ j ≤ k_{i}, \\ 0 if j ≥ 1 + k_{i}, \end{matrix}

with e_{i}x^{i} = (0, ..., x_{i}, ..., 0), e_{i}y^{i} = (0, ..., y_{i}, ..., 0) and (x_{i}, y_{i}) = (−α_{i}, 1). Calculating for the other choices of Π_{i}(a_{i}) we see that for all i ∈ I_{0}, (x_{i}, y_{i}) is as in formula 5.10. Since Fil^{j}(ℕ_{k}^{i}(a)/πℕ_{k}^{i}(a)) = ⨁_{i=0}^{f−1} e_{i}Fil^{j}(ℕ_{k}^{i}(a)/πℕ_{k}^{i}(a)), arguing as in the proof of Proposition 3.4 we get

Fil^{j}(ℕ_{k}^{i}(a)/πℕ_{k}^{i}(a)) = \{ \begin{matrix}(O_{E}^{|τ|})η_{1} ⨁(O_{E}^{|τ|})η_{2}  if j ≤ 0, \\ (O_{E}^{|τ|})f_{I_{0}}(xη_{1} + yη_{2}) if 1 ≤ j ≤ w_{0}, \\ (O_{E}^{|τ|})f_{I_{1}}(xη_{1} + yη_{2}) if 1 + w_{0} ≤ j ≤ w_{1}, \\ ⋯ & ⋯ \\ (O_{E}^{|τ|})f_{I_{t−1}}(xη_{1} + yη_{2}) if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0  if j ≥ 1 + w_{t−1}, \end{matrix}

with x = (x_{0}, x_{1}, ..., x_{f−1}) and y = (y_{0}, y_{1}, ..., y_{f−1}) and (x_{i}, y_{i}) as in formula 5.10. The isomorphism 𝔻_{k}^{i} (a) ≃ E^{|τ|} ⨂_{O_{E}^{|τ|}}(ℕ_{k}^{i}(a)/πℕ_{k}^{i}(a)) is now obvious.

The crystalline representation corresponding to 𝔻_{k}^{i} (a) is denoted by V_{k,a}^{i}.

6 Reductions of crystalline representations

In this section we explicitly compute the semisimplified modulo p reductions of the families of crystalline representations constructed in §5. We will need the following lemma.

Lemma 6.1. Let F be any field, G any group and H any finite index subgroup. Let V be an irreducible finite-dimensional F G-module whose restriction to H contains some F H-submodule W with dim_{F} V = [G : H] dim_{F} W. Then V ≃ Ind_{H}^{G} (W) .

Proof. By Frobenius reciprocity there exists some nonzero α ∈ Hom_{F G}(Ind_{H}^{G} (W) , V). It is an isomorphism because V is irreducible and Ind_{H}^{G} (W) and V have the same dimension over F.

We start with the reductions of crystalline characters and reducible twodimensional crystalline representations of G_{K}. The embeddings τ_{i} of K_{f} into E fixed in the introduction induce embeddings of residue fields k_{K_{f}} →^{τ̄_{i}} k_{E}. The level f fundamental characters ω_{f,τ̄_{i}} of I_{K_{f}} are defined by composing the embeddings τ̄_{i} with the homomorphism I_{K_{f}} → k_{K_{f}}^{×} obtained from local class field theory, with uniformizers corresponding to geometric Frobenius elements. We recall the following lemma which follows immediately from [BDJ, Lemma 3.8], where the χ_{i} are as in §3.

Lemma 6.2. (i) (χ̄_{i})_{|I_{K_{f}}} = ω_{f,τ̄_{i+1}}^{−1} for i = 0, 1, ..., f −1; (ii) ω_{f,τ̄_{i}} = ω_{f,τ̄_{0}}^{p^{i}} for all

i; (iii) ω_{2f,τ̄_{0}}^{1+p^{f}} = ω_{f,τ̄_{0}}; (iv) ω = ∏_{i∈I_{0}} ω_{f,τ̄_{i}}, where ω is the cyclotomic character

modulo 𝔪_{E}.

Our next goal is to compute the determinant of a two-dimensional crystalline representations in terms of its labeled Hodge-Tate weights. To do this, we will need some facts about weakly admissible filtered φ-modules which we briefly recall. For the missing details we refer to [Dou10]. We remark that similar results for odd p have been obtained by Imai in [Ima09].

Proposition 6.3. Let (𝔻, φ) be a rank two F-semisimple, nonscalar filtered φ-module over E^{|τ|} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}. After enlarging E if necessary, there exists an ordered basis \underline{η} of 𝔻 over E^{|τ|} with respect to which the matrix of Frobenius takes the form Mat\underline{η}(φ) = diag(α, δ) for some vectors α, δ ∈ (E^{×})^{|τ|} with Nm_{φ}(α) ≠ Nm_{φ}(δ). The filtration in the same basis has the form of formula 5.10 for some vectors x, y ∈ E^{|τ|} with (x_{i}, y_{i}) ≠ (0, 0) for all i ∈ I_{0}. We call such a basis \underline{η} a standard basis of (𝔻, φ). The Frobeniusfixed submodules are 0, 𝔻, 𝔻_{1} := (E^{|τ|}) η_{1} and 𝔻_{2} := (E^{|τ|}) η_{2}. The module 𝔻 is weakly admissible if and only if

(1) v_{p}(Nm_{φ}(α)Nm_{φ}(δ)) = ∑_{i∈I_{0}} k_{i};

(2) v_{p}(Nm_{φ}(α)) ≥ ∑_{{i∈I_{0}: y_{i}=0}} k_{i};

(3) v_{p}(Nm_{φ}(δ)) ≥ ∑_{{i∈I_{0}: x_{i}=0}} k_{i}.

Assuming that 𝔻 is weakly admissible,

(i) The filtered φ-module 𝔻 is irreducible if and only if both the inequalities (2) and (3) above are strict;

(ii) The filtered φ-module 𝔻 is split-reducible if and only if both inequalities (2) and (3) are equalities, or equivalently I_{0}^{+} ∩ J_{x} ∩ J_{y} = ∅. In this case, the only nontrivial weakly admissible submodules are 𝔻_{i}, i = 1, 2, and we have 𝔻 = 𝔻_{1} ⨁ 𝔻_{2};

(iii) In any other case the filtered φ-module 𝔻 is reducible, non-split.

In the Proposition above, if v_{p}(Nm_{φ}(α)) = ∑_{{i∈I_{0}: y_{i}=0}} k_{i}, the only nontriv-

ial weakly admissible submodule is (𝔻_{1}, φ). If v_{p}(Nm_{φ}(δ)) = ∑_{{i∈I_{0}: x_{i}=0}} k_{i},

the only nontrivial weakly admissible submodule is (𝔻_{2}, φ). If (𝔻, φ) is not F-semisimple, after extending E if necessary, there exists an ordered basis \underline{η} = (η_{1}, η_{2}) of 𝔻 over E^{|τ|} with respect to which the matrix of Frobenius takes the form

Mat\underline{η}(φ) = ( \begin{matrix}α & 0 \\ γ & α \end{matrix} )

for some vectors α ∈ (E^{×})^{|τ|} and γ ∈ E (see [Dou10, §2.1]). The filtration in this basis has the shape of formula 5.10. The filtered φ-module (𝔻, φ) is weakly admissible if any only if 2 · v_{p}(Nm_{φ}(α)) = ∑_{i∈I_{0}} k_{i} and v_{p}(Nm_{φ}(α)) ≥

∑_{{i∈I_{0}:x_{i}=0}} k_{i}. The corresponding crystalline representation is irreducible if and

only if the latter inequality is strict and reducible, non-split otherwise. In this case, the only φ-stable weakly admissible submodule is (𝔻_{2}, φ) (see also [Dou10, § 5.4]). If (𝔻, φ) is F-scalar, there exists an ordered basis \underline{η} = (η_{1}, η_{2}) of 𝔻 over

E^{|τ|} with respect to which Mat\underline{η}(φ) = diag (α · 1, α · 1) for some α ∈ E^{×} and the filtration is as in formula 5.10. The only φ-stable submodules are the 𝔻_{i}, i = 1, 2 and 𝔻(c) = (E^{|τ|}) (η_{1} + c · 1 · η_{2}) for any c ∈ E^{×} (cf. [Dou10, §5.3]). To summarize, we have the following.

Proposition 6.4. Let (𝔻, φ) be a reducible weakly admissible rank two filtered φ-module over E^{|τ|} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}. After enlarging E if necessary, there exists an ordered basis \underline{η} = (η_{1}, η_{2}) of 𝔻 over E^{|τ|} with

respect to which the matrix of Frobenius takes the form Mat\underline{η}(φ) = ( \begin{matrix}α & 0 \\ ∗ & δ \end{matrix} )

and is such that 𝔻_{2} = (E^{|τ|}) η_{2} is a φ-stable, weakly admissible submodule.

The following propositions which will be used in §§ 6.2 and 6.3.

Proposition 6.5. A rank two weakly admissible effective filtered φ-module (𝔻, φ) with labeled Hodge-Tate weights {−k_{i}, 0}_{τ_{i}} and v_{p} (Tr(φ^{f})) = 0 is reducible.

Proof. If 𝔻 is F-semisimple and nonscalar, see [Dou10, Corollary 7.2]. Suppose that this is not the case. Since we assume that k_{i} > 0 for at least one i, for any F-scalar or non-F-semisimple filtered φ-module with labeled weights {−k_{i}, 0}_{τ_{i}}, v_{p} (Tr(φ^{f})) ≠ 0. Indeed, in this case there exists an ordered basis \underline{η} of 𝔻 over E^{|τ|} with respect to which the matrix of Frobenius takes the form

Mat\underline{η}(φ) = ( \begin{matrix}α & 0 \\ γ & α \end{matrix} )

for some vectors α ∈ (E^{×})^{|τ|} and γ ∈ E (see [Dou10, §2.1]). Weak admissibility implies that 2 · v_{p}(Nm_{φ}(α)) = ∑_{i∈I_{0}} k_{i} > 0 (see [Dou10, Propositions 4.3 and

4.4]), therefore v_{p} (Tr(φ^{f})) = v_{p} (2 · Nm_{φ}(α)) > 0.

The following lemma allows us to compute determinants of two-dimensional crystalline representations in terms of their labeled Hodge-Tate weights.

Lemma 6.6. If (𝔻, φ) is a rank two weakly admissible filtered φ-module over K with E-coefficients and labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, then (∧_{E⊗K}^{2}𝔻, ∧_{E⊗K}^{2}φ) is weakly admissible with labeled Hodge-Tate weights {−k_{i}}_{τ_{i}}.

Proof. Let \underline{η} = (η_{1}, η_{2}) be a standard basis of (𝔻, φ) such that Mat\underline{η}(φ) is as in Proposition 6.4 and Fil^{j}𝔻 as in Formula 5.10. Clearly (∧^{2}φ)(η_{1} ∧ η_{2}) =

α · δ(η_{1} ∧ η_{2}). Since Fil^{j}(𝔻 ∧ 𝔻) = ∑_{j_{1}+j_{2}=j}(Fil^{j_{1}}𝔻 ∧_{E⊗K} Fil^{j_{2}}𝔻) and J_{x} ∪ J_{y} = I_{0},

a simple computation yields

Fil^{j}(𝔻 ∧ 𝔻) = \{ \begin{matrix}E^{|τ_{I_{0}} |}(η_{1} ∧ η_{2}) if j ≤ w_{0}, \\ E^{|τ_{I_{1}} |}(η_{1} ∧ η_{2}) if 1 + w_{0} ≤ j ≤ w_{1}, \\ ⋯ & ⋯ \\ E^{|τ_{I_{t−1}} |}(η_{1} ∧ η_{2}) if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0 if j ≥ 1 + w_{t−1}, \end{matrix}

from which the statement about the labeled Hodge-Tate weights follows immediately. Weak admissibility is clear.

Corollary 6.7. If V is the crystalline representation corresponding to 𝔻, then

det V ≃ η · χ_{e_{0}}^{k_{1}} · χ_{e_{1}}^{k_{2}} · · · · · χ_{e_{f−2}}^{k_{f−1}} · χ_{e_{f−1}}^{k_{0}} and (det \overline{V})_{|I_{K}} ≃ ω_{f,τ̄_{0}}^{α},

where η is an unramified character of G_{K} and α = − ∑_{i=0}^{f−1} p^{i}k_{i}.

Proof. By Proposition 3.4 and Lemma 6.6 the crystalline character det V ⊗

(χ_{e_{0}}^{k_{1}} · χ_{e_{1}}^{k_{2}} · · · · · χ_{e_{f−2}}^{k_{f−1}} · χ_{e_{f−1}}^{k_{0}})^{−1} has labeled Hodge-Tate weights {0}_{τ_{i}}. If the corresponding filtered φ-module has Frobenius endomorphism φ(η) = a·η, then by Proposition 3.5 Nm_{φ}(a) = c · 1 for some c ∈ E^{×} with v_{p}(c) = 0. Lemma 3.7

implies that det V ⊗ (χ_{e_{0}}^{k_{1}} · χ_{e_{1}}^{k_{2}} · · · · · χ_{e_{f−2}}^{k_{f−1}} · χ_{e_{f−1}}^{k_{0}})^{−1} is the unramified character of G_{K} which maps Frob_{K} to c. The rest of the corollary follows from Lemma 6.2.

We recall the following well-known proposition in which the field K has absolute inertia degree f and is not assumed to be unramified over ℚ_{p}.

Proposition 6.8. [Bre07, Prop. 2.7] Let ρ̄ : G_{K} → GL_{2}(𝔽̄_{p}) be a continuous representation. Then

ρ̄_{|I_{K}} ≃ ( \begin{matrix}ω_{2f}^{m} & ∗ \\ 0 & ω_{2f}^{mp^{f}} \end{matrix} )

for some integer m. The representation ρ̄ is irreducible if and only 1 + p^{f} ∤ m, and in this case ∗ = 0.

Corollary 6.9. Let χ be a crystalline character of G_{K_{2f}} with labeled Hodge-Tate weights {−k_{i}}_{τ_{i}}, where the k_{i} are arbitrary integers for all i = 0, 1, ..., 2f − 1, and let V = Ind_{K_{2f}}^{K_{f}} (χ) . The residual representation \overline{V} is irreducible if and

only if 1 + p^{f} ∤ ∑_{i=0}^{2f−1} p^{i}k_{i}.

Proof. Follows immediately from Lemma 6.2 and Proposition 6.8.

6.1 Reductions of reducible two-dimensional crystalline representations

In this section we compute the semisimplified modulo p reduction of any reducible two-dimensional crystalline representation of G_{K_{f}} .

Lemma 6.10. Let k_{0}, k_{1}, ..., k_{f−1} be arbitrary integers and let

Fil^{j}𝔻 = \{ \begin{matrix}E^{|τ_{I_{0}} |}η if j ≤ w_{0}, \\ E^{|τ_{I_{1}} |}η if 1 + w_{0} ≤ j ≤ w_{1}, \\ ⋯ & ⋯ \\ E^{|τ_{I_{t−1}} |}η if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0  if j ≥ 1 + w_{t−1}. \end{matrix}  (6.1)

For each i ∈ I_{0},

e_{i}Fil^{j}𝔻 = \{ \begin{matrix}e_{i}E^{|τ_{I_{0}} |}η if j ≤ k_{i}, \\ 0 if r ≥ 1 + k_{i}. \end{matrix}

Proof. Let k_{i} = w_{r} for some r ∈ {1, ..., t − 1}. Since w_{r} > w_{r−1} we have i ∈ I_{r} from the definition of I_{r}. Similarly, since k_{i} = w_{r} we have i ∉ I_{r+1}. The same is clear for r = 0. Hence e_{i}f_{I_{r}} = e_{i} and e_{i}f_{I_{r+1}} = 0 for all r. Multiplying formula 6.1 by e_{i}, we get

e_{i}Fil^{j}𝔻 = \{ \begin{matrix}e_{i}E^{|τ_{I_{0}} |}η if j ≤ w_{r}, \\ 0 if r ≥ 1 + w_{r}. \end{matrix}

Let 𝔻 be as in Proposition 6.4 and let Mat\underline{η}(φ) = ( \begin{matrix}α & 0 \\ ∗ & δ \end{matrix} ) . The filtration is

as in formula 5.10 for some vectors x, y ∈ E^{|τ|}. By Proposition 2.10 in [Dou10] (or by a direct computation),

Fil^{j}(𝔻_{2}) = 𝔻_{2} ∩ Fil^{j}𝔻 = \{ \begin{matrix}𝔻_{2}  if j ≤ 0, \\ E^{|τ_{I_{0,x}} |}η_{2} if 1 ≤ j ≤ w_{0}, \\ ⋯ & ⋯ \\ E^{|τ_{I_{t−1,x}} |}η_{2} if 1 + w_{t−2} ≤ j ≤ w_{t−1}, \\ 0  if j ≥ 1 + w_{t−1}, \end{matrix}

where I_{r,x} = I_{r} ∩ J′_{x} = {i ∈ I_{r} : x_{i} = 0}. Let δ = (δ_{0}, δ_{1}, ..., δ_{f−1}). By Lemma 6.10,

e_{i}Fil^{j}(𝔻_{2}) = \{ \begin{matrix}e_{i}E^{|τ|}η_{2} if j ≤ 0 \\ e_{i}E^{|τ|}f_{J′_{x}}η_{2} if 1 ≤ j ≤ k_{i}, \\ 0 if j ≥ 1 + k_{i}, \end{matrix}

therefore the labeled Hodge-Tate weight of 𝔻_{2} with respect to the embedding τ_{i} is

m_{i} = \{ \begin{matrix}0 if x_{i} ≠ 0 , \\ k_{i} if x_{i} = 0, \end{matrix}

and (𝔻_{2}, φ_{2}) corresponds to the effective crystalline character χ_{c,0} · χ_{e_{f−1}}^{m_{0}} ·

χ_{e_{0}}^{m_{1}} · · · · · χ_{e_{f−2}}^{m_{f−1}}, where c = ( ∏_{i∈I_{0}} δ_{i}) · p^{− ∑_{i∈I_{0}} k_{i}}. The following theorem follows

immediately from Corollary 6.7.

Theorem 6.11. (i)

V ≃ ( \begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix} ) ,

where ψ_{1} = η_{1} · χ_{e_{f−1}}^{m_{0}} · χ_{e_{0}}^{m_{1}} · · · · · χ_{e_{f−2}}^{m_{f−1}} and ψ_{2} = η_{2} · χ_{e_{0}}^{k_{1}−m_{1}} · χ_{e_{1}}^{k_{2}−m_{2}} ·

⋯ χ_{e_{f−2}}^{k_{f−1}−m_{f−1}} · χ_{e_{f−1}}^{k_{0}−m_{0}}, where η_{i} are unramified characters of G_{K_{f}} .

(ii)

(\overline{V}_{|I_{K}} )^{s.s.} = ω_{f,τ̄_{0}}^{α_{1}} ⊕ ω_{f,τ̄_{0}}^{α_{2}},

where α_{1} = − ∑_{i=0}^{f−1} m_{i}p^{i} and α_{2} = ∑_{i=0}^{f−1} (m_{i} − k_{i}) p^{i}.

Notice that for an ordered basis is in Proposition 6.4, (\overline{V}_{|I_{K_{f}}} )^{s.s.} only depends on the filtration with respect to that basis.

6.2 Proof of theorem 1.5

Let {ℓ_{i}, ℓ_{i+f}} = {0, k_{i}} for i = 0, 1, ..., f − 1 and assume that at least one k_{i} is strictly positive. In this section we construct infinite families of crystalline representations of Hodge-Tate type {0, −k_{i}}_{τ_{i}} which contain the irreducible representations V_{ℓ} = Ind_{G_{K_{2f}}}^{G_{K_{f}}} (χ_{e_{0}}^{ℓ_{1}} · χ_{e_{1}}^{ℓ_{2}} · · · · · χ_{e_{2f−2}}^{ℓ_{2f−1}} · χ_{e_{2f−1}}^{ℓ_{0}}) of Proposition 3.11, and have the same mod p reductions with V_{ℓ}. We choose f-tuples of matrices (Π_{1}, Π_{2}, ..., Π_{f}) (with Π_{f} = Π_{0}), where the types of the matrices Π_{i} (recall Definition 5.4) are chosen as follows:

(1) If ℓ_{1} = 0, Π_{1} ∈ {t_{2}, t_{3}};

(2) If ℓ_{1} = k_{1} > 0, Π_{1} ∈ {t_{1}, t_{4}}. For i = 2, 3, ..., f − 1, we choose the type of the matrix Π_{i} as follows:

(1) If ℓ_{i} = 0, then:

• If an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{i−1}) is of even type, Π_{i} ∈ {t_{2}, t_{3}};

• If an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{i−1}) is of even type, Π_{i} ∈ {t_{1}, t_{4}}.

(2) If ℓ_{i} = k_{i} > 0, then:

• If an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{i−1}) is of even type, Π_{i} ∈ {t_{1}, t_{4}};

• If an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{i−1}) is of even type, Π_{i} ∈ {t_{2}, t_{3}}.

Finally, we choose the type of the matrix Π_{0} as follows:

(1) If ℓ_{0} = 0, then:

• If an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f−1}) is of even type, Π_{0} = t_{4};

• If an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f−1}) is of even type, Π_{0} = t_{3}.

(2) If ℓ_{0} = k_{0} > 0, then:

• If an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f−1}) is of even type, Π_{0} = t_{2};

• If an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f−1}) is of even type, Π_{0} = t_{1}.

Notice that from the choice of Π_{0}, an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f}) is of even type. Let i = (i_{1}, i_{2}, ..., i_{0}) ∈ {1, 2, 3, 4}^{f} be the type-vector attached to (Π_{1}, Π_{2}, ..., Π_{f}). For the matrices Π_{i}, we assume that c_{i} = 1 for all i. Let P_{i} = Π_{i}mod π for each i and notice that from the choice of the matrices Π_{i} it follows that (P_{1}, P_{2}, ..., P_{f}) ∉ C_{1} ∪ C_{2}. The type of P_{i} is defined to be the type of Π_{i}. For any a ∈ 𝔪_{E}^{f} we consider the families of crystalline E-representations V_{k}^{i} (a) of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}} constructed in §5.2. We prove the following.

Proposition 6.12. (i) V_{k}^{i}(0) = Ind_{K_{2f}}^{K_{f}} (χ_{e_{0}}^{ℓ_{1}} · χ_{e_{1}}^{ℓ_{2}} · · · · · χ_{e_{2f−2}}^{ℓ_{2f−1}} · χ_{e_{2f−1}}^{ℓ_{0}}) and

V_{k}^{i}(0) is irreducible;

(ii) For any a ∈ 𝔪_{E}^{f}, \overline{V}_{k}^{i} (a) = \overline{V}_{k}^{i}(0);

(iii) For any a ∈ 𝔪_{E}^{f}, (\overline{V}_{k}^{i} (a)_{|I_{K_{f}}} )^{s.s.} = ω_{2f,τ̄_{0}}^{β} ⨁ ω_{2f,τ̄_{0}}^{p^{f} β}, where β =

− ∑_{i=0}^{2f−1} p^{i}ℓ_{i};

(iv) \overline{V}_{k}^{i} (a) is irreducible if and only if 1 + p^{f} ∤ β;

(v) Any irreducible member of the family {V_{k}^{i} (a) , a ∈ 𝔪_{E}^{f}} , other than

V_{k}^{i}(0), is non-induced.

Proof. We restrict V_{k}^{i}(0) to G_{K_{2f}} . By the construction of the representa-

tion V_{k}^{i}(0) in §5.1, there exists some G_{K_{f}} -stable lattice (T_{k}^{i}(0))_{G_{K_{f}}} in-

side V_{k}^{i}(0) whose Wach module has φ-action defined by (φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2}) Π(0), where Π(0) = (Π_{1} (0) , Π_{2} (0) , ..., Π_{f−1} (0) , Π_{0} (0)) . By Proposition 2.6, the Wach module of the G_{K_{2f}} -stable lattice (T_{k}^{i}(0))_{|G_{K_{2f}}} in-

side (V_{k}^{i}(0))_{|G_{K_{2f}}} is defined by (φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2}) Π (0)^{⊗2} , therefore

the filtered φ-module corresponding to (V_{k,0}^{i})_{|G_{K_{2f}}} has Frobenius endomor-

phism (φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2}) P (0)^{⊗2}. By Corollary 2.7 the restricted representation (V_{k}^{i}(0))_{|G_{K_{2f}}} has labeled Hodge-Tate weights ({0, −k_{i}})_{τ_{i}} , i = 0, 1, ..., 2f − 1, with k_{i+f} = k_{i} for i = 0, 1, ..., f − 1, and filtration as in formula 5.10 for some vectors x, y, with the sets I_{j} being defined with respect to the 2f weights above. We prove that (V_{k}^{i}(0))_{|G_{K_{2f}}} is reducible and determine its

irreducible constituents. First, we change the basis to diagonalize the matrix of Frobenius. We see that

P_{i} (0) = \{ \begin{matrix}R (β̄_{i}, γ̄_{i}) , with \{β̄_{i}, γ̄_{i}\} = \{1, p^{k_{i}}\} if P_{i} has type 2 or 4, \\ diag (ᾱ_{i}, δ̄_{i}) , with \{ᾱ_{i}, δ̄_{i}\} = \{1, p^{k_{i}}\} if P_{i} has type 1 or 3, \end{matrix}

where R (β̄_{i}, γ̄_{i}) is the 2 × 2 matrix with β̄_{i} in the (1, 2) entry, γ̄_{i} in the (2, 1) entry, and zero on the diagonal. Let Q_{0} = Id,

Q_{1} = \{ \begin{matrix}Id if P_{1} ∈ \{t_{1}, t_{3}\}, \\ R if P_{1} ∈ \{t_{2}, t_{4}\}, \end{matrix}

where R := R (1, 1) ,

Q_{i} = \{ \begin{matrix}Id if Q_{i−1} = Id and P_{i} ∈ \{t_{1}, t_{3}\}, \\ R if Q_{i−1} = Id and P_{i} ∈ \{t_{2}, t_{4}\}, \\ R if Q_{i−1} = R and P_{i} ∈ \{t_{1}, t_{3}\}, \\ Id if Q_{i−1} = R and P_{i} ∈ \{t_{2}, t_{4}\} \end{matrix}  (6.2)

for i = 2, 3, ..., 2f − 1. Let Q = (Q_{0}, Q_{1}, ..., Q_{2f−1}) . By the definition of the matrices Q_{i}, the matrix Q · P (0)^{⊗2} · φ (Q^{−1}) is diagonal. By induction, Q_{0} = Id and

Q_{i} = \{ \begin{matrix}Id if an even number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type, \\ R if an odd number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type, \end{matrix}  (6.3)

for i = 1, 2, ..., 2f − 1, where P_{i+f} = P_{i} for i = 0, 1, ..., f − 1. We claim that for each i = 0, 1, ..., f − 1, Q_{i} = Id if and only if Q_{i+f} = R. Indeed, for i = 0, Q_{0} = Id. Since an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f}) is of even type, Q_{f} = R. Let q_{ij}^{r} be the r-th coordinate of the (i, j)-entry q_{ij} of Q for each i, j ∈ {1, 2} and r ∈ {0, 1, ..., 2f − 1}. Assume that i ∈ {1, 2, ..., f − 1}. From the definition of the matrices Q_{i} we see that

q_{11}^{i} = \{ \begin{matrix}1 if an even number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type, \\ 0 if an odd number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type. \end{matrix}  (6.4)

For any i = 0, 1, ..., f − 1 we have

q_{11}^{i+f} = \{ \begin{matrix}1 if an even number of coordinates of \\ (P_{1}, P_{2}, ..., P_{f}, ...P_{i+f}) & is of even type, \\ 0 if an odd number of coordinates of \\ (P_{1}, P_{2}, ..., P_{f}, ...P_{i+f}) & is of even type. \end{matrix}  (6.5)

Since an odd number of coordinates of (P_{1}, P_{2}, ..., P_{f}) is of even type and P_{i} = P_{i+f} for all i, this is equivalent to

q_{11}^{i+f} = \{ \begin{matrix}1 if an odd number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type, \\ 0 if an even number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type, \end{matrix}  (6.6)

which implies that q_{11}^{i+f} = 1−q_{11}^{i} for all i = 0, 1, ..., f−1. Similarly q_{ij}^{r+f} = 1−q_{ij}^{r} for all entries ij. Consider the ordered basis \underline{ζ} = (ζ_{1}, ζ_{2}) defined by (ζ_{1}, ζ_{2}) := (η_{1}, η_{2}) Q^{−1}. In the ordered basis \underline{ζ} the filtration is as in formula 5.10 with the vector xη_{1} + yη_{2} replaced by x · (q_{11} · ζ_{1} + q_{12} · ζ_{2}) + y · (q_{12} · ζ_{1} + q_{22} · ζ_{2}) . Let z = x · q_{11} + y · q_{12} and w = x · q_{12} + y · q_{22}. From the definition of the matrices Q_{i}, the matrix of Frobenius in this new basis is the diagonal matrix

diag (λ, μ) :=

(Q_{0} · P_{1} · Q_{1}^{−1}, ..., Q_{f−1} · P_{f} · Q_{f}^{−1}, Q_{f} · P_{f+1} · Q_{f+1}^{−1}, ..., Q_{2f−1} · P_{0} · Q_{0}^{−1}) .

We prove that Nm_{φ}(λ) = Nm_{φ} (μ) = p^{∑_{i=0}^{f−1} k_{i}} · 1. First we see that λ_{i}μ_{i} = p^{k_{i}} for all i. Since Q_{i} = Id if and only if Q_{i+f} = R, a case by case analysis for the choices of Q_{i} and Q_{i+1}, bearing in mind that P_{i+f} = P_{i}, implies that Q_{i} · P_{i+1} · Q_{i+1}^{−1} = diag (λ_{i+1}, μ_{i+1}) if and only if Q_{i+f} · P_{i+f+1} · Q_{i+f+1}^{−1} = diag (μ_{i+1}, λ_{i+1}) . Therefore,

∏_{i=0}^{2f−1} (Q_{i} · P_{i+1} · Q_{i+1}^{−1})

= ∏_{i=0}^{f−1} (Q_{i} · P_{i+1} · Q_{i+1}^{−1}) · ∏_{i=0}^{f−1} (Q_{i+f} · P_{i} · Q_{i+f+1}^{−1})

= ∏_{i=0}^{f−1} diag (λ_{i+1}, μ_{i+1}) · ∏_{i=0}^{f−1} diag (μ_{i+1}, λ_{i+1}) = p^{∑_{i=0}^{f−1} k_{i}} · Id.

Next we notice that y = 1 − x and q_{12} = 1 − q_{11}, so z = x · q_{11} + (1 − x) ·

(1 − q_{11}) = 1 + 2 · x · q_{11} − q_{11} − x. Since x_{i} and q_{11}^{i} ∈ {0, 1} for all i, z_{i} = 0 if and only if q_{11}^{i} = 1 and x_{i} = 0, or q_{11}^{i} = 0 and x_{i} = 1. Recall from formula 5.11 that x_{i} = 0 if and only if P_{i} ∈ {t_{3}, t_{4}} and x_{i} = 1 if and only if P_{i} ∈ {t_{1}, t_{2}}. This combined with the definition of the matrices Q_{i} gives

z_{i} = 0 ⇔ \{ \begin{matrix}i = 0 and P_{0} ∈ \{t_{3}, t_{4}\}, or \\ i ≥ 1, P_{i} ∈ \{t_{3}, t_{4}\} and an even number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type, or \\ i ≥ 1, P_{i} ∈ \{t_{1}, t_{2}\} and an odd number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type. \end{matrix} (6.7)

Similarly,

z_{i} = 1 ⇔ \{ \begin{matrix}i = 0 and P_{0} ∈ \{t_{1}, t_{2}\}, or \\ i ≥ 1, P_{i} ∈ \{t_{1}, t_{2}\} and an even number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type, \\ i ≥ 1, P_{i} ∈ \{t_{3}, t_{4}\} and an odd number of coordinates \\ of & (P_{1}, P_{2}, ..., P_{i}) & is of even type. \end{matrix} (6.8)

We claim that z_{i+f} = 1−z_{i} for all i = 0, 1, ..., f −1. Indeed, z_{i+f} = 1+2·x_{i+f} · q_{11}^{i+f} − q_{11}^{i+f} − x_{i+f}. Since P_{i} = P_{i+f}, we have x_{i} = x_{i+f} for all i, and since q_{11}^{i+f} = 1 − q_{11}^{f} we get z_{i+f} = 1 − z_{i}. Since z_{i} ∈ {0, 1} for all i,

∑_{\begin{matrix}i=0 \\ z_{i}=0 \end{matrix}}^{2f−1} k_{i} = ∑_{\begin{matrix}i=0 \\ z_{i}=0 \end{matrix}}^{f−1} k_{i} + ∑_{\begin{matrix}i=0 \\ z_{i+f} =0 \end{matrix}}^{f−1} k_{i} = ∑_{i=0}^{f−1} k_{i}.

Since v_{p} (Nm_{φ} (μ)) = ∑_{i=0}^{f−1} k_{i} = ∑_{\begin{matrix}i=0 \\ z_{i}=0 \end{matrix}}^{2f−1} k_{i}, Proposition 6.3^{1} implies that the rep-

resentation (V_{k}^{i}(0))_{|G_{K_{2f}}} is reducible. If 𝔻_{2} := (E^{|τ_{K_{2f}} |}) ζ_{2}, by [Dou10, proof

of Prop. 4.3] (or by a direct computation),

Fil^{j}𝔻_{2} = \{ \begin{matrix}𝔻_{2}  if j ≤ 0, \\ (E^{|τ_{K_{2f}} |}f_{I_{i,z}}) ζ_{2} if 1 + w_{i−1} ≤ j ≤ w_{i}, i = 0, 1, ..., t − 1, \\ 0  if j ≥ 1 + w_{t−1}, \end{matrix}

(6.9)

where I_{i,z} = I_{i} ∩ {j ∈ {0, 1, ..., 2f − 1} : z_{j} = 0}. Let i ∈ {0, 1, ..., 2f − 1}.

1F-semisimplicity is not assumed here, but the part of Proposition 6.3 used still holds.

Documenta Mathematica 15 (2010) 873–938

Arguing as in Lemma 6.10 we see that

e_{i}Fil^{j}𝔻_{2} = \{ \begin{matrix}e_{i}E^{|τ_{K_{2f}} |}ζ_{2}  if j ≤ 0, \\ e_{i} ( ∑_{\begin{matrix}r=0 \\ z_{r}=0 \end{matrix}}^{2f−1} e_{i}) E^{|τ_{K_{2f}} |}ζ_{2} if 1 ≤ j ≤ k_{i}, \\ 0  if j ≥ 1 + k_{i}. \end{matrix}

Hence

e_{i}Fil^{j}𝔻_{2} = \{ \begin{matrix}e_{i}E^{|τ_{K_{2f}} |}ζ_{2} if j ≤ 0, \\ 0 if j ≥ 1 \end{matrix}

if z_{i} = 1 and

e_{i}Fil^{j}𝔻_{2} = \{ \begin{matrix}e_{i}E^{|τ_{K_{2f}} |}ζ_{2} if j ≤ k_{i}, \\ 0 if j ≥ 1 + k_{i} \end{matrix}

if z_{i} = 0. The labeled Hodge-Tate weight of 𝔻_{2} with respect to the embedding τ_{i} of K_{2f} into E is 0 if z_{i} = 1 and −k_{i} if z_{i} = 0. Next we prove that

z_{i} = \{ \begin{matrix}0 if ℓ_{i} = 0, \\ 1 if ℓ_{i} = k_{i} > 0 \end{matrix}

for i = 0, 1, ..., f − 1, and

z_{i+f} = \{ \begin{matrix}1 if ℓ_{i} = 0, \\ 0 if ℓ_{i} = k_{i} > 0. \end{matrix}

Since z_{i+f} = 1 − z_{i} for all i = 0, 1, ..., f − 1, it suffices to prove the first formula. Suppose that ℓ_{1} = 0. Then P_{1} ∈ {t_{2}, t_{3}} and by formula 6.7, z_{1} = 0. If ℓ_{1} = k_{1} > 0, then P_{1} ∈ {t_{1}, t_{4}} and formula 6.7 implies that z_{1} = 1. Let i ∈ {1, 2, ..., f − 2} and assume that ℓ_{i} = 0. If an even number of coordinates of (P_{1}, P_{2}, ..., P_{i−1}) is of even type, then P_{i} ∈ {t_{2}, t_{3}} and formula 6.7 implies z_{i} = 0. Arguing similarly we see that if ℓ_{i} = k_{i} > 0, formula 6.8 implies z_{i} = 1. Finally, assume that i = f and ℓ_{f} = 0. If an even number of coordinates of (P_{1}, P_{2}, ..., P_{f−1}) is of even type, then P_{0} = P_{f} = t_{4} and formula 6.7 implies that z_{0} = z_{f} = 0. We finish the proof by verifying similarly the remaining cases. By the formulas above, the labeled Hodge-Tate weight of 𝔻_{2} with respect to the embedding τ_{i} is

\{ \begin{matrix}−k_{i} if ℓ_{i} = 0, \\ 0 if ℓ_{i} = k_{i} > 0 \end{matrix}

for i = 0, 1, ..., f − 1 and

\{ \begin{matrix}−k_{i} if ℓ_{i} = k_{i} > 0, \\ 0 if ℓ_{i} = 0 \end{matrix}

for i = f, f + 1, ..., 2f − 1. Therefore the labeled Hodge-Tate weight of 𝔻_{2} with respect to the embedding τ_{i} is

\{ \begin{matrix}− (k_{i} − ℓ_{i}) if i = 0, 1, ..., f − 1 \\ −ℓ_{i}  if i = f, f + 1, ..., 2f − 1. \end{matrix}

Since {ℓ_{i}, ℓ_{i+f}} = {0, k_{i}} for all i = 0, 1, ..., f − 1, the labeled Hodge-Tate weights of 𝔻_{2} are (−ℓ_{0}, −ℓ_{1}, ..., −ℓ_{f−1}, −ℓ_{f}, −ℓ_{f+1}, ..., −ℓ_{2f−1}) . Since

Nm_{φ} (μ) = p^{∑_{i=0}^{f−1} k_{i}} · 1, Proposition 3.5 implies that the crystalline character corresponding to 𝔻_{2} is χ_{e_{0}}^{ℓ_{1}} · χ_{e_{1}}^{ℓ_{2}} · · · · · χ_{e_{2f−2}}^{ℓ_{2f−1}} · χ_{e_{2f−1}}^{ℓ_{0}}. Suppose that V_{k}^{i}(0) is re-

ducible. Then there exists some irreducible constituent of V_{k}^{i}(0) whose restric-

tion to G_{K_{2f}} is χ_{e_{0}}^{ℓ_{1}} ·χ_{e_{1}}^{ℓ_{2}} ·· · ··χ_{e_{2f−2}}^{ℓ_{2f−1}} ·χ_{e_{2f−1}}^{ℓ_{0}}. Since the labeled weights of the latter character are (−ℓ_{0}, −ℓ_{1}, ..., −ℓ_{f−1}, −ℓ_{f}, ..., −ℓ_{2f−1}) , Corollary 2.7 implies that ℓ_{i} = ℓ_{i+f} for all i = 0, 1, ..., f −1. Since {ℓ_{i}, ℓ_{i+f}} = {0, k_{i}} for i = 0, 1, ..., f −1, and since some labeled weight is strictly positive this is absurd. Hence V_{k}^{i}(0)

is irreducible and its restriction to G_{K_{2f}} contains χ_{e_{0}}^{ℓ_{1}} · χ_{e_{1}}^{ℓ_{2}} · · · · · χ_{e_{2f−2}}^{ℓ_{2f−1}} · χ_{e_{2f−1}}^{ℓ_{0}} as an irreducible constituent. By Frobenius reciprocity,

V_{k}^{i}(0) = Ind_{K_{2f}}^{K_{f}} (χ_{e_{0}}^{ℓ_{1}} · χ_{e_{1}}^{ℓ_{2}} · · · · · χ_{e_{2f−2}}^{ℓ_{2f−1}} · χ_{e_{2f−1}}^{ℓ_{0}}) .

This finishes the proof of part (i). Part (ii) follows from Theorem 4.7 and parts (iii) and (4) follow from Corollary 6.9. For part (iv), notice that any irreducible induced member of V_{k}^{i} (a) has the form η_{c} ·

Ind_{K_{2f}}^{K_{f}} (χ_{e_{0}}^{ℓ′_{1}} · χ_{e_{1}}^{ℓ′_{2}} · · · · · χ_{e_{2f−2}}^{ℓ′_{2f−1}} · χ_{e_{2f−1}}^{ℓ′_{0}}) for some unramified character η_{c} and some nonnegative integers with {ℓ′_{i}, ℓ′_{i+f}} = {0, k_{i}} for all i (see Proposi-

tion 3.11). All the members of V_{k}^{i} (a) have determinant (−1)^{t} p^{∑_{i=0}^{f−1} k_{i}}, where t is the number of even coordinates of i. This equals the determinant of Ind_{K_{2f}}^{K_{f}} (χ_{e_{0}}^{ℓ′_{1}} · χ_{e_{1}}^{ℓ′_{2}} · · · · · χ_{e_{2f−2}}^{ℓ′_{2f−1}} · χ_{e_{2f−1}}^{ℓ′_{0}}) and forces the unramified character η_{c} to be trivial. Hence the only irreducible induced member of our family is V_{k}^{i}(0).

Remark 6.13. Let R be as in the proof of Proposition 6.12. If A is a set of 2×2 matrices, let RA := {R·A : A ∈ A} and AR := {A·R : A ∈ A}. We write {t_{i}, t_{j}} for a set which contains matrices of type t_{i} and t_{j}. Then R{t_{1}, t_{2}} = {t_{1}, t_{2}}, R{t_{3}, t_{4}} = {t_{3}, t_{4}}, {t_{1}, t_{2}}R = {t_{3}, t_{4}} and {t_{3}, t_{4}}R = {t_{1}, t_{2}}. In the definition of the matrices P_{i} we may always assume that P_{i} ∈ {t_{1}, t_{2}} for all i = 1, 2, ...f − 1. Indeed, let Q_{0} = R, and for i = 1, 2, ..., f − 1 let

Q_{i} = \{ \begin{matrix}Id if P_{i} ∈ \{t_{1}, t_{2}\}, \\ R if P_{i} ∈ \{t_{3}, t_{4}\}. \end{matrix}

After changing the basis by the matrix Q = (Q_{0}, Q_{1}, ..., Q_{f−1}) we have P_{i} ∈ {t_{1}, t_{2}} for all i = 1, 2, ...f − 1. By the definition preceding Proposition 6.12, the type of the matrix P_{0} ∈ {t_{1}, t_{2}, t_{3}, t_{4}} is uniquely determined by (P_{1}, P_{2}, ..., P_{f−1}) .

Theorem 6.14. Theorem 1.5 holds.

Proof. Follows from Proposition 6.12 and Remark 6.13.

Example 6.15. Let f = 2 and k_{i} > 0 for i = 0, 1. Up to twist by some unramified character, there exist two distinct isomorphism classes of irreducible two-dimensional crystalline E-representations of G_{K_{2}} with labeled Hodge-Tate weights ({0, −k_{0}}, {0, −k_{1}}) induced from crystalline characters of G_{K_{4}}.

(i) If ℓ_{0} = k_{0} and ℓ_{1} = k_{1}, then from the definition of the matrices Π_{i} preceding Proposition 6.12 and Remark 6.13, (Π_{1}, Π_{0}) = (t_{1}, t_{2}) . Let P_{i} = Π_{i}mod π. The polynomials z_{i} in the definition of the matrices Π_{i} are such that z_{i} ≡ p^{m}mod π, where m := ⌊\frac{max\{k_{0},k_{1}\}−1}{p−1}⌋, unless k_{0} = k_{1} = p in which case we define m = 0. For any a = (a_{0}, a_{1}) ∈ 𝔪_{E}^{2} we consider the family of crystalline representations V_{k,a}^{(1,2)} constructed in §5.2. The corresponding family of filtered φ-modules is

(𝔻_{k,a}^{(1,2)}, φ) , with (φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2}) P^{(1,2)}(a), where

P^{(1,2)}(a) = ( \begin{matrix}(p^{k_{1}}, a_{0}p^{m}) & (0, 1) \\ (a_{1}p^{m}, p^{k_{0}}) & (1, 0) \end{matrix} ) ,

and the filtrations are

Fil^{j}(𝔻_{k,a}^{(1,2)}) = \{ \begin{matrix}E^{2}η_{1} ⨁ E^{2}η_{2}  if j ≤ 0, \\ E^{2} (x · η_{1} + y · η_{2}) if 1 ≤ j ≤ w_{0}, \\ E^{1}f_{I_{1}} (x · η_{1} + y · η_{2}) if 1 + w_{0} ≤ j ≤ w_{1}, \\ 0  if j ≥ 1 + w_{1}, \end{matrix}

(6.10)

with w_{0} = min{k_{0}, k_{1}} and w_{1} = max{k_{0}, k_{1}},

f_{I_{1}} = \{ \begin{matrix}(0, 1) if k_{0} < k_{1}, \\ (1, 0) if k_{1} < k_{0}, \\ (0, 0) if k_{0} = k_{1}, \end{matrix}

and (x, y) = ((1, 1) , (0, 0)) . We have

V_{k,0}^{(1,2)} ≃ Ind_{K_{4}}^{K_{2}} (χ_{e_{0}}^{k_{1}} · χ_{e_{3}}^{k_{0}}) ,

and for any a ∈ 𝔪_{E}^{2},

((\overline{V} _{k,a}^{(1,2)})_{|I_{K_{2}}} )^{s.s.} ≃ ω_{4,τ̄_{0}}^{−(k_{0}+pk_{1})} ⨁ ω_{4,τ̄_{0}}^{−(k_{0}+pk_{1})p^{2}}.

Let α_{i} = a_{i}p^{m} and A = α_{1}+p^{k_{1}}α_{0}. Assume that A^{2} ≠ −4p^{k_{0}+k_{1}} and let ε_{0} ≠ ε_{1} be the distinct roots of the characteristic polynomial X^{2}−A·X+p^{k_{0}+k_{1}}. Arguing as in the proof of Proposition 2.2 in [Dou10], we get the following “standard parametrization” for the family V_{k,a}^{(1,2)},

φ (η_{1}) = (1, ε_{0}) η_{1}, φ (η_{2}) = (λ, \frac{ε_{1}}{λ} ) η_{2},

where

λ = λ (α_{0}) = \frac{ε_{1}}{ε_{0}} · \frac{(ε_{1} − A + p^{k_{1}}α_{0})}{(ε_{0} − A + p^{k_{1}}α_{0})}

(notice that ε_{i} ≠ A − α_{0}p^{k_{1}}), and filtrations are as in Formula 6.10 with x = y = 1. (ii) If ℓ_{0} = ℓ_{1} = 0. Again, taking into account Remark 6.13, we may only consider the case (Π_{1}, Π_{0}) = (t_{2}, t_{3}) . For each a ∈ 𝔪_{E}^{2} consider the family V_{k,a}^{(2,3)} of two-dimensional crystalline E-representations of G_{K_{2}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, i = 0, 1. We have

V_{k,0}^{(2,3)} ≃ Ind_{K_{4}}^{K_{2}} (χ_{e_{2}}^{k_{1}} · χ_{e_{1}}^{k_{0}}) ≃ Ind_{K_{4}}^{K_{2}} (χ_{e_{0}}^{k_{1}} · χ_{e_{3}}^{k_{0}}) .

For any a ∈ 𝔪_{E}^{2},

((\overline{V} _{k,a}^{(2,3)})_{|I_{K_{2}}} )^{s.s.} ≃ ω_{4,τ_{0}}^{−(k_{0}+pk_{1})} ⨁ ω_{4,τ_{0}}^{−(k_{0}+pk_{1})p^{2}}.

Notice that the family {V_{k,a}^{(1,2)}, a ∈ 𝔪_{E}^{2}} of case (i) coincides with the fam-

ily {V_{k,a}^{(2,3)}, a ∈ 𝔪_{E}^{2}} , as the second family is obtained from the first one by changing the basis by the matrix Q = (R, R) . (iii) Let f = 2, ℓ_{0} = 0 and ℓ_{1} = k_{1} > 0. Then (Π_{1}, Π_{0}) = (t_{1}, t_{4}) . For each a ∈ 𝔪_{E}^{2} consider the family V_{k,a}^{(1,4)} of two-dimensional crystalline E-representations of G_{K_{2}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, i = 0, 1. Then

V_{k,0}^{(1,4)} ≃ Ind_{K_{4}}^{K_{2}} (χ_{e_{0}}^{k_{1}} · χ_{e_{1}}^{k_{0}}) ,

and for any a ∈ 𝔪_{E}^{2},

((\overline{V} _{k,a}^{(1,4)})_{|I_{K_{2}}} )^{s.s.} ≃ ω_{4,τ̄_{0}}^{−(pk_{1}+p^{2}k_{0})} ⨁ ω_{4,τ̄_{0}}^{−(pk_{1}+p^{2}k_{0})p^{2}}.

Let α_{i} = a_{i}p^{m} and A = α_{0}+p^{k_{0}}α_{1}. Assume that A^{2} ≠ −4p^{k_{0}+k_{1}} and let ε_{0} ≠ ε_{1} be the distinct roots of the characteristic polynomial X^{2}−A·X+p^{k_{0}+k_{1}}. Arguing as in the proof of Proposition 2.2 in [Dou10], we get the following “standard parametrization” for the family V_{k,a}^{(1,4)},

φ (η_{1}) = (1, ε_{0}) η_{1}, φ (η_{2}) = (λ, \frac{ε_{1}}{λ} ) η_{2},

where

λ = λ (α_{1}) = (\frac{ε_{1}}{ε_{0}} )^{2} · \frac{(A − p^{k_{0}}α_{1} − ε_{0})}{(A − p^{k_{0}}α_{1} − ε_{1})}

(notice that ε_{i} ≠ A−α_{1}p^{k_{0}}), and filtrations as in Formula 6.10 with x = y = 1. (iv) If f = 2 ℓ_{0} = k_{0} > 0 and ℓ_{1} = 0. Then (Π_{1}, Π_{0}) = (t_{2}, t_{1}) and this gives the family obtained in case (iii).

Example 6.16. If f = 2, k_{0} > 0 and k_{1} = 0. Then up to unramified twist, Ind_{K_{4}}^{K_{2}} (χ_{e_{3}}^{k_{0}}) is a unique isomorphism class of two-dimensional crystalline irreducible E-representations of G_{K_{2}} with labeled weights ({0, −k_{0}}, {0, 0}) induced from G_{K_{4}}. We have

V_{k,0}^{(2,3)} ≃ Ind_{K_{4}}^{K_{2}} (χ_{e_{3}}^{k_{0}}) ≃ Ind_{K_{4}}^{K_{2}} (χ_{e_{1}}^{k_{0}}) ,

and for any a ∈ 𝔪_{E}^{2},

((\overline{V} _{k,a}^{(2,3)})_{|I_{K_{2}}} )^{s.s.} ≃ ω_{4,τ̄_{0}}^{−k_{0}} ⨁ ω_{4,τ̄_{0}}^{−p^{2}k_{0}}.

Example 6.17. Let f = 3, k_{i} > 0 for all i = 0, 1, 2. Up to twist by some unramified character, there exist 4 distinct isomorphism classes of irreducible two-dimensional crystalline E-representations of G_{K_{3}} with labeled Hodge-Tate weights ({0, −k_{0}}, {0, −k_{1}}, {0, −k_{2}}) induced from G_{K_{6}}. One of those classes is represented by Ind_{K_{6}}^{K_{3}} (χ_{e_{0}}^{k_{1}} · χ_{e_{1}}^{k_{2}} · χ_{e_{2}}^{k_{0}}) . For the families containing it we have ℓ_{i} = k_{i} > 0 for all i = 0, 1, 2. Since k_{0} > 0, Π_{0} = t_{2} if Π_{2} = t_{4} and Π_{0} = t_{1} if Π_{2} = t_{1}. Hence (Π_{1}, Π_{2}, Π_{0}) ∈ {(t_{4}, t_{4}, t_{2}) , (t_{4}, t_{1}, t_{1}) , (t_{1}, t_{2}, t_{1}) , (t_{1}, t_{3}, t_{2})}. By Remark 6.13 we may only consider the case (Π_{1}, Π_{2}, Π_{0}) = (t_{1}, t_{2}, t_{1}) . For any a ∈ 𝔪_{E}^{3}, consider the the families V_{k,a}^{(1,2,1)} of two-dimensional crystalline Erepresentations of G_{K_{3}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}}, i = 0, 1, 2. We have

V_{k,0}^{(1,2,1)} ≃ Ind_{K_{6}}^{K_{3}} (χ_{e_{0}}^{k_{1}} · χ_{e_{1}}^{k_{2}} · χ_{e_{2}}^{k_{0}}) ,

and for any a ∈ 𝔪_{E}^{3},

((\overline{V} _{k,a}^{(1,2,1)})_{|I_{K_{3}}} )^{s.s.} ≃ ω_{6,τ̄_{0}}^{−(k_{0}+pk_{1}+p^{2}k_{2})} ⨁ ω_{6,τ̄_{0}}^{−(k_{0}+pk_{1}+p^{2}k_{2})p^{3}}.

6.3 Proof of theorem 1.7

Let V_{ℓ,ℓ′} (η) = η ·χ_{e_{0}}^{ℓ_{1}} ·χ_{e_{1}}^{ℓ_{2}} ·· · ··χ_{e_{f−1}}^{ℓ_{0}} ⨁ χ_{e_{0}}^{ℓ′_{1}} ·χ_{e_{1}}^{ℓ′_{2}} ·· · ··χ_{e_{f−1}}^{ℓ′_{0}} with {ℓ_{i}, ℓ′_{i}} = {0, k_{i}} for all i, where η = η_{c} is the unramified character of G_{K_{f}} which maps the geometric Frobenius Frob_{K_{f}} of G_{K_{f}} to c ∈ O_{E}^{×}. As usual, we assume that at least one k_{i} is strictly positive. We choose f-tuples of matrices (Π_{1}, Π_{2}, ..., Π_{f}) (with Π_{f} = Π_{0}) as follows:

(1) If ℓ_{1} = 0, Π_{1} ∈ {t_{2}, t_{3}};

(2) If ℓ_{1} = k_{1} > 0, Π_{1} ∈ {t_{1}, t_{4}}. For i = 2, 3, ..., f − 1, we choose the type of the matrix Π_{i} as follows:

(1) If ℓ_{i} = 0, then:

• If an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{i−1}) is of even type, Π_{i} ∈ {t_{2}, t_{3}};

• If an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{i−1}) is of even type, Π_{i} ∈ {t_{1}, t_{4}}.

(2) If ℓ_{i} = k_{i} > 0, then:

• If an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{i−1}) is of even type, Π_{i} ∈ {t_{1}, t_{4}};

• If an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{i−1}) is of even type, Π_{i} ∈ {t_{2}, t_{3}}.

Finally, we choose the type of the matrix Π_{0} as follows:

(1) If ℓ_{0} = 0, then:

• If an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f−1}) is of even type, Π_{0} = t_{3};

• If an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f−1}) is of even type, Π_{0} = t_{4}.

(2) If ℓ_{0} = k_{i} > 0, then:

• If an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f−1}) is of even type, Π_{0} = t_{1};

• If an odd number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f−1}) is of even type, Π_{0} = t_{2}.

Notice that the type of Π_{0} has been chosen so that an even number of coordinates of (Π_{1}, Π_{2}, ..., Π_{f}) is of even type. We choose the units c_{i} appearing in the entries of the matrices Π_{i} in Definition 5.4 so that c_{i} = 1 for all i = 1, 2, ..., f −1, and c_{0} = c. Let i be the type-vector attached to (Π_{1}, Π_{2}, ..., Π_{f}) . We exclude those vectors i for which (Π_{1}, Π_{2}, ..., Π_{f}) ∈ C_{1} ∪ C_{2}, which is to exclude the cases where ℓ = 0 or ℓ′ = 0. For any a ∈ 𝔪_{E}^{f} we consider the families of crystalline E-representations V_{k}^{i} (a) of G_{K_{f}} with labeled Hodge-Tate weights {0, −k_{i}}_{τ_{i}} constructed in §5.2.

Proposition 6.18. (i) For any type vector i chosen as above there exists some unramified character μ such that μ ⊗ V_{k}^{i}(0) ≃ V_{ℓ,ℓ′}(η);

(ii) For any a ∈ 𝔪_{E}^{f}, \overline{V}_{k}^{i} (a) ≃ \overline{V}_{k}^{i}(0) and

(\overline{V}_{k}^{i} (a))_{|I_{K_{f}}} ≃ (\overline{V}_{ℓ,ℓ′} (η))_{|I_{K_{f}}} ≃ ω_{f,τ̄_{0}}^{α} ⊕ ω_{f,τ̄_{0}}^{α′},

where α = − ∑_{i=i}^{f−1} ℓ_{i}p^{i} and α′ = − ∑_{i=0}^{f−1} ℓ′_{i}p^{i}.

Proof. For simplicity assume that η = 1. The general case follows by choosing the unit c_{0} in the definition of Π_{0} appropriately. We restrict V_{k}^{i}(0) to

G_{K_{2f}} . By the construction of the representation V_{k}^{i}(0) in §5.1, there ex-

ists some G_{K_{f}} -stable lattice (T_{k}^{i}(0))_{G_{K_{f}}} inside V_{k}^{i}(0) whose Wach mod-

ule has φ-action defined by (φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2}) · Π(0). By Proposition 2.6, the Wach module of the G_{K_{2f}} -stable lattice (T_{k}^{i}(0))_{|G_{K_{2f}}} inside

(V_{k}^{i}(0))_{|G_{K_{2f}}} is defined by (φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2}) · Π (0)^{⊗2} , therefore the

filtered φ-module corresponding to (V_{k}^{i}(0))_{|G_{K_{2f}}} has Frobenius endomorphism

(φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2})·P (0)^{⊗2} . The restricted representation (V_{k}^{i}(0))_{|G_{K_{2f}}}

has labeled weights ({0, −k_{i}})_{τ_{i}} , i = 0, 1, ..., 2f − 1, with k_{i+f} = k_{i} for i = 0, 1, ..., f − 1, and filtration as in formula 5.10 for some vectors x, y, with the sets I_{j} being defined with respect to the 2f weights above. We prove that (V_{k}^{i}(0))_{|G_{K_{2f}}} is reducible and determine its irreducible constituents. First we

change the basis to diagonalize the matrix of Frobenius. We define matrices Q_{i} as in the proof of Proposition 6.12, and we let Q = (Q_{0}, Q_{1}, ..., Q_{2f−1}) . By the definition of the matrices Q_{i}, the matrix Q · P (0)^{⊗2} · φ (Q^{−1}) is diagonal. By the proof of Proposition 6.12, Q_{0} = Id and for i = 1, 2, ..., 2f − 1, Q_{i} is as in formula 6.3. We claim that for each i = 0, 1, ..., f − 1, Q_{i} = Q_{i+f}. Indeed, from the definition of the matrices Q_{i} we see that q_{11}^{i} and q_{11}^{i+f} are as in formulas 6.4 and 6.5 respectively in the proof of Proposition 6.12. Since an even number of coordinates of (P_{1}, P_{2}, ..., P_{f}) are of even type, q_{11}^{i+f} = q_{11}^{i}. Similarly, q_{ij}^{r+f} = q_{ij}^{r} for any entry (i, j) . Consider the ordered basis \underline{ζ} = (ζ_{1}, ζ_{2}) defined by (ζ_{1}, ζ_{2}) := (η_{1}, η_{2}) · Q^{−1}. Let q_{ij} be th (i, j)-entry of Q. In the new basis \underline{ζ} the filtration is as in formula 5.10 with the vector xη_{1} + yη_{2} replaced by x · (q_{11} · ζ_{1} + q_{12} · ζ_{2}) + y · (q_{12} · ζ_{1} + q_{22} · ζ_{2}) . Let z = x · q_{11} + y · q_{12} and w = x·q_{12}+y·q_{22}. The matrix of Frobenius in this new basis is the diagonal matrix diag (λ, μ) . Arguing as in Proposition 6.12, and taking into account that

q_{ij}^{r+f} = q_{ij}^{r} for all r = 0, 1, ..., f − 1 and all entries (i, j) we see that z_{r+f} = z_{r} for all r. From the proof of the same proposition, z_{i} = 0 if and only if q_{11}^{i} = 1 and x_{i} = 0 or q_{11}^{i} = 0 and x_{i} = 1. Formula 5.11 implies that x_{i} = 0 if and only if P_{i} ∈ {t_{4}, t_{3}} and x_{i} = 1 if and only if P_{i} ∈ {t_{2}, t_{1}}. Since z_{i} = z_{i+f} and k_{i} = k_{i+f} for all i = 0, 1, ..., f − 1,

∑_{\begin{matrix}i=0 \\ z_{i}=0 \end{matrix}}^{2f−1} k_{i} = 2 ∑_{\begin{matrix}i=0 \\ z_{i}=0 \end{matrix}}^{f−1} k_{i} = 2 ∑_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i}=t_{1} \end{matrix}}^{f−1} k_{i} + 2 ∑_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i}=t_{2} \end{matrix}}^{f−1} k_{i} + 2 ∑_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i}=t_{3} \end{matrix}}^{f−1} k_{i} + 2 ∑_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i}=t_{4} \end{matrix}}^{f−1} k_{i}.

We now show that the (2, 2) entry of ∏_{i=0}^{2f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) is the p^{n}, where

n = 2 ∑_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i}=t_{1} \end{matrix}}^{f−1} k_{i} + 2 ∑_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i}=t_{2} \end{matrix}}^{f−1} k_{i} + 2 ∑_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i}=t_{3} \end{matrix}}^{f−1} k_{i} + 2 ∑_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i}=t_{4} \end{matrix}}^{f−1} k_{i}.  (6.11)

Since the matrices Q_{i}P_{i+1}Q_{i+1}^{−1} are diagonal, and since Q_{i+f} = Q_{i} and P_{i+f} = P_{i} for all i,

∏_{i=0}^{2f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) = ∏_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i+1}=t_{4} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})^{2}  · ∏_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i+1}=t_{3} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})^{2} ·

∏_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i+1}=t_{1} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})^{2} · ∏_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i+1}=t_{2} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})^{2}  · ∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i+1}=t_{4} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})^{2} ·

∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i+1}=t_{3} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})^{2} · ∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i+1}=t_{1} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})^{2}  · ∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i+1}=t_{2} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})^{2} .

We notice that when Q_{i} = Id and P_{i+1} = t_{4}, then by formula 6.2, Q_{i+1} = R

and Q_{i}P_{i+1}Q_{i+1}^{−1} = diag (p^{k_{i+1}}, 1) . Therefore the product ∏_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i+1}=t_{4} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})

has no contribution to the (2, 2) entry of ∏_{i=0}^{2f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) . Similarly, the

products

∏_{\begin{matrix}i=0 \\ Q_{i}=Id \\ P_{i+1}=t_{1} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) , ∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i+1}=t_{3} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) and ∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i+1}=t_{2} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1})

have no contribution to the (2, 2) entry of ∏_{i=0}^{2f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) . We now compute

the product ∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i+1}=t_{1} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) . Formula 6.2 implies that if Q_{i} = R and

P_{i+1} = t_{1} then Q_{i+1} = R, therefore Q_{i}P_{i+1}Q_{i+1}^{−1} = diag (1, p^{k_{i+1}}) . Again, by formula 6.2, Q_{i} = R and P_{i+1} = t_{1} is equivalent to Q_{i+1} = R and P_{i+1} = t_{1}. Hence

∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i+1}=t_{1} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) = ∏_{\begin{matrix}i=0 \\ Q_{i+1}=R \\ P_{i+1}=t_{1} \end{matrix}}^{f−1} (Q_{i}P_{i+1}Q_{i+1}^{−1}) = ∏_{\begin{matrix}i=0 \\ Q_{i}=R \\ P_{i}=t_{1} \end{matrix}}^{f−1} diag (1, p^{k_{i+1}})

which contributes the fourth summand of the right hand side of equation 6.11. The claim made before formula 6.11 follows arguing similarly for the remaining

cases. Hence v_{p} (Nm_{φ} (μ)) = ∑_{\begin{matrix}i=0 \\ z_{i}=0 \end{matrix}}^{2f−1} k_{i}. Proposition 6.3 implies that (V_{k,0}^{i})_{|G_{K_{2f}}}

is reducible and (𝔻_{2}, φ) is a weakly admissible submodule, where 𝔻_{2} = (E^{2f}) · ζ_{2}. By [Dou10, proof of Prop. 4.3] (or by a direct computation),

Fil^{j}𝔻_{2} = \{ \begin{matrix}𝔻_{2}  if j ≤ 0, \\ (E^{|τ_{K_{2f}} |}) f_{I_{i,z}}ζ_{2} if 1 + w_{i−1} ≤ j ≤ w_{i} for all i = 0, 1, ..., t − 1, \\ 0  if j ≥ 1 + w_{t−1}, \end{matrix}

(6.12)

where I_{i,z} = I_{i} ∩ {j ∈ {0, 1, ..., 2f − 1} : z_{j} = 0}. As in the proof of Proposition 6.12, the labeled weight for the embedding τ_{i} is 0 if z_{i} = 1 and −k_{i} if z_{i} = 0. Next, we prove that for i = 0, 1, ..., f − 1,

z_{i} = z_{i+f} = \{ \begin{matrix}0 if ℓ_{i} = 0, \\ 1 if ℓ_{i} = k_{i} > 0. \end{matrix}  (6.13)

This is proved exactly as in Proposition 6.12, taking into account that an even number of the coordinates of (P_{1}, P_{2}, ..., P_{f}) is of even type. We have z_{i} = 0 for all i if and only if ℓ_{i} = 0 for all i if and only if (Π_{1}, Π_{2}, ..., Π_{f}) ∈ C_{1} and z_{i} = 1 for all i if and only if ℓ_{i} = k_{i} > 0 for all i if and only if (Π_{1}, Π_{2}, ..., Π_{f}) ∈ C_{2}, cases excluded. Therefore neither of the summands of V_{k}^{i}(0) is unramified. By the discussion above the labeled weights of 𝔻_{2} are

(−ℓ′_{0}, −ℓ′_{1}, ..., −ℓ′_{f−1}, −ℓ′_{0}, −ℓ′_{1}, ..., −ℓ′_{f−1}) . By formula 6.13, v_{p} (Nm_{φ} (μ)) =

∑_{\begin{matrix}i=0 \\ z_{i}=0 \end{matrix}}^{2f−1} k_{i} = ∑_{i=0}^{2f−1} ℓ′_{i}. By Proposition 3.5 and Lemma 3.7, the corresponding crys-

talline character is ψ = χ_{e_{0}}^{ℓ′_{1}} · χ_{e_{1}}^{ℓ′_{2}} · · · · · χ_{e_{f−2}}^{ℓ′_{f−1}} · χ_{e_{f−1}}^{ℓ′_{0}} · χ_{e_{0}}^{ℓ′_{1}} · χ_{e_{1}}^{ℓ′_{2}} · · · · · χ_{e_{f−2}}^{ℓ′_{f−1}} · χ_{e_{f−1}}^{ℓ′_{0}}. If V_{k}^{i}(0) is irreducible, then by Frobenius reciprocity V_{k}^{i}(0) = Ind_{K_{2f}}^{K_{f}} (ψ) , which

is absurd by Corollary 3.10. Hence V_{k}^{i}(0) is reducible and contains an irreducible constituent which restricts to ψ. By Lemma 3.7(iv) the only choices are η_{±1} · χ_{e_{0}}^{ℓ_{1}} · χ_{e_{1}}^{ℓ_{2}} · · · · · χ_{e_{f−2}}^{ℓ_{f−1}} · χ_{e_{f−1}}^{ℓ_{0}}, and we are done after twisting by η_{∓1}. The rest of the proposition follows as in the proof of Proposition 6.12.

Theorem 6.19. Theorem 1.7 holds.

Proof. Follows from Proposition 6.18, taking into account Remark 6.13.

Example 6.20. Let f = 2, ℓ_{0} = 0 and ℓ_{1} = k_{1.} Let (Π_{1}, Π_{0}) = (t_{1}, t_{3}) with c_{0} = c_{1} = 1. After possibly twisting by η_{±1} we have V_{k}^{(1,3)}(0) ≃ χ_{e_{0}}^{k_{1}} ⊕ χ_{e_{1}}^{k_{0}}.

In the next proposition we study closer the F-semisimple members of this family assuming that c = 1.

Proposition 6.21. Assume that V_{k}^{(1,3)} (α) is F-semisimple.

(i) V_{k}^{(1,3)} ( α) is irreducible if and only if α_{0}α_{1} ≠ 0, and is non-induced for all but finitely many such α;

(ii) V_{k}^{(1,3)} (α) is non-split reducible if and only if precisely one of the coordinates α_{i} of α is zero;

(iii) The families {V_{k}^{(1,3)} ((α_{0}, 0)) , α_{0} ∈ p^{m}𝔪_{E} \ {0}} and

{V_{k}^{(1,3)} ((0, α_{1})) , α_{1} ∈ p^{m}𝔪_{E} \ {0}} are disjoint;

(iv) V_{k}^{(1,3)}(0) is split-reducible.

Proof. The weakly admissible filtered φ-module corresponding to V_{k}^{(1,3)} (α) has Frobenius endomorphism

(φ (η_{1}) , φ (η_{2})) = (η_{1}, η_{2}) ( \begin{matrix}(p^{k_{1}}, 1) & (0, α_{0}) \\ (α_{1}, 0) & (1, p^{k_{0}}) \end{matrix} )

and filtration

Fil^{j} (𝔻) = \{ \begin{matrix}(E × E)η_{1} ⨁(E × E)η_{2} if j ≤ 0, \\ (E × E)f_{I_{0}}(xη_{1} + yη_{2}) if 1 ≤ j ≤ w_{0}, \\ (E × E)f_{I_{1}}(xη_{1} + yη_{2}) if 1 + w_{0} ≤ j ≤ w_{1}, \\ 0  if j ≥ 1 + w_{1}, \end{matrix}  (6.14)

with x = (−α_{0}, 1) and y = (1, −α_{1}) . We diagonalize the matrix of Frobenius arguing as in the proof of Proposition 2.2 in [Dou10]. The characteristic polynomial is X^{2} − (p^{k_{0}} + p^{k_{1}} + α_{0}α_{1}) X + p^{k_{0}+k_{1}}, and we assume that (p^{k_{0}} + p^{k_{1}} + α_{0}α_{1})^{2} ≠ 4p^{k_{0}+k_{1}} so that its roots ε_{0} and ε_{1} are distinct. We have the following cases. Case (1) . α_{0}α_{1} ≠ 0. We change the basis to \underline{ξ} = (ξ_{1}, ξ_{2}) , where ξ_{1} =

((ε_{0} − p^{k_{1}} − α_{0}α_{1}) α_{1}, \frac{α_{0}(ε_{0}−ε_{1})(ε_{0}−p^{k_{0}})(ε_{0}−p^{k_{0}} −α_{0}α_{1})(ε_{1}−p^{k_{1}} −α_{0}α_{1})}{(2ε_{0}ε_{1}−p^{k_{0}} ε_{1}−p^{k_{1}} ε_{0}−α_{0}α_{1}ε_{1})(ε_{1}−p^{k_{0}} −α_{0}α_{1})}) η_{1}

+ ((ε_{0} − p^{k_{1}} − α_{0}α_{1}) α_{1}, \frac{α_{0}(ε_{0}−ε_{1})(ε_{1}−p^{k_{0}})(ε_{0}−p^{k_{0}} −α_{0}α_{1})(ε_{1}−p^{k_{1}} −α_{0}α_{1})}{(2ε_{0}ε_{1}−p^{k_{0}} ε_{1}−p^{k_{1}} ε_{0}−α_{0}α_{1}ε_{1})(ε_{1}−p^{k_{0}} −α_{0}α_{1})}) η_{2}

and ξ_{2} =

Documenta Mathematica 15 (2010) 873–938

((ε_{1} − p^{k_{1}} − α_{0}α_{1}) (ε_{0} − p^{k_{1}}) , \frac{α_{0}^{2}(ε_{0}−ε_{1})(ε_{1}−p^{k_{1}} −α_{0}α_{1})(ε_{1}−p^{k_{0}} −α_{0}α_{1})}{(2ε_{0}ε_{1}−p^{k_{0}} ε_{1}−p^{k_{1}} ε_{0}−α_{0}α_{1}ε_{1})(ε_{1}−p^{k_{0}} −α_{0}α_{1})}) η_{1}

+ ((ε_{1} − p^{k_{1}} − α_{0}α_{1}) (ε_{1} − p^{k_{1}}) , \frac{α_{0}^{2}(ε_{0}−ε_{1})(ε_{1}−p^{k_{1}} −α_{0}α_{1})(ε_{1}−p^{k_{0}} −α_{0}α_{1})}{(2ε_{0}ε_{1}−p^{k_{0}} ε_{1}−p^{k_{1}} ε_{0}−α_{0}α_{1}ε_{1})(ε_{1}−p^{k_{0}} −α_{0}α_{1})}) η_{2}

In the ordered basis \underline{ξ} we have φ (ξ_{1}) = (1, ε_{0}) ξ_{1} and φ (ξ_{2}) = (λ, \frac{ε_{1}}{λ} ) ξ_{2}, where

λ=−\frac{(ε_{0} − p^{k_{1}} − α_{0}α_{1})}{(ε_{1} − p^{k_{1}} − α_{0}α_{1})} · \frac{(ε_{1} − p^{k_{0}} − α_{0}α_{1})}{(ε_{0} − p^{k_{0}} − α_{0}α_{1})} · \frac{(2ε_{0}ε_{1} − p^{k_{0}}ε_{1} − p^{k_{1}}ε_{0} − α_{0}α_{1}ε_{1})}{(2ε_{0}ε_{1} − p^{k_{0}}ε_{0} − p^{k_{1}}ε_{1} − α_{0}α_{1}ε_{0})},

and the filtration is as in formula 6.14, with xη_{1} + yη_{2} replaced by ξ_{1} + ξ_{2}. By Proposition 6.3, the representation V_{k}^{(1,3)} ( α) is irreducible. Arguing as in

the proof of Proposition 6.12(iv) we see that the representations V_{k}^{(1,3)} ( α) are non-induced with the possibility of at most finitely many exceptions. Case (2). α_{0} = 0, α_{1} ≠ 0. We argue as above and see that in the ordered basis \underline{ξ} = (ξ_{1}, ξ_{2}) , where

ξ_{1} = η_{2} and ξ_{2} = (1, \frac{p^{k_{0}} − p^{k_{1}}}{α_{1}p^{k_{1}}} ) η_{1} − (\frac{α_{1}}{p^{k_{0}} − p^{k_{1}}} , p^{k_{0}−k_{1}}) η_{2}

we have φ (ξ_{1}) = (1, p^{k_{0}}) ξ_{1} and φ (ξ_{2}) = (λ (α_{1}) , \frac{p^{k_{1}}}{λ(α_{1})}) ξ_{2}, with λ (α_{1}) =

α_{1}^{−1} (p^{k_{0}} − p^{k_{1}}) . The filtration in this basis is given by formula 6.14, with xη_{1} + yη_{2} replaced by ξ_{1} + (0, 1) ξ_{2}. By Proposition 6.3 the representation V_{k}^{(1,3)} ((0, α_{1})) is reducible and non-split. Case (3). α_{1} = 0, α_{0} ≠ 0. In the ordered basis \underline{ξ} = (ξ_{1}, ξ_{2}) , where

ξ_{1} = η_{2} − (\frac{p^{k_{1}}α_{0}}{p^{k_{1}} − p^{k_{0}}} , \frac{α_{0}}{p^{k_{1}} − p^{k_{0}}} ) η_{1} and ξ_{2} = (\frac{α_{0}p^{k_{0}}}{p^{k_{1}} − p^{k_{0}}} , 1) η_{1},

we have φ (ξ_{1}) = (1, p^{k_{0}}) ξ_{1} and φ (ξ_{2}) = (λ (α_{0}) , \frac{p^{k_{1}}}{λ(α_{0})}) ξ_{2}, with

λ (α_{0}) = α_{0}^{−1} (p^{k_{1}} − p^{k_{0}}) p^{k_{1}−k_{0}}. The filtration in the basis ξ is given by formula 6.14, with xη_{1} + yη_{2} replaced by (1, 0) ξ_{1} + ξ_{2}. By Proposition 6.3, V_{k}^{(1,3)} ((α_{0}, 0)) is reducible, non-split. By [Dou10, Proposition 7.1] it follows that there are no isomorphisms between members of the families {V_{k}^{(1,3)} ((α_{0}, 0)) , α_{0} ∈ p^{m}𝔪_{E} \ {0}} and

{V_{k}^{(1,3)} ((0, α_{1})) , α_{1} ∈ p^{m}𝔪_{E} \ {0}} .

Case (4) . α_{0} = α_{1} = 0. Then φ (η_{1}) = (p^{k_{1}}, 1) η_{1} and φ (η_{2}) = (1, p^{k_{0}}) η_{2}, while the filtration is as in formula 6.14,with x = (0, 1) and y = (1, 0) . Since J_{x} ∩ J_{y} = ∅, Proposition 6.3 implies that V_{k}^{(1,3)}(0) is split-reducible.

Proposition 6.22. Let 0 < v_{p} (ε_{i}) < k_{0} + k_{1} with ε_{0} ≠ ε_{1} such that ε_{0}ε_{1} = p^{k_{0}+k_{1}} and assume that 0 ≤ k_{i} ≤ p−1. Define the families of filtered φ-modules 𝔻 (λ) with

φ (η_{1}) = (1, ε_{0}) η_{1}, φ (η_{2}) = (λ, \frac{ε_{1}}{λ} ) η_{2},

and filtrations as in formula 6.10 with x = y = 1. These filtered modules are weakly admissible, irreducible, sharing the same characteristic polynomial and filtration. Let V (λ) be the corresponding to 𝔻 (λ) crystalline representations of G_{ℚ_{p^{2}}} .

(i) If λ = \frac{ε_{1}}{ε_{0}} (\frac{p^{k_{1}} α−ε_{0}}{p^{k_{1}} α−ε_{1}} ) , where α ∈ m_{E}, then (\overline{V (λ)}_{|I_{ℚ_{p^{2}}}} )^{ss} =

ω_{4,τ̄_{0}}^{−(k_{0}+pk_{1})} ⨁ ω_{4,τ̄_{0}}^{−(k_{0}+pk_{1})p^{2}} and \overline{V (λ)} is irreducible;

(ii) If λ = (\frac{ε_{1}}{ε_{0}} )^{2} (\frac{p^{k_{1}} α−ε_{1}}{p^{k_{1}} α−ε_{0}} ) , where α ∈ m_{E}, then (\overline{V (λ)}_{|I_{ℚ_{p^{2}}}} )^{ss} =

ω_{4,τ̄_{0}}^{−(pk_{1}+p^{2}k_{0})} ⨁ ω_{4,τ̄_{0}}^{−(pk_{1}+p^{2}k_{0})p^{2}} and \overline{V (λ)} is irreducible;

(iii) If λ = 1, then \overline{V (λ)} is reducible and \overline{V (λ)}_{|I_{ℚ_{p^{2}}}} = ω_{2,τ̄_{0}}^{−k_{1}} ⨁ ω_{2,τ̄_{0}}^{−pk_{0}}.

Proof. The common characteristic polynomial is X^{2} − (ε_{0} + ε_{1}) X + p^{k_{0}+k_{1}}. Parts (i) and (ii) follow from Examples 6.15 (i) and (iii) using the “standard parametrization” for the families V_{k,a}^{(1,2)} and V_{k,a}^{(1,4)}, and taking into account that m = 0 and Proposition 6.8. Part (iii) follows from Proposition 6.21(i) and a little computation to prove that if p^{k_{0}} + p^{k_{1}} + α_{0}α_{1} = ε_{0} + ε_{1} and ε_{0}ε_{1} = p^{k_{0}+k_{1}}, then λ = 1.

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Makis Dousmanis Université de Lyon UMPA ENS Lyon 46 allée d’Italie 69007 Lyon France Makis.Dousmanis@umpa.ens-lyon.fr