Daniel Le
email: le@math.toronto.edu

AND Stefano Morra
email: smorra@umontpellier.fr

Chol Park
email: cpark@kias.re.kr

ON MOD p LOCAL-GLOBAL COMPATIBILITY FOR GL 3 (Q p ) IN THE NON-ORDINARY CASE

Let F/Q be a CM field where p splits completely and r : Gal(Q/F ) → GL 3 (Fp) a continuous modular Galois representation. Assume that r is non-ordinary and nonsplit reducible (niveau 2) at a place w above p. We show that the isomorphism class of r| Gal(F w /Fw ) is determined by the GL 3 (Fw)-action on the space of mod p algebraic automorphic forms by using the refined Hecke action of [HLM]. We also give a nearly optimal weight elimination result for niveau two Galois representations compatible with the explicit conjectures of [Her09] and [GHS]. Moreover, we prove the modularity of certain Serre weights, in particular, when the Fontaine-Laffaille invariant takes special value ∞, our methods provide with the modularity of a certain shadow weight.

1. Introduction 1.1. Notation 2. The local Galois side 2.1. The Fontaine-Laffaille parameter 2.2. p-adic Hodge theory: Preliminaries 2.3. Classification of simple Breuil modules of rank 2 2.4. Crystalline lifts 3. Elimination of Galois types 3.1. Elimination of Galois types of niveau 1 3.2. Elimination of Galois types of niveau 2 4. Fontaine-Laffaille parameter and crystalline Frobenius 4.1. Filtration on strongly divisible modules 4.2. From Frobenius eigenvalues to Fontaine-Laffaille parameters 5. The local automorphic side 5.1. Basic set up 5.2. Group algebra operators and the automorphic parameter 6. Local-global compatibility 6.1. Automorphic forms on unitary groups 6.2. Serre weights 6.3. Weight elimination 6.4. Local-global compatibility 6.5. Freeness over the Hecke algebra 1. Introduction Let p be a prime. In this paper, we address a problem about local-global compatibility in the mod p Langlands program for GL 3 (Q p ). In [START_REF] Serre | Sur les représentations modulaires de degré 2 de Gal(Q/Q)[END_REF], J-P. Serre conjectured that if r : Gal(Q/Q) → GL 2 (F p ) is a modular Galois representation, then the minimal weight of a modular form giving rise to r is determined (in an explicit way) from the local datum r| Ip , where I p denotes the inertia group at p. From the explicit description, one easily sees that the conjectured minimal weight actually determines the isomorphism class of r| Ip . Serre interpreted this as evidence for compatible mod p local and global Langlands correspondences (cf. loc. cit., Section 3.4). These correspondences were established along with their p-adic analogues in several works of many authors-Breuil, Berger, Colmez, Dospinescu, Emerton, Kisin, and Paskunas to name a few (see [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Qp). I[END_REF][START_REF] Colmez | Représentations de GL 2 (Qp) et (φ, Γ)-modules[END_REF][START_REF] Emerton | Local-global compatibility in the p-adic langlands programm for GL 2/Q[END_REF]). In particular, r| Gal(Q p /Qp) can be recovered from the minimal weight and the Hecke action on it.

One would hope for analogous correspondences in greater generality. For a CM extension F/F + in which p splits completely, fix a place w|p. For a modular Galois representation r : Gal(Q/F ) → GL 3 (F p ), one could consider the GL 3 (F w )-representation Π(r) coming from the space of mod p automorphic forms on a definite unitary group. It is not known whether Π(r) depends only on r| Gal(F w /Fw) . It is expected that if r| Gal(F w /Fw) is tamely ramified, then it is determined by the set of modular Serre weights (the GL 3 (Z p )-socle of Π(r)) and the Hecke action on its constituents. However, this is not true if r| Gal(F w /Fw) is wildly ramified, and the question of determining r| Gal(F w /Fw) from Π(r) lies deeper than the weight part of Serre's conjecture. Using a refined Hecke action, we show that the GL 3 (Q p )action on Π(r) determines r| Gal(F w /Fw) in many non-ordinary cases following the work in the ordinary cases of [HLM] for GL 3 (Q p ) and [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extensions entre caractères galoisiens[END_REF] for GL 2 over unramified extensions of Q p .

In order to present the main results in more detail we need to fix some notation. We let E/Q p be a finite extension, O E its ring of integers and F its residue field. These are the rings of coefficients of our representations and are always assumed to be sufficiently large. Let ρ : G Qp → GL 3 (F) be a continuous reducible indecomposable Galois representation. It is not hard to see from the results of [START_REF] Gee | Companion forms for unitary and symplectic groups[END_REF] that the semisimplification of ρ is often determined by the modular Serre weights of ρ and the Hecke actions on them. If ρ is Fontaine-Laffaille, the extension class, and hence the isomorphism class of ρ, is determined by an invariant FL(ρ) ∈ P 1 (F) generalizing the one in [HLM] (cf. Definition 2.8).

One can also define a parameter on the automorphic side. Let I 1 denote the standard pro-p Iwahori subgroup. If π p is a smooth F-valued representation of GL 3 (Q p ), which verifies certain multiplicity one properties with respect to its GL 3 (Z p )-socle, then there is a natural action of certain group algebra operators S, S on (a 2 , a 1 , a 0 )-isotypic parts of π I1 p (isotypic with respect to the residual action of the finite torus) and one can associate a non-zero parameter to the pair (S, S ) (see Section 5 for the precise definition of the operators and their properties).

The main result of this paper is to prove that the two local parameters defined above coincide when the local representations are obtained from the cohomology of unitary arithmetic manifolds (cf. Theorem 6.13). Let F/Q be a CM field with F + its maximal totally real subfield and let r : G F def = Gal(Q/F ) → GL 3 (F p ) be a continuous Galois representation. Assume that p is totally split in F and fix a place w 0 |v 0 of F, F + respectively, above p. We assume that r is modular: for the purpose of this introduction this means that there exists a totally definite unitary group G defined over F + (outer form of GL 3 and split at places above p), a tame level U p ≤ G(A ∞,p F + ) away from p and a maximal ideal m r associated to r in the anemic Hecke algebra acting on S sm (U p , F) (the space of algebraic automorphic forms with infinite level at p and coefficients in F) such that S sm (U p , F)[m r ] = 0.

We write W (r) for the irreducible smooth GL 3 (O F + ,p )-representations V over F such that Hom G(O F + ,p ) V ∨ , S sm (U p , F)[m r ] = 0.

(the set of Serre weights of r) and we fix a Fontaine-Laffaille set of weights V v0 away from v 0 (i.e. V v0 is an irreducible smooth representation of v|p, v =v0 GL 3 (O F + v ) and there exists an irreducible smooth GL 3 (O F + v 0

)-representation V v0 such that V v0 ⊗ V v0 ∈ W (r); see Definition 6.5 for details on the definition of V v0 ). In particular, we define the space S sm (U v0 , V v0 )[m r ] of algebraic automorphic forms of infinite level at v 0 and coefficients in

V v0
; it is a G(F + v0 )-representation. Theorem 1.1. In the previous hypothesis and settings, let U = U v0 × U v0 ≤ G(A ∞,p F ) × G(O F + ,p ) be a sufficiently small compact open, where U v0 ⊂ G(A ∞,v0 F + ). We make the following assumptions:

(i) r| G Fw 0
is indecomposable of residual niveau 2 as in (2.1.1) with genericity condition (2.1.2); (ii) FL(r| G Fw 0

) / ∈ {0, ∞}; (iii) r is Fontaine-Laffaille at all places dividing p; (iv) r is unramified at places away from p; (v) r has an image containing GL 3 (k) for some k ⊂ F with #k > 9;

(vi) F ker (adr) does not contain F (ζ p ).

Let S, S be the group algebra operators defined in Section 5 (associated to the triple of integers (-a 0 , -a 1 , -a 2 ). Then S •   0 1 0 0 0 1 p 0 0   = (-1) a2-a1 • a 1 -a 0 a 2 -a 1

• FL(r| G Fw 0 ) • S (1.0.1) on S sm (U v0 , V v0 )[m r ] I,(-a1,-a0,-a2) [U 2 ], where the notation (•) I,(-a1,-a0,-a2) denotes the (-a 1 , -a 0 , -a 2 )-isotypic part, for the residual action of the finite torus, of the pro-p Iwahori fixed vectors of S sm (U v0 , V v0 )[m r ], and U 2 is a carefully chosen U p -operator.

In the theorem above, the global assumption (iii)-(vi) are needed in order to obtain a freeness result for a Hecke algebra acting on S sm (U v0 , V v0 )[m r ] (cf. Theorem 6.16).

As mentioned before, in order to obtain Theorem 1.1 one needs a certain multiplicity one condition on the G(O F + v 0 )-socle. This is obtained by a thorough type elimination in niveau 2, which highlights that the set of Serre weights for r depends on the associated Fontaine-Laffaille parameter.

When r| G Fw 0 is semisimple, there is a conjectural description of the set W ? w0 (r) of irreducible smooth representations V v0 of G(O F + ,p ) such that V v0 ⊗ V v0 ∈ W (r) (cf. [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF]). When r| G Fw 0 is not semisimple, we define here an explicit set W ? w0 (r), which depends on the Fontaine-Laffaille parameter associated to r| G Fw 0 (cf. Definition 6.3). We remark that in the set W ? w0 (r) we can distinguish an explicit subset W ?,obv w0 (r) of obvious weights (related to "obvious" crystalline lifts of r| G Fw 0

). Our main result on Serre weights for r is contained in the following theorem: Theorem 1.2. Assume that r verifies assumption (i) of Theorem 1.1. Then W w0 (r) ⊆ W ? w0 (r). Moreover, the obvious weights F (a 2 -1, a 1 , a 0 + 1) and F (a 2 -1, a 0 + 1, a 1 -p + 1) are always modular, while, if the Fontaine-Laffaille parameter at w 0 verifies FL(r| G Fw 0 ) = ∞, the shadow weight F (a 2 , a 0 , a 1 -(p -1)) is modular.

Finally, assume that F is unramified at all finite places and that there is a RACSDC automorphic representation Π of GL 3 (A F ) of level prime to p such that (i) r rp,i (Π);

(ii) For each place w|p of F , r p,i (Π)| G Fw is potentially diagonalizable;

(iii) r(G F (ζp) ) is adequate. Then we have the following inclusion: W ?,obv w0 (r) ⊆ W w0 (r).

Remark 1.3. If r| G Fw is split, and r verifies items (i)-(iii) of Theorem 1.2 we can always prove that W ?,obv w0 (r) ∩ W L ⊆ W w0 (r) where W ?,obv w0 (r) ∩ W L is the set of obvious lower weights of r at w 0 (cf. §6.3) For the weight elimination results, we classify rank 2 simple Breuil modules with descent data of niveau 2 corresponding to the 2-dimensional irreducible quotient of r| G Fw 0

. The classification of the rank 2 simple Breuil modules is also heavily used to show the connection between the Fontaine-Laffaille parameter and a Frobenius eigenvalue of certain potentially crystalline lift of r| G Fw 0 (cf. Proposition 4.3 and Theorem 4.5). The proof of weight existence is here performed by purely Galois cohomology arguments. We remark that along the proof of Theorem 1.2, we obtain a potential diagonalizability result, which lets us infer that representations satisfying the hypotheses of Theorem 1.1 do exist (cf. Theorem 6.17).

We conclude this introduction with an overview of the sections of this paper. In the remainder of this introduction, we introduce the notation that will be used throughout the paper. In Section 2, we analyze the local mod p Galois representation ρ 0 in terms of Fontaine-Laffaille theory. We also classify rank 2 simple Breuil modules with tame descent data, and show the existence of crystalline lifts with certain Hodge-Tate weights of the representation ρ 0 . In Section 3, we perform elimination of Galois types, by determining the structure of possible Breuil modules with descent data corresponding to the representation ρ 0 . In Section 4 we completely determine the filtration of strongly divisible modules lifting the Breuil modules, with a carefully chosen descent datum, corresponding to the representation ρ 0 . The filtration on strongly divisible modules gives information of the eigenvalues of the Frobenius map of the corresponding weakly admissible filtered (φ, N )-modules, and we find an explicit relation between certain Frobenius eigenvalues and the Fontaine-Laffaille parameter. In Section 5, we quickly review certain group algebra operators and their properties, developed in [HLM]. Our main results are stated and proved in Section 6. We establish weight elimination result in Section 6.3, and prove mod p local-global compatability and modularity of certain weights in Section 6.4. A freeness result for a Hecke algebra acting on S sm (U v0 , V v0 )[m r ] is proved in Section 6.5.

1.1. Notation. Let Q be an algebraic closure of Q. All number fields F/Q will be considered as subfields in Q and we write G F def = Gal(Q/F ) to denote the absolute Galois group of F . For any rational prime ∈ Q, we fix an algebraic closure Q of Q and an embedding Q → Q (and so an inclusion G Q → G Q ). In a similar fashion, we fix an algebraic closure F for the residue field F of Q . As above, all algebraic extensions of Q (resp. F ) will be considered as subfields in the fixed algebraic closure Q (resp. F ).

Let f ≥ 1. We let K 0 def = W (k)[ 1 p ] be the unramified extension of degree f of Q p . We consider the Eisenstein polynomial E(u) def = u e + p ∈ Z p [u] where e = p f -1. We fix a root = e √ -p ∈ Q p and set K def = K 0 ( ). In particular, K/K 0 is a tamely, totally ramified extension of K 0 of degree e and a uniformizer .

Let E be a finite extension of Q p . We write O E for its ring of integers, F for its residue field and E ∈ O E to denote an uniformizer. From now on, we fix an embedding σ 0 : K → E, hence an embedding σ 0 : k → F.

The choice of ∈ K provides us with a map:

ω : Gal(K/Q p ) -→ W (F p f ) × g -→ g( )
whose reduction mod will be denoted as ω . Note that the choice of the embedding σ 0 : k → F provides us with a fundamental character of niveau f , namely ω

f def = σ 0 •ω | Gal(K/K0
) . Write ϕ for the absolute Frobenius on k. By extension of scalars, the ring k ⊗ Fp F is equipped with a Frobenius endomorphism ϕ ⊗ 1 and with a Gal(K/Q p )-action via ω ⊗ 1. In particular, we recall the standard idempotent elements e σ ∈ k ⊗ Fp F defined for σ ∈ Hom(k, F), which verify ϕ(e σ ) = e σ•ϕ -1 and (λ ⊗ 1)e σ = (1 ⊗ σ(λ))e σ . We write e σ ∈ W (k) ⊗ Zp O E for the standard idempotent elements; they reduce to e σ modulo p.

Given a p-adic Galois representation ρ : G Qp → GL n (E), we write ρ ∨ to denote the linear dual representation. Given a potentially semistable representation ρ : G Qp → GL n (E), we write WD(ρ) to denote the associated Weil-Deligne representation as defined in [START_REF] Conrad | Modularity of certain potentially Barsotti-Tate Galois representations[END_REF], Appendix B.1. We refer to WD(ρ)| I Qp as to the inertial type associated to ρ. Note that, in particular, WD(ρ) is defined via the (covariant) filtered (ϕ, N )-module D

Qp st (ρ) def = lim -→ H/Qp (B st ⊗ Qp ρ) G H (and D
for some integers a 0 , a 1 , a 2 ∈ N. It is obvious that it can be rewritten as follows:

ρ 0 | I Qp ∼ =    ω (a2-a0-1)+1 * * 0 ω (a1-a0-1)+1 2 0 0 0 ω p((a1-a0-1)+1) 2    ⊗ ω a0+1 .
We let ρ 2 be the one-dimensional subrepresentation such that ρ 2 | I Qp ∼ = ω a2+1 and ρ 10 the two-dimensional irreducible quotient such that ρ 10

| I Qp ∼ = ω a0+1+p(a1+1) 2 ⊕ ω (a1+1)+p(a0+1) 2
.

2.1.1. Preliminaries on Fontaine-Laffaille theory. We briefly recall the theory of Fontaine-Laffaille modules with F-coefficients and its relation with mod-p Galois representations. The main reference will be [HLM], Section 2.1.

A Fontaine-Laffaille module (M, Fil • M, φ • ) over k ⊗ Fp F is the datum of (i) a finite k ⊗ Fp F-module M , free over k;
(ii) a separated, exhaustive and decreasing filtration {Fil j M } j∈Z on M by k ⊗ Fp F submodules (the Hodge filtration), which are k-direct summands; (iii) A ϕ-semilinear Frobenius isomorphism φ • : gr • M → M Note that, by property (iii), a Fontaine-Laffaille module is indeed free over k ⊗ Fp F. Defining the morphisms in the obvious way, we obtain the abelian category F-FL k of Fontaine-Laffaille modules over k ⊗ Fp F. If the field k is clear from the context, we simply write F-FL to lighten the notation.

Given a Fontaine-Laffaille module M , the set of its Hodge-Tate weights in the direction of σ ∈ Gal(k/F p ) is defined as

HT σ def = i ∈ N, dim F e σ Fil i M e σ Fil i+1 M = 0 .
In the remainder of this paper we will be focused on Fontaine-Laffaille modules in parallel Hodge-Tate weights, i.e. we will assume that for all i ∈ N, the submodules Fil i M are free over k ⊗ Fp F. Definition 2.1. Let M be a Fontaine-Laffaille module in parallel Hodge-Tate weights. A k ⊗ Fp F basis f = (f 1 , . . . , f n ) on M is compatible with the filtration if for all i ∈ N there exists

j i ∈ N such that Fil i M = n j=ji k ⊗ Fp F • f j .
In particular, the principal symbols (gr(f 1 ), . . . , gr(f n )) provide a k ⊗ Fp F basis for gr • M .

Note that if the graded pieces of the Hodge filtration have rank at most one then any two compatible basis on M are related by a lower triangular matrix in GL n (k ⊗ Fp F). Given a Fontaine-Laffaille module and a compatible basis f , it is convenient to describe the Frobenius action via a matrix Mat f (φ • ) ∈ GL 3 (k ⊗ Fp F), defined in the obvious way using the principal symbols (gr(f 1 ), . . . , gr(f n )) as a basis on gr • M .

It is customary to write F-FL [0,p-2] to denote the full subcategory of F-FL formed by those modules M verifying Fil 0 M = M and Fil p-1 M = 0 (it is again an abelian category). We have the following description of mod p Galois representations of G K0 via Fontaine-Laffaille modules: Theorem 2.2. There is an exact fully faithful contravariant functor

T * cris,K0 : F-FL [0,p-2] k → Rep F (G K0 )
which is moreover compatible with the restriction over unramified extensions: if K 0 /K 0 is unramified, with residue field k /k, then

T * cris,K 0 (k ⊗ k M ) ∼ = T * cris,K0 (M )| G K 0 .
Proof. The statement with F p -coefficients is in [START_REF] Fontaine | Construction de représentations p-adiques[END_REF], Théorème 6.1; its analogue with F-coefficient is a formal argument which is left to the reader (cf. also [START_REF] Gao | A note on potential diagonalizability of crystalline representations[END_REF], Theorem 2.2.1).

We will simply write T * cris if the base field K 0 is clear from the context. It is well known, (for instance [START_REF] Gee | Companion forms for unitary and symplectic groups[END_REF], Lemma 3.1.5), that under mild condition on the inertial weights, ρ 0 is Fontaine-Laffaille:

Proposition 2.3. Let ρ 0 : G Qp → GL 3 (F) be as in (2.1.1). If the triple (a 2 , a 1 , a 0 ) ∈ Z 3 verifies p -2 ≥ (a 2 -a 0 -1) ≥ a 1 -a 0 ≥ 2 then ρ 0 is Fontaine-Laffaille.
In order to obtain results on local-global compatibility and to perform weight elimination (cf. Section 3), we shall assume a stronger genericity condition on the integers a i , Definition 2.4. We say that a niveau 2 Galois representation ρ 0 :

G Qp → GL 3 (F) as in (2.1.1) is generic if the triple (a 2 , a 1 , a 0 ) satisfy the condition p -3 > (a 2 -a 0 -1) > (a 1 -a 0 ) > 3. (2.1.2)
2.1.2. The Fontaine-Laffaille parameter. Let ρ 0 be as in (2.1.1) and assume that the integers a i ∈ N verify the generic condition (2.1.2). By Proposition 2.3 there is a Fontaine-Laffaille module M such that T * cris (M ) ∼ = ρ 0 ⊗ ω -a0-1 and which is moreover endowed with a filtration by Fontaine-Laffaille submodules M 0 M 1 M 2 = M induced via T * cris from the cosocle filtration on ρ 0 (cf. Theorem 2.2).

Lemma 2.5. Assume (2.1.2) and let M ∈ F-FL be such that T * cris (M ) ∼ = ρ 0 ⊗ ω -a0-1 . Then there exists a basis f = (f 0 , f 1 , f 2 ) on M which is compatible with the Hodge filtration Fil • M and with the filtration by Fontaine-Laffaille submodules on M , and such that

Mat f (φ • ) =   0 µ -1 1 x µ -1 0 z y 0 0 µ -1 2   (2.1.3) for some µ i ∈ F × , x, y, z ∈ F.
Proof. For the rest of this proof, we set

c def = a 2 -a 1 -1, r def = a 1 -a 0 -1. In particular M has Hodge-Tate weights {0, r + 1, c + 1}.
Let N be the rank two irreducible Fontaine-Laffaille submodule of M corresponding to

T * cris (N ) ∼ = ρ 0 ω c+1 . Then we have Fil i N = N ∩Fil i M for all i ∈ N. As N is irreducible, we can find a basis (f 0 , f 1 ) on N , such that Fil 1 N = • • • = Fil r+1 N = f 1 and Mat (f0,f1) (φ • ) = 0 µ -1 1 µ -1 0 z
. Let f 2 be a generator of Fil r+2 M . As Fil r+2 N = 0 and the Frobenius on N is induced from the Frobenius on M it is obvious that the element Mat (f0,f1,f2) (φ • ) ∈ GL 3 (F) has the desired shape (2.1.3).

Remark 2.6. Keep the notation in the proof of Lemma 2.5. As N is a rank two irreducible Fontaine-Laffaille module, it is easy to show that it is always possible to choose (f 0 , f 1 ) so that z = 0.

The Fontaine-Laffaille invariant FL(ρ 0 ) associated to ρ 0 is defined in terms of Mat f (φ • ).

Lemma 2.7. Keep the hypotheses and the notation of Lemma 2.5. Assume moreover that x, y are not both zero, so that [x : y] ∈ P 1 (F) is well defined. Then the elements

µ 0 µ 1 , µ 2 , -x : det µ -1 1 x z y
deduced from Mat f (φ • ) do not depend on the choice of a basis which is compatible with both the Hodge and the submodule filtration on M .

Proof. The proof is an elementary computation in GL 3 (F). Indeed, let f be a basis on M as in the statement of Lemma 2.5. Then the matrix B ∈ GL 3 (F) associated to a change of basis (compatible with the Hodge filtration) on M is lower triangular and the requirement that the new basis is compatible with the submodule filtration on M provides us the following equation:

B • Mat f (φ • ) • gr(B) -1 = 0 λ -1 1 x λ -1 0 z y 0 0 λ -1 2
where the diagonal matrix gr(B) is defined by gr(B) i,i = (B) i,i , and the left hand side is in an element in GL 3 (F).

By letting

B = α 0 0 δ β 0 η γ
, an easy computation provides us with

  0 λ -1 1 x λ -1 0 z y 0 0 λ -1 2   =   0 µ -1 1 β -1 α αxγ -1 α -1 µ -1 0 β µ -1 1 β -1 δ + z xγ -1 δ + yγ -1 β 0 0 µ -1 2   .
We have

-αxγ -1 : det µ -1 1 β -1 α αxγ -1 µ -1 1 β -1 δ + z xγ -1 δ + yγ -1 β = -x : det µ -1 1 x z y
and the conclusion is now clear.

Definition 2.8. Keep the hypothesis and notation of Lemma 2.5, and let M be the Fontaine-Laffaille module associated to ρ 0 ⊗ ω -a0-1 , f = (f 0 , f 1 , f 2 ) a basis on M as in Lemma 2.7, and µ -1 1 , x, y, z ∈ F be the elements defined by Mat f (φ • ) as in (2.1.3). The Fontaine-Laffaille parameter associated to ρ 0 is defined as

FL(ρ 0 ) = -x : det µ -1 1 x z y ∈ P 1 (F).
Remark 2.9. Let ρ 0 be as in (2.8). The isomorphism class of ρ 0 is completely determined by the pair (µ 0 µ 1 , µ 2 ) and the Fontaine-Laffaille parameter FL(ρ 0 ) as well as their Hodge-Tate weights.

p-adic

Hodge theory: Preliminaries. We place ourselves in the framework of strongly divisible lattices, Breuil module, étale ϕ-modules with coefficients and descent data, having [EGH13] Section 3.1 and [HLM] Section 2 as a main reference.

S W (k) = ∞ i=0 w i E(u) i i! , w i ∈ W (k)[u], lim i→∞ w i = 0
where W (k)[u] is endowed with the topology of the pointwise convergence. The ring S W (k) is endowed with a continuous, semilinear Frobenius endomorphism ϕ : S W (k) → S W (k) (semilinear with respect to the absolute Frobenius on W (k)), uniquely characterized by u → u p and a W (k)-linear derivation N , uniquely determined by N (u) = -u (hence N ϕ = pϕN ). This ring is naturally endowed with a filtration {Fil i S W (k) } i∈N , where Fil i S W (k) is the closure of the ideal generated by E(u) j j! , j ≥ i, and with a residual Galois action by W (k)-algebra endomorphisms, defined by g(u) = ω (g)u for any g ∈ Gal(K/Q p ). In particular, the action of any g ∈ Gal(K/Q p ) is compatible with the Frobenius, the filtration and the monodromy on S. Note that, by extension of scalars, the ring S Qp def = S W (k) ⊗ Zp Q p is endowed with the evident additional structures inherited from S W (k) .

We will be mainly concerned with objects having E-coefficients. Concretely, we write

S def = S W (k) ⊗ Zp O E , S E def = S ⊗ Zp Q p ,
so that the additional structures on S W (k) induce, by O E and E-linearity respectively, a Frobenius, a derivation, a filtration and a compatible residual Galois action on S, S E .

Recall that a strongly divisible lattice in weights (0, r) is the datum of a free S-module of finite type M, an S-submodule Fil r M ⊆ M, together with additive morphisms ϕ r , N such that:

(i) Fil r S • M ⊆ Fil r M and M/ Fil r M is E -torsion free;
(ii) the morphism ϕ r : Fil r M → M is semilinear with respect to the Frobenius on S and its image contains a family of S-generators for M;

(iii) the morphism N : M → M is W (k) ⊗ Zp O E -linear and verifies (a) N (sx) = N (s)x + sN (x) for all x ∈ M, s ∈ S; (b) E(u)N (Fil r M) ⊆ Fil r M; (c) ϕ r (E(u) • N ) = cN • ϕ r where c def = ϕ(E(u)) p ∈ S × .
Let K ∈ {K 0 , Q p }. A descent data from K to K on M is the datum of an action of Gal(K/K ) by additive automorphisms on M, which are semilinear (with respect to the descent data on S) and compatible with the additional structures on M (i.e. with the Frobenius, monodromy, and the filtration). We write O E -Mod r dd to denote the category of strongly divisible lattices in weights (0, r), with descent data from K to K .

We have a contravariant functor

T * ,K st : O E -Mod r dd → Rep K-st,[-r,0] O E (G K )
where Rep

K-st,[-r,0] O E (G K ) is the category of G K -stable O E -lattices inside E-
= (k ⊗ Fp F)[u]/(u ep ) is equipped with an action of Gal(K/Q p ) by k ⊗ Fp F-semilinear automorphisms. Explicitly if g ∈ Gal(K/Q p ) and a ∈ k ⊗ Fp F, we have g(au) def = (g • a)(ω (g) ⊗ 1)u
where g • a denotes the natural Gal(K/Q p ) action on k ⊗ Fp F.

We recall that S is equipped with an k ⊗ Fp F-linear derivation N defined by N (u) = -u and with a semilinear Frobenius ϕ defined by u → u p (semilinear with respect to the absolute Frobenius on k ⊗ Fp F). (c) ϕ r (u e N (x)) = N (ϕ r (x)) for all x ∈ Fil r M. A morphism of Breuil modules is defined as an S-linear morphism which is compatible, in the evident sense, with the additional structures (monodromy, Frobenius, filtration).

As above, we let K ∈ {Q p , K 0 }. A descent data relative to K on a Breuil module M is the datum of an action of Gal(K/K ) on M by F-linear automorphisms which are semilinear with respect to the residual Galois action on S and which are compatible, in the evident sense, with the additional structures on M. We write F-BrMod r dd to denote the category of Breuil modules over F with descent data to K .

We recall that we have an exact, faithful, contravariant functor

T * st : F-BrMod r dd → Rep F (G K ) M → T * st (M) def = Hom(M, A)
where A is a certain period ring (cf. [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 3.2 building on [Bre99a], Section 2.2; see also [HLM], appendix A). The functor T * st respects the rank on both sides, i.e. dim F T * st (M) = rank S M (cf. [START_REF]Fp-représentations semi-stables[END_REF], Théorème 4.2.4 and the Remarque following it, see also [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF] Lemma 3.2.2)

We have a natural compatibility between strongly divisible lattices and Breuil modules:

Proposition 2.10. Let M be an object in O E -Mod r dd . Then M ⊗ S S/( E , Fil p S) is an object in F-BrMod r dd in a natural way and one has a natural isomorphism:

T * ,K st ( M) ⊗ O E F ∼ = T * st ( M ⊗ S S/( E , Fil p S))
. Proof. This is contained in [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 3.2 (Lemma 3.2.2 and Definition 3.2.8).

In the rest of this paper we will be mainly interested in the covariant version of the previous functors toward Galois representations. For this reason we define T K ,r st

: O E -Mod r dd → Rep K-st,[-r,0] O E (G K ) and T r st : F-BrMod r dd → Rep F (G K ) via T K ,r st ( M) def = T * ,K st ( M) ∨ ⊗ ε r p , T r st (M) def = (T * st (M)) ∨ ⊗ ω r
(where we write • ∨ to denote the usual linear dual for an F-linear space •). We remark that this definition is compatible with the notion of duality on Breuil and strongly divisible modules as defined in [START_REF] Caruso | Conjecture de l'inertie modérée de Serre[END_REF] and [START_REF]Fp-représentations semi-stables[END_REF], namely T Definition 2.11. Let n ∈ N and let (a 0 , . . . , a n-1 ) ∈ Z n be an n-tuple. A rank n Breuil module M ∈ F-BrMod r dd is of (framed) type ω a0 ⊕ • • • ⊕ ω an-1 if M has an S-basis (e 0 , . . . , e n-1 ) such that ge i = (ω ai (g) ⊗ 1)e i for all i and all g ∈ Gal(K/K 0 ). We call such a basis a framed basis of M.

We also say that (f 0 , . . . , f n-1 ) is a framed system of generators of Fil r M if (f 0 , . . . , f n-1 ) is a system of S-generators for Fil r M and gf i = (ω p -1 ai (g) ⊗ 1)f i for all i and all g ∈ Gal(K/K 0 ).

A key tool in local to global compatibility is that the inertial type on a Breuil module M is closely related to the Weil-Deligne representation associated to a potentially crystalline lift of T r st (M).

Proposition 2.12. Let M be an object in O E -Mod r dd and let M def = M ⊗ S S/( E , Fil p S) be the Breuil module associated to M via the base change S S. Assume that T Qp,r st ( M) has inertial type ⊕ n-1 i=0 ω ai f . Then the Breuil module M is of type ⊕ n-1 i=0 ω ai and Fil r M admits a framed system of generators.

Proof. This can be spelled out from, e.g. [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 3.3 (proof of Theorem 3.3.13).

See also [HLM], Lemma 2.3.8.

2.2.3.

Comparison between Breuil and Fontaine-Laffaille modules. We now recall the following categories of étale ϕ-modules, first introduced by Fontaine ([Fon90]). Let k((p)) be the field of norms associated to (K 0 , p). In particular, p is identified with a sequence (p n ) n ∈ Q p N verifying p p n = p n-1 for all n. We define the category ϕ, F ⊗ Fp k((p)) -Mod of étale (ϕ, F⊗ Fp k((p)))-modules as the category of free F⊗ Fp k((p))modules of finite rank D endowed with a semilinear map ϕ : D → D (semilinear with respect to the Frobenius on k((p))) and inducing an isomorphism ϕ * D → D (with obvious morphisms between objects).

By work of Fontaine [START_REF] Fontaine | Représentations p-adiques des corps locaux. II[END_REF], we have an anti-equivalence

ϕ, F ⊗ Fp k((p)) -Mod ∼ -→ Rep F (G (Qp)∞ ) D -→ Hom D, k((p)) sep .
Let us consider def = e √ -p ∈ K. We can fix a sequence ( n ) n ∈ Q p N such that e n = p n for all n ∈ N and which is compatible with the norm maps

K( n+1 ) → K( n ) (cf. [Bre14], Appendix A). By letting K ∞ def = ∪ n∈N K( n ) and (K 0 ) ∞ def = ∪ n∈N K 0 (p n ),
we have a canonical isomorphism Gal(K ∞ /(K 0 ) ∞ ) → Gal(K/K 0 ) and we will identify ω as a character on Gal(K ∞ /(K 0 ) ∞ ).

The field of norms k(( )) associated to (K, ) is then endowed with a residual action of Gal(K ∞ /(K 0 ) ∞ ), which is completely determined by g( ) = ω (g) .

We can therefore define the category (ϕ, F ⊗ Fp k(( )))-Mod dd of étale (ϕ, F ⊗ Fp k(( )))modules with descent data: an object D is defined in the analogous, evident way as for the category (ϕ, F ⊗ Fp k((p)))-Mod, but we moreover require that D is endowed with a semilinear action of Gal(K ∞ /(K 0 ) ∞ ) (semilinear with respect to the residual action on F⊗ Fp k(( )), where F is endowed with the trivial Gal(K ∞ /(K 0 ) ∞ )-action) and the Frobenius

ϕ is Gal(K ∞ /(K 0 ) ∞ )-equivariant.
From [HLM], Appendix A.2 (which builds on the classical result of Fontaine) we have an anti-equivalence

ϕ, F ⊗ Fp k(( )) -Mod dd ∼ -→ Rep F (G (K0)∞ ) D → Hom (D, k(( )) sep ) .
The main result concerning the relations between the various categories and functors introduced so far is summarized by the following proposition ([HLM], Proposition 2.2.9). Proposition 2.13. There exist faithful functors

M k(( )) : F-BrMod r dd → ϕ, F ⊗ Fp k(( )) -Mod dd and F : F-FL [0,p-2] → ϕ, F ⊗ Fp k((p)) -Mod fitting in the following commutative diagram: (2.2.1) F-BrMod r dd T * st M k(( )) / / ϕ, F ⊗ Fp k(( )) -Mod dd Hom(-,k(( )) sep ) y y Rep F (G K0 ) Res / / Rep F (G (K0)∞ ) F-FL [0,p-2] T * cris O O F / / ϕ, F ⊗ Fp k((p)) -Mod -⊗ k((p)) k(( )) O O Hom(-,k((p)) sep ) e e
where the descent data is relative to K 0 and the functor Res • T cris is fully faithful.

The functors M k(( )) , F are defined in [HLM], Appendix A, building on the classical work of Breuil [START_REF] Breuil | Une application de corps des normes[END_REF] and Caruso-Liu [START_REF] Caruso | Quasi-semi-stable representations[END_REF].

Corollary 2.14. Let r ≤ p -2 and let M, M be objects in F-BrMod r dd and F-FL [0,p-2] respectively. Assume that

T * st (M) is Fontaine-Laffaille. If M k(( )) (M) ∼ = F(M ) ⊗ k((p)) k(( ))
then one has an isomorphism of G K0 -representations

T * st (M) ∼ = T * cris (M ). 2.2.4
. Linear algebra with descent data. We recall here some formalism on linear algebra with descent data which was introduced in [HLM]. In what follows we fix a residual Galois type τ :

I Qp → GL n (F), with a framing τ = ⊕ n-1 i=0 ω ai f . Definition 2.15. Let M ∈ F-BrMod r dd be of type ⊕ n-1 i=0 ω ai . Let e def
= (e 0 , . . . , e n-1 ), f = (f 0 , . . . , f n-1 ) be a framed basis and a framed system of generators of M, Fil r M respectively.

The matrix of the filtration, with respect to e, f , is the element Mat e,f (Fil

r M) ∈ M n (S) verifying f = e • Mat e,f (Fil r M).
Similarly, we define the matrix of the Frobenius with respect to e, f as the element Mat e,f (ϕ r ) ∈ GL n (S) characterized by

ϕ r (f ) = e • Mat e,f (ϕ r ).
As we require e, f to be compatible with the framing, the coefficients in the matrix of the filtration verify important additional properties:

Mat e,f (Fil r M) i,j ∈ S ω p -1 a j -a i f . Concretely, one has Mat e,f (Fil r M) i,j = u [p -1 aj -ai] s i,j where for any x ∈ Z we define [x] ∈ {0, . . . , e -1} by [x] ≡ a j -a i mod e and s i,j ∈ S ω 0 = k ⊗ Fp F[u e ]/(u ep ).
We can therefore introduce the subspace M n (S) of "matrices with framed type τ ": Definition 2.16. Let τ be a framed tame Galois type.

The space M n (S) is defined as

M n (S) def = V ∈ M n (S), V i,j ∈ S ω a j -a i for all 0 ≤ i, j ≤ n -1 .
Similarly, we set

GL n (S) def = GL n (S) ∩ M n (S)
which is a subgroup in GL n (S).

As τ is a residual Galois type, there exists an element w τ ∈ S n such that gf wτ (j) = (ω aj ⊗ 1)f wτ (j) for all g ∈ Gal(K/K 0 ) and 0 ≤ j ≤ n -1. Moreover as ϕ r (f i ) is a ω ai f eigenvector for the residual Galois action we deduce that

Mat e,f (Fil r M) • w τ ∈ Mat n (S), Mat e,f (ϕ r ) ∈ GL n (S)
where we used the same notation w τ for the permutation matrix associated to w τ . Given A, B ∈ M n+1 (S) and x ∈ S ω 0 we write , with a slight abuse of notation,

A ≡ B mod x
meaning that there exists an element C ∈ M n+1 (S) such that A = B + xC.

Lemma 2.17. Let M be a Breuil module of framed type ⊕ n-1 i=0 ω ai , and let e, f be a framed basis for M and a framed system of generators for Fil r M respectively.

Let

V = Mat e,f (Fil r M) ∈ M n (S) and A def = Mat e,f (ϕ r ) ∈ GL n (S)
be the matrices for the filtration and the Frobenius action respectively.

Then there exists a basis e for M k(( )) (M * ), framed with respect to the type ⊕ n-1 i=0 ω -ai , such that the Frobenius action is described by

Mat e (φ) = V t A -1 t ∈ M n (F ⊗ Fp k[[ ]]) where V , A are lifts of V, A in M n (F ⊗ Fp k[[ ]]) via the reduction morphism F ⊗ Fp k[[ ]] S F and Mat e (φ) ij ∈ F ⊗ Fp k[[ ]] ω a i -pa j .
Proof. This is Lemma 2.2.12 in [HLM] Lemma 2.18. Let M ∈ F-FL [0,p-2] be a rank n Fontaine-Laffaille module in parallel Hodge-Tate weights Proof. This is Lemma 2.2.13 in [HLM].

0 ≤ m 0 ≤ • • • ≤ m n-1 ≤ p -2 (counted
Finally, we need a technical result which lets us keep track of base changes on Breuil modules with descent data.

Lemma 2.19. Let M ∈ F-BrMod r dd be of type ⊕ n-1 i=0 ω ai and let e, f be respectively a framed basis for M and a framed system of generators for Fil r M.

Write V = Mat e,f (Fil r M), A def = Mat e,f (ϕ r ) to denote the matrix of the filtration and of the Frobenius respectively.

Assume that there exists an element r+1) .

V ∈ M n (S) such that (2.2.2) A • V ≡ V • w τ • B mod u e(
for some B ∈ GL n (S).

Then the element e def = e • A. defines a framed basis on M. Moreover:

(i) V • w -1 τ = Mat e ,f (Fil r M
) is a matrix of the filtration with respect to e and a system f of generators for Fil r M; (ii) ϕ(B) is the matrix of the Frobenius with respect to e , f . Proof. It easily follows from Lemma 2.2.14 in [HLM].

2.3. Classification of simple Breuil modules of rank 2. By [START_REF]Fp-représentations semi-stables[END_REF], Théorème 4.2.4 and the Remarque following it, the category F-BrMod r dd is additive and admits kernels and cokernels. In particular a complex

0 → M 0 f0 → M 1 f1 → M 2 → 0 in F-BrMod r
dd is exact if the morphisms f i induce exact sequences on the underlying Smodules M j and Fil r M j (j ∈ {0, 1, 2}). This endows F-BrMod r dd with the structure of an exact category.

In what follows, we give a slight improvement of a technical result in [HLM] (loc. cit., Lemma 2.2.2) concerning the submodule structure of a given Breuil module M ∈ F-BrMod r dd and which will be crucial to provide the classification of rank two irreducible objects in F-BrMod r dd . We recall the definition of Breuil submodule: Definition 2.20. Let M be an object in F-BrMod r

dd . An S-submodule N ⊆ M is said to be a Breuil submodule if N fulfills the following conditions:

(i) N is an S k -direct summand in M;

(ii) N is stable under the descent data action and the monodromy operator on M;

(iii) the Frobenius ϕ r on Fil r M restricts to a ϕ-semilinear morphism N ∩ Fil r M → N.

The importance of Definition 2.20 is explained in the following two propositions.

Lemma 2.21 ([HLM], Lemma 2.2.8). Let

0 → M 1 f → M → M 2 → 0 be an exact sequence in F-BrMod r dd . Then the S-module f (M 1 ) is a Breuil submodule of M.
Conversely if M is an object in F-BrMod r dd and N ⊆ M is a Breuil submodule of M, the pair (N, Fil r N def = Fil r M ∩ N) with the induced structures is an object of F-BrMod r dd in a natural way and the complex

0 → N → M → M/N → 0 is an exact sequence in F-BrMod r dd . In particular, if N is a Breuil submodule in M, then N is an S-direct summand of M.
Recall that we have a faithful, covariant functor T r st :

F-BrMod r dd → Rep F (G Qp ) (cf. Section 2.2.2) Proposition 2.22 ([HLM], Proposition 2.2.5). Let K ∈ {K 0 , Q p }.
With the above notion of exact sequence, the category F-BrMod r dd is an exact category in the sense of [Kel90] and T r st is an exact functor. Moreover, if M an object in F-BrMod r dd the functor T r st induces an order preserving bijection

Θ : {Breuil submodules in M} ∼ -→ {G K subrepresentations of T r st (M)} sending N ⊆ M to the image of T r st (N) → T r st (M) and canonically identifying Θ(M)/Θ(N) with T r st (M)/T r st (N)
. We now establish the main result of this section, namely the complete classification of rank 2 Breuil modules with descent data of niveau 2 relative to Q p . We start with a preliminary lemma:

Lemma 2.23. Let e = p 2 -1, K 0 = Q p 2 , K = K 0 ( e √ -p), and S = (F p 2 ⊗ Fp F)[u]/u ep . Let M ∈ F-BrMod s
dd be a rank two Breuil module, with descent data relative to

K 0 . Assume that T s st (M)| I K 0 ∼ = ω r+1 2 ⊕ ω p(r+1) 2
and the integers r, s ∈ N satisfy n(p + 1) + (s + 1) < r + 1 < (n + 1)(p + 1) -(s + 1) for some n ∈ Z.

Then we have a decomposition of Breuil modules

M ∼ = M k ⊕M l where T s st (M k )| I K 0 = ω r+1 2 and T s st (M l )| I K 0 = ω p(r+1) 2 .
Note that the numerical assumption on r, s implies s < p-1 2 . Proof. By Proposition 2.22, there exist Breuil submodules

M k and M l in M such that T s st (M k )| I Qp ω r+1 2 and T s st (M l )| I Qp ω p(r+1) 2 . Let us write M k = Sm k (resp. M l = Sm l ) with descent data ĝ(m k ) = 1 i=0 (ω 2 (g) ki ⊗ 1)m k (resp. ĝ(m l ) = 1 i=0 (ω 2 (g) li ⊗ 1)m l ), filtration Fil s M k = (u r0 e 0 + u r1 e 1 )m k (resp. Fil s M l = (u s0 e 0 + u s1 e 1 )m l ), Frobenius map ϕ s : (u r0 e 0 + u r1 e 1 )m k → λm k (resp. ϕ s : (u s0 e 0 + u s1 e 1 )m l → ηm l ),

and monodromy operator

N : m k → 0 (resp. N : m l → 0). Note that the integers k i , l i , r i , s i satisfy r i ≡ pk i+1 -k i mod (e) and s i ≡ pl i+1 -l i mod e (cf. [EGH13], Lemma 3.3.2).
Assume first that {m k , m l } is linearly independent in M over S. By comparing the cardinalities, it is clear that S(m k , m l ) = M, and so it is obvious that the Frobenius map ϕ s and the monodromy operator N on M are immediately determined by the ones on M k and M l . We have Fil s M ⊃ (u r0 e 0 + u r1 e 1 )m k , (u s0 e 0 + u s1 e 1 )m l . As the Frobenius on Fil s M k , Fil s M l is induced from the Frobenius on Fil s M, and since the Frobenius acts via λ, η ∈ F p 2 ⊗ Fp F on Fil s M k , Fil s M l , the previous inclusion is an equality. Hence, the Breuil module M is a direct sum of these two Breuil submodules in the obvious way.

We now check that {m k , m l } is linearly independent over S. Assume on the contrary that α m l = β m l for α , β ∈ S \ {0}. Then the minimal degree of α and β should be the same (if not, M k and M l would not have the same cardinality): more precisely, u i αe 0 m k = u i βe 0 m l , u j αe 1 m k = u j βe 1 m l , or both, for α, β ∈ S × and for i, j ∈ [0, ep).

Say, u i αe 0 m k = u i βe 0 m l . Then this immediately implies that k 0 ≡ l 0 mod (e). We check that this violates our numerical assumption on r and s. Since pr 0 + r 1 ≡ 0 mod (e) and ps 0 + s 1 ≡ 0 mod (e), we let pr 0 + r 1 = ae and ps 0 +

s 1 = be for 0 ≤ a, b ≤ s(p + 1). Since T s st (M k )| I Qp ω r+1 2 and T s st (M l )| I Qp ω p(r+1) 2
, we also have

k 0 + pa ≡ r + 1 mod (e); l 0 + pb ≡ p(r + 1) mod (e).
Subtracting the first one from the second one, (p -1)(r + 1) ≡ p(b -a) mod (e) and so we may let b -a = (p -1), and -(s + 1) ≤ ≤ s + 1 since s < p -1. Hence, r + 1 ≡ -mod (p + 1) and so we may let r + 1 = -+ δ(p + 1) for δ ∈ Z. Our assumption on r and s implies that n(p + 1) < δ(p + 1) = r + 1 + < (n + 1)(p + 1), which is obviously impossible.

Proposition 2.24.

Let e = p 2 -1, K 0 = Q p 2 , K = K 0 ( e √ -p
), and S = (F p 2 ⊗ Fp F)[u]/u ep . We let x and y be integers with x ≡ y mod (e) and M ∈ F-BrMod s dd be a Breuil module of type τ ω x ⊕ω y such that

T s st (M) is an absolutely irreducible 2-dimensional representation of G Qp , i.e, T s st (M)| I Qp ω r+1 2 ⊕ ω p(r+1) 2 . Assume further that n(p + 1) + (s + 1) < r + 1 < (n + 1)(p + 1) -(s + 1) for some n ∈ Z.
Then there exists a framed basis e = (e x , e y ) for M and a framed system of generators f = (f px , f py ) for Fil 2 M such that

• Mat e,f (Fil s M) = 0 u rx u ry 0 where 0 ≤ r x , r y ≤ es with r x ≡ py -x mod (e) and r y ≡ px -y mod (e);

• Mat e,f (ϕ s ) = λ x 0 0 λ y where λ x , λ y ∈ (F p 2 ⊗ Fp F) × ; • Mat e (ĝ) = ω x (g) ⊗ 1 0 0 ω y (g) ⊗ 1 for all g ∈ G(K/K 0 ); • N (e x ) = 0 = N (e y ); • T s st (M)| I Qp ω x+p prx +ry e 2 ⊕ ω y+p pry +rx e 2 .
Proof. By Lemma 2.23, we deduce that M has a basis e = (m k , m l ) over S, and a system of generators f = (f k , f l ) for Fil 2 M such that:

• Mat e,f (Fil s M) = u r0 e 0 + u r1 e 1 0 0 u s0 e 0 + u s1 e 1 where 0 ≤ r i , s i ≤ es with r i ≡ pk i-1 -k i mod (e) and s i ≡ pl i-1 -l i mod (e); • Mat e,f (ϕ s ) = λ 0 0 η where λ, η ∈ (F p 2 ⊗ Fp F) × ; • Mat e (ĝ) = 1 i=0 (ω ki (g) ⊗ 1)e i 0 0 1 i=0 (ω li (g) ⊗ 1)e i for all g ∈ G(K/K 0 ); • N (m k ) = 0 = N (m l ). Let σ be the unique lift in G(K/Q p ) of the arithmetic Frobenius in G(K 0 /Q p ) such that σ( e √ -p) = e √ -p
, and let us try to recover the action of

σ on M. Let σ(m k ) = α k m k + α l m l and σ(m l ) = β k m k + β l m l where α * , β * ∈ S. The identity σgσ -1 = g p for g ∈ G(K/K 0 )
gives rise to the following two identities:

•: [(ω pk0 (g)⊗1)e 1 +(ω pk1 (g)⊗1)e 0 ](α k m k +α l m l ) = σĝ(m k ) = ĝp σ(m k ) = ĝp (α k )[(ω pk0 (g)⊗ 1)e 0 + (ω pk1 (g) ⊗ 1)e 1 ]m k + ĝp (α l )[(ω pl0 (g) ⊗ 1)e 0 + (ω pl1 (g) ⊗ 1)e 1 ]m l ; •: [(ω pl0 (g)⊗1)e 1 + (ω pl1 (g)⊗1)e 0 ](β k m k +β l m l ) = σĝ(m l ) = ĝp σ(m l ) = ĝp (β k )[(ω pk0 (g)⊗ 1)e 0 + (ω pk1 (g) ⊗ 1)e 1 ]m k + ĝp (β l )[(ω pl0 (g) ⊗ 1)e 0 + (ω pl1 (g) ⊗ 1)e 1 ]m l .
Comparing the coefficients in these two identities, we have the following relations of descent data:

(i) k 1 ≡ a 0 + k 0 mod (e) and e 0 α k ∈ e 0 u a0 (S 0 ) × if e 0 α k = 0; k 0 ≡ a 1 + k 1 mod (e) and e 1 α k ∈ e 1 u a1 (S 0 ) × if e 1 α k = 0, (ii) k 1 ≡ b 0 + l 0 mod (e) and e 0 α l ∈ e 0 u b0 (S 0 ) × if e 0 α l = 0; k 0 ≡ b 1 + l 1 mod (e) and e 1 α l ∈ e 1 u b1 (S 0 ) × if e 1 α l = 0, (iii) l 1 ≡ c 0 + k 0 mod (e) and e 0 β k ∈ e 0 u c0 (S 0 ) × if e 0 β k = 0; l 0 ≡ c 1 + k 1 mod (e) and e 1 β k ∈ e 1 u c1 (S 0 ) × if e 1 β k = 0, (iv) l 1 ≡ d 0 + l 0 mod (e) and e 0 β l ∈ e 0 u d0 (S 0 ) × if e 0 β l = 0; l 0 ≡ d 1 + l 1 mod (e) and e 1 β l ∈ e 1 u d1 (S 0 ) × if e 1 β l = 0.
It is immediate that a 0 + a 1 ≡ 0 mod (e), b 0 + c 1 ≡ 0 mod (e), b 1 + c 0 ≡ 0 mod (e), and

d 0 + d 1 ≡ 0 mod (e).
Since Fil r M is stable under the action of σ, we have

σ(Fil r M) = (u r0 e 1 + u r1 e 0 )(α k m k + α l m l ), (u s0 e 1 + u s1 e 0 )(β k m k + β l m l )
⊂ Fil r M = (u r0 e 0 + u r1 e 1 )m k , (u s0 e 0 + u s1 e 1 )m l , which immediately implies the following inequalities: (a) r 1 + a 0 ≥ r 0 and

r 0 + a 1 ≥ r 1 ; (b) r 1 + b 0 ≥ s 0 and r 0 + b 1 ≥ s 1 ; (c) s 1 + c 0 ≥ r 0 and s 0 + c 1 ≥ r 1 ; (d) s 1 + d 0 ≥ s 0 and s 0 + d 1 ≥ s 1 . Since σ 2 = 1, we have 1 0 0 1 = α k σ(α k ) + β k σ(α l ) α k σ(β r ) + β k σ(β l ) α l σ(α k ) + β l σ(α l ) α l σ(β k ) + β l σ(β l ) .
From the (1, 1)-and (2, 2)-entries, we have the equations:

(2.3.1)

α k σ(α k ) = β l σ(β l ) and β k σ(α l ) = α l σ(β k ),
and so at least one of α k σ(α k ) and

β k σ(α l ) are in S × 0 . (Note that σ fixes the quantities in (2.3.1).) Assume that α k σ(α k ) ∈ S × 0 , i.e., a 0 + a 1 = 0. By the identity (2.3.1), d 0 + d 1 = 0. Hence, we have a 0 = a 1 = d 0 = d 1 = 0.
Then, by (i) and (iv), k 0 ≡ k 1 mod (e) and l 0 ≡ l 1 mod (e), and we also have r 0 = r 1 and s 0 = s 1 by (a) and (d). But this is impossible since we assume that the Breuil submodules Sm k and Sm l correspond to characters of niveau 2. Hence,

α k σ(α k ) ∈ S × 0 , i.e., either α k σ(α k ) = 0 or a 0 + a 1 > 0. Assume now that β k σ(α pr ) ∈ S × 0 , i.e., b 0 + c 1 = 0 = b 1 + c 0 . Thus, b 0 = b 1 = c 0 = c 1 = 0.
Then, by (ii) and (iii), k 0 ≡ l 1 mod (e) and k 1 ≡ l 0 mod (e), and we also have r 0 = s 1 and r 1 = s 0 by (b) and (c). We let x = k 0 , y = l 0 , r x = r 0 , and r y = s 0 . Then, by change of basis: e x = e 0 m k + e 1 m l and e y = e 1 m k + e 0 m l , we get the description in the statement.

The following lemma lets us specialize the result of Proposition 2.24 to a niveau 1 descent data:

Lemma 2.25. For i ∈ {1, 2}, let e i def = p i -1, K i def = Q p i ( e i √ -p) and S i def = F⊗ Fp F p i [u]/(u pei
). Let ι : S 1 → S 2 be the morphism defined by the embedding F p → F p 2 and u → u p+1 .

If M ∈ F-BrMod s dd is a Breuil module of niveau one of niveau one type, then M ⊗ S1,ι S 2 has a natural structure of a Breuil module of niveau 2 of niveau two type and the functor M → M ⊗ S1,ι S 2 is fully faithful. Moreover, one has

T * st (M) ∼ = T * st (M ⊗ S1,ι S 2 )
Proof. Just for the duration of this proof, let us write F-BrMod s,i dd to denote the category of Breuil modules with F-coefficients and descent data from

K i to Q p .
The exact sequence

1 → Gal(K 2 /K 1 ) → Gal(K 2 /Q p ) → Gal(K 1 /Q p ) → 0
shows that any object in F-BrMod s,i dd is naturally endowed, by inflation, with a niveau two descent datum. In particular, the natural morphism S 1 → S 2 factors through (S 2 ) Gal(K2/K1) ; by the explicit definition of the descent data action on S 2 , one checks that the previous factorization is indeed a isomorphism: S 1 ∼ -→ (S 2 ) Gal(K2/K1) . Hence, by endowing M⊗ S1 S 2 with the diagonal residual action of Gal(K 2 /Q p ), we deduce that the natural morphism

M → M ⊗ S1 S 2 factors through a (functorial) isomorphism M ∼ -→ (M ⊗ S1 S 2 ) Gal(K2/K1) . It follows that the functor M → M ⊗ S1 S 2 , defined on F-BrMod s,1
dd is fully faithful. As for the last statement, we recall the functor T * ,i st :

F-BrMod s,i dd → Gal F (G Qp ) is defined by M → Hom(M, A Ki ⊗ F p i F), where A Ki = F p i ⊗ O Q p /p X
is a certain a period ring described in [START_REF]Fp-représentations semi-stables[END_REF], Section 2.1 (where is simply noted as A, as in loc. cit. the extension F p i /F p has been fixed).

More importantly, one has

A Ki ∼ = A st /p ⊗ F p i u F p i [u]/u eip (cf.
[HLM], Section A.2). By virtue of the fully faithfulness of M → M ⊗ S1 S 2 , the last statement follows once we show that

A K1 ⊗ S1 S 2 → A K2
is an isomorphism, which can be verified by a direct computation on the definition of A Ki .

We deduce:

Corollary 2.26. Let e = p -1, K = Q p ( e √ -p
), and S = F[u]/u ep . We also let x and y be integers with x ≡ y mod (e), and let M ∈ F-BrMod s dd be a Breuil module of type

τ ω x ⊕ ω y such that T s st (M) is an absolutely irreducible 2-dimensional representation of G Qp , i.e, T s st (M)| I Qp ω r+1 2 ⊕ ω p(r+1) 2
. Assume further that n(p + 1) + (s + 1) < r + 1 < (n + 1)(p + 1) -(s + 1) for some n ∈ Z.

Then there exists a framed basis e = (e x , e y ) for M and a framed system of generators f = (f x , f y ) for Fil 2 M such that

• Mat e,f (Fil s M) = 0 u rx u ry 0 where 0 ≤ r x , r y ≤ es with r x ≡ y -x mod (e) and r y ≡ x -y mod (e);

• Mat e,f (ϕ s ) = λ x 0 0 λ y where λ x , λ y ∈ F × ;

• Mat e (ĝ) = ω x (g) ⊗ 1 0 0 ω y (g) ⊗ 1 for all g ∈ G(K/Q p );
• N (e x ) = 0 = N (e y ).

• T s st (M)| I Qp ω (p+1)x+p prx +ry e 2 ⊕ ω (p+1)y+p pry +rx e 2 .
Proof. Using the notation of Lemma 2.25, it suffices to apply Proposition 2.24 to M ⊗ S1 S 2 and then take the Gal(K 2 /K 1 )-fixed part.

2.4. Crystalline lifts. We end this section with certain results for crystalline lifts of ρ 0 . The results in this subsection will be used in Section 6.5.

Proposition 2.27. Let ρ 0 be as in Definition 2.4. Then ρ 0 admits a crystalline lift ρ :

G Qp → GL 3 (Q p ) such that ρ| G Q p 2
is ordinary crystalline, with parallel Hodge-Tate weights {a 2 + 1, a 1 + 1, a 0 + 1}. In particular ρ is potentially diagonalizable. Moreover, if FL(ρ 0 ) = [0 : 1] then ρ 0 admits a crystalline lift with Hodge-Tate weights

{p + a 0 + 1, a 2 + 1, a 1 }.
Finally if ρ 0 is split then then ρ 0 admits further crystalline lift with Hodge-Tate weights {p + a 1 , p + a 0 , a 2 + 1}.

The proof of Proposition 2.27 will occupy the reminder of this section. Let α, β ∈ Z. By [GS] Lemma 6.2, there is a crystalline character ε (α,β) : Proof. Indeed, we have

G Q p 2 → O × E , unique up to unramified twist such that HT σ0 (ε (α,β) ) = α, HT σ1 (ε (α,β) ) = β; such a character verifies moreover ε (α,β) | I Qp = ω α+pβ 2 . If V (α,β) def = Ind G Qp G Q p 2 ε (α,β) then V (α,β) ⊗ O E F = Ind G Qp G Q p 2
V (α,β) | G Q p 2 = ε (α,β) ⊕ ε (1) (α,β) , where we have defined the G Q p 2 - character ε (1) (α,β) by g → ε (α,β) (Frob -1 p •g • Frob p )
where Frob p denotes a geometric Frobenius. By [GHS], Lemma 7.1.2 we have that HT σ0 (ε

(1) (α,β) ) = β, HT σ1 (ε (1) (α,β) ) = α. The representation V (α,β) | G Q p 2 is crystalline, as crystalline property is insensitive to unramified base change. If γ ∈ Z we define the space of O E -valued crystalline extensions Ext 1 O E [G Qp ],cris (V (α,β) , ε γ p ) as the inverse image (under base change O E → E) of Ext 1 E[G Qp ],cris (V (α,β) ⊗ O E E, ε γ p ⊗ O E E)
. By an immediate application of Hochschild-Serre spectral sequence and since the crystalline condition is insensitive with respect to restriction to unramified base change, we have the following commutative diagram:

(2.4.1) Ext 1 O E [G Qp ],cris (V (α,β) , ε γ p ) ∼ / / _ Ext 1 O E [G Q p 2 ],cris (ε (α,β) ⊕ ε (1) (α,β) , ε (γ,γ) G2 _ Ext 1 O E [G Qp ] (V (α,β) , ε γ p ) ∼ / / Ext 1 O E [G Q p 2 ] (ε (α,β) ⊕ ε (1) (α,β) , ε (γ,γ) ) G2 Ext 1 F[G Qp ] (Ind G Qp G Q p 2 ω α+pβ 2 , ω γ ) ∼ / / Ext 1 F[G Q p 2 ] (ω α+pβ 2 ⊕ ω β+pα 2 , ω (p+1)γ 2 ) G2
where the bottom vertical arrows are the mod E -reduction maps and

G 2 def = Gal(Q p 2 /Q p ).
The following technical lemma is a simple manipulation with Fontaine-Laffaille modules. In its statement, we set e 0 def = e σ0 , e 1 def = e σ0•Frobp for the standard idempotent elements of F p 2 ⊗ Fp F, following the notation of Section 1.1.

Lemma 2.29. Let M ∈ F-FL [0,p-2] be a Fontaine-Laffaille module over F p ⊗ Fp F, with Hodge-Tate weghts (β, α, γ). Assume that

Mat f (φ • ) =   0 λ 1 x λ 0 0 y 0 0 λ 2   (2.4.2) in a basis f = (f 0 , f 1 , f 2 )
which is compatible with the Hodge filtration on M . Then if we write M for the induced Breuil module F p 2 ⊗ Fp M , we have two Fontaine-Laffaille quotients M N , M N (1) of rank two over F p 2 ⊗ Fp F. Explicitly, we have N = N e 0 ⊕ N e 1 where N e i are F-linear spaces, with Hodge-Tate weights (α, γ) and (β, γ) for i = 0 and i = 1 respectively, and

Mat(N e 1 φ0 → N e 0 ) = λ 0 y 0 λ 2 & Mat(N e 0 φ1 → N e 1 ) = λ 1 x 0 λ 2
We have a similar description for N (1) = N (1) e 0 ⊕ N (1) e 1 :

Mat(N (1) e 1 φ0 → N (1) e 0 ) = λ 1 x 0 λ 2 & Mat(N (1) e 0 φ1 → N (1) e 1 ) = λ 0 y 0 λ 2
and N (1) e 0 , N (1) e 1 have Hodge-Tate weights (β, γ), (α, γ) respectively.

Proof. This is elementary. Let f = (f 0 , f 1 , f 2 ) be a basis on M , compatible with the Hodge filtration, such that the matrix of the Frobenius on M is given by (2.4.2). In particular, we have

Fil i+1 M =        M if i < β f 1 , f 2 F if β ≤ i < α f 2 F if α ≤ i < γ 0 if i ≥ γ Then, considering the change of basis we get 1 ⊗ f def = (1 ⊗ f 0 , 1 ⊗ f 1 , 1 ⊗ f 2 ) •   e 0 e 1 0 e 1 e 0 0 0 0 1   we obtain Mat 1⊗f (φ • ) =   λ 1 e 0 + λ 0 e 1 0 xe 0 + ye 1 0 λ 0 e 0 + λ 1 e 1 ye 0 + xe 1 0 0 λ 2   .
We define N to be the Fontaine-Laffaille quotient characterized by

ker(M N ) = (1 ⊗ f 0 ) • e 0 + (1 ⊗ f 1 ) • e 1 .
This is well-defined since the kernel is a rank one submodule. Note that, by construction, we have

Fil i+1 N e 0 def = Fil i+1 M e 0 + (1 ⊗ f 0 ) • e 0 (1 ⊗ f 0 ) • e 0 =              M e0 (1⊗f0)•e0 = N e 0 if i < α (1⊗f2)•e0,(1⊗f0)•e0 (1⊗f0)•e0 if α ≤ i < γ 0 if i ≥ γ Fil i+1 N e 1 def = Fil i+1 M e 1 + (1 ⊗ f 1 ) • e 1 (1 ⊗ f 1 ) • e 1 =              M e1 (1⊗f1)•e1 = N e 1 if i < β (1⊗f1)•e1,(1⊗f2)•e1 (1⊗f1)•e1 if β ≤ i < γ 0 if i ≥ γ.
Hence, N has Hodge-Tate weights HT σ0 = {α, γ} and HT σ0•Frobp = {β, γ}.

Similarly, one takes N (1) to be the Fontaine-Laffaille quotient of M characterized by ker(M

N (1) ) = (1 ⊗ f 0 ) • e 1 + (1 ⊗ f 1 ) • e 0 .
This is well-defined by the same reason as N .

We deduce from Lemma 2.29:

Lemma 2.30. Assume that ρ 0 is as in Definition 2.4. Let M ∈ F-FL [0,p-2] be the associated Fontaine-Laffaille module and fix a basis on it in such a way that Mat f (φ • ) has the form (2.1.3), with moreover z = 0. Let τ be the image of ρ 0 | G Q p 2 via the projection map

(2.4.3) Ext 1 F[G Q p 2 ] ω (a1+1)+p(a0+1) 2 ⊕ ω (a0+1)+p(a1+1) 2
, ω

(p+1)(a2+1) 2 Ext 1 F[G Q p 2 ] ω (a1+1)+p(a0+1) 2 , ω (p+1)(a2+1) 2
.

Then τ has a crystalline lift with Hodge-Tate weights HT σ0 = {a 2 + 1, a 1 + 1}, HT σ1 = {a 2 + 1, a 0 + 1}.

If moreover FL(ρ 0 ) = [0 : 1] then τ has also a crystalline lift with Hodge-Tate weights

HT σ0 = {a 2 + 1, a 1 }, HT σ1 = {p + a 0 + 1, a 2 + 1}.
If finally ρ 0 is split then τ admits further a crystalline lift with the following Hodge-Tate weights HT σ0 = {p + a 1 , a 2 + 1}, HT σ1 = {p + a 0 , a 2 + 1}.

Proof. We can assume that a 0 = -1 and set

c def = a 2 -a 0 -1, r def = a 1 -a 0 -1.
By Lemma 2.29 we see that the Fontaine-Laffaille module N = N e 0 + N e 1 associated to τ has Hodge-Tate weights HT σ0 (N e 0 ) = {r + 1, c + 1}, HT σ1 (N e 1 ) = {0, c + 1} and Frobenius described by

Mat(N e 1 φ0 → N e 0 ) = µ -1 0 y 0 µ -1 2 (2.4.4) Mat(N e 0 φ1 → N e 1 ) = µ -1 1 x 0 µ -1 2 (2.4.5)
We now use the explicit description of the set of modular weights for τ , given in [Bre14] pag. 26. Following the notation in loc. cit. we deduce from (2.4.4) that the weight (c -r -1, c) ⊗ det r+1 (which would be written as σ (c,c),(r+1,0) in the notation of [GLS], Definition 4.1.1) is always modular, while the weight (c -r, p -2 -c) ⊗ det r+p(c+1) (i.e. σ (c,p-1),(r,c+1) in the notation of [GLS]) is modular when x = 0. For sake of completeness, the weight (p -2 -c + r, p -3 -c) ⊗ det c+1+p(c+1) i.e. σ (p-1+r,p-2),(c+1,c+1) in the notation of [GLS] is modular when x = y = 0. We now can globalize τ : by [GK], Corollary A.3 there is a totally real field F + such that F + v ∼ = Q p for all places v|p, and a RAESDC automorphic representation π of GL 2 (A F + ) such that the mod p reduction of the associated p-adic Galois representation rp,ı (π) :

G F + → GL 2 (F) (cf. [BLGGT] §2.1) is absolutely irreducible (modular) and verifies rp,ı (π)| G F + v
∼ = τ for all places v|p. The conclusion follows from [GLS],

Theorem A.

Proof of Proposition 2.27. The existence of the crystalline lifts as in the statement of Proposition 2.27 follows now from Lemma 2.30 and the diagram (2.4.1). More precisely, let

τ ⊕ τ (1) be the image of ρ 0 in Ext 1 F[G Q p 2 ] ω (a1+1)+p(a0+1) 2 , ω (p+1)
G Q p 2 → GL 2 (O E ) be defined by τ (1) (g) def = τ (Frob -1 p g Frob p )
we see that τ (1) is a crystalline lift of τ (1) with Hodge-Tate weights HT σ1 = {α, γ}, HT σ0 = {β, γ}. By con-

struction τ ⊕ τ (1) ∈ Ext 1 O E [G Q p 2 ],cris (ε (α,β) ⊕ ε (1) (α,β) , ε (γ,γ)
) is fixed under the G 2 -action on the Ext 1 -space. Its inverse image via the isomorphism in the first line of the diagram (2.4.1) provides the required crystalline lift.

Moreover, any element of Ext 1

E[G Qp ],cris (V (α,β) ⊗ O E E, ε γ p )
becomes ordinary when restricted to G Q p 2 , as it can be directly checked on the associated filtered φ-module.

Remark 2.31. The existence of the crystalline lift for ρ 0 with Hodge-Tate weights {a 2 + a 1 + 1, a 0 +1} can be obtained without invoking [GLS]. Indeed if (α 0 , . . . , α f -1 ), (α 0 , . . . , α f -1 ) ∈ Z f are such that α i -α i > 1 for all i, then we are in the setting of [Nak], Lemma 4.2(1) and Lemma 4.3(3), so that

dim E Ext 1 E[G Q p f ],cris (ε (α0,...,α f -1 ) , ε (α 0 ,...,α f -1 ) ) = f
(cf. also loc. cit., Definition 2.4 and Remark 2.5). On the other hand, under the previous hypotheses on α i -α i , we have also dim

E Ext 1 E[G Q p f ] (ε (α0,...,α f -1 ) , ε (α 0 ,...,α f -1 ) ) = f ([Nak], Proposition 2.15).

Elimination of Galois types

The aim of this section is to perform elimination of Galois types for a niveau 2, generic representation ρ 0 : G Qp → GL 3 (F) (cf. Definition 2.4), by means of integral p-adic Hodge theory.

For K ∈ {Q p , K 0 } we recall the category Mod w.a. E (ϕ, N, K/K ) of weakly admissible filtered (ϕ, N, K/K , E)-modules (see e.g. [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 3.1). We have a contravariant equivalence of categories D * ,K st

: Rep K-st E (G K ) → Mod w.a. E (ϕ, N, K/K ), where Rep K-st E (G K ) denotes the category of finite dimensional E-representations of G K that be- come semistable over K. If ρ ∈ Rep K-cris E (G K ) has Hodge-Tate weights in {-r, 0}, we define D K ,r st (ρ) def = D * ,K st ρ ∨ ⊗ ε r p .
The following result will be particularly useful to us:

Proposition 3.1. Let ρ : G Qp → GL 3 (O E
) be a potentially semistable Galois representation, becoming crystalline over K in Hodge-Tate weights in {-r, 0}. Let M be a strongly divisible

O E -module in O E -Mod r dd such that T Qp,r st ( M) ⊗ O E E ∼ = ρ. Then D Qp,r st (ρ) ∼ = M[ 1 p ] ⊗ S Qp ,s0 Q p and and M has inertial type WD(ρ ⊗ ε -r p )| I Qp = WD(ρ)| I Qp (where s 0 : S Qp → Q p is the morphism defined by "u → 0"). Proof. The isomorphism D Qp,r st (ρ) ∼ = M[ 1 p ] ⊗ S Qp ,s0 Q p is proved in [EGH13], proof of Propo- sition 3.1.4.
As for the second part of the proposition, let us write WD(ρ

)| I Qp ∼ = χ 1 ⊕ • • • ⊕ χ n for the inertial type associated to ρ.
By definition of type on a strongly divisible lattice M, we have to prove that there exists a basis ( e 1 , . . . , e n ) of M such that g • e i = 1⊗χ i (g) e i for all g ∈ Gal(K/K 0 ) and i = 1, . . . , n.

For r = 1 this is proved in [START_REF] Gee | Serre weights for quaternion algebras[END_REF], Proposition 5.1 (note that the functors M → T Recall that the restriction functor ρ 0 → ρ 0 | G K 0 is not full. The following elementary lemma shows that in our situation, the Fontaine-Laffaille invariant FL(ρ 0 ) can be deduced from ρ 0 | G K 0 . Lemma 3.2. Let ρ 0 be as in Definition 2.4 and let F ∈ GL 3 (F) be the matrix describing the Frobenius action on the associated Fontaine-Laffaille module as in (2.1.3).

Assume that the Fontaine-Laffaille module M associated to ρ 0 | G K 0 has parallel Hodge-Tate weights {0, r + 1, c + 1} and Frobenius action described by

F def =   0 λ 1 X λ 0 0 Y 0 0 λ 2   ∈ GL 3 (k ⊗ Fp F).
Then X = 0 if and only if x = 0, and Y = 0 if and only if y = 0.

Proof. In the given hypotheses, we have an isomorphism of Fontaine-Laffaille modules (in parallel Hodge-Tate weights {0, r + 1, c + 1}) over k ⊗ Fp F. This means that there exists a lower triangular matrix

B ∈ B opp (k ⊗ Fp F) such that (3.0.6) B • F • (ϕ ⊗ 1)(gr(B)) = F ⊗ Fp k,
where gr(B) ∈ T(k ⊗ Fp F) is defined by (gr(B)) ii = (B) ii for i = 0, 1, 2 and ϕ ⊗ 1 denotes the induced Frobenius automorphism on k ⊗ Fp F.

By an immediate computation we deduce that condition (3.0.6) forces B to be diagonal. In particular, there exists units α, β, γ ∈ k ⊗ Fp F such that 1 ⊗ x = ασ(γ)X and βσ(γ)Y .

As the natural morphism F → k ⊗ Fp F is injective, the result follows.

For the reminder of this section, we assume that a 0 = -1 and define c def = a 2 -a 0 + 1, r def = a 1 -a 0 -1 (it is always possible to reduce to this case by twisting by ω -(a0+1) ).

3.1. Elimination of Galois types of niveau 1. We start this subsection by recalling the following (cf. [MP], Lemma 3.3): let i, j, k be integers , and let ρ be a potentially crystalline representation with Hodge-Tate weights {-2, -1, 0} and of inertial type ω i ⊕ ω j ⊕ ω k such that ρ ss 0 ρ ss . Then we have the identity

(3.1.1) ω 3+i+j+k = det ρ| I Qp = ω (r+1)+(c+1) .
In this subsection, we fix e = p -1 and K = Q p ( e √ -p). We also let S = F[u]/u ep and S 0 = F[u e ]/u ep . Recall that by [m] 1 for an integer m we mean the unique integer in [0, e) congruent to m mod (e).

Proposition 3.3. Let M ∈ F-BrMod 2 dd be a Breuil module of type τ ∼ = ω z ⊕ ω x ⊕ ω y such that T 2 st (M) ss ∼ = ρ ss 0 and ρ 2 ⊂ T 2 st (M)
, where ρ 2 is the one-dimensional subrepresentation of ρ 0 . Assume moreover that the submodule corresponding to ρ 2 is of type ω z .

Then there exists a framed basis e = (e z , e x , e y ) and a framed system of generators f for Fil 2 M such that

(3.1.2) Mat e,f (Fil 2 M) =   u se u [x-z]1 • v y u [y-z]1 • v x 0 0 u rx 0 u ry 0   ; (3.1.3) Mat e,f (ϕ 2 ) =   α z u [x-z]1 • η x u [y-z]1 • η y 0 α x 0 0 0 α y   ,
where α x , α y , α z ∈ F × , v x , v y , η x , η y ∈ S 0 , and the tuple (x, y, z, r x , r y , s) satisfies one of the following properties:

(a):

x ≡ r + 1 -m 0 mod (p -1), y ≡ 0 mod (p -1), z ≡ c + 1 -m 2 mod (p -1), and
   r x = (p -1)m 0 -(r + 1 -m 0 ); r y = r + 1 -m 0 ; s = m 2 ,
where m 0 , m 2 ∈ {1, 2} satisfy m 0 + m 2 = 3; (b): x ≡ r -m 0 mod (p -1), y ≡ p -2 mod (p -1), z ≡ c + 1 -m 2 mod (p -1), and

   r x = (p -1)(m 0 + 1) -(r + 1 -m 0 ); r y = (p -1) + (r + 1 -m 0 ); s = m 2 ,
where m 0 , m 2 ∈ {0, 1} satisfy m 0 + m 2 = 1.

Proof. Since ρ 0 is an extension of a two-dimensional irreducible representation by a character of niveau 1, M is also an extension of a simple Breuil module of rank 2 by a Breuil module of rank 1 by Proposition 2.22. Hence, it is immediate that the filtration and the Frobenius map ϕ 2 of M are described as in (3.1.2) and (3.1.3) respectively, by using the classification of simple Breuil modules of rank 2 in Corollary 2.26 and the classification of simple Breuil modules of rank 1 in [MP], Lemma 3.1.

By Corollary 2.26 we have r x ≡ y -x mod e and r y ≡ x -y mod e, r x + r y ≡ 0 mod e. We let r x + r y = ae for a ∈ {0, 1, 2, 3, 4}. Again by Corollary 2.26, we have (3.1.4)    (p + 1)x + pr x + pa ≡ r + 1 mod (p 2 -1); (p + 1)y + pr y + pa ≡ p(r + 1) mod (p 2 -1); z + s ≡ c + 1 mod (p -1).

By the determinant condition (3.1.1), 3(p + 1) + (r + 1) -p(r x + a) + p(r + 1) -p(r y + a) + (p + 1)(c + 1 -s) ≡ (p + 1)(c + 1 + r + 1) mod (p 2 -1). Hence, we get a + s = 3, and so a ∈ {1, 2, 3} since s ∈ {0, 1, 2}.

Via the equations (3.1.4) we now write r y in terms of a and the inertial weights z, x, y. We have (p + 1)r y ≡ (p + 1)(x -y) ≡ (1 -p)(r + 1) -p(r x -r y ) mod (p 2 -1). So r y ≡ -(p -1)(r + 1) -p(ae -r y ) mod (p 2 -1). Solving this for r y , we get r y ≡ r + 1 -a mod (p + 1). We let r y = r + 1 -a + (p + 1) for ∈ {0, 1} (since 0 ≤ r y ≤ 2e). Then r x = ae-(r+1-a)-(p+1). Moreover, by the equations (3.1.4), we also have x ≡ r+1-a+ mod e and y ≡ e -mod e. We let s = m 2 . Then we have a + m 2 = 3.

Assume that = 0. If m 2 = 0, then a = 3, and so r x = 3e -(r + 1 -3) > 2e, which contradicts to r x ∈ [0, 2e]. Hence, a, m 2 ∈ {1, 2} and this gives rise to the case (a), letting m 0 = a.

Assume that = 1. If m 2 = 2, then a = 1, and so r x = (p -1) -r -(p + 1) < 0, which contradicts to r x ∈ [0, 2e]. Hence, m 2 ∈ {0, 1} and a ∈ {2, 3}. Letting m 0 = a -2, this gives rise to the case (b).

Lemma 3.4. Keep the notation as in Proposition 3.3 (in particular, recall the elements v x and v y in the matrix (3.1.2)) and let s = 1.

(i) If r x -[y -z] 1 > e then there is a framed basis for which v x = 0. (ii) If r y -[x -z] 1 > e then there is a framed basis for which v y = 0.

Proof. Since s = 1, we may assume that v x , v y ∈ F. We only give a proof for (i), but one can prove (ii) by the same argument. Assume that v x = 0. Then the matrix (3.1.2) is column-equivalent to

  0 u [x-z]1 • v y u [y-z]1 • v x u rx+e-[y-z]1 0 u rx 0 u ry 0   , which implies that Fil 2 M ⊗ S S/u ∼ = ω x ⊕ ω x ⊕ ω y , since r x + e -[y -z] 1 > 2e.
But this is impossible unless x ≡ z mod (p -1).

Lemma 3.5. Keep the notation as in Proposition 3.3. If

p([y -z] 1 + r y -se) > [x -z] 1 and p([x -z] 1 + r x -se) > [y -z] 1
then there is a framed basis such that η x = 0 = η y in the matrix (3.1.3). Moreover, this change of basis does not affect the vanishing of v x and v y .

Proof. We let V 0 be the matrix in (3.1.2) and A 0 the matrix in (3.1.3). We also let

V 1 =   u se u [x-z]1 • v y u [y-z]1 • v x 0 0 u rx 0 u ry 0   and B 1 =   α z u [x-z]1 • η x u [y-z]1 • η y 0 α y 0 0 0 α x   .
One can easily check that the equation (3.1.5)

A 0 V 1 = V 0 B 1
holds if and only if the following two equalities hold:

α z u [x-z]1 v y + u [y-z]1+ry η y = u se+[x-z]1 η x + α y u [x-z]1 v y ; α z u [y-z]1 v x + u [x-z]1+rx η x = u se+[y-z]1 η y + α x u [y-z]1 v x .
Hence, the equation (3.1.5) holds true if we let

v x = α x α -1 z v x , v y = α y α -1 z v y , u [x-z]1 η x = u [y-z]1+ry-se η y ∈ S, and u [y-z]1 η y = u [x-z]1+rx-se η x ∈ S.
Note that our assumption implies that [x -z] 1 + r x -se ≥ 0 and [y -z] 1 + r y -se ≥ 0. Now let us consider the new basis

e def = eA 0 . Then V 1 = Mat e ,f (Fil 2 M) and A 1 def = ϕ(B 1 ) = Mat e ,f (ϕ 2 )
, where f is the system of generators given by the column vectors of V 1 . By our hypothesis the (1, 2)-entry and (1, 3)-entry of A 1 can be written as follows:

ϕ(u [x-z]1 η x ) = u [x-z]1 u p([y-z]1+ry-se)-[x-z]1 ϕ(η y ) and ϕ(u [y-z]1 η y ) = u [y-z]1 u p([x-z]1+rx-se)-[y-z]1 ϕ(η x ).
As p([y -z] 1 +r y -se)-[x-z] 1 , p([x-z] 1 +r x -se)-[y -z] 1 > 0, by iterating the previous procedure, we end up with a basis with the required properties. For the last statement, it is obvious that v x = 0 (resp. v y = 0) if and only if v x = 0 (resp. v y = 0). Proposition 3.6. Keep the notation as in Proposition 3.3 and assume ρ 0 ∼ = T 2 st (M). (i) If s = 1 in the case (a) and ρ 0 is non-split, then FL(ρ 0 ) = [0 : 1]. (ii) If s = 0 in the case (b), then ρ 0 splits as a sum of a two-dimensional irreducible representation and a character.

Proof. Assume that s = 1 in the case (a), Proposition 3.3, i.e., (m 2 , m 0 ) = (1, 2). Then x ≡ r -1 mod e, y ≡ 0 mod e, z ≡ c mod e, r x = 2e -(r -1), r y = r -1, and s = 1.

Clearly, [x -z] 1 = e -c + (r -1) and [y -z] 1 = e -c. Then by Lemma 3.5, we can assume v x = 0 in the matrix (3.1.2), and by the Lemma 3.5, we can assume η x = 0 = η y in the matrix (3.1.3). We can also assume that v y ∈ F as s = 1. Let V be the matrix (3.1.2) and A the matrix (3.1.3). By Proposition 2.13, the φ-module over F ⊗ Fp F p (( )) defined by

M def = M Fp(( )) (M * ) is described by (3.1.6) Mat e (φ) = V t ( Â-1 ) t =   α -1 z e 0 0 α -1 z [x-z]1 • v y 0 α -1 y ry 0 α -1 x rx 0  
in an appropriate basis e = (e z , e x , e y ). By considering the change of basis e = ( c e z , r-1 e x , e y ) we have:

Mat e (φ) =   α -1 z e(c+1) 0 0 α -1 z v y e(c+1) 0 α -1 y 0 α -1 x e(r+1) 0   .
We easily see that the φ-module M is the base change via

F ⊗ Fp F p ((p)) → F ⊗ Fp F p (( )) of the φ-module M 0 over F ⊗ Fp F p ((p)) described by Mat(φ 0 ) =   α -1 z p (c+1) 0 0 α -1 z v y p (c+1) 0 α -1 y 0 α -1 x p (r+1) 0   .
Now we can find a basis for M 0 such that

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   0 α -1 x 0 α -1 y 0 α -1 z v y 0 0 α -1 z   ,
and so FL(ρ 0 ) = [0 : 1] as ρ 0 is non-split.

Assume that s = 0 in the case (b), Proposition 3.3, i.e., (m 2 , m 0 ) = (0, 1). Since s = 0, we can assume v x = 0 = v y . One can readily check that we can assume η x = 0 = η y as well, using Lemma 3.5. By the same argument as above, it is easy to check that

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   0 α -1 x 0 α -1 y 0 0 0 0 α -1 z  
(the only difference is the base change: e = ( c+1 e z , r-1 e x , -1 e y )). Hence, the corresponding representation ρ 0 splits as a sum of a two-dimensional irreducible representation and a character.

3.2. Elimination of Galois types of niveau 2. We start this subsection by recalling the following (cf. [MP], Lemma 3.3): let j, k be integers with k ≡ 0 mod (p + 1), and let ρ be a potentially crystalline representation with Hodge-Tate weights {-2, -1, 0} and inertial type ω j ⊕ ω k 2 ⊕ ω pk 2 such that ρ ss 0 ρ ss . Then we have the identity (3.2.1)

ω 3+j+k = det ρ| I Qp = ω (r+1)+(c+1) .
In this section, we fix e = p 2 -1, K 0 = Q p 2 , and K = K 0 ( e √ -p). We also let S = (F p 2 ⊗ Fp F)[u]/u ep and S 0 = (F p 2 ⊗ Fp F)[u e ]/u ep . Recall that by [m] 2 for an integer m we mean the unique integer in [0, e) congruent to m mod (e).

Proposition 3.7. Let M ∈ F-BrMod 2 dd be a Breuil module over S of type τ ω z ⊕ ω x ⊕ ω y such that T 2 st (M) ss ∼ = ρ ss 0 and ρ 2 ⊂ T 2 st (M), where ρ 2 is the one-dimensional subrepresentation of ρ 0 . Assume that the submodule corresponding to ρ 2 has descent data ω z 2 . Then there exists a framed basis e = (e z , e x , e y ) and a framed system of generators f such that

(3.2.2) Mat e,f (Fil 2 M) =   u s(p-1) u [px-z]2 • v y u [py-z]2 • v x 0 0 u rx 0 u ry 0   ; (3.2.3) Mat e,f (ϕ 2 ) =   α z u [x-z]2 • η x u [y-z]2 • η y 0 α x 0 0 0 α y   ,
where α x , α y , α z ∈ (F p 2 ⊗ F) × , v x , v y ∈ S 0 , and the tuple (x, y, z, r x , r y , s) satisfies the following properties:

(a): if x ≡ k mod (e), y ≡ pk mod (e), and z ≡ (p + 1)j mod (e), then Proof. Since ρ 0 is an extension of a two-dimensional irreducible representation by a character of niveau 1, M is also an extension of a simple Breuil module of rank 2 by a Breuil module of rank 1 by Proposition 2.22. Hence, it is immediate that the filtration and the Frobenius map ϕ 2 of M are described as in (3.2.2) and (3.2.3) respectively, by using the classification of simple Breuil modules of rank 2 in Proposition 2.24 and the classification of simple Breuil modules of rank 1 in [MP], Lemma 3.1. Recall from Proposition 2.24 that (3.2.4) r x ≡ py -x mod (e), r y ≡ px -y mod (e), and z + ps ≡ 0 mod (p + 1).

j ≡ c + 1 -m 2 mod (p -1), k ≡ r + 1 -m 0 -pm 1 mod (e),
We also recall that 0 ≤ r x , r y ≤ 2e, 0 ≤ s ≤ 2(p + 1), and by Lemma 3.3.2 in [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF] and by Proposition 2.24 we have:

(3.2.5)

x + p prx+ry e ≡ r + 1 mod (e); z + ps ≡ (p + 1)(c + 1) mod (e).

For case (a), assume that x ≡ k mod (e), y ≡ pk mod (e), and z ≡ (p + 1)j mod (e). We let r x = m 0 e, r y = m 1 e, and s = (p + 1)m 2 for m i ∈ {0, 1, 2}, due to the equation (3.2.4).

(since 0 ≤ r x , r y ≤ 2e). We have ae = pr x + r y = (2δ + -c -4 + pm 0 + m 1 )e so that δ -(c + 1) + p = a = 2δ + -c -4 + pm 0 + m 1 . Hence, we have δ + -3 + m 1 = p( -m 0 ) which easily implies m 0 = and m 1 = 3 -δ -. The requirement m 1 ∈ {0, 1} implies that (δ, ) = (0, 1). Lemma 3.8. Keep the notation as in Proposition 3.7 (in particular, recall the elements v x and v y in the matrix (3.2.2)) and assume s ≤ p + 1.

(i) If r x + s(p -1) -[py -z] 2 > 2e then there is a basis such that v x = 0. (ii) If r y + s(p -1) -[px -z] 2 > 2e then there is a basis such that v y = 0.
Proof. The same argument as in Lemma 3.4 works. Lemma 3.9. Keep the notation as in Proposition 3.7 (in particular, recall the elements η x and η y in the matrix (3.2.3)).

(i) If [x -z] 2 + r x -s(p -1) + e ≥ 0 and [y -z] 2 + r y -s(p -1) -e ≥ 0 then there is a basis such that η x ∈ F p 2 ⊗ Fp F and η y = 0. (ii) If [x -z] 2 + r x -s(p -1) -e ≥ 0 and [y -z] 2 + r y -s(p -1) + e ≥ 0 then there is a basis such that η x = 0 and η y ∈ F p 2 ⊗ Fp F. (iii) If p([x -z] 2 + r x -s(p -1)) > [y -z] 2 and p([y -z] 2 + r y -s(p -1)) > [x -z] 2
then there is a basis such that η x = 0 and η y = 0.

Moreover, the change of basis does not affect the vanishing of v x and v y .

Proof. One can prove case (iii) by the same argument as in Lemma 3.5, and case (i) is similar to case (ii). We only provide with a proof for case (ii). Let V 0 be the matrix (3.2.2) and A 0 the matrix (3.2.3). We define ηy ∈ u e • S 0 by η y = η 0 y + ηy with η 0 y ∈ F p 2 ⊗ Fp F and let Ã0 be the matrix obtained from A 0 by replacing η y in A 0 by ηy . We also let

B 1 =   α z u [p -1 (x-z)]2 • η x u [p -1 (y-z)]2 • η y 0 α y 0 0 0 α x  
for some η x , η y ∈ S 0 . One can easily check that the equation

(3.2.7) Ã0 V 1 = V 0 B 1
holds true if and only if the following two equalities hold:

α z u [px-z]2 v y + u [y-z]2+ry ηy = u s(p-1)+[p -1 (x-z)]2 η x + α y u [px-z]2 v y ; α z u [py-z]2 v x + u [x-z]2+rx η x = u s(p-1)+[p -1 (y-z)]2 η y + α x u [py-z]2 v x .
Hence, the equation (3.2.7) holds if we choose

v x = α x α -1 z v x , v y = α y α -1 z v y , u [p -1 (x-z)]2 η x = u [y-z]2+ry-s(p-1) ηy , and u [p -1 (y-z)]2 η y = u [x-z]1+rx-s(p-1) η x .
Here, both u [p -1 (x-z)]2 η x and u [p -1 (y-z)]2 η y are well-defined elements in S by our assumption on (x, y, z) and (s, r x , r y ). Now let us consider the new basis e def = e Ã0 . Then V 1 = Mat e ,f (Fil 2 M), where f be the system of generators given by the column vectors of V 1 . Note that ϕ(u [p -1 (y-z)]2 η y ) = 0, again by our assumption. We compute Mat e ,f (ϕ 2 ) as follows:

ϕ 2 (e V 1 ) = eA 0 ϕ(B 1 ) = e   Ã0 +   0 0 η 0 y u [y-z] 0 0 0 0 0 0       ϕ(αz) u p[p -1 (x-z)] 2 ϕ(η x ) 0 0 ϕ(αy) 0 0 0 ϕ(αx)   = e   Ã0 + Ã0   0 0 η 0 y αz u [y-z] 0 0 0 0 0 0       ϕ(αz) u p[p -1 (x-z)] 2 ϕ(η x ) 0 0 ϕ(αy) 0 0 0 ϕ(αx)   = e Ã0   ϕ(αz) u p[p -1 (x-z)] 2 ϕ(η x ) η 0 y ϕ(αx) αz u [y-z] 0 ϕ(αy) 0 0 0 ϕ(αx)   = e    ϕ(αz) ϕ(u [y-z] 2 +ry -s(p-1) ηy) η 0 y ϕ(αx) αz u [y-z] 0 ϕ(αy) 0 0 0 ϕ(αx)    =Mat e ,f (ϕ2)
.

Hence, for Mat e ,f (ϕ 2 ), we see that η y = η 0 y , i.e., ηy = 0. Performing the above procedure one more time, we see that η x = 0 and η y ∈ F p 2 ⊗ Fp F. It is obvious that the above procedure does not affect the vanishing of v x and v y . Proof. Let V be the matrix (3.2.2) and A the matrix (3.2.3), and assume that s(p -1) ≤ e. Since s ≤ (p+1), we may assume that v x , v y ∈ F p 2 ⊗ Fp F. By Proposition 2.13, the φ-module

over F ⊗ Fp F p 2 (( )) defined by M def = M F p 2 (( )) (M * ) is described by (3.2.8) Mat e (φ) = V t ( Â-1 ) t =    1 αz s(p-1) 0 0 vy αz [px-z]2 + ηy αzαy ry+[y-z]2 0 1 αy ry vx αz [py-z]2 + ηx αzαx rx+[x-z]2 1 αx rx 0   
in an appropriate basis e = (e z , e x , e y ). We now prove case (iii). Assume that (m 2 , m 1 , m 0 ) = (1, 2, 0). Then we have x ≡ r+1-2p mod e, y ≡ p(r + 1) -2 mod e, z ≡ (p + 1)c, s = (p + 1), r x = 0, and r y = 2e. So we have [x -z] 2 = e + r + 1 -2p -(p + 1)c and [y -z] 2 = e + p(r + 1) -2 -(p + 1)c. By lemma 3.9, case (i), we may assume that η y = 0 and η x ∈ F p 2 ⊗ Fp F, and, by Lemma 3.8, case (ii), we may assume that v y = 0 as well. Hence, in this specific case, we have We easily see that the φ-module M is the base change via

F ⊗ Fp F p 2 ((p)) → F ⊗ Fp F p 2 (( )) of the φ-module M 0 over F ⊗ Fp F p 2 ((p)) described by Mat(φ 0 ) =    1 αz p (c+1) 0 0 0 0 1 αy vx αz p (c+1) + ηx αzαx p (c+1) 1 αx p (r+1) 0    .
Now we can find a basis for M 0 such that

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   0 1 αx vx αz + ηx αzαx 1 αy 0 0 0 0 1 αz   ,
and so FL(ρ 0 ) = [1 : 0], by Lemma 3.2, as ρ 0 is non-split. Case (ii) is very similar to the previous one. We now have v x = 0 = η x and η y ∈ F p 2 ⊗ Fp F. By the same argument as above, one can check that

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   0 1 αx 0 1 αy 0 vy αz + ηy αzαy 0 0 1 αz   ,
and so FL(ρ 0 ) = [0 : 1], by Lemma 3.2, as ρ 0 is non-split.

Assume that s = 0, i.e., m 2 = 0. Since s = 0, we may let v x = 0 = v y . One can readily check η x = 0 = η y as well, using Lemma 3.9, case (iii). By the same argument as above, it is easy to check that

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   0 1 αx 0 1 αy 0 0 0 0 1 αz   .
Hence, the corresponding representation ρ 0 splits as a sum of a two-dimensional irreducible representation and a character. Assume that ( , δ) = (2, 0) in the case (b). By Lemma (3.8), case (i), we have v x = 0, and, by Lemma (3.9), case (iii), η x = η y = 0. By the same argument as above, one can check that

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   0 1 αx 0 1 αy 0 vy αz 0 0 1 αz   ,
and so FL(ρ 0 ) = [0 : 1], by Lemma 3.2, as ρ 0 is non-split. Assume that ( , δ) = (2, 0) in the case (c). In this case, we may let v x = 0 since s(p -1) ≤ [py -z] 2 . By Lemma (3.9), case (iii), η x = η y = 0. By the same argument as above, one can check that

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   0 1 αx 0 1 αy 0 vy αz 0 0 1 αz   ,
and so FL(ρ 0 ) = [0 : 1], by Lemma 3.2, as ρ 0 is non-split.

Fontaine-Laffaille parameter and crystalline Frobenius

The aim of this section is to explicitly determine the Fontaine-Laffaille module associated to the mod-p reduction of a potentially crystalline lift of ρ 0 , with a carefully chosen inertial type. The main result is Theorem 4.5, whose proof relies on some direct manipulation in semilinear algebra (cf. Section 2.2.4, Lemmas 2.17, 2.18).

As we did in Section 3, in the reminder of this section we may and do assume a 0 = -1 and define

c def = a 2 -a 0 -1, r def = a 1 -a 0 -1.
4.1. Filtration on strongly divisible modules. We go back to the setting of section 2.1 and we let ρ 0 : G Qp → GL 3 (F) be as in (2.1.1) with the genericity condition as in Definition 2.4.

Proposition 4.1. Let M ∈ F-BrMod 2 dd be a Breuil module of type τ = ω c ⊕ ω r ⊕ ω -1 such that T 2 st (M) ∼ = ρ 0 .
Then there exists a framed basis e = (e c , e r , e -1 ) on M and a framed system of generators

f = (f c , f r , f -1 ) for Fil 2 M such that Mat e,f (Fil M) =   u e u e-(c-r) λ u e-(c+1) µ 0 0 u e-(r+1) 0 u e+(r+1) 0   and Mat e,f (ϕ 2 ) =   α c 0 0 0 α r 0 0 0 α -1  
where λ i ∈ F × and λ, µ ∈ F. Moreover, we have the following properties: (i) λ = 0 = µ if and only if ρ 0 splits;

(ii) if ρ 0 is non-split, then FL(ρ 0 ) = [µα r : -λ] ∈ P 1 (F).

Proof. From Proposition 3.3, (b) for m 0 = 0 and m 2 = 1, it is immediate to get Mat e (Fil 2 M) as above. By Lemma 3.5, it is also easy to check that η x = 0 = η y in the matrix (3.1.3), and so we get Mat e,f (ϕ 2 ) as above.

By the same argument as in Proposition 3.6, one can readily compute the following φ-module over F ⊗ Fp F p ((p)) from the Breuil module structure as above:

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   0 1 αr µ αc 1 α-1 0 λ αc 0 0 1 αc   .
The second part is immediate from this matrix.

From now on in this section, we restrict our attention to ρ 0 that is non-split. We easily deduce the following:

Lemma 4.2. Let M ∈ F-BrMod 2
dd and λ, µ ∈ F as in the statement of Proposition 4.1. Assume that ρ 0 is non-split, i.e., not both λ and µ are zero.

Then the elementary divisors for M/ Fil 2 M are described by one of the following possibilities:

(i) if λµ = 0, by (u e-(c+1) , u e , u e+(c+1) );

(ii) if λ = 0, by (u e-(c+1) , u e+(c-r) , u e+(r+1) );

(iii) if µ = 0, by (u e-(c-r) , u e-(r+1) , u e+(c+1) ). In particular, one has:

(iv) Fil 2 M ω -1 ⊆ u e-(c+1) M; moreover, Fil 2 M ω -1 ⊆ u e-(r+1) M holds true if and only if µ = 0; (v) Fil 2 M ∩ u e M ω -1 ⊆ u 2e-(c+1) M; (vi) Fil 2 M ω c ⊆ u e M.
Proof. The elementary divisors are immediately deduced from the Smith normal forms of Mat e (Fil 2 M) in Proposition 4.1.

It is easy to check the following computation:

(Fil 2 M) ω c = u e e c , u e λe c + u e+(c-1) e -1 , u e µe c + u e+(c-r) e -1 ;

(Fil 2 M) ω r = u e+r e c , u e-(c-r) λe c + u e+(r+1) e -1 , u e-(c-r) µe c + u e e -1 ;

(Fil 2 M) ω -1 = u 2e-(c+1) e c , u 2e-(c+1) λe c + u 2e e -1 , u e-(c+1) µe c + u e-(r+1) e -1 .
The second part is also immediate from the computation above.

Proposition 4.3. Let ρ : G Qp → GL 3 (O E ) be a p-adic Galois representation becoming crystalline over K, with inertial type τ = ω c ⊕ ω r ⊕ ω -1 and Hodge-Tate weights {-2, -1, 0} such that ρ ∼ = ρ 0 . Let M ∈ O E -Mod 2 dd be a strongly divisible lattice such that T Qp,2 st ( M) = ρ. Then there exists a framed basis ( e c , e r , e -1 ) for M and a framed system of generators ( f c , f r , f -1 ) for Fil 2 M/ Fil 2 S • M whose coordinates are described as follows:

A : if FL(ρ 0 ) ∈ P 1 (F) \ {[0 : 1], [1 : 0]} then f c =   -p 2 α 0 pu c+1   + E(u)   0 0 u c+1   f r =E(u)   0 1 0   f -1 =   u e-(c+1) 0 α   where 0 < v p (α) < 2. B : if FL(ρ 0 ) = [1 : 0] then f c =   -pβ α 0 βu c+1   + E(u)   0 u c-r 0   f r = E(u)   0 -p β u r+1   f -1 =   u e-(c+1) 0 α   where 0 < v p (β) and 0 < v p (α) < v p (β) + 1 < 2. C : if FL(ρ 0 ) = [0 : 1] then f c =    -p 2 α β -p 2 β u c-r pu c+1    + E(u)   0 0 u c+1   f r =   u e-(c-r) -p α β α u r+1   + E(u)   0 0 γu r+1   f -1 =   αu e-(c+1) u e-(r+1) β   where 0 < v p (α) < 1, 0 < v p (γ), and 0 < v p (α) < v p (β) < 2.
Proof. Let e def = (e c , e r , e -1 ) be a framed basis for M. We write the elements of M in terms of coordinates with respect to e. Moreover, we let

M def = M ⊗ S S/( E , Fil p S) denote the Breuil module associated to M, define D def = M⊗ O E S E and, if χ : F p × → O E is a tame character, we write X χ def = Fil 2 D/ Fil 2 S • D χ , which is a E[E(u)]/(E(u) 2 )-module explicitly described in [HLM], Proposition 2.2.24.
By [HLM], Proposition 2.2.24 we have an element

f -1 ∈ X ω -1 ∩ M of the form f -1 :   xu e-(c+1) yu e-(r+1) z   + E(u)   x u e-(c+1) y u e-(r+1) z  
where x, y, z, x , y , z ∈ O E and (x, y, z) = (0, 0, 0). By Lemma 4.2-(iv) we necessarily have z ≡ 0 modulo E . Case A : Assume that FL(ρ 0 ) = [1 : 0], [0 : 1], or equivalently, by Proposition 4.1, that λµ = 0. Then v p (x) = 0 as u e-(c+1) is an elementary divisor for M/ Fil 2 M and v p (y) > 0 by Lemma 4.2-(iv). We define e c ∈ M as follows:

e c :   x + x E(u) u c-r (y + y E(u)) u c+1 z   .
As v p (x) = 0, e def = (e c , e r , e -1 ) is again a framed basis for M. By letting α def = z + pz we therefore have the following coordinates for f -1 in the basis e :

f -1 :   u e-(c+1) 0 α  
where v p (α) > 0. From now onwards we use the basis e to write the coordinates of the elements in M. By [HLM], Proposition 2.2.24 we easily deduce:

Fil 2 M Fil 2 S M ω -1 =   u e-(c+1) 0 α   , E(u)   u e-(c+1) 0 α   , E(u)   0 γu e-(r+1) β   O E
where β, γ ∈ O E . Moreover, by Lemma 4.2-(v) we necessarily have v p (β) > 0 so that, without loss of generality, we can assume γ = 1.

By [HLM], Proposition 2.2.24 we have

X ω r =   u e-(c-r) 0 αu r+1   , E(u)   u e-(c-r) 0 αu r+1   , E(u)   0 p βu r+1   E .
If 0 < v p (β) < 1, then one can easily check that it violates Lemma 4.2-(i). Assume that v p (β) ≥ 1. Then the element e r defined by e r :

  0 1 -β p u r+1   is in Fil 2 M
Fil 2 S M ω r and the family e def = (e c , e r , e -1 ) is again a framed basis for M. Until the end of case A we use the basis e to write the coordinates of the elements in M. Hence,

Fil 2 M Fil 2 S M ω -1 is generated by   u e-(c+1) 0 α   , E(u)   u e-(c+1) 0 α   , E(u)   0 u e-(r+1) 0   over O E , and Fil 2 M Fil 2 S M ω r by   u e-(c-r) 0 αu r+1   , E(u)   u e-(c-r) 0 αu r+1   , E(u)   0 1 0   over O E .
Again by [HLM], Proposition 2.2.24 we further deduce

X ω c =   -p 0 αu c+1   + E(u)   1 0 0   , E(u)   -p 0 αu c+1   , E(u)   0 u c-r 0   E ,
and an immediate manipulation provides us with:

  -p 2 α 0 pu c+1   + E(u)   0 0 u c+1   ∈ X ω c . By Lemma 4.2-(vi) we necessarily have v p ( p 2 α ) > 0, in particular   -p 2 α 0 pu c+1   + E(u)   0 0 u c+1   ∈ Fil 2 M Fil 2 S M ω c .
Hence, we obtain the following inclusion:

  u e-(c+1) 0 α   , E(u)   0 1 0   ,   -p 2 α 0 pu c+1   + E(u)   0 0 u c+1   O E ⊆ Fil 2 M Fil 2 S M
. By Nakayama's lemma and noticing that the elementary divisors of M/ Fil 2 M are described by Lemma 4.2-(i) we conclude that the inclusion is indeed an equality. Case B : Assume that FL(ρ 0 ) = [1 : 0], or equivalently, by Proposition 4.1, that λ = 0 and µ = 0. By exactly the same argument as in the proof of case A, we get the same

Fil 2 M Fil 2 S M ω -1
as well as X ω r as in case A. If v p (β) ≥ 1, then one can easily check that it violates Lemma 4.2-(ii). Assume 0 < v p (β) < 1.

As in case A we easily deduce

E(u)   0 -p β u r+1   ∈ Fil 2 M Fil 2 S M ω r and X ω c =   -p 0 αu c+1   + E(u)   1 0 0   , E(u)   -p 0 αu c+1   , E(u)   0 -p β u c-r u c+1   E .
In particular,  

-pβ α 0 βu c+1   + E(u)   0 u c-r 0   ∈ Fil 2 M Fil 2 S M ω c
and, by Lemma 4.2-(vi) we necessarily have v p (β) > 0 and v p (β) + 1 > v p (α). Hence, we obtain the following inclusion:

  u e-(c+1) 0 α   , E(u)   0 -p β u r+1   ,   -pβ α 0 βu c+1   + E(u)   0 u c-r 0   O E ⊆ Fil 2 M Fil 2 S M .
which implies that the elementary divisors for M/ Fil 2 M are necessarily of the form described by Lemma 4.2-(ii). It follows, as for case A, that the inclusion is actually an equality and the case B claimed in the statement of the proposition follows.

Case C :Assume that FL(ρ 0 ) = [0 : 1], or equivalently, by Proposition 4.1, that λ = 0 and µ = 0. We may assume that y = 1 as u e-(r+1) is an elementary divisor for M/ Fil 2 M and v p (x), v p (y) > 0 by Lemma 4.2-(iv). We define e r ∈ M as follows:

e r :   x u e-(c-r) 1 + y E(u) z u r+1   .
Then e def = (e c , e r , e -1 ) is again a framed basis for M. By letting α def = x+px and β def = z +pz we therefore have the following coordinates for f -1 in the basis e :

f -1 :   αu e-(c+1) u e-(r+1) β  
where v p (α) > 0 and v p (β) > 0. From now onwards we use the basis e to write the coordinates of the elements in M. By [HLM], Proposition 2.2.24 we easily deduce:

Fil 2 M Fil 2 S M ω -1 =   αu e-(c+1) u e-(r+1) β   , E(u)   αu e-(c+1) u e-(r+1) β   , E(u)   δu e-(c+1) 0 γ   O E
where γ, δ ∈ O E . Moreover, by Lemma 4.2-(v) we necessarily have v p (γ) > 0 so that, without loss of generality, we can assume δ = 1.

By [HLM], Proposition 2.2.24 we have

X ω r =   αu e-(c-r) -p βu r+1   + E(u)   0 1 0   , E(u)   αu e-(c-r) -p βu r+1   , E(u)   u e-(c-r) 0 γu r+1   E . If min{1, v p (β)} ≤ v p (α)
, then one can easily check that it violates Lemma 4.2-(iii). Assume that 0 < v p (α) < min{1, v p (β)}. Then easy manipulations provide us with

E(u)   0 1 -β-αγ p u r+1   ,   u e-(c-r) -p α β α u r+1   + E(u)   0 0 β-αγ pα u r+1   ∈ X ω r .
Again by [HLM], Proposition 2.2.24 we further deduce

X ω c =   -p -p α u c-r β α u c+1   + E(u)   1 0 β-αγ pα u c+1   , E(u)   0 u c-r -β-αγ p u c+1   , E(u)   -p 0 γu c+1   E ,
and an immediate manipulation provides us with:

   -p 2 α β -p 2 β u c-r pu c+1    + E(u)   0 0 u c+1   ∈ X ω c . By Lemma 4.2-(vi) we necessarily have v p ( p 2 β ) > 0, in particular    -p 2 α β -p 2 β u c-r pu c+1    + E(u)   0 0 u c+1   ∈ Fil 2 M Fil 2 S M ω c . Hence, we obtain that Fil 2 M Fil 2 S M contains    -p 2 α β -p 2 β u c-r pu c+1   +E(u)   0 0 u c+1   ,   u e-(c-r) -p α β α u r+1   +E(u)   0 0 β-αγ pα u r+1   , E(u)   αu e-(c+1) u e-(r+1) β   .
By Nakayama's lemma and noticing that the elementary divisors of M/ Fil 2 M are described by Lemma 4.2-(iii) we conclude that the inclusion is indeed an equality. Note that v p (βαγ) > 1 + v p (α) by Lemma 4.2-(iii).

Corollary 4.4. Let ρ and M be respectively a Galois representation and a strongly divisible lattice as in Proposition 4.3. Write (λ c , λ r , λ -1 ) for the Frobenius eigenvalue on the

( ω c , ω r , ω -1 )-isotypic component of the filtered (ϕ, N )-module D Qp,2
st (ρ). Then the valuation of the Frobenius eigenvalues on D Qp,2 st (ρ) is described as follows:

A : if FL(ρ 0 ) ∈ P 1 (F) \ {[0 : 1], [1 : 0]} then (v p (λ c ), v p (λ r ), v p (λ -1 )) = (v p (α), 1, 2 -v p (α)) where 0 < v p (α) < 2. B : if FL(ρ 0 ) = [1 : 0] then (v p (λ c ), v p (λ r ), v p (λ -1 )) = (1 + v p (α) -v p (β), v p (β), 2 -v p (α)) where 0 < v p (β) and 0 < v p (α) < v p (β) + 1 < 2. C : if FL(ρ 0 ) = [0 : 1] then (v p (λ c ), v p (λ r ), v p (λ -1 )) = (v p (β) -v p (α), 1 + v p (α), 2 -v p (β)) where 0 < v p (α) < 1 and 0 < v p (α) < v p (β) < 2.
Proof. Let us write s 0 : S Qp → E to denote the morphism defined by u → 0. Then one has D

Qp,2 st (ρ) ∼ = M[ 1 p ] ⊗ S Qp ,s0 E. Moreover, the Frobenius ϕ on M[ 1 p ] ⊗ S Qp ,s0 E is uniquely determined by the condition ϕ( e i ⊗ s0 1) = p 2 (ϕ 2 ⊗ s0 1)( f i ⊗ s0 κ i )
for i ∈ {c, r, -1}, where the elements e i , f i can be chosen to be as in Proposition 4.3 and the

κ i ∈ E are such that f i ⊗ s0 κ i = e i ⊗ s0 1.
The result is therefore immediate from the explicit description of the elements f i given in Proposition 4.3.

4.2.

From Frobenius eigenvalues to Fontaine-Laffaille parameters. We are now ready to state the main local result on the Galois. Let red : P 1 (O E ) → P 1 (F) be the natural reduction map on the rational points of the projective lines over O E . Namely, red([x : y]) is defined as [(x/y) : 1] if v p (x) ≥ v p (y) and [1 : (y/x)] if v p (x) ≤ v p (y). We fix a coordinate on P 1 (O E ) (hence on P 1 (F)).

Theorem 4.5. Let ρ : G Qp → GL 3 (O E ) be a potentially crystalline Galois representation with parallel Hodge-Tate weights {-2, -1, 0} and inertial type WD(ρ

)| I Qp ∼ = τ def = ω c ⊕ ω r ⊕ ω -1 such that ρ ∼ = ρ 0 . We also let (λ c , λ r , λ -1 ) ∈ (O E ) 3 be the Frobenius eigenvalues on the ( ω c , ω r , ω -1 )-isotypic component of D Qp,2 st (ρ).
Then the Fontaine-Laffaille parameter associated to ρ 0 is computed by:

FL(ρ 0 ) = red [λ r : p] .
The rest of this subsection is devoted to the proof of Theorem 4.5. In the case where FL(ρ 0 ) = [0 : 1] or FL(ρ 0 ) = [1 : 0], it is straightforward to prove it from the results in the previous subsection (see the end of this subsection) and in what follows we will be firstly interested in the case where FL(ρ 0 ) / ∈ {[1 : 0], [0 : 1]}.

Lemma 4.6. Keep the notation of Proposition 4.3. Define α • ∈ F × by the condition

α • e • = λ• p 2 f • modulo ( E , u
) for all • ∈ {c, r, -1} (note that the α i here is not necessarily the same as the ones in Proposition 4.1), and assume that FL(ρ 0 ) /

∈ {[1 : 0], [0 : 1]}. If M ∈ F-BrMod 2
dd denotes the associated Breuil module to M, then there exists a framed basis e = (e c , e r , e -1 ) on M and a framed system of generators f = (f c , f r , f -1 ) for Fil 2 M such that Mat e,f (ϕ 2 ) = Diag(α c , α r , α -1 ) and

Mat e,f (Fil M) =   0 0 u e-(c+1) 0 u e u e-(r+1) y u e+(c+1) u e+(r+1) x u e z  
for some x, y, z ∈ F.

Proof. The proof follows closely the argument of [HLM], Lemma 2.3.2, which we outline here for the comfort of the reader. Let M ∈ O E -Mod 2 dd be a strongly divisible lattice as in the statement of Proposition 4.3. In particular we have a framed basis e on M and a framed family f of generators for Fil 2 M/ Fil 2 S • M which is explicitly described in terms of e-coordinates according to the value of FL(ρ 0 ). Write e 0 , f 0 for the base change of e, f via S S and set

V 0 def = Mat e 0 ,f 0 (Fil 2 M), A 0 def = Mat e 0 ,f 0 (ϕ 2 ).
Note that, by construction, we have (A 0 ) 00 e c ≡ α c e c ≡ λc p 2 f c modulo (u, E ), and, similarly, (A 0 ) 11 e r ≡ α r e r ≡ λr p 2 f r , (A 0 ) 22 e -1 ≡ α -1 e -1 ≡ λ-1 p 2 f -1 . Moreover, by the height condition, we can write V adj 0 = u e W 0 where W 0 ∈ Mat 3 (S) is well defined modulo u e(p-1) . We deduce from Proposition 4.3-Case A that the matrix of the filtration for Fil 2 M has the form

V 0 =   0 0 u e-(c+1) 0 u e 0 u e+(c+1) 0 0   .
Then there exists b 12 , b 21 , b 22 ∈ F such that (4.2.1) for some β ij ∈ F. Indeed, an elementary computation shows that it suffices to take b 12 ≡ -α -1 r a 10 , b 21 ≡ -α -1 a 21 and b 22 ≡ -α -1 -1 (a 21 b 12 + a 20 ) modulo u e , where the a ij 's denote the corresponding entries of A 0 .

-W 0 • A 0 •   0 0 u e-(c+1) 0 u e u e-
By Lemma 2.19 we deduce that V 1 describes the coordinates of a framed system of generators f 1 for Fil 2 M with respect to the basis e 1 def = e 0 •A 0 and moreover A 1 def = Mat e 1 ,f 1 (ϕ 2 ) = ϕ(B 0 ) is the matrix for the associated Frobenius action.

We now iterate the previous procedure: as A 1 ∈ Diag(α -1 , α r , α c ) + u 3 Mat 3 (S) (by the genericity assumption (2.1.2)), we easily find V 2 ∈ Mat 3 (S) as in the statement, and B 1 ∈ Diag(α c , α r , α -1 ) + u Mat 3 (S) verifying:

A 1 V 2 ≡ B 1 V 1 mod u 3e .
By virtue of Lemma 2.19, this completes the proof.

Lemma 4.7. Keep the notation of Lemma 4.6 and assume that FL(ρ 0 ) / ∈ {[1 : 0], [0 : 1]}. Let M ∈ F-FL [0,p-2] be the contravariant Fontaine-Laffaille module associated to ρ 0 .

Then there exists a basis f on M , compatible with its Hodge filtration, such that the Frobenius action on M is described by

Mat f (φ • ) =     0 yα -1 r α -1 c α -1 -1 x 0 -α -1 c y 0 0 α -1 c xy    
for some x, y ∈ F × .

Proof. By Lemma 4.6 and Lemma 2.17, the Frobenius action on the (φ, F(( )))-module

M def = M Fp(( )) (M * ) is described by Mat e (φ) =   0 0 e+(c+1) α -1 -1 0 e α -1 r e+(r+1) xα -1 -1 e-(c+1) α -1 c e-(r+1) yα -1 r e zα -1 -1  
where e = (e -c , e -r , e 1 ) is a framed basis for the dual type τ ∨ and x, y, z ∈ F. By performing the change of basis e def = ( c e c , r e r , -1 e 1 ), it can be easily checked that M = M 0 ⊗ F((p)) F(( )) where the (φ, F((p)))-module M 0 is described by

Mat(φ 0 ) =   0 0 α -1 -1 0 α -1 r xα -1 -1 α -1 c yα -1 r zα -1 -1   Diag(p c+1 , p r+1 , 1)
i.e., by an evident change of basis over F,

Mat(φ 0 ) = Diag(1, p r+1 , p c+1 )   zα -1 yα -1 r α -1 c xα -1 -1 α -1 r 0 α -1 -1 0 0   def = F
.

By Lemma 2.14 we deduce that M 0 ∼ = F(M ) for a rank 3 Fontaine-Laffaille module M ∈ F-FL [0,p-2] with Hodge-Tate weights {0, r + 1, c + 1} and Mat f (φ • ) = F for a basis f on M compatible with the Hodge filtration.

On the other hand the condition T * cris (M ) = ρ 0 implies, by Lemma 2.5, the existence of another basis f on M such that Mat f (φ • ) is the one described in (2.1.3). Equivalently, there exists of a change of basis A ∈ GL 3 (F) from f to f , compatible with the Hodge filtration (i.e. A = (a ij ) i,j is lower unipotent) and such that

A • F =   0 η -1 1 γ η -1 0 0 δ 0 0 η -1 2   (4.2.3) for some γ, δ ∈ F, η i ∈ F × .
This can be easily verified: condition (4.

The local automorphic side

We now need to recall certain group algebra operators for O E [GL 3 (F p )], F[GL 3 (F p )] which are needed to obtain local-global compatibility in terms of Hecke action. In order to introduce such operators, we need some notation. In what follows, we have [START_REF] Carsten | Representations of algebraic groups[END_REF] as a main reference for the notation and terminology. 5.1. Basic set up. We let G def = GL 3/Z p , T be the maximal split torus consisting of diagonal matrices and B ⊃ T the Borel subgroup of upper triangular matrices. The character and cocharacter groups X * (T ), X * (T ) are identified with Z 3 in the usual way. In particular the positive simple roots {α 1 , α 2 } for the pair (B, T ) become α 1 = (1, -1, 0), α 2 = (0, 1, -1). = soc G H 0 (λ) for its irreducible socle. If the weight λ is p-restricted, i.e. if 0 ≤ λ, α ∨ i ≤ p -1 for i = 1, 2, then F (λ) is irreducible as a G(F p )representation (see for example [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Corollary 3.17).

As in [HLM] we let I be the Iwahori subgroup of G(Z p ) (preimage of B(F p ) under the reduction map G(Z p ) G(F p )) and I 1 ≤ I for its maximal pro-p subgroup. If V is a smooth representation of G(Z p ) over O E and a i ∈ Z we write V I,(a2,a1,a0) to denote the ω a2 ⊗ ω a1 ⊗ ω a0 -isotypic component for the I-action on V I1 . 5.2. Group algebra operators and the automorphic parameter. Let (a, b, c) ∈ Z 3 be a triple satisfying condition (2.1.2) (when specialized at (a 2 , a 1 , a 0 ) = (a, b, c)). In this case the weight (a, b, c) is in particular restricted. In [HLM] the following elements of F[ Ḡ(F p )] are defined:

S def = x,y,z∈Fp x p-(a-c) z p-(b-c) 1 x y 0 1 z 0 0 1 ẇ0 S def =
x,y,z∈Fp

x p-(a-b) z p-(a-c) 1 x y 0 1 z 0 0 1 ẇ0 as well as their characteristic zero counterparts

S def = x,y,z∈Fp x p-(a-c) z p-(b-c) 1 x y 0 1 z 0 0 1 ẇ0 S def = x,y,z∈Fp x p-(a-b) z p-(a-c) 1 x y 0 1 z 0 0 1 ẇ0 .
The behavior of such operators is described in [HLM], §3 and we include here the statements for the comfort of the reader. , the U i operator is defined as the double coset operator [I 1 t i I 1 ], i.e. In characteristic zero, we have: we have U 1 = pχ a (p) -1 and U 2 = p 2 χ a (p) -1 χ c (p) -1 .

U i (v) = x∈I1/(tiI1t -1 i ∩I1) xt i v.

Local-global compatibility

This section contains the main global application of the local results obtained in Section 4. We follow closely the setup of [HLM], which we reproduce in Section 6.1, Section 6.2 for the convenience of the reader. 6.1. Automorphic forms on unitary groups. Let F/Q be a CM field, F + = Q its maximal totally real subfield. We write c for the generator of Gal(F/F + ) and assume that all places v of F + above p further decompose as v = ww c in F . We let S + p (resp. S p ) the set of places of F + (resp. F ) above p. For v (resp. w) a finite place of F + (resp. F ) we write k v (resp. k w ) for the residue field of F + v (resp. F w ). We let G /F + be a reductive group, which is an outer form for GL 3 , and which splits over F . We assume that G(F + v ) U 3 (R) for all v|∞. By [CHT08], Section 3.3, G admits an integral model G such that G × O F + v is reductive if v is a finite place of F + which splits in F . If v is such a place and w|v is a place of F , we obtain and fix an isomorphism (6.1.1)

ι w : G(O F + v ) ∼ → G(O Fw ) ∼ → GL 3 (O Fw ). Define F + p def = F + ⊗ Q Q p and O F + ,p def = O F + ⊗ Z Z p . If W is a O E -module endowed with an action of G(O F + ,p ) and U ≤ G(A ∞,p F + ) × G(O F + ,p ) is a compact open subgroup, the space of algebraic automorphic forms on G of level U and coefficients in W is the following O E -module: (6.1.2) S(U, W ) def = f : G(F + )\G(A ∞ F + ) → W | f (gu) = u -1 p f (g) ∀ g ∈ G(A ∞ F + ), u ∈ U (with the usual notation u = u p u p for the elements in U ).
Recall that the level U is sufficiently small if t -1 G(F + )t ∩ U has order prime to p for all t ∈ G(A ∞ F + ). For a finite place v of F + we say that U is unramified at

v if one has a decomposition U = G(O F + v )U v for some compact open U v ≤ G(A ∞,v F + ).
If w is a finite place of F we say, with an abuse, that w is an unramified place for U if its restriction w| F + is unramified for U .

Let P U denote the set consisting of finite places w of F such that v def = w| F + is split in F , v / ∈ S + p and U is unramified at v. For a subset P ⊆ P U of finite complement and closed with respect to complex conjugation we write

T P = O E [T (i)
w , w ∈ P, i ∈ {0, 1, 2, 3}] for the universal Hecke algebra on P, where the Hecke operator T (i) w acts on S(U, W ) as the usual double coset operator We briefly recall the relation between the space A of classical automorphic forms and the previous spaces of algebraic automorphic forms, in the particular case which is relevant to us.

ι -1 w GL 3 (O Fw ) w Id i 0 0 Id 3-i GL 3 (O Fw ) .
Let S def = Hom(F, Q p ) and, for any place w|p, let S w def = Hom(F w , Q p ), S w def = Hom(k w , F p ). Following [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 7.3 we consider the subset (Z 3 + ) S 0 of dominant weights λ = (λ σ ) σ verifying the condition (6.1.3) λ 1,σc + λ 3,σ = 0, λ 2,σ + λ 2,σc = 0, λ 3,σc + λ 1,σ = 0 for all triples λ σ = (λ 1,σ , λ 2,σ , λ 3,σ ) and all σ ∈ S. If w|p and λ ∈ (Z 3 + ) S 0 we write λ w for the projection of λ on (Z 3 + ) Sw 0 and W λ w for the O Fw -specialization of the dual Weyl module associated to λ w (cf. [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 4.1.1); by condition (6.1.3) and Remark 6.1 one deduces an isomorphism of G

(O F + v )-representations W λ w • ι w ∼ = W λ w c • ι w c . Therefore, by letting W λ v def = W λ w • ι w for any place w|v, the G(O F + ,p )-representation W λ def = v|p W λ v is well defined.
For a weight λ ∈ (Z 3 + ) S 0 and an irreducible smooth G(O F + ,p )-representation τ over Q p , let us write S λ,τ (Q p ) to denote the inductive limit of the spaces S(U, W

λ ⊗ O E τ ) over the compact open subgroups U ≤ G(A ∞,p F + ) × G(O F + ,p
) (note that the latter is an inductive system in a natural way, with injective transition maps induced from the inclusions between levels). Then S λ,τ (Q p ) has a natural left action of G(A ∞ F + ) induced by right translation of functions.

Fix an isomorphism ı : Q p ∼ → C. As we did for the O Fw -specialization of the dual Weyl modules, we define a smooth G(

F + ⊗ Q R)-representation σ λ ∼ = v|∞ σ λ v with C-coefficients,
where σ λ v depends only on λ w for a place w|v (we invite the reader to refer to [EGH13], Section 7.1.4 for the precise definition of σ λ ). Lemma 6.2. The isomorphism ı : In particular, if V v is a Serre weight at v, the Serre weights at w c defined by where the V v are Serre weights at v verifying V v • ι -1 w ∼ = F (a w , b w , c w ). Again, thanks to condition (6.2.1) and Remark 6.1 we deduce that V v is well defined. Definition 6.4. Let r : G F → GL 3 (F) be a continuous, absolutely irreducible Galois representation and let V be a Serre weight for G. We say that r is automorphic of weight V (or that V is a Serre weight of r) if there exists a compact open subset U in G(A ∞,p F )×G(O F + ,p ) unramified above p and a cofinite subset P ⊆ P U such that r is unramified at each place of P and S(U, V ) mr = 0 where m r is the kernel of the system of Hecke eigenvalues α :

Q p ∼ → C induces an injective morphism of smooth G(A ∞ F + )-representations S λ,τ (Q p ) ⊗ Q p ,ı C ı -→ Hom G(F + ⊗ Q R) (σ ∨ λ , A). If Π is an irreducible automorphic representation of G(A F + ), Π p contains τ ⊗ Q p ,ı C if and only if the isotypic space Hom G(F + ⊗ Q R) (σ ∨ λ , Π)
V v • ι -1 w • c, V v • ι -
, c w ) ∨ • ι w c ∼ = F (a w , b w , c w ) • ι w as G(k v )-representations (i.e. F (a w c , b w c , c w c ) • ι w c ∼ = F (a w , b w , c w ) • ι w ) and the smooth G(O F + v )-representation F a v def = F a w • ι w is well defined. We set F a def = v|p F a v
T P → F associated to r, i.e. det (1 -r ∨ (Frob w )X) = 3 j=0 (-1) j (N Fw/Qp (w)) ( j 2 ) α(T (j) w )X j for all w ∈ P.
In what follows (sections 6.3, 6.4) we will be needing the notion of Serre weight above a specific place w|p. That is the reason for the following: Definition 6.5. Let r : G F → GL 3 (F) be a continuous Galois representation and let w 0 |v 0 be places of F , F + respectively, above p.

If V w0 is a Serre weight at w 0 , we say that r is automorphic of weight V w0 at w 0 (or that V w0 is a Serre weight of r at w 0 ) if for all v|p, v = v 0 there exist Serre weights V v such that by letting

V v0 def = v|p, v =v0 V v , the smooth G(O F + p ) representation V v0 ⊗ V v0
is a Serre weight of r as in Definition 6.4, where V v0 = V w0 • ι w0 .

As above, we write W w (r) for the set of all Serre weights of r at a place w|p. Note that condition 6.2.1 implies that W w (r) and W w c (r) are in natural bijection via the involution c ∈ Gal(F/F + ):

V w ∈ W w (r) if and only if (V w ) ∨ • c ∈ W w c (r).
We recall some formalism related to Deligne-Lusztig representations and potentially crystalline lifts for r| G Fw 0

. We refer the reader to [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Section 4 for a precise reference. Let w|p be a place of F , n ∈ {1, 2, 3} and let k w,n /k w be an extension verifying [k w,n : k w ] = n. Let T be a maximal torus in GL 3/k w . Following [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Lemma 4.7 we have an identification (6.2.2)

T (k w ) ∼ -→ j k × w,nj
where 3 ≥ n j > 0 and j n j = 3; the isomorphism is unique up to j Gal(k w,nj /k w )conjugacy. In particular, any character θ : T (k w ) → Q × p can be written as θ = ⊗ j θ j where θ

j : k × w,nj → Q × p .
We say that θ is primitive if θ j is primitive as in [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Section 4.2 for all j.

Given a maximal torus T and a primitive character θ we consider the Deligne-Lusztig representation R θ T of GL 3 (k w ). By letting Θ(θ j ) be the cuspidal representation of GL nj (k w ) associated to the primitive character θ j via [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Lemma 4.7, we have

R θ T ∼ = (-1) n-r Ind GL3(kw)
Pn(kw) (⊗ j Θ(θ j )) where P n is the standard parabolic subgroup containing the Levi j GL nj and r denotes the number of its Levi factors.

Let F w,n def = W (k w,n )[ 1 p ]; we consider θ j as a character on O × Fw,n j by inflation and we define the following character rec(θ):

(i) rec(θ) def = 3 j=1 θ j • Art -1 Fw if θ j : k w → Q × p are niveau one characters; (ii) rec(θ) def = θ 1 • Art -1 Fw ⊕ σ∈Gal(kw,2/kw) σ θ 2 • Art -1 Fw,2 if θ 1 is a niveau one character and θ 2 is a niveau 2, primitive character on k × w,2 ; (iii) rec(θ) def = σ∈Gal(kw,3/kw) σ θ 1 • Art -1
Fw,3 if θ 1 is a niveau three, primitive character.

From now on we assume that p is unramified in F + . In particular, the set of embeddings S w , S w are in natural bijection. Theorem 6.6. Assume that p is unramified in F + and let w be a place of F above p. Let V w be a Serre weight at w for the Galois representation r : G F → GL 3 (F) and assume that V w is a Jordan-Hölder constituent in the mod-p reduction of a Deligne-Lusztig representation R θ T of GL 3 (k w ), where T is a maximal torus in GL 3/k w and θ : T (k w ) → Q × p is a primitive character. If rec(θ) is as in item (i) above, we assume the characters θ j are pairwise distinct.

Then r| G Fw has a potentially crystalline lift with parallel Hodge-Tate weights {-2, -1, 0} and Galois type rec(θ).

Proof. This is the statement of [MP], Theorem 5.5. Note that in loc. cit. one assumes further that p splits completely in F , but this condition is unnecessary as long as p is unramified in F + (the statement of loc. cit., Proposition 5.2 holds true for p unramified in F + ). 6.3. Weight elimination. Let w 0 |v 0 be places above p of F and F + respectively with F w0 ∼ = Q p . We define a predicted set of Serre weights W ? w0 (r) for r at w 0 . Assume that r| G Fw 0 is of the form (2.1.1). We write ρ 0 for r| G Fw 0 in this subsection. Recall that we defined in Section 2.1 the Fontaine-Laffaille parameter FL(ρ 0 ) ∈ P 1 (F). From now onwards, we fix an affine coordinate in P 1 (F) ∼ = A 1 (F) ∪ {∞} via [x 0 : x 1 ] → x1 x0 if x 0 = 0 and [0 : 1] → ∞. If ρ 0 is split, then we let W ? w0 (r) = W L ∪ W U ∪ W S where W L def = {F (a 1 -1, a 0 , a 2 + 2 -p), F ((p -1) + a 0 , a 2 , a 1 ), F (a 2 -1, a 1 , a 0 + 1)}; W U def = {F ((p-1)+a 0 , a 1 -1, a 2 +2-p), F ((p-1)+a 1 , a 2 , a 0 ), F (a 2 -1, a 0 +1, a 1 -(p-1))}; W S def = {F (a 2 , a 0 , a 1 -(p -1)), F (p -2 + a 1 , a 2 , a 0 + 1), F (p -1 + a 0 , a 1 , a 2 -(p -1))}. If ρ 0 is non-split, then W ? w0 (r) = F (a 2 -1, a 1 , a 0 + 1), F ((p -1) + a 0 , a 1 , a 2 -(p -1)), F (a 2 -1, a 0 + 1, a 1 -(p -1)) W

where

W def =               
F (p -1 + a 0 , a 2 , a 1 ), F (p -2 + a 1 , a 2 , a 0 + 1), F (a 2 , a 0 , a 1 -(p -1)) if FL(ρ 0 ) = ∞; F ((p -1) + a 1 , a 2 , a 0 ) if FL(ρ 0 ) = 0; ∅ otherwise.

Moreover, we define the set of obvious weights at w 0 as W ?,obv w0 (r) def = W ? w0 (r) ∩ (W L ∪ W U ) . Theorem 6.7. Let w 0 |v 0 be a place above p on F and F + respectively with F w0 ∼ = Q p , and assume that r| G Fw 0 is of the form (2.1.1) with the generic condition (2.1.2). If V w0 is a modular weight for r at w 0 , then V w0 ∈ W ? w0 (r). In what follows, we prove the inclusion W w0 (r) ⊆ W ? w0 (r) under the assumption a 0 = -1, c = a 2 -a 0 -1, and r = a 1 -a 0 -1. This assumption is harmless since W w0 (r ⊗ ω a ) = W w0 (r) ⊗ ω a . The proof is performed case by case, by series of lemmas. Lemma 6.8. Keep the assumption as in Theorem 6.7 and assume that ρ 0 is semisimple. If V w0 is a Serre weight of r at w 0 , then V w0 ∈ W ? w0 (r). Proof. Proposition 3.3 tells us the possible Galois types of niveau 1 for the potentially crystalline lifts with Hodge-Tate weights {-2, -1, 0} of ρ 0 . Hence, the Serre weights of ρ 0 are constituents of R θ T for θ determined in Proposition 3.3. Moreover, we can restrict our attention to the obvious weights in JH R θ T since a shadow weight is either non-modular or an obvious weight of R θ T for another θ. Thus, there are 24 weights we need to consider.

It is easy to check the following:

for θ = ω p-2 ⊗ ω c+2+p(r-2) 2 .
None of the Galois types θ of niveau 2 above appears in Proposition 3.7. Hence, we can further eliminate the weights listed above so that there are 9 weights survived, which are exactly the same as the set W ? w0 (r) for ρ 0 split. This completes the proof.

Lemma 6.9. Keep the assumption as in Theorem 6.7 and assume that ρ 0 is non-split with FL(ρ 0 ) = 0. If V w0 is a Serre weight of r at w 0 , then V w0 is isomorphic to one of the weights in the following list: Proof. It is enough to consider in the set of Serre weights listed in Lemma 6.8. Proposition 3.6, (ii) tells us that we can further eliminate the Galois type ω c+1 ⊕ ω r-1 ⊕ ω -1 . It is for θ = ω c ⊗ ω r-1 ⊗ ω 0 .

Hence, we can eliminate the four weights above. Proposition 3.10, (v) tells us that we can further eliminate the Galois type ω p-2 ⊕ ω c+1+p(r-1) 2 ⊕ ω r-1+p(c+1) 2

. It is easy to check the following:

F ((p -1) + c, p -2, r) ∈ JH R θ T for θ = ω p-2 ⊗ ω c+1+p(r-1) 2 .
Hence, we can further eliminate this weight as well in this case, so that there are only the four weights in the statement of this lemma remaining.

Proof of Theorem 6.7. The lemma 6.8 provides with a complete proof for the case ρ 0 split. If FL(ρ 0 ) = ∞ then it holds by Lemma 6.9, and if FL(ρ 0 ) = 0 then it holds by Lemma 6.10. Finally, if FL(ρ 0 ) = 0, ∞ then, by Lemmas 6.9 and 6.10, the Serre weights are isomorphic to the weights listed in both lemmas. via T 1 ). Finally, F (a 1 + p -2, a 1 , a 0 + 1) cycles to F (a 0 + p -1, a 2 , a 1 ) via both T 1 and T 2 . Remark 6.12. In the semisimple case it is easy to prove, along the argument of Lemma 6.11, that either {F (a 2 -1, a 1 , a 0 + 1), F (a 2 -1, a 0 + 1, a 1 -p + 1)} ⊆ W w0 (r) or {F (a 0 + p -1, a 1 -1, a 2 + 2 -p), F (a 1 -1, a 0 , a 2 + 2 -p)} ⊆ W w0 (r).

Indeed, the only weights where T 1 , T 2 need not both act by zero are F (a 2 -1, a 1 , a 0 + 1), F (a 2 -1, a 0 + 1, a 1 -p + 1) (where T 1 may be non-zero, according to the normalizations) and F (a 0 + p -1, a 1 -1, a 2 + 2 -p), F (a 1 -1, a 0 , a 2 + 2 -p) (where T 2 may be non-zero).

By weight cycling an easy but tedious check, using [EGH13] Proposition 6.1.3 and Theorem 6.7 shows that:

(i) F (a 2 -1, a 1 , a 0 + 1), F (a 2 -1, a 0 + 1, a 1 -p + 1) (resp. F (a 0 + p -1, a 1 -1, a 2 + 2 -p), F (a 1 -1, a 0 , a 2 + 2 -p)) cycle to each other via T 1 (resp. via T 2 ); (ii) F (a 1 -2 + p, a 2 , a 0 + 1) cycles to F (a 0 + p -1, a 2 , a 1 ) (via both T 1 and T 2 ); (iii) F (a 0 + p -1, a 2 , a 1 ) can be cycled to either F (a 0 + p -1, a 1 , a 2 -p + 1) (via T 1 ) and F (a 2 , a 0 , a 1 -p + 1) (via T 2 ); (iv) both F (a 2 , a 0 , a 1 -p + 1) and F (a 0 + p -1, a 1 , a 2 -p + 1)) can be cycled to one of the weights in {F (a 1 -1, a 0 , a 2 -p + 2), F (a 2 -1, a 1 , a 0 + 1)}, via T 2 and T 1 respectively. (v) F (a 1 + p -1, a 2 , a 0 ) can be cycled to one of the weights in {F (a 1 -1, a 0 , a 2 -p + 2), F (a 2 -1, a 1 , a 0 + 1), F (a 0 + p -1, a 1 , a 2 -p + 1)} via T 1 (resp. to one of the weights in {F (a 1 -1, a 0 , a 2 -p + 2), F (a 2 -1, a 1 , a 0 + 1), F (a 2 , a 0 , a 1 -p + 1)} via T 2 ).

In the following picture, we draw the Hasse diagram of the cosocle filtration in the principal series Ind GL3(Fp) B(Fp) ω a2 ⊗ ω a1 ⊗ ω a0 : letting e def = p -1 for brevity, F (a 2 , a 1 , a 0 ) F (a 1 + e, a 2 , a 0 ) F (a 2 , a 0 , a 1 -e) F (a 1 , a 0 , a 2 -e) F (a 0 -1 + e, a 2 , a 1 + 1) F (a 2 -1, a 1 , a 0 + 1) F (a 1 -1, a 0 , a 2 + 1 -e) F (a 0 + e, a 2 , a 1 )

F (a 0 + e, a 1 , a 2 -e)
Provided that ρ 0 is non-semisimple as in the statement of Theorem 6.7, the weights in red are always in W ? w0 (r); the weight in blue is in W ? w0 (r) if and only if FL(ρ 0 ) = 0 and the weights in green are in W ? w0 (r) if and only if FL(ρ 0 ) = ∞. Using the notation and convention of Section 6.2 we define

S(U v0 , V v0 ) def = f : G(F + )\G(A ∞ F + ) → V v0 | f (gu) = u -1 p f (g) ∀ g ∈ G(A ∞ F + ), u ∈ U v0 which is G(F + v )
representation by right translation of functions. We write S sm (U v0 , V v0 ) to denote the submodule of S(U v0 , V v0 ) consisting of locally constant functions (i.e. the submodule of smooth vectors for the G(F + v0 ) action on S(U v0 , V v0 )). Theorem 6.13. Let F be a CM field in which p splits completely and let r : G F → GL 3 (F) be an absolutely irreducible and automorphic Galois representation. Let w 0 |p be a place of F + ). We make the following two assumptions: (i) r| G Fw 0 is indecomposable of the form (2.1.1) with the strongly generic condition (2.1.2); (ii) FL(r| G Fw 0

) / ∈ {0, ∞};

(iii) The O E -dual of S sm (U v0 , V v0 ) I,(-a1,-a0,-a2) mr is free over T, where T denotes the O E -subalgebra of End S sm (U v0 , V v0 ) I,(-a1,-a0,-a2) mr generated by T P , U 1 and U 2 .

Let S, S be the operators defined in Section 5 specialised to (a, b, c) = (-a 0 , -a 1 , -a 2 ). Then is embedded into S sm (U v0 , V v0 )[m r ] I,(-a0-1,-a1,-a2+1) under the map S.

Proof. The proof follows closely the proof of the local-global compatibility statement of [HLM] (Theorem 4.4.1 in loc. cit.). We sketch here the argument. We identify G(F + v0 ) with GL 3 (Q p ) via ι w0 without further comment. Let θ : T (F p ) → O × E be the character ω a1 ⊗ ω a0 ⊗ ω a2 , (where T is the maximal split torus in GL 3 ) and consider the Deligne-Lusztig representation R θ T (which will be considered as a smooth GL 3 (Z p )representation by inflation).

Recall that we have fixed at the beginning of Section 6.4 the weights λ w = (a w,2 , a w,1 , a w,0 ) for places w|v above p with v = v 0 . By letting λ w0 We write M E , M F , T E etc. to denote the extension of scalars of M , T to E, F etc.

By Lemma 6.11 we have that S(U, F (a 2 -1, a 1 , a 0 +1)) mr = 0. As F (a 2 -1, a 1 , a 0 +1)⊗ F F is a constituent of R θ T we can lift the system of Hecke eigenvalues associated to m r to deduce the following: (iv) for π as in item (ii), r ∨ π | G Fw 0 is potentially crystalline, with Hodge-Tate weights {-2, -1, 0} lifting r, and moreover WD(r π | G Fw 0

) F-ss ∼ = ψ a1 ⊕ ψ a2 ⊕ ψ a0 .

Fix

  r ∈ {0, . . . , p -2} and let S k def = k[u]/u ep . A Breuil module over F is the datum of a quadruple (M, Fil r M, ϕ r , N ) where (i) M is a finitely generated S-module which is free over S k ; (ii) Fil r M is a S-submodule of M, verifying u er M ⊆ Fil r M; (iii) the morphism ϕ r : Fil r M → M is ϕ-semilinear and the associated fibered product S ⊗ k⊗ Fp F Fil r M → M is surjective; (iv) the operator N : M → M is k ⊗ Fp F-linear and satisfies the following properties: (a) N (P (u)x) = P (u)N (x) + N (P (u))x for all x ∈ M, P (u) ∈ S; (b) u e N (Fil r M) ⊆ Fil r M;

  with multiplicity). Let e = (e 0 , . . . , e n-1 ) be a k ⊗ Fp F basis for M i , compatible with the Hodge filtration Fil • M and let F ∈ M n (k ⊗ Fp F ) be the associated matrix of the Frobenius φ • : gr • M → M There exists a basis e for M def = F(M ) such that the Frobenius φ on M is described by Mat e (φ) = Diag(p m0 . . . p mn-1 )F.

ω α+pβ 2

 2 up to an unramified twist and we have the following particular case of[GHS], Corollary 7.1.3:Lemma 2.28. The representation V (α,β) | G Q p 2 is crystalline with parallel Hodge-Tate weights {α, β}.

  (a2+1) 2 via the isomorphism in the bottom line of the diagram (2.4.1). By Lemma 2.30, τ admits a crystalline lift τ : G Q p 2 → GL 2 (O E ) with Hodge-Tate weights HT σ0 = {α, γ}, HT σ1 = {β, γ} where the integers α, β, γ are suitably specialized according to ρ 0 (e.g. specialized at α = a 1 + 1, β = a 0 + 1, γ = a 2 + 1 for the first case of Proposition 2.27). By letting τ (1) :

  st (ρ) would be written as T Qp st,r+1 , D Qp st,r+1 (ρ) in loc. cit.). But the proof in loc. cit. generalizes verbatim for higher Hodge-Tate weights. See also[START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Proof of Proposition 3.3.1.

  m 0 e; r y = m 1 e; s = m 2 (p + 1), where m i ∈ {0, 1, 2} satisfy m 0 + m 1 + m 2 = 3. (b): if x ≡ (p + 1)j mod (e), y ≡ k mod (e), and z ≡ pk mod (e), thenj ≡ r + 1 -mod (p -1), k ≡ (δ + -3) + p(c + 1 -δ) mod (e),and   r x = (c -r -δ + ) + p(δ + 2 -r -4) + e; r y = (r + 4 -δ -2 ) + p(r -c + δ -) + (3 -δ -)e; s = (c + 4 --δ) + pδ,where ∈ {1, 2} and δ ∈ {0, 1} with + δ = 3. (c): if x ≡ k mod (e), y ≡ (p + 1)j mod (e), and z ≡ pk mod (e), thenj ≡ + δ -3 mod (p -1), k ≡ (r + 1 -) + p(c + 1 -δ) mod (e),and    r x = (2 + δ -r -4) + p( + 2δ -c -4) + e; r y = (c + 4 --2δ) + p(r + 4 -δ -2 ) + (3 -δ -)e; s = (c -r + ) + pδ, where ∈ {1, 2} and δ ∈ {0, 1} with + δ = 1.

Proposition 3. 10 .

 10 Keep the notation as in Proposition 3.7 and assume that ρ 0 ∼ = T 2 st (M). (i) If m 2 = 0 in the case (a), then ρ 0 splits as a sum of a two-dimensional irreducible representation and a character. (ii) If (m 2 , m 1 , m 0 ) = (1, 0, 2) in the case (a) and ρ 0 is non-split, then FL(ρ 0 ) = [0 : 1]. (iii) If (m 2 , m 1 , m 0 ) = (1, 2, 0) in the case (a) and ρ 0 is non-split, then FL(ρ 0 ) = [1 : 0]. (iv) If ( , δ) = (2, 0) in the case (b) and ρ 0 is non-split, then FL(ρ 0 ) = [0 : 1]. (v) If ( , δ) = (2, 0) in the case (c) and ρ 0 is non-split, then FL(ρ 0 ) = [0 : 1].

  By considering the change of basis e = ( (p+1)c e z , p(r+1)-2 e x , r+1-2p e y ) we have: Mat e (φ) =

=

  (r+1) b 12 u e+(c+1) u e+(r+1) b 21 u e b 22 u 2e B 0 (4.2.2)where B 0 ∈ GL 3 (S) verifies moreoverB 0 ≡   α -1 u e-(c-r) β 01 u e-(c+1)

  2.3) holds true if and only if one has z = 0, x, y, a 20 ∈ F × , 1 + ya 10 = 0 1 + xa 21 = 0 a 21 + ya 20 = 0, which completes the proof. Proof of Theorem 4.5. First of all, note that Proposition 4.3 and its corollary apply in our context. If FL(ρ 0 ) = [1 : 0], then it is immediate that FL(ρ 0 ) = [1 : 0] = red([λ r : p]), since v p (λ r ) < 1 by Corollary 4.4, Case B. Similarly, one can prove the case FL(ρ 0 ) = [0 : 1] by Corollary 4.4, Case C. For the case that FL(ρ 0 ) / ∈ {[1 : 0], [0 : 1]} it is also easy to check that FL(ρ 0 ) = [α r : 1] = red([λ r : p]), by Lemma 4.7 and by Definition 2.8.

  Finally, we let G, B, . . . denote the base change of G, B, . . . via Z p F p . The Weyl group W G of G is canonically isomorphic to the Weyl group of G. We write w 0 ∈ W G for the longest element and define ṡ1 in G(Z p ) of the simple reflections s 1 , s 2 ∈ W G corresponding to α 1 , α 2 . In particular ẇ0 def = ṡ1 ṡ2 ṡ1 is a lift of w 0 ∈ W G . For any dominant character λ ∈ X * (T ) we let H 0 (λ) def = Ind G B w 0 λ alg ⊗ Fp F be the associated dual Weyl module. It is an algebraic representation of G (or more precisely of G /F ) and we write F (λ) def



  Proposition 5.1. Let (a, b, c) ∈ Z 3 be a triple satisfying (2.1.2)) (when specialized at (a 2 , a 1 , a 0 ) = (a, b, c)) and consider the associated operators S, S ∈ F[ Ḡ(F p )].(i) There is a unique non-split extension of irreducible Ḡ(F p )-representations0 → F (a -1, b, c + 1) → V → F (b + (p -1), a, c) → 0 and S induces an isomorphism S : V I,(b,a,c) ∼ -→ V I,(a-1,b,c+1) of one-dimensional vector spaces. (ii) There is a unique non-split extension of irreducible Ḡ(F p )-representations 0 → F (a -1, b, c + 1) → V → F (a,c, b -(p -1)) → 0 and S induces an isomorphism S : (V ) I,(a,c,b) ∼-→ (V ) I,(a-1,b,c+1) of onedimensional vector spaces.In characteristic zero, we have: Proposition 5.2. Let (a, b, c) ∈ Z 3 be a triple satisfying (2.1.2)) (when specialized at (a 2 , a 1 , a 0 ) = (a, b, c)). Let π p def = Ind G(Qp) B(Qp) χ b ⊗χ a ⊗χ c be a principal series representation, where the smooth characters χ• : Q × p → E × verify χ • | Z × p = ω • for • ∈ {a, b, c}. On the one-dimensional isotypic component π I,(b,a,c) p = pχ b (p)η S,where the element η ∈ Z × p verifies η ≡ (-1) b-c • a-b b-c mod p.Recall that if σ is a smooth representation of G(Q p ) we can define certain U p -operators on isotypic components of σ I1 . Concretely, by letting t 1

  Lemma 5.3 ([HLM] Lemma 3.1.8). Let (a, b, c) ∈ Z 3 be a triple with a -b > 0, b -c > 0, a -c < p -1 and defineτ def = Ind K I ω b ⊗ ω a ⊗ ω c . Let σ be a representation of G(Q p ) over F. Then Hom K (τ , σ)[U i ] = Hom K (τ /M i , σ)for i ∈ {1, 2}, where M 1 (resp. M 2 ) is the minimal subrepresentation of τ containing F (a, c, b -p + 1) (resp. F (c + p -1, b, a -p + 1)) as subquotient.

  ) χ b ⊗ χ a ⊗ χ c be a principal series representation, where the smooth characters χ• : Q × p → E × verify χ • | Z × p = ω • for • ∈ {a, b,c} and where a, b, c are distinct modulo p -1. (i) On the one-dimensional isotypic component π I,(b,a,c) p we have U 1 = χ b (p) -1 and U 2 = χ b (p) -1 χ a (p) -1 . (ii) On the one-dimensional isotypic component π I,(a,c,b) p

Remark 6. 1 .

 1 It important to note that for places v which split as v = ww c in F the composite c • ι w is conjugate by an element of GL 3 (O F w c ) to the transpose inverse of ι w c (cf. [EGH13], Section 7.1.1).

  1 w c are dual to each other by Remark 6.1. As explained in [EGH13], Section 7.3, a Serre weight V admits an explicit description in terms of GL 3 (k w )-representations. More precisely, let w be a place of F above p and write v def = w| F + . The element c ∈ Gal(F/F + ) induces an involution S w ∼ → S w c and we define the set ⊕ w|p (Z 3 + ) Sw 0 as the set of tuples (a w , b w , c w ) w (where each triple (a w , b w , c w ) is dominant) verifying: a w,σ + c w c ,σc = 0, b w,σ + b w c ,σc = 0, c w,σ + a w c ,σc = 0 (6.2.1) for all σ ∈ S w . If the triple a w def = (a w , b w , c w ) ∈ Z 3 + is restricted (i.e. 0 ≤ a w,σ -b w,σ , b w,σc w,σ ≤ p -1 for all w|p, σ ∈ S w ) we consider the Serre weight F a w = F (a w , b w , c w ) as defined in [EGH13], Section 4.1.2. It is an irreducible representation of GL 3 (k w ), hence of G(k v ) and (by inflation) of G(O F + v ) via the morphism ι w . As above, condition (6.2.1) implies that F (a w , b w

which is a

  Serre weight for G(O F + ,p ). From [EGH13], Lemma 7.3.4 if V is a Serre weight for G, there exists a tuple a = (a w , b w , c w ) w ∈ w|p (Z 3 + ) Sw 0 and a decomposition V ∼ = v|p V v

F

  (c -1, r, 0), F (p -2, r, c -(p -1)), F (c -1, 0, r -(p -1)), F (p -2, c, r), F (p -2 + r, c, 0), F ((p -1) + c, p -2, r).

F with v 0 def=

 0 w 0 | F + and fix a sufficiently small compact open U = U v0 × U v0 ≤ G(A ∞,p F ) × G(O F + ,p ) where U v0 ⊂ G(A ∞,v0

  -1) a2-a1 • a 1 -a 0 a 2 -a 1 • FL(r| G Fw 0 ) • S (6.4.1) on S sm (U v0 , V v0 )[m r ] I,(-a1,-a0,-a2) [U 1 , U 2 ]. Moreover, S sm (U v0 , V v0 )[m r ] I,(-a1,-a0,-a2) [U 1 , U 2 ]

  def = (0, 0, 0) we define the tuple λ def = (λ w ) v|p,v =v0 , λ w0 ) ∈ ⊕ v|p (Z 3 ) and set M def = S sm (U v , V v0 ) I,(-a1,-a0,-a2) mr .

  (i) M E = p M E [p E ]where the direct sum runs over the minimal primes of T;(ii) For each minimal prime p of T we have M E [p E ] = π m(π)π I,(-a1,-a0,-a2) v0 ⊗ π ∞,v0 U v 0, where π ⊗ E C runs among the cuspidal automorphic representations such that the representation π ∞ ⊗ E C is algebraic, of weight determined by ( V v0 ) ∨ , r ∨ π lifts r, and the Satake parameters of the base change of π v to G(F w ) (for v = w| F + with w ∈ P) are determined by p E ; (iii) there are smooth, E-valued characters ψ ai :Q × p → Q × p such that ψ ai | Z × p = ω -i for i ∈ {0, 1, 2} and such that for any π as in item (ii) we haveπ v0 ∼ = Ind GL3(Qp) B(Qp) ψ a1 | • | 2 ⊗ ψ a2 | • | ⊗ ψ a0 ;

  is in the image of ı. 6.2. Serre weights. We recall the notion of Serre weights of r : G F → GL 3 (F) and relate constituents of GL 3 (O Fw )-types and potentially crystalline lifts of r| G Fw . Definition 6.3. A Serre weight for G (or just Serre weight if G is clear from the context) is an isomorphism class of a smooth, absolutely irreducible representation V of G(O F + ,p ). If v|p is a place of F + , a Serre weight at v is an isomorphism class of a smooth, absolutely irreducible representation V v of G(O F + v ). Finally, if w|p is a place of F , a Serre weight at w is an isomorphism class of a smooth, absolutely irreducible representation V w of GL 3 (O Fw ).

Acknowledgements References

Acknowledgements

The authors express their deepest gratitude to Florian Herzig for his kind suggestion to work on this problem, his encouragements, his many remarks, and his constant interest.

Then it is immediate from the equation (3.2.5) that k + p(pm 0 + m 1 ) ≡ r + 1 mod (e); j + pm 2 ≡ (c + 1) mod (p -1).

Hence, j ≡ c + 1 -m 2 mod (p -1) and k ≡ r + 1 -m 0 -pm 1 mod (e). The determinant condition (3.2.1) gives rise to the condition m 0 + m 1 + m 2 ≡ 3 mod (p -1) and so m 0 + m 1 + m 2 = 3 since p > 5.

For case (b), assume that x ≡ (p + 1)j mod (e), y ≡ k mod (e), and z ≡ pk mod (e). From equation (3.2.4) we can write pr x + r y = ae for 0 ≤ a ≤ 2(p + 1). From the equation (3.2.5) we get (p + 1)j + pa ≡ r + 1 mod (e); pk + ps ≡ (p + 1)(c + 1) mod (e).

From the determinant condition (3.2.1), we have (3.2.6) (p + 1)(c -r + 3) -(p + 1)s ≡ pa -(r + 1) mod (e), and so a ≡ -(r + 1) mod (p + 1). We let a = (p + 1) -(r + 1) where ∈ {1, 2} (recall that 0 ≤ a ≤ 2(p + 1)). We now determine j, k, and s in terms of a = (p + 1) -(r + 1) and the inertial weights. We have (p + 1)j ≡ (r + 1) -pa ≡ (r + 1) -p[ (p + 1) -(r + 1)] ≡ (p + 1)(r + 1 -) mod (e) and hence j ≡ r + 1 -mod (p -1). From equation (3.2.6) we have (p + 1)s ≡ (p + 1)(c -r + 3) -p[ (p + 1) -(r + 1)] + (r + 1) ≡ (p + 1)(c + 4 -) mod (e) and so we have s ≡ c + 4 -mod (p -1). We write s = c + 4 -+ δ(p -1) for δ ∈ {0, 1} (again, since 0 ≤ s ≤ 2(p + 1)). Finally k is immediately deduced from s: k ≡ (p + 1)(c + 1) -s ≡ (p + 1)(c + 1)

We now describe r x , r y in the filtration. From the equation (3.2.4), r x ≡ pk -(p + 1)j ≡ (c -r + -δ) + p(δ + 2 -r -4) mod (e) and r y ≡ (p + 1)j -k ≡ (r + 4 -δ -2 ) + p(δ -+ r -c) mod (e). Hence we have r x = (c -r + -δ) + p(δ + 2 -r -4) + m 0 e and r y = (r + 4 -δ -2 ) + p(δ -+ r -c) + m 1 e for some m 0 , m 1 ∈ {1, 2} (since 0 ≤ r x , r y ≤ 2e).

We finally determine m 0 , m 1 . We have ae = pr x + r y = (δ + 2 -r -4 + pm 0 + m 1 )e and so (p+1)-(r +1) = a = δ +2 -r -4+pm 0 +m 1 . Hence, we have δ + -3+m 1 = p( -m 0 ) which immediately implies that m 0 = and m 1 = 3 -δ -. The requirement m 1 ∈ {1, 2} implies that (δ, ) = (1, 2).

For case (c), assume that x ≡ k mod (e), y ≡ (p + 1)j mod (e), and z ≡ pk mod (e). We write pr x + r y = ae for 0 ≤ a ≤ 2(p + 1) from the equation (3.2.4). From the equation (3.2.5) we get k + pa ≡ r + 1 mod (e); pk + ps ≡ (p + 1)(c + 1) mod (e). We now determine j, k, s in terms of a and the inertial weights. From the determinant condition (3.2.1), we have

We also have p(r+1-pa) ≡ pk ≡ (p+1)(c+1)-ps which gives s ≡ (p+1)(c+1)-(r+1)+pa ≡ (c -r) + p(c + 1 + a) mod (e). Hence we can write s = (c -r) + p(c + 1 + a) -e = (c-r+ )+p(c+1+a-p ) where ∈ {1, 2} since 1 ≤ s, a ≤ 2(p+1). Define δ := c+1+a-p . Then δ ∈ {0, 1} (since 0 ≤ s ≤ 2(p + 1)) and we have a = δ + p -(c + 1). We finally obtain j ≡ + δ -3 mod (p -1) and k ≡ r + 1 -pa ≡ r + 1 -+ p(c + 1 -δ) mod (e).

We now describe r x , r y in the filtration. From the equation (3.2.4), r x ≡ (p + 1)j -k ≡ (2 + δ -r -4) + p(2δ + -c -4) mod (e) and r y ≡ pk -(p + 1)j ≡ (c + 4 -2δ -) + p(r + 4 -δ -2 ) mod (e). So we can write r x = (2 + δ -r -4) + p(2δ + -c -4) + m 0 e and r y = (c + 4 -2δ -) + p(r + 4 -δ -2 ) + m 1 e for some m 0 ∈ {1, 2} and m 1 ∈ {0, 1} 6.4. Local-global compatibility. From now on we assume that p is totally split in the CM field F . We fix a place w 0 of F above p and let v 0 def = w 0 | F + . The aim of this section is to prove that under suitable local hypotheses, the Fontaine-Laffaille invariant FL(ρ 0 ) defined in Section 2.1 can be recovered from a refined Hecke action when ρ 0 : G Qp → GL 3 (F) is realized as a local parameter in an automorphic Galois representation r : G F → GL 3 (F).

From now on we assume that the Galois representation r : G F → GL 3 (F) is automorphic of weight V w0 = F (a 2,w0 , a 1,w0 , a 0,w0 ) at w 0 (cf. Definition 6.5) Let V v0 def = v|p, v =v0 W λv where W λv def = W (a2,w,a1,w,a0,w) • ι w for any w|v (cf. Section 6.1). We fix a sufficiently small subgroup U of G(A ∞,p F ) × G(O F + ,p ), unramified at all places dividing p, and such that

where m r is the system of Hecke eigenvalues associated to r in the Hecke algebra T P as in Section 6.1 (such a subgroup exists, cf. [EGH13], Remark 7.3.6.). Note that we can write

) is compact open. We first prove the modularity of certain Serre weights, which will be needed to prove Theorem 6.13. We introduce the following useful notation. If

Lemma 6.11. Assume that r : G F → GL 3 (F) is absolutely irreducible and automorphic, and that ρ

Proof. The argument is not new: it is a "weight cycling" technique for GL 3 , first used in [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Theorem 6.2.3 for a niveau three Galois representation, and recently adapted in the niveau two semisimple case in upcoming work by Hui Gao [Gao].

As in [Gao] one proves that the commuting operators T 1 , T 2 (acting on S(V ) mr for any V ∈ W w (r) and defined as in [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 4.2) act nilpotently on S(V ) mr whenever V / ∈ {F (a 2 -1, a 1 , a 0 + 1), F (a 2 -1, a 0 + 1, a 1 -p + 1)} (as ρ is not semisimple, it can not have ω a2+1 as a quotient); if else V ∈ {F (a 2 -1, a 1 , a 0 + 1), F (a 2 -1, a 0 + 1, a 1 -p + 1)} then T 2 = 0 (while T 1 still acts nilpotently).

As T 1 acts nilpotently on both S(F (a 2 -1, a 0 + 1, a 1 -p + 1)) mr and S(F (a 2 -1, a 1 , a 0 + 1)) mr we deduce from [EGH13], Proposition 6.1.3 and the upper bound on W w0 (r) (Theorem

i.e. that these two weights cycle to each other (this is independent on the value of FL(ρ 0 )).

Assume that FL(ρ 0 ) / ∈ {0, ∞} and that F (p -1 + a 0 , a 1 , a 2 -p + 1) ∈ W w0 (r). As T i act nilpotently on S(F (p -1 + a 0 , a 1 , a 2 -p + 1)) mr for i = 1, 2 we conclude by [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Proposition 6.1.3 and the weight elimination above that F (a 2 -1, a 1 , a 0 + 1) ∈ W w0 (r).

Assume that FL(ρ 0 ) = 0 and that one of F (p -1 + a 0 , a 1 , a 2 -p + 1), F (a 1 + p -1, a 2 , a 0 ) is modular. By Theorem 6.7 (and again [EGH13] Proposition 6.1.3(ii)) we deduce that F (a 1 + p -1, a 2 , a 0 ) can be cycled to F (a 2 -1, a 1 , a 0 + 1) via T 2 (cf. Remark 6.12 (iv) and (v)). Similarly, F (p -1 + a 0 , a 1 , a 2 -p + 1) can be cycled to F (a 2 -1, a 1 , a 0 + 1) via T 1 .

Finally, consider the case FL(ρ 0 ) = ∞. As above, the weight F (a 2 , a 0 , a 1 -p + 1) (resp. F (p -1 + a 0 , a 1 , a 2 -p + 1)) cycles to F (a 2 -1, a 1 , a 0 + 1) via T 2 (resp. T 1 ). Similarly, F (a 0 + p -1, a 2 , a 1 ) cycles to F (a 2 , a 0 , a 1 -p + 1) via T 2 (resp. to F (p -1 + a 0 , a 1 , a 2 -p + 1) From (iii)-(iv) above and Corollary 4.4 we deduce (cf. Lemma 5.4) that the eigenvalues of the U p -operators have positive valuation. In particular T is a finite reduced, local O E -algebra, with maximal ideal m generated by the image of m r , U 1 , and U 2 .

Moreover, from (iii)-(iv) above and Corollary 4.4 the ϕ-eigenvalue on D

Qp,2 st (r ∨ π ) I Fw 0 = ω a 1 is given by p 2 ψ a1 (p) -1 and hence FL(r|

By Proposition 5.2 specialized at (a, b, c) = (-a 0 , -a 1 , -a 2 ) we have (6.4.2)

The argument of [HLM], Theorem 4.4.1 shows that M [p] is free of rank d and we have an isomorphism

which implies the desired relation (6.4.1) on

Let

Let v ∈ M F [m] be non-zero. Then by the upper bound of Theorem 6.7 we see by Lemma 5.3 and [Le], Proposition 2.2.2 that K • v is uniserial, of shape F (-a 0 -1, -a 1 , -a 2 + 1)-F (-a 1 + p -1, -a 0 , -a 2 ) and K • Πv is uniserial, of shape F (-a 0 -1, -a 1 , -a 2 + 1)-F (-a 0 , -a 2 , -a 1 -p + 1). Hence Sv, S • Πv are non-zero by Proposition 5.1 and the result follows.

Remark 6.14. There is a symmetry under the involution w 0 → w c 0 . Indeed, if w 0 is a place where ρ w0 def = r| G Fw 0 admits a Fontaine-Laffaille parameter (in particular, it is non-semisimple, and maximally non-split if its niveau is moreover one) then FL(ρ w0 ) = ι FL(ρ w c 0 ) where ι : P 1 (F) → P 1 (F) denotes the standard involution on the projective line. Similarly, the role of the group algebra operators is exchanged: one has S w0 = S w c 0 and S w c 0 = S w0 (in the obvious notation).

From the proof of Theorem 6.13, we deduce the following modularity result:

Corollary 6.15. Assume that r satisfies the assumption (i) in Theorem 6.13. Then

Furthermore,

Assume moreover that F is unramified at all finite places of F + and that there is a RACSDC automorphic representation Π of GL 3 (A F ) of level prime to p such that • r rp,i (Π);

Then W ?,obv w0 (r) ⊆ W w0 (r).

Proof. The first part is immediate from Lemma 6.11. Assume now that FL(r| G Fw 0 ) = ∞. The argument is now similar to [HLM], Proposition 4.5.8.

We claim that F (a 2 , a 0 , a 1 -p + 1) ∈ W w0 (r). Suppose that K • v contains the weight F (-a 1 , -a 2 , -a 0 -p + 1). Then an easy check (as in the proof of Lemma 6.11) shows that both Hecke operators T 1 and T 2 act by zero on F (a 0 + p -1, a 2 , a 1 ), which implies, by weight cycling and Theorem 6.7 above, that F (a 2 , a 0 , a 1 -p + 1) is in W w0 (r).

We now suppose that K • v does not contain the weight F (-a 1 , -a 2 , -a 0 -p + 1). Then both K •v and K •Πv are quotients of the uniserial representations F (-a 0 -1, -a 1 , -a 2 + 1)-F (-a 1 + p -1, -a 0 , -a 2 ) and F (-a 0 -1, -a 1 , -a 2 + 1)-F (-a 0 , -a 2 , -a 1 -p + 1), respectively (by Lemma [HLM], 3.1.8 and Theorem 6.7 above). As

. By Proposition 5.1 (cf. [HLM] Proposition 3.1.2) and the previous observation on K • v this forces K • v to have length one, i.e. F (a 2 , a 0 , a 1 -p + 1) is modular. The case FL(r| G Fw 0 ) = 0 is easier and treated similarly. As for the last statement (which needs to be proved only if FL(ρ 0 ) = ∞), it is enough to remark that for FL(ρ 0 ) = ∞, the representation ρ 0 admits a potentially diagonalizable lift with Hodge-Tate weights {p + 1 + a 0 , a 2 + 1, a 1 } by Proposition 2.27, and the conclusion follows from [BLGG], Theorem 4.1.9 and Lemma 5.1.1. 6.5. Freeness over the Hecke algebra. In this section, we prove Theorem 6.16, which states that the dual

, O E ) of the space of automorphic forms is free over a Hecke algebra for certain choices of compact open subgroup U v0 ( V v0 and m r are as defined in Section 6.2).

We keep the notation of Section 6.4. Hence F/Q is a CM field in which p splits, F + its maximal totally real field, with F/F + unramified at all finite places and [F : F + ] ≡ 0 mod 4. Fix a place w|p of F , and let v def = w| F + . Let r : G F → GL 3 (F) be a Galois representation with r| G Fw niveau two non-split as in Theorem 6.13 (i) satisfying the following additional properties.

(i) r is unramified at places away from p.

(ii) r is Fontaine-Laffaille and regular at all places dividing p.

(iii) r has image containing GL 3 (k) for some k ⊂ F with #k > 9.

(iv) F ker adr does not contain F (ζ p ).

By condition (iii) (stronger than the usual condition of adequacy (see Definition 2.3 of [Tho12])) we can choose a place v 1 of F + which is prime to p satisfying the following properties (see Section 2.3 of [CEG + ]).

• v 1 splits in

• ρ(Frob w1 ) has distinct F-rational eigenvalues, no two of which have ratio (Nv 1 ) ±1 . We now fix an unitary group G \F + and a model G over O F + as in Section 6.1. We require moreover that G is quasi-split at all finite places (which is possible by the choice of F ). Let U v0 ≤ G(A ∞,v0 F + ) be a compact open subgroup satisfying the following properties. (v) U v0 = G(O v0 ) for all places v which split in F other than v 1 and those dividing p; (vi) U v1 is the preimage of the upper triangular matrices under the map

The choice of the compact open set U v1 implies that U v0 U v0 is sufficiently small in the sense of Section 6.1 for any compact open subgroup U v0 of G(F v0 ).

Let P denote the set consisting of finite places w of F such that v def = w | F + is split in F and w does not divide p or v 1 . We define the maximal ideal m r of T P as in 6.4. Recall the space of automorphic forms S sm (U v0 , V v0 ) I,(-a1,-a0,-a2) mr defined in Section 6.1, which carries a natural action of the algebra T P and the operators U 1 , U 2 . From now on, we assume that the highest weights λ w ∈ (Z 3 + ) Sp 0 appearing in the constituents of V v0 ∼ = v|p, v =v0 W λv all lie in the lowest alcove (i.e. for all w|v, v ∈ S + p \ {v 0 } we have a 2,w -a 0,w < p -2). We make finally the following assumption:

I,(-a1,-a0,-a2) mr

) generated by T P (resp. T P , U 1 , and U 2 , resp. U i ). Here the subscript a stands for the "anemic" Hecke algebra. See Section 5.2 of [HLM] for the definitions of M ∞ and R ∞ . As in [HLM], we let R i be the R ∞ -subalgebra of End R∞ (M ∞ (τ )) generated by U i .

Theorem 6.16. Let r be as in Theorem 6.13 (i).Assume Proof. The proof is exactly as in Section 5 of [HLM]. The key point is that Lemma 5.3.3 of [HLM] still holds using Theorem 6.7 in place of Theorem 4.3.1 of [HLM].

Note that by combining Proposition 2.27, Theorems 6.13, 6.16 and [EG] Corollary A.7 we can infer the following: Theorem 6.17. Let ρ 0 be as in Definition 2.4.Then there is a CM field F , an automorphic Galois representation r : G F → GL 3 (F), verifying r| Fw ∼ = ρ 0 for all w|p, such that all the hypotheses in the setup of Section 6.5 are satisfied.

In particular Theorem 6.13 applies to r: if FL(r| G Fw ) = ∞ (resp. FL(r| G Fw ) = 0) then S sm (U v0 , V v0 )[m r ] I,(-a1,-a0,-a2) [U 2 ] (resp. S sm (U v0 , V v0 )[m r ] I,(-a0,-a2,-a1) [U 1 ]) is free over T and if moreover FL(r| G Fw ) / ∈ {0, ∞} then the equality (6.4.1) of refined Hecke operators on S sm (U v0 , V v0 )[m r ] I,(-a1,-a0,-a2) [U 1 , U 2 ] holds true.