V Main Result 1
In this section, we review the result of Dousmanis [1] to find all admissible filtered (φ,N)-modules corresponding to the all 2-dimensional semi-stable representations ofGQp with parallel labeled Hodge–
Tate weights {0,1}. To use the result, we find its strongly divisible modules and its Breuil modules.
LetK =Qpf, hence the maximal unramified subextension K0=Qpf. LetE be a large enough finite extension ofQpf. Letτ be the absolute Frobenius ofQpf, and fix an embeddingιf :Qpf ,→E. Define τj=ιf◦τj for all j=0,1,· · ·,f−1.
∏i=0f−1ai =∏i=0f−1bi, then ∏i=0f−1ai∏i=0f−1xi =∏i=0f−1bi∏i=0f−1xi, which implies af−1xf−1 =bf−1x0, i.e., a·x=b·φ(x).
LetDbe a rank twoφ-module overEf and letη andebe ordered bases. Then we can get a matrix M∈GL2(Ef)such that
η1 η2
=
e1 e2
M.
UsingM, we can easily get the equation
[φ]e=M[φ]ηφ(M)−1.
Lemma V.3. LetDbe a rank twoφ-module overEf.
(1) Ifφf is not anE×-scalar times the identity map, then there exists an ordered basisη ofDsuch that[φ]η = ⃗ε ⃗0
⃗η ⃗θ
!
, with the additional properties that:
(a) If Nmφ(⃗ε)̸=Nmφ(⃗θ), then⃗η=⃗0,
(b) If Nmφ(⃗ε) =Nmφ(⃗θ), then⃗ε =⃗θ and⃗ηφ =⃗1, where⃗ηφ is the(2,1) entry of the matrix Nmφ([φ]η).
(2) If φf =α·⃗id for some α ∈E×, then there exists an ordered basis η of D such that [φ]η = diag((α,1,· · ·,1),(α,1,· · ·,1)).
Proof. (The proof of Lemma 2.3 in Dousmanis [1])
(1) Sinceφf is anEf-linear isomorphism, then we can have an ordered basiseofDsuch that[φf]e=
⃗α ⃗0
⃗γ ⃗λ
!
withγi=0 whenαi̸=λi andγi∈ {0,1}whenαi=λi. Also, it impliesαiλi̸=0 for alli. Let P:= [φ]e = (P0,· · ·,Pf−1) andQ:=Nmφ(P) = (Q0,· · ·,Qf−1)(= [φf]e). SinceQ=Pφ(Q)P−1, i.e., Qi=PiQi+1P−1, then{αi,λi}={αi+1λi+1}for alli, which impliesαiλi=α0λ0for alli.
We will get another ordered basis η of D having better conditions rather than e. Define the matrix R= (R0,· · ·,Rf−1), whereRi= 0 1
1 0
!
ifαi=λ0orRi= 1 0 0 1
!
ifαi̸=λ0which impliesαi=α0. Therefore, whenα0=λ0, sinceαi=λ0for alli, thenRQR−1= α0·⃗1 ⃗γ
⃗0 α0·⃗1
!
, and, whenα0̸=λ0, sinceγi=0, thenRQR−1= α0·⃗1 ⃗0
⃗0 λ0·⃗1
!
. By the way, inα0=λ0 case,⃗γ =⃗1 because, ifri =0, by the equationQi=PiQi+1Pi−1, thenri+1=0, i.e.,⃗γ=⃗0, which is contradiction because the assump- tion that φf is not an E×-scalar times the identity map. Therefore, we can get an ordered basis η such that [φf]η = λ0·⃗1 ⃗0
γ·⃗1 α0·⃗1
!
with γ=1 if α0 =λ0 and γ=0 ifα0̸=λ0. Using the equation
[φ]ηφ([φf]η) = [φf]η[φ]η, we can check (a), (b) in the statement.
(2) It suffices to prove the claim that, if Nmφ(P) =diag(α·⃗1,α·⃗1)for someP∈GL2(Ef)andα∈E×, then there exists some matrixQ∗∈GL2(Ef)such that
Q∗Pφ(Q∗)−1=diag((α,1,· · ·,1),(α,1,· · ·,1)).
Proposition V.4. LetDbe a rank twoφ-module overEf. Then there exists an ordered basisηofDwith respect to[φ]η which takes one of the following forms:
(1) [φ]η =diag(α·⃗1,λ·⃗1)for someα,λ∈E×withαf ̸=λf, (2) [φ]η =diag(α·⃗1,α·⃗1)for someα ∈E×, or
(3) [φ]η = α·⃗1 ⃗0
⃗1 α·⃗1
!
for someα∈E×.
Proof. Dousmanis [1] proved the proposition (Proposition 2.2).
Again, the notation in the proof of this lemma has independent meaning and should not be confused with that of previous sections. Chooseηas in Lemma 5.3. in case (1)(a) so that[φ]η =diag(⃗ε,⃗θ)with Nmφ(⃗ε)̸=Nmφ(⃗θ), letα1,λ1∈E×be such that Nmφ(⃗ε) =α1f·⃗1 and Nmφ(⃗θ) =λ1f·⃗1. By Lemma 5.2 there exists a matrixM∈GL2(Ef)such thatM([φ]η)φ(M)−1=diag(α1·⃗1,λ1·⃗1), and clearlyα1f ̸=λ1f. This gives the first possibility of the proposition.
In case (1)(b) of Lemma 5.3, letα1 be an f-th root of α. By Lemma 5.2 there exists a matrixM∈ GL2(Ef)such that
M([φ]η)φ(M)−1= α1·⃗1 ⃗0
⃗γ α1·⃗1
! .
Since
[φf]η= α1f·⃗1 ⃗0 α1f−1Trφ(⃗γ) α1f·⃗1
!
, [φf]e= α·⃗1 ⃗0
⃗1 α·⃗1
! ,
we have Trφ(⃗γ)̸=⃗0. Let
M∗= f·⃗1 ⃗0
⃗z Trφ(⃗γ)
! , where
⃗z= (0,1,· · ·,f−1)Trφ(⃗γ)−f(γ0,γ0+γ1,· · ·,γ0+· · ·+γf−2).
Then
α1·⃗1 ⃗0
⃗γ α1·⃗1
!
φ(M∗) =M∗ α1·⃗1 ⃗0
⃗1 α1·⃗1
! . This gives the third possibility of the proposition.
Finally, in case (2)(b) of Lemma 5.3, letα1∈E×be an f-th root ofα and proceed as in case(1). This gives the second possibility of the proposition and concludes the proof.
Definition. (1) A filtered(φ,N)-moduleDis calledF-semisimpleif there exists an ordered basisη ofDwith respect to[φ]η which takes case (1) or (2) in Proposition 5.4.
(2) A filtered(φ,N)-moduleDis calledF-scalarif there exists an ordered basisηofDwith respect to[φ]η which takes case (2) in Proposition 5.4.
(3) A filtered(φ,N)-moduleDis called non-F-scalar if there exists an ordered basisη ofDwith respect to[φ]η which takes case (1) in Proposition 5.4.
Proposition V.5. LetDbe a rank two(φ,N)-module overEf.
(1) IfDisF-semisimple and[φ]η =diag(α·⃗1,λ·⃗1), then the monodromy operatorNis as follows:
(a) Ifαf ̸=p±fλ±f, thenN=0, (b) Ifαf =pfλf, then[N]η = ⃗0 ⃗0
⃗n ⃗0
!
, where⃗n=n(1,ζ,· · ·,ζf−1)withζ = pλα andn∈E,
(c) Ifλf =pfαf, then[N]η =
⃗0 ⃗n
⃗0 ⃗0
!
, where⃗n=n(1,ε,· · ·,εf−1)withε= pαλ andn∈E.
(2) IfDis non-F-semisimple, thenN=0.
Proof. (The proof of Proposition 2.5 in Dousmanis [1])
Use[N]η[φ]η =p[φ]ηφ([N]η)and Lemma 5.2, then we get this proposition.
Corollary V.6. LetDbe a rank two filtered (φ,N)-module with nontrivial monodromy. There exists an ordered basisηofDwith respect to[φ]η =diag(α·⃗1,λ·⃗1)for someα,λ ∈E× withα =pλ and [N]η = ⃗0 ⃗0
⃗1 ⃗0
! .
Proof. (The proof of Corollary 2.6 in Dousmanis [1])
Ifαf =pfδf, then change the basis toη′= (η1,⃗n·η2). Ifδf =pfαf, then change the basis toη′= (⃗n·η1,η2).
We want to find semi-stable noncrystalline case. By Corollary 5.6, we can choose a basisη forD such that[φ]η =diag(pλ·⃗1,λ·⃗1)and[N]η =
⃗0 ⃗0
⃗1 ⃗0
! .
Labeled Hodge–Tate weights
LetDbe a rank two filtered(φ,N)-module overEf and defineDτi :=eτiD. We have the decomposition D=
f−1
M
i=1
Dτi.
SinceDis free of rank two overEf, the componentsDτi are equidimensional overE, each of dimension two. They can have the filtration by defining FiljDτi :=eτiFiljD. An integer jis calledlabeled Hodge–
Tate weightofDwith respect to the embeddingτi if FiljDτi ̸=Filj+1Dτi, counting multiplicities. The labeled Hodge–Tate weights ofDis defined(Wi)τi, whereWi is the set containing the labeled Hodge–
Tate weights ofDwith respect toτi. Since the dimension ofDτi is 2 for alli, thenWi has 2 elements.
From now on we will focus only on the({0,1})τi case.
Letη be an ordered basis ofDoverEf. We can have
FiljDτi =
( Dτi if j≤0 Di if j=1 0 if j≥2
where Di =Ef ·(⃗xi·η1+⃗yi·η2)eτi for some⃗xi := (xi0,· · ·,xif−1),⃗yi:= (yi0,· · ·,yif−1)∈Ef with the additional condition that(xii,yii)̸= (0,0). Since FiljD=Li=0f−1 FiljDτi, then Fil1D=Li=0f−1Di and we have
FiljD=
( D if j≤0 Ef·(⃗x·η1+⃗y·η2) if j=1
0 if j≥2
where⃗xi:= (x00,· · ·,xff−1−1),⃗yi:= (y00,· · ·,yff−1−1)∈Ef with(xii,yii)̸= (0,0).
Admissibility
Proposition V.7. LetDbe a rank two filteredφ-module with an ordered basisη ofDwith respect to [φ]η =diag(pλ·⃗1,λ·⃗1)for someλ∈E×. All theφ-stableEf-submodules ofDare 0,D,D1=Ef·η1, or of the formD2=Ef·η2.
Proof. Obvious.
Corollary V.8. LetDbe a rank two filtered(φ,N)-module with an ordered basisηofDwith respect to [φ]η =diag(pλ·⃗1,λ·⃗1)for someλ ∈E×and[N]η =
⃗0 ⃗0
⃗1 ⃗0
!
. All the(φ,N)-stableEf-submodules ofDare 0,D, andD2.
Proof. Obvious.
Proposition V.9. LetDbe a rank two admissible filtered (φ,N)-module with an ordered basisη ofD with respect to [φ]η = diag(pλ·⃗1,λ·⃗1) for someλ ∈E× and[N]η = ⃗0 ⃗0
⃗1 ⃗0
!
and parallel labeled Hodge–Tate weights(0,1). Then
FiljD=
( D if j≤0 Ef·(⃗x·η1+⃗y·η2) if j=1
0 if j≥2
withxi̸=0 for alliandvp(λ) =0.
Proof. By Proposition 4.3 in Dousmanis [1], 2e f vp(λ) +e f =
∑
i∈I0
ki, e f vp(λ)≥
∑
{i∈I0:xi=0}
ki, hencevp(λ) =0 andxi̸=0 for alli.
To sum up, an admissible filtered(φ,N)-moduleD= (Qpf ⊗QpE)·(η1,η2)together with (1) [φ]η = pλ·⃗1 ⃗0
⃗0 λ·⃗1
!
, whereλ ∈E×andvp(λ) =0;
(2) [N]η= ⃗0 ⃗0
⃗1 ⃗0
!
;
(3) FiljD=
( D if j≤0, (Qpf ⊗QpE)·(η1+⃗yη2) if j=1,
0 if j≥2
represents all rank two semi-stable noncrystallinep-adicE-representations ofGQ
p f with parallel labeled Hodge–Tate weights(0,1).
5.2 Another admissible filtered(φ,N)-modules