RAMIFICATION OF CRYSTALLINE REPRESENTATIONSSHIN HATTORI
Abstract. This is a survey on integral p-adic Hodge theory, especiallyon the Fontaine-Laﬀaille theory, and a ramiﬁcation bound for crystalline
representations due to Abrashkin and Fontaine, based on the author’s
talks at the spring school “Classical and p-adic Hodge theories”.

Contents
1. Introduction
2. Fontaine-Laﬀaille modules
2.1. Deﬁnition of Fontaine-Laﬀaille modules
2.2. Associated Galois representations
2.3. Relation to crystalline cohomology
2.4. Ramiﬁcation of torsion crystalline representations
2.5. Classiﬁcation of ﬁnite ﬂat group schemes
3. Further developments
3.1. Applications and generalizations of the Fontaine-Laﬀailletheory3.2. Wach modules
3.3. Breuil modules
3.4. Breuil-Kisin classiﬁcation of ﬁnite ﬂat group schemes
3.5. Kisin modules and (φ, Ĝ)-modules
3.6. Ramiﬁcation theory
4. Sketch of proofs
4.1. Cutting oﬀ higher divided powers
4.2. Classiﬁcation of simple objects∗
4.3. The fullness of Tcrys
4.4. Ramiﬁcation bound
References

1
3
4
8
12
13
16
17
17
18
19
20
21
22
24
24
28
32
35
37

1. Introduction
Let p be a rational prime, k a perfect ﬁeld of characteristic p and W =W (k) the Witt ring for k. Put Wn = W/pn W for any n ∈ Z>0 . We write
σ for the p-th power Frobenius maps of various p-torsion rings, and also
Date: September 17, 2014.
1

2

SHIN HATTORI

for the natural lift (x0 , x1 , . . .) 7→ (xp0 , xp1 , . . .) to W of the Frobenius map
on k. Put K0 = Frac(W ). Let K be a ﬁnite totally ramiﬁed extension of
K0 of degree e with integer ring OK , maximal ideal mK and uniformizer
πK . We ﬁx an algebraic closure K̄ of K and let k̄ be the residue ﬁeld of K̄,
which is an algebraic closure of k. Let vp and vK be the p-adic valuations
on K̄ normalized as vp (p) = 1 and vK (πK ) = 1, respectively. Put GK =Gal(K̄/K). Let C be the completion of K̄ and OC its integer ring. Let K ur
be the maximal unramiﬁed extension of K inside K̄ and IK = Gal(K̄/K ur )the inertia subgroup of GK . We denote the category of Zp -modules with acontinuous GK -action for the p-adic topology by RepZ (GK ).
p
Recall that p-adic Hodge theory has the following two aspects:(i) to study, or classify if possible, nice p-adic GK -representations (suchas Hodge-Tate, de Rham, semi-stable and crystalline representations), using semi-linear algebraic data (such as ﬁltered (φ, N )modules).
(ii) to study a relation between p-adic etale cohomology of algebraic
varieties over K and other cohomology theories on the diﬀerentialside, such as de Rham, log-crystalline and crystalline cohomology.
Integral p-adic Hodge theory is its variant of Zp - or p-power torsion coefﬁcients. Namely, its aim can be summarized as:(i) to study, or classify if possible, subquotients of nice p-adic representations, using semi-linear algebraic data which are in a sense
subquotients of the data attached to p-adic representations (suchas ﬁltered (φ, N )-modules).
(ii) to study a relation between etale cohomology of Zp - or p-power torsion coeﬃcients of algebraic varieties over K and other cohomology
theories on the diﬀerential side, such as de Rham, log-crystalline
and crystalline cohomology.
Example 1.1.
(i) Let A be an Abelian variety over K of dimension
g. Let Tp (A) = limn A[pn ](K̄) be the Tate module of A and put←−
Vp (A) = Tp (A) ⊗Zp Qp . The module Tp (A) is a free Zp -module ofrank 2g on which GK acts continuously (for the p-adic topology).
Similarly, the module Vp (A) is a p-adic GK -representation of dimension 2g, and moreover it is de Rham with Hodge-Tate weights in[0, 1]. The module Tp (A) is a GK -stable Zp -lattice in Vp (A), and the
GK -module A[pn ](K̄) is isomorphic to the quotient Tp (A)/pn Tp (A).
These subquotients of Vp (A) are typical examples of what integral
p-adic Hodge theory studies on the aspect (i).
More generally, let V be a de Rham GK -representation. Thenwe want to study any GK -stable Zp -lattice T in V and the quotient
T /T ′ for any such lattices T ⊇ T ′ in V .

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

3(ii) Let X be a proper smooth scheme over K. Then the etale cohomoli (X , Z ) and H i (X , Z/pn Z) are what we want toogy groups Het
p
et
K̄
K̄
study on the aspect (ii).
The integral theory captures more subtle information than the p-adic
counterpart, such as ramiﬁcation of Galois representations. We can passfrom the theory of Zp -coeﬃcients both to that of Qp -coeﬃcients by invertingp, and to that of Fp -coeﬃcients by reducing modulo p. For example, let V
be a semi-stable GK -representation with associated ﬁltered (φ, N )-module
D and T a GK -stable Zp -lattice in V . By integral p-adic Hodge theory,
we can associate with T a semilinear algebraic data M . The data M is a
ﬁnitely generated module over a coeﬃcient ring which is a W -algebra, andM is endowed with additional structures (such as a Frobenius map). We canrecover the ﬁltered (φ, N )-module D via M ⊗W K0 , and the data M/pM
has information of the GK -representation T /pT . In some cases M is deﬁned
as a W -lattice in D as we will see in §2, while it is not the case in general.
Downsides are the following: In the integral case, the coeﬃcient ring ofsemi-linear algebraic data is not a ﬁeld, unlike the coeﬃcient ﬁeld K0 of ﬁltered (φ, N )-modules. Actually, it is not even a complete discrete valuation
ring but a larger W -algebra, unless the base ﬁeld K is absolutely unramiﬁed(namely, K = K0 ), and the theory becomes more complicated compared tothe p-adic theory. Moreover, it is often very hard to describe explicitly the
semilinear algebraic data associated with a given integral GK -representation.
In this article, we will focus on the classical integral theory of FontaineLaﬀaille [FL82], which treats the subquotients of crystalline GK -representations
with Hodge-Tate weights in [0, p − 1] for the case where K is absolutely unramiﬁed. We introduce basic deﬁnitions and state main theorems of thetheory in §2. We also give a sketch of proofs for some of the main theorems(Theorem 2.9 and Theorem 2.19) in §4. In fact, now we have an integral theory without these three restrictions (crystalline representations, small rangeof Hodge-Tate weights, absolutely unramiﬁed base ﬁeld) thanks to works ofmany people including Breuil, Kisin and Liu. We will discuss such recent
developments of the integral theory in §3.
The author would like to thank Yoshiyasu Ozeki, Takeshi Saito andYuichiro Taguchi for helpful comments on an earlier draft of the article.
The interested reader is recommended also to consult other survey articles[Bre02, BMs02, BC14+], to which this article owes very much.

2. Fontaine-Laffaille modules
Let the notation be as in §1. Let V be a p-adic GK -representation and∗ (V ) = Hom
D = Dcrys
Qp [GK ] (V, Bcrys ) the associated ﬁltered φ-module. The
module D is a K0 -vector space satisfying dimK0 (D) ≤ dimQp (V ). We say
V is crystalline if the equality holds. Put DK = D ⊗K0 K. The Hodge-Tate• ̸= 0. Note that here we adopt theweights are the integers i such that gri DK

4SHIN HATTORI

convention such that the cyclotomic character Qp (1) has the Hodge-Tateweight one.
Remark 2.1. This dual convention to the usual Dcrys (V ) = (V ⊗Qp Bcrys )GK
is preferred in integral p-adic Hodge theory. A reason for this dual preference∗ (V ) is a Hom group, and so isseems to be the following: The module Dcrys∗ (D) = Homits quasi-inverse Vcrys
K0 ,φ,Fil (D, Bcrys ). In the integral theory, we
use a similar Hom group to construct Galois representations from semilinear
algebraic data, as we will see later. One of the advantages of Hom and Ext
compared to ⊗ and Tor is that we can study them in a more explicit way, byusing the Yoneda extension. This makes the analysis of the associated Galois
representations easier especially in the integral theory, where the coeﬃcientring of semilinear algebraic data is not a ﬁeld and we cannot always resort
to dimension calculation or ﬂatness. However, in many cases we can develop
covariant integral theories and they are also useful (for example, see §2.3).
Now we concentrate on the case of K = K0 . Let V be a crystalline GK0 ∗ (V ).
representation with Hodge-Tate weights in [0, p − 1] and put D = Dcrys
The module D is a K0 -vector space of dimension dimQp (V ) endowed witha decreasing ﬁltration {Di }i∈Z and a σ-semilinear bijection φ : D → D
satisfying the admissibility condition.
The idea of the integral p-adic Hodge theory of Fontaine-Laﬀaille is to classify GK0 -stable Zp -lattices in V and their quotients by φ-stable W -lattices
M in D and their quotients. We have to impose on M an integral version ofadmissibility;
namely, the conditions that φ(M ∩ Di ) ⊆ pi M for any i and∑−ithat i∈Z p φ(M ∩ Di ) = M . In fact, these conditions originated from astructure of integral crystalline cohomology [Fon83, 1.3, Proposition]. Thisidea leads to the following deﬁnitions.
2.1. Deﬁnition of Fontaine-Laﬀaille modules.
Deﬁnition 2.2.
(i) A Fontaine-Laﬀaille module over W is a triplet(M, {M i }i∈Z , {φiM }i∈Z ), which we denote abusively by M , consisting of• a W -module M ,• a decreasing∪ﬁltration {M∩i }i∈Z of M by W -submodules satisfying M = i∈Z M i and i∈Z M i = 0,
• a family of σ-semilinear maps {φiM : M i → M }i∈Z which makes
the diagram
M i+1
_

φi+1
M



Mi

/M


φiM

×p

/M

commutative for any i. We often drop the subscript M in φiMif no confusion occurs.

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

5(ii) A morphism of Fontaine-Laﬀaille modules f : M → N over W isa W -linear map f : M → N between the underlying W -modules
satisfying f (M i ) ⊆ N i and f ◦ φiM = φiN ◦ f for any i.
(iii) A sequence of Fontaine-Laﬀaille modules 0 → L → M → N → 0 isexact if the sequence on the underlying W -modules 0 → L → M →N → 0 is exact, and also the sequence induced on the i-th ﬁltration
0 → Li → M i → N i → 0 is exact for any i.
We denote the category of Fontaine-Laﬀaille modules over W by MFW .
We also denote by MFk its full subcategory consisting of those killed by
p. With the above notion of exact sequence, the categories MFW and MFk
are exact categories in the sense of Quillen [Qui73]. Thus we can deﬁneExtn groups via the Yoneda extension and we have long exact sequences of
Hom and Ext for these categories. Though Yoneda Extn groups in an exact
category may be a proper class in general, we can see that Ext1MFW (M, N )and Ext1MFk (M, N ) are sets, since any equivalence class of such an extension
is represented by a Fontaine-Laﬀaille module whose underlying set is N ×M .
For any W -module M , put Mσ = M ⊗W,σ−1 W . We identify the data
{φiM }i∈Z in the above deﬁnition with a family of W -linear maps {φiM :M i → Mσ }i∈Z making the diagram there commutative. Note that if M isof ﬁnite length, then so is Mσ and lgW (M ) = lgW (Mσ ).
We deﬁne a category of torsion Fontaine-Laﬀaille modules satisfying atorsion version of the admissibility condition, as follows.
[a,b]

Deﬁnition 2.3.
(i) For any integers a ≤ b, we denote by MFW thefull subcategory of MFW consisting of M such that M a = M andM b+1 = 0.
(ii) We denote by MFfW,tor the full subcategory of MFW consisting of
M such that• M
of ﬁnite length,∑ is a iW -modulei) = M .
•
φ
(M
σ
i∈Z
[a,b]

The full subcategory of MFW
f,[a,b]denoted by MFW,tor .
f,[0,p−1]′

deﬁned by the same conditions isf,[0,p−1](iii) We denote by MFW,tor
the full subcategory of MFW,tor consisting of M such thatf,[0,p−1]• There exists no non-zero quotient M → N in MFW,tor satisfying N = N p−1 .
′This category is denoted in [FL82, Théorème 6.1] by MFf,ptor . Note′f,[0,p−2]
f,[0,p−1]that MFW,tor ⊆ MFW,tor .
[a,b](iv) Similar full subcategories of MFk are denoted by MFk
f,[a,b]MFk

f,[0,p−1]′

and MFk, MFfk ,.

Finally, we deﬁne the category of free objects over W , which classiﬁes
GK0 -stable Zp -lattices in crystalline GK0 -representations.

6

SHIN HATTORI
fd,[a,b]Deﬁnition 2.4. For any integers a ≤ b, we denote by MFW
the full[a,b]subcategory of MFW consisting of M such that
• M is a free W -module of ﬁnite rank,i• Msummand of the W -module M for any i,∑ is ai directi) = M .
•
φ
(M
σ
i∈Z
The sign fd stands for the French word fortement divisible, or strongly divisible (Deﬁnition 2.10).
f,[a,b]What is striking for the category MFW,tor is the following.
Proposition 2.5 ([FL82], 1.10 (b)). Any morphism f : M → N of thef,[a,b]category MFW,tor is strict with ﬁltrations. Namely, we havef (M i ) = f (M ) ∩ N i
for any i.
[a,b]Proof. First let M be any object of MFW and deﬁne a W -module M̄ by
the exact sequence⊕b⊕bl θM /
l
/
/ M̄
/ 0,
0
l=a+1 M
l=a M
where θM is given by(xa+1 , . . . , xb ) 7→ (xa+1 , −pxa+1 + xa+2 , . . . , −pxb−1 + xb , −pxb ).
Then the mapb⊕

M l → Mσ ; (ya , . . . , yb ) 7→

l=a

b
∑

φl (yl )l=ainduces a W -linear map φM : M̄ → Mσ . These constructions are functorial,
and the functor M 7→ M̄ is exact by the snake lemma. If the underlying
W -module M is of ﬁnite length, then we have the equalitylgW (M ) = lgW (M̄ ).
f,[a,b]Moreover, such M is contained in MFW,tor if and only if φM is surjective,
and from the above length equality, it is equivalent to saying that φM is
injective or bijective.
To prove the proposition, ﬁrst we assume that f is injective, and identify
M with a submodule of N . By the commutative diagram
/ N̄

M̄
≀



Mσ





≀

/ Nσ

the induced map M̄ → N̄ is injective (note that here we cannot use theexactness of M 7→ M̄ , since the cokernel object N/M is well-deﬁned only if

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

7

M l = M ∩ N l for any l). On the other hand, suppose that there exists anelement x ∈ (N i ∩ M ) \ M i . Take j < i such that x ∈ M j \ M j+1 . Since M
is p-power torsion, we have ps x ∈/ M j+1 and ps+1 x ∈ M j+1 for some integers ≥ 0. Replacing x by ps x, we may assume px ∈ M j+1 . Put
y = (0, . . . , 0, x, −px, 0, . . . , 0) ∈

b
⊕

M l,

l=a

where x is at the j-th entry. Then y ∈/ Im(θM ) since x ∈/ M j+1 , whileij+1y ∈ Im(θN ) since x ∈ N ⊆ N. Thus the image of y in M̄ is non-zero,
while the image in N̄ is zero. This contradicts the injectivity of M̄ → N̄ .
For a general f , put L = f (M ) and Li = f (M i ). Since Ker(M i →iL ) = Ker(f ) ∩ M i and f ◦ φiM = φiN ◦ f , the map φiM induces a map
φiL : Li → L for any i. Then the triplet L = (L, {Li }i∈Z , {φiL }i∈Z ) is an[a,b]object of MFW whose underlying W -module is of ﬁnite length, and thenatural maps M → L → N are morphisms of this category. Now we havethe commutative diagram
/ L̄
M̄
≀



Mσ


/ / Lσ ,which yields the surjectivity of the map L̄ → Lσ . This implies that L ∈f,[a,b]
MFW,tor . Applying the former half of the proof to the injection L → N , weobtain f (M i ) = f (M ) ∩ N i .
□f,[a,b]Corollary 2.6.
(i) ([FL82], Proposition 1.8) The category MFW,tor is
Abelian.
f,[a,b](ii) ([Win84], Proposition 1.4.1 (ii)) For any object M ∈ MFW,tor , theW -submodule M i ⊆ M is a direct summand for any i.
f,[a,b]

Proof. For the ﬁrst assertion, let f : M → N be a morphism of MFW,tor andput Ker(f )i = Ker(f ) ∩ M i . Then, by using the exact sequence0/ Ker(f )

/M

/L

/0

with L as in the proof of Proposition 2.5, we can show that the triplet(Ker(f ), {Ker(f )i }i∈Z , {φiM |Ker(f )i }i∈Z )f,[a,b]is an object of MFW,tor , just as in the proof of the latter part of Proposition
2.5. Moreover, since the map φiN induces a map φ̄iN : N i /f (M i ) → N/f (M ),Proposition 2.5 implies that the Fontaine-Laﬀaille module(Coker(f ), {N i /f (M i )}i∈Z , {φ̄iN }i∈Z )f,[a,b]is well-deﬁned, and it is an object of MFW,tor similarly. These construcf,[a,b]tions give the kernel and cokernel of f in MFW,tor . To show this category

8

SHIN HATTORI

is Abelian amounts to showing that the ﬁltrations {f (M i )}i∈Z (coimage ﬁltration) and {f (M ) ∩ N i }i∈Z (image ﬁltration) on f (M ) coincide, which is
Proposition 2.5.
On the other hand, applying Proposition 2.5 to the map pn : M → M
yields pn M i = M i ∩ pn M for any n, which implies the second assertion by
an Ext calculation.
□
2.2. Associated Galois representations. In this subsection, we will atf,[0,p−1]∗ (M ) in a similar way
tach to any object M ∈ MFW,tor a GK0 -module Tcrys∗ (D), using A
to Vcrys
crys instead of Bcrys . Recall that we have the projective
limit ringR = lim(OC /pOC ← OC /pOC ← · · · ),←−where every transition map is the p-th power map. For any element x =l(x0 , x1 , . . .) ∈ R, take any lift x̂l ∈ OC of xl and put x(m) = liml→∞ x̂pm+l ∈OC , which is independent of the choice of lifts. Then the ring R is a complete valuation ring of characteristic p with algebraically closed fraction ﬁeld
whose valuation is given by vR (x) = vp (x(0) ) for any element x ∈ R. Thisring admits a natural action of GK0 induced by the action on each entry.
Moreover, using the natural inclusion W (k̄) → OC , we consider R as a k2
algebra by the map k → R given by a 7→ ([a], [a1/p ], [a1/p ], . . .). Then the
Witt ring W (R) has a natural W -algebra structure.
We also have a natural W -linear, GK0 -equivariant surjection θ : W (R) →∑l (l)OC sending (r0 , r1 , . . .) ∈ W (R) to ∞l=0 p rl . Choose a system of p-powerl+1
lroots of −p in OC satisfying ((−p)1/p )p = (−p)1/p for any l. Put
2

β = (−p, (−p)1/p , (−p)1/p , · · · ) ∈ R
and ξ = p + [β] ∈ W (R). Then we have Ker(θ) = (ξ). The ring Acrys is bydeﬁnitionξnAcrys = W (R)[ | n ∈ Z≥0 ]∧ ,n!
where ∧ means the p-adic completion.
The natural GK0 -action on R induces a GK0 -action on Acrys . Other additional structures on Acrys are deﬁned as follows: The p-th power map R → R
induces a natural map φ : W (R) → W (R) by (r0 , r1 , . . .) 7→ (r0p , r1p , . . .) andit extends to a ring endomorphism φ : Acrys → Acrys . Note thatφ(ξ) = p + [β ] = p + (ξ − p) = p(1 + (−1) p
p

p

p p−1

p−1 ( )
∑
1 p l
ξp
+
ξ (−p)p−l + )
p l
p
l=1

ξp
+ p(· · · )).
= p(1 +p
Let Fili Acrys be the closure of the ideal of Acrys generated by ξ n /n! for any
n ≥ i. Then we have φ(Fili Acrys ) ⊆ pi Acrys for i = 0, 1, . . . , p − 1. Put

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

9

φi = p−i φ|Fili Acrys for such i. The above equality impliesξp
mod pAcrys .
p
All these structures are compatible with the GK0 -action. By putting −i(i < 0) Acrys p φ0 (i < 0)φi(0 ≤ i ≤ p − 1)
Aicrys =Fili Acrys (0 ≤ i ≤ p − 1) and φiAcrys =0(i ≥ p)0(i ≥ p)(1)

φ1 (ξ) = 1 +we consider the ring Acrys as a Fontaine-Laﬀaille module over W with anatural GK0 -action. On the other hand, since Acrys /Fili Acrys is p-torsion free
for any i, the natural map Fili Acrys /pn Fili Acrys → Acrys /pn Acrys is injective.
Put Fili (Acrys /pn Acrys ) = Fili Acrys /pn Fili Acrys . We also have the induced
map φi : Fili (Acrys /pn Acrys ) → Acrys /pn Acrys from φi : Fili Acrys → Acrys .
Using these induced structures, we consider Acrys /pn Acrys andAcrys,∞ = Acrys ⊗W (K0 /W ) ≃ lim Acrys /pn Acrys−→nas Fontaine-Laﬀaille modules over W with a natural GK0 -action similarly,
by setting i-th ﬁltration to be zero for any i ≥ p.
f,[0,p−1]
Let M be an object of MFW,tor . We deﬁne∗
Tcrys(M ) = HomMFW (M, Acrys,∞ ).

This is a W -module with a GK0 -action induced by the natural action on
Acrys,∞ . This construction is functorial on M , and we obtain a functor∗
Tcrys: MFW,tor

f,[0,p−1]→ RepZ (GK0 ).
pRemark 2.7. In [FL82, Théorème 3.3], an associated GK0 -module with
M is deﬁned as U (M ) = Ext1MFW (M, SFL ) by using the subring SFL =p

W (R)[ ξp ] of Acrys which they denote by S [FL82, Lemme 5.4] and it isidentiﬁed by [FL82, Lemme 3.8] with HomMFW (M, SFL,∞ ), where SFL,∞ =SFL ⊗W (K0 /W ). In fact, we can show that the natural map SFL,∞ →∗ (M ) (Remark
Acrys,∞ induces an isomorphism of GK0 -modules U (M ) ≃ Tcrys
4.13).
Note that M 7→ M ⊗W W (k̄) deﬁnes a functor
f,[0,p−1]

MFW,torf,[0,p−1]→ MFW (k̄),tor .

The eﬀect of this base extension on the associated Galois representation is
the following. Let IK0 be the inertia subgroup of GK0 , which we identify
with the absolute Galois group of K̂0ur = Frac(W (k̄)). By using the natural∗ (M ′ )embedding W (k̄) → OC , we identify the associated IK0 -module Tcrysof an object M ′ ∈ MFW (k̄),tor with
f,[0,p−1]

HomMFW (k̄) (M ′ , Acrys,∞ ).

10SHIN HATTORI

From this we can see:f,[0,p−1]

Lemma 2.8 ([FL82], 3.11). For any object M ∈ MFW,tor
map
∗
∗
Tcrys
(M )|IK0 → Tcrys(M ⊗W W (k̄)), the naturalis an isomorphism of IK0 -modules.
Now the ﬁrst main theorem of this article is the following:∗
Theorem 2.9.
(i) ([FL82], Théorème 3.3 (ii)) The functor Tcrys
isexact and faithful.
f,[0,p−1](ii) ([FL82], Théorème 3.3 (i)) For any M ∈ MFW,tor , the Zp -module∗ (M ) is of ﬁnite length. Moreover, the invariant factors of the
Tcrys∗ (M ) and the W -module M are the same.
Zp -module Tcrys∗(iii) ([FL82], Théorème 6.1 (ii)) The restriction of Tcrys
to the full subf,[0,p−1]′

category MFW,tor

is full.
f,[0,p−1]′∗(iv) The essential image of the restricted functor Tcrys: MFW,tor
RepZ (GK0 ) is stable under subquotients.

→p

Next we consider the case where M is free over W . Let r be an integer
fd,[0,r]. Then the associatedsatisfying 0 ≤ r ≤ p − 1 and M an object of MFW
GK0 -representation with M is deﬁned by∗T̂crys(M ) = HomMFW (M, Acrys ).
f,[0,r]

Note that M/pn M is an object of MFW,tor . From Theorem 2.9 (ii) and∗ (M ) is a free Z -module of rank
M ≃ limn M/pn M , we can show that T̂crysp←−equal to rankW (M ).
Moreover, put D = M ⊗W K0 , Di = M i ⊗W K0 and φ = φ0 ⊗ 1. Thesedeﬁne on D a structure of a ﬁltered φ-module over K0 , which is admissible
by a theorem of Laﬀaille [Laf80, Théorème 3.2]. From the deﬁnition, we canshow that the natural map∗∗+
T̂crys(M ) ⊗Zp Qp → Vcrys(D) = HomK0 ,φ,Fil (D, Bcrys)

is an isomorphism of GK0 -representations.
Conversely, let us start from a crystalline GK0 -representation V with∗ (V ), which is a ﬁltered φ-module
Hodge-Tate weights in [0, r]. Put D = Dcrys
over K0 .
Deﬁnition 2.10. A strongly divisible lattice in D is a W -submodule M of
D such that
• M is free of ﬁnite rank and M ⊗W K0 = D,
i
i
• φ(M
∑ ∩1 D ) ⊆ p Mi for any i,•i∈Z pi φ(M ∩ D ) = Mσ .

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

11

For such M , put M i = M ∩ Di and φi = p1i φ|M i . With these structures
fd,[0,r]we consider M as an object of MFW, and the structure of a ﬁltered φmodule on D can be recovered by Di = M i ⊗W K0 and φ = φ0 ⊗ 1. Then
we can show the following.
Theorem 2.11.
(i) ([FL82], Théorème 8.4 (i)) For any r ∈ {0, . . . , p−fd,[0,r]1} and any M ∈ MFW, the p-adic GK0 -representation V =∗T̂crys (M ) ⊗Zp Qp is crystalline with Hodge-Tate weights in [0, r] sat∗ (V ) ≃ M ⊗isfying Dcrys
W K0 as ﬁltered φ-modules over K0 .
(ii) ([Bre99], Proposition 3) Suppose r < p − 1. Let V be a crystalline GK0 -representation with Hodge-Tate weights in [0, r] and put∗ (V ). Then the functor T̂ ∗D = Dcrys
crys induces an anti-equivalence
between the category of strongly divisible lattices in D and that of
GK0 -stable Zp -lattices in V .
In fact, the ﬁrst assertion is now a part of the “weakly admissible implies
admissible” theorem of Colmez-Fontaine [CF00, Théorème A]. The second
assertion follows from Theorem 2.9, except the essential surjectivity. For
this, by the theorem of Laﬀaille [Laf80, Théorème 3.2], D contains a strongly
divisible lattice M . Using Theorem 2.9 (iv), we can modify M to produce thestrongly divisible lattice corresponding with a given GK0 -stable Zp -latticein V .
Deﬁnition 2.12. A GK0 -representation T̄ ∈ RepZ (GK0 ) is said to be torp
sion crystalline with Hodge-Tate weights in [0, r] if there exist a crystalline
GK0 -representation V with Hodge-Tate weights in [0, r] and GK0 -stable Zp lattices T ⊇ T ′ in V such that T̄ is isomorphic as a GK0 -module to T /T ′ .
Corollary 2.13 ([BMs02], Theorem 3.1.3.3). For any r ∈ {0, . . . , p−2}, thef,[0,r]∗functor Tcrys
induces an anti-equivalence between the category MFW,tor andthe full subcategory of RepZ (GK0 ) consisting of torsion crystalline GK0 p
representations with Hodge-Tate weights in [0, r].
For this, ﬁrst we can show by using Corollary 2.6 (ii) that any object
f,[a,b]M ∈ MFW,tor has a resolution in MFW
0

/ L′

/L

/M

/0

[Win84, Proposition 1.6.3]. This and Theorem
with objects L, L′ ∈ MFW
2.11 show that the functor in Corollary 2.13 is well-deﬁned, since we have
∗ (M ) ≃ T̂ ∗ (L′ )/T̂ ∗ (L). Theorem 2.9 and Theorem 2.11 imply that
Tcrys
crys
crys
it is an anti-equivalence.
As we will see in §4, the proof in [FL82] of Theorem 2.9 is based on af,[0,p−1]classiﬁcation of the simple objects of the category MFk
and an explicit
description of their associated Galois representations, for the case where k is
algebraically closed (Propositions 4.7 and 4.9). They also have the following
fd,[a,b]

12

SHIN HATTORI

interesting corollary. Let h be a positive integer. We deﬁne a characterθh : IK0 → F×byphg(p1/(p −1) )mod mK̄ ∈ µph −1 (k̄) = F×,θh (g) =php1/(ph −1)h1which is independent of the choice of a (ph − 1)-st root p ph −1 of p. It
factors through the quotient IK0 /PK0 by the wild inertia subgroup PK0 .
The character θh is called the fundamental character of level h.
Theorem 2.14 ([FL82], Théorème 5.3). Let r be an integer satisfying 0 ≤f,[0,r]r ≤ p − 1 and M an object of MFW,tor . Then the semi-simpliﬁcation of the∗ (M )|
IK0 -module Tcrys
IK0 is a direct sum of characters of the form
i +pi1 +···+ph−1 ih−1

θh0with integers h ≥ 1 and ij ∈ [0, r].
In fact, each character appearing in the semi-simpliﬁcation comes from a
Jordan-Hölder factor N of the Fontaine-Laﬀaille module M ⊗W W (k̄) overW (k̄), and the ij ’s in the theorem are exactly the integers i satisfying N i ̸=
N i+1 . Thus we have a complete understanding of the semi-simpliﬁcation∗ (M )|of Tcrys
IK0 in terms of M . As for the extension structure of these∗ (M ) (Theorem
characters, we will prove an estimate of ramiﬁcation of Tcrys
2.19).
2.3. Relation to crystalline cohomology. Let X be a proper smooth
scheme over W and put Xn = X ×W Spec(Wn ). For any r ∈ {0, 1, . . . , p−1},
r (X ) = H r (X , Ω• ) a
we can give the r-th de Rham cohomology group HdR
n
n
Xn
[0,r]

structure as an object of MFW , as follows. The i-th ﬁltration for 0 ≤ i ≤ r
is the usual Hodge ﬁltration:r
HdR(Xn )i = H r (Xn , σ≥i Ω•Xn ),where σ≥i Ω•Xn is the truncation (0 → · · · → 0 → ΩiXn → Ωi+1
Xn → · · · ) of thede Rham complex of Xn over Wn . Note that we have the isomorphism
H r (Xn , σ≥i Ω•Xn ) ≃ H r ((Xn /Wn )crys , JXn /Wn ),[i]where the right-hand side is the r-th crystalline cohomology group of the ith divided power of the natural ideal sheaf JXn /Wn on the crystalline site of
Xn /Wn [Bert74, Chapitre V, Corollaire 2.3.7]. Hence we obtain a crystalliner (X ) → H r (X ), and by [FMe87, II, 2.3] we can
Frobenius map φ0 : HdRnndRalso deﬁne a σ-semilinear map
rφi : H r (Xn , σ≥i Ω•Xn ) → HdR(Xn ).

Theorem 2.15 ([FMe87], II, Corollary 2.7 (ii)). With these structures,r (X ) is an object of MFf,[0,r] .
HdR
n
W,tor

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

13

f,[0,r]For any M ∈ MFW,tor , the map Fill Acrys ⊗W M i−l → Acrys ⊗W M is
injective, since Acrys /Fill Acrys is p-torsion free and the W -submodule M i−l of
M is a direct summand by Corollary 2.6 (ii). Using this, we give Acrys ⊗W M
the tensor product structure as a Fontaine-Laﬀaille module. Namely, we put∑(Acrys ⊗W M )i =Alcrys ⊗W M i−land φiAcrys ⊗W M =

l∈Z

∑

i−l
l
l∈Z φAcrys ⊗ φM . We deﬁnerTcrys (M ) = ((Acrys ⊗W M )r )φ =1 (−r),
where (−r) means the Tate twist. Note that we can show Tcrys (M ) ≃∗ (M )∨ [Bre98b, Proposition 3.2.1.7]. Then we can state a torsion version
Tcrysof the comparison isomorphism between etale and de Rham cohomology.
Theorem 2.16 ([FMe87], III, Remark 6.4). For any r ∈ {0, . . . , p − 2},there exists a natural isomorphism of GK0 -modulesrr
Tcrys (HdR
(Xn )) ≃ Het(X ×W Spec(K̄), Z/pn Z).

From this and Theorem 2.14, we obtain the following corollary, which
settles a conjecture of Serre [Ser72, 1.13] on tame characters appearing in
torsion etale cohomology of proper smooth schemes over OK , for the casewhere K is absolutely unramiﬁed.
Corollary 2.17 ([FMe87], I, 3.2). Let X be as above. Then the semir (X × Spec(K̄), Z/pn Z)|
simpliﬁcation of the IK0 -module Het
W
IK0 is a direct
sum of characters of the form−(i0 +pi1 +···+ph−1 ih−1 )θhwith integers h ≥ 1 and ij ∈ [0, r].
2.4. Ramiﬁcation of torsion crystalline representations. For any ﬁnite Galois extension L/K in K̄ and any j ∈ R≥−1 , we have the j-th upper numbering ramiﬁcation subgroups Gal(L/K)j ⊆ Gal(L/K) and GjK =(j)j−1
limL/K Gal(L/K)j ⊆ GK [Ser68, Chapitre IV]. Put GK = GK
. We also
←−
∪
′
(j+)
j
deﬁne Gj+j ′ >j GK , and similarly for GK .
K to be the closure of the union(0+)

Then GK is equal to the inertia subgroup IK , and G0+
K is equal to the wildinertia subgroup PK . For any ﬁnite Galois extension L/K in K̄, the value(j)uL/K = inf{j ∈ R>0 | GK ⊆ GL }measures the extent of ramiﬁcation of L/K. For example, uL/K = 0 if andonly if L/K is unramiﬁed and uL/K = 1 if and only if it is tamely ramiﬁed and not unramiﬁed. It is also related to the diﬀerent DL/K [Fon85,
Proposition 1.3], and used as a local term to calculate the diﬀerent andthe discriminant of a ﬁnite extension of number ﬁelds. In algebraic number theory, imposing local constraints in terms of ramiﬁcation everywhere

14

SHIN HATTORI

often yields the ﬁniteness, and even the non-existence, of number ﬁelds. For
example, the Hermite-Minkowski theorem states that for any integer d andany ﬁnite set S of primes, there exist only ﬁnitely many number ﬁelds ofdegree d which is unramiﬁed outside S, and that there exists no nontrivial
unramiﬁed extension over Q.
More generally, a GK -representation V is said to be unramiﬁed (resp.
tamely ramiﬁed) if IK (resp. PK ) acts trivially on V . We want to study
ramiﬁcation (resp. wild ramiﬁcation) of V , namely how far V is from unramiﬁed (resp. tamely ramiﬁed) ones. For any p-adic GK -representation V
of ﬁnite local monodromy (in other words, when IK acts on V through aﬁnite quotient), the Artin and Swan conductors
∑
∑
(j+)
(j)
j+
j
j dimQp V GK /V GK ,Art(V ) =j dimQp V GK /V GK , Sw(V ) =
j>0

j>0

which are integers by the Hasse-Arf theorem, measure the extent of ramiﬁcation and wild ramiﬁcation of V , respectively. They are also used, for
any number ﬁeld F with absolute Galois group GF and any p-adic GF representation V which factors through a ﬁnite quotient, to compute theglobal Artin conductor of V∏
Art(V |GF )p ,Art(V ) =pp∈Spec(OF )which appears in the functional equation of the Artin L-function of V .
If V is not of ﬁnite local monodromy (say V is Hodge-Tate with a non-zero
Hodge-Tate weight), these deﬁnitions of conductors do not work well. When
V is de Rham, one way to study ramiﬁcation is to pass to the Weil-Deligne
representation Dpst (V ), by which any crystalline representation is consideredunramiﬁed. For any number ﬁeld F and any p-adic GF -representation V
which is unramiﬁed outside a ﬁnite set of places and de Rham at any place
dividing p, by using Dpst (V ) at places over p and Grothendieck’s monodromy
theorem at places over primes l ̸= p, we can attach to V a GFv -representationof ﬁnite local monodromy for any ﬁnite place v of F and can deﬁne the global
Artin conductor Art(V ) for V .
For the case where F = Q and V is of dimension two over a coeﬃcient ﬁeld
which is a ﬁnite extension of Qp , Fontaine-Mazur [FMa95] conjectured thatsuch V is modular, namely it is isomorphic to the p-adic GQ -representation
attached to an elliptic modular form of level equal to the global Artin conductor Art(V ) and weight determined by the Hodge-Tate weights of V |GQp .
Such modularity theorems often give ﬁniteness or non-existence results for
global objects, since the set of normalized Hecke eigenforms of ﬁxed level
and weight is ﬁnite, and even empty for some cases. One of the examples ofsuch results is the proof of Fermat’s last theorem [TW95, Wil95] via proving modularity of elliptic curves over Q (the Shimura-Taniyama conjecture).
The Fontaine-Mazur conjecture was proved in many cases by works of Kisin[Kis10] and Emerton [Eme11+].

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

15Another way to study ramiﬁcation of a p-adic GK -representation V , which
we will consider in this article, is to take a GK -stable Zp -lattice T in V
and consider its quotients T /pn T . Deﬁne the ﬁnite extension Ln /K by
GLn = Ker(GK → Aut(T /pn T )). Then by a theorem of Sen [Sen72], there
exists a constant c such that for any n we havee(n − c) ≤ uLn /K ≤ e(n + c).
Though uLn /K itself depends on the choice of a lattice T , we may consider
that such an estimate of ramiﬁcation reﬂects how good V is. In fact, for thecase where V is crystalline with Hodge-Tate weights in [0, 1], the GK -module
T /pn T is isomorphic to the one attached to a ﬁnite ﬂat (commutative) groupscheme over OK , and we have the following upper bound of ramiﬁcation:
Theorem 2.18. ([Fon85], Théorème A and Corollaire) Let G be a ﬁnite ﬂat(j)group scheme over OK which is killed by pn . Then the subgroup GK acts
trivially on the GK -module G(K̄) for any j > e(n + 1/(p − 1)). Namely, if
we deﬁne a ﬁnite Galois extension L/K by GL = Ker(GK → Aut(G(K̄))),then we have1
uL/K ≤ e(n +).
p−11Moreover, we have vK (DL/K ) < e(n + p−1), where DL/K is the diﬀerent of
L/K.

Fontaine’s proof makes an essential use of the fact that G is locally ofcomplete intersection over OK , while the special case of n = 1 and e = 1
was obtained independently by Abrashkin [Abr87] via the theory of Honda
systems (§2.5). Combining this theorem with the Odlyzko bound of the root
discriminant of a number ﬁeld, Abrashkin and Fontaine deduced that there
exists no Abelian variety with everywhere good reduction over Q, and evenover a couple of other small number ﬁelds [Abr87, Fon85]. Note that Faltings[Fal83] proved that for any number ﬁeld F , any ﬁnite set S of places of F
and any positive integer g, there exist ﬁnitely many isomorphism classes ofAbelian varieties over F of dimension g which has good reduction outside
S. These generalizations of the Hermite-Minkowski theorem to Abelianvarieties had originally been conjectured by Shafarevich.
On the other hand, Fontaine also conjectured that, for any proper smooth(j)scheme X over OK , the subgroup GK acts trivially on the GK -modulei
Het(X ×OK Spec(K̄), Z/pn Z)

for any j > e(n + i/(p − 1)) [Fon85, Remarques 2.2]. Though this conjectureis still wide open, Abrashkin and Fontaine independently gave a proof forthe case where K = K0 , i < p − 1 and n = 1. Theorem 2.16 reduces it to
the following theorem on ramiﬁcation of Galois representations associatedwith Fontaine-Laﬀaille modules, which is the second main theorem of thisarticle.

16

SHIN HATTORI

Theorem 2.19 ([Abr89b], §2, Assertion 8.1 and [Fon93], Théorème 2). Letf,[0,r]r ∈ {0, . . . , p − 2} be an integer and M an object of the category MFk.
(j)∗Then the subgroup GK0 acts trivially on the GK0 -module Tcrys (M ) for any
j > 1 + r/(p − 1). Namely, if we deﬁne a ﬁnite extension L/K0 by GL =∗ (M ))), then we haveKer(GK0 → Aut(Tcrys
uL/K0 ≤ 1 +r
r, vK0 (DL/K0 ) < 1 +
.
p−1
p−1
f,[0,p−1]′

Moreover, this also holds for r = p − 1 and M ∈ MFk.

For 0 ≤ r ≤ p − 2, Corollary 2.13 implies that the same assertions as
Theorem 2.19 hold for any torsion crystalline GK0 -representation T̄ with
Hodge-Tate weights in [0, r] such that pT̄ = 0. Note that uL/K0 = 0 for∗ (M ) is unramiﬁed
r = 0, since Corollary 4.6 shows that the GK0 -module Tcrys
f,[0,0]for any M ∈ MFk.

2.5. Classiﬁcation of ﬁnite ﬂat group schemes. Another applicationof Fontaine-Laﬀaille modules is a classiﬁcation theory of ﬁnite ﬂat groupschemes killed by some p-power over W . First recall the classiﬁcation ofﬁnite group schemes over k via Dieudonné modules. Let Ḡ be a ﬁnite groupscheme over k which is killed by some p-power. Let Ḡ (p) be the pull-backof Ḡ by the Frobenius map Spec(σ) : Spec(k) → Spec(k). Let FḠ : Ḡ →Ḡ (p) and VḠ : Ḡ (p) → Ḡ be the (relative) Frobenius and the Verschiebung
homomorphisms of Ḡ. On the other hand, a Dieudonné module over W isa W -module M endowed with a σ-semilinear endomorphism F and a σ −1 semilinear endomorphism V satisfying F V = V F = p. Then there exists ananti-equivalenceḠ 7→ M ∗k (Ḡ)
from the category of ﬁnite group schemes over k which are killed by somep-power to that of Dieudonné modules over W whose underlying W -module
is of ﬁnite length, in a way that F and V correspond with FḠ and VḠ , respectively [Fon77, III, Théoremè 1]. Moreover, M ∗k induces an anti-equivalence
between the category of p-divisible groups over W and that of Dieudonnémodules over W whose underlying W -module is free of ﬁnite rank [Fon77,
Proposition 6.1].
The crystalline Dieudonné theory of Grothendieck-Messing [Gro74, Mes72]
suggested that ﬁnite ﬂat group schemes and p-divisible groups G over W
should be classiﬁed by a pair of the Dieudonné module associated with thespecial ﬁber Ḡ = G ×W Spec(k) and its submodule (the Hodge ﬁltration).
Fontaine constructed such a classiﬁcation via Honda systems, which werenamed after a work of Honda [Hon70] (see [Fon77, IV, Théorème 1] for pdivisible groups, and [Con99, Fon75] for ﬁnite ﬂat group schemes, where weneed the additional assumption that group schemes are unipotent or p ̸= 2).
Using this, Fontaine-Laﬀaille also obtained the following:

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

17

Theorem 2.20 ([FL82], 9.11 and Proposition 9.12). Suppose p ̸= 2. Then
there exists an anti-equivalence ILM from the category of ﬁnite ﬂat groupf,[0,1]schemes over W killed by some p-power to the category MFW,tor , with anatural isomorphism of GK0 -modules∗
G(K̄) ≃ Tcrys(ILM(G)).

The same assertion holds for any p if we restrict to the category of ﬁnite
f,[0,1]′
ﬂat unipotent group schemes and MFW,tor .
The Fontaine-Laﬀaille module M = ILM(G) for a ﬁnite ﬂat group scheme
G over W is deﬁned as follows: the underlying W -module M is M =M ∗k (Ḡ) ⊗W,σ W for the special ﬁber Ḡ. The ﬁrst ﬁltration M 1 is the image by V : M ∗k (Ḡ) → M ∗k (Ḡ) ⊗W,σ W of the Hodge ﬁltration L ⊆ M ∗k (Ḡ).
The restricted map V |L is injective, and the Frobenius structure on M isdeﬁned by φ0 = F and φ1 = V −1 .
3. Further developments
3.1. Applications and generalizations of the Fontaine-Laﬀaille theory. Typical applications of the Fontaine-Laﬀaille theory are as follows:(i) Studying crystalline deformation rings, especially to prove modularity of Galois representations for number ﬁelds unramiﬁed over p[BeK13, CHT08, DFG04, FMa95, Fuj06+, Ram93, Tay03, Tay06,
Tay08, TW95, Wes04, Wil95].
(ii) Studying the reduction modulo p of GK0 -stable Zp -lattices in crystalline GK0 -representations, and of Galois representations attached
to modular forms of level prime to p [BMz02, Dim05], including ageneralization of Serre’s conjecture [Ser87] on modularity of mod p
GQ -representations to totally real number ﬁelds unramiﬁed over p[Gee11].
(iii) Proving a rareness of global objects satisfying strong local constraints everywhere [Jos99], including a vanishing of Hodge numbers for algebraic varieties over Q with everywhere good reduction[Abr89b, Fon93], which will be explained in this volume.
Integral p-adic Hodge theory itself has also been highly developed andused for applications where original theories stated above are not available—
under ramiﬁcation of the base ﬁeld K or with larger range of Hodge-Tateweights.
First we remark that the theory of Honda systems was generalized to thecase where the base ﬁeld is ramiﬁed by Fontaine himself [Fon77] to obtain aclassiﬁcation of p-divisible groups over OK for the case of e < p − 1, and also
their classiﬁcation up to isogeny for any e. For ﬁnite ﬂat group schemes over
OK , a classiﬁcation via Honda systems was obtained by Conrad [Con99] forthe case of e < p − 1 and applied to the proof of a case of the ShimuraTaniyama conjecture which was not treated by Taylor-Wiles [CDT99].

18

SHIN HATTORI

Note that Theorem 2.9 does not hold for r = p − 1, a counterexample
with p = 2 being given by the isomorphism of GK0 -modules µ2 → Z/2Z.
By considering a category of ﬁltered GK0 -modules, Abrashkin constructed∗a modiﬁed version of the functor Tcrys
which is fully faithful [Abr89a] anddeduced the above vanishing of Hodge numbers for weight p − 1 [Abr89b].
The assumption r ≤ p−1 is crucial for the Fontaine-Laﬀaille theory, since
the map p−i φ on Fili Acrys is well-deﬁned only for i ≤ p − 1. This was one ofthe diﬃculties for studying subquotients of crystalline representations withlarger range of Hodge-Tate weights. Integral theories without any restriction
on Hodge-Tate weights have been developed in two ways: theories of Wachmodules and Kisin modules.

∪
3.2. Wach modules. Consider the inﬁnite extension K(ζp∞ ) = n K(ζpn )of K obtained by adjoining all the p-power roots of unity. It is a Galois extension over K which is arithmetically proﬁnite with Galois group ΓK , and thus
the theory of ﬁelds of norms of Fontaine-Wintenberger [Win83] can be applied. This theory shows that the Galois group GK(ζp∞ ) = Gal(K̄/K(ζp∞ ))is isomorphic to the absolute Galois group of the ﬁeld of Laurent power series l((π)) over the residue ﬁeld l of K(ζp∞ ). Thus the category of GK(ζp∞ ) modules is equivalent to that of Gl((π)) -modules.
Fontaine [Fon90] showed that the latter is also equivalent to a category of
φ-modules over the p-adic completion OE of the ring of Laurent power series
W (l)[[π]][1/π]. Here, for any ring B endowed with a ring endomorphism
φ : B → B, a φ-module over B is a B-module M with a φ-semilinear
map φM : M → M which we often denote abusively by φ. We can deﬁne anatural Frobenius map φ on the ring OE , and we consider φ-modules over OE
with respect to this map. The ring OE has a natural ΓK -action, and he also
showed that the category of GK -modules which are ﬁnite over Zp (resp. Qp )is equivalent to a category of such φ-modules over OE (resp. E = Frac(OE ))with a compatible ΓK -action, which are called (φ, ΓK )-modules.
For the case of K = K0 , we can study crystalline GK0 -representations
and their subquotients with this construction: The theory of Wach modules [Berg04, Col99, Wac96, Wac97] tells us that for any crystalline GK0 representation V with non-negative Hodge-Tate weights, there exists a nice(φ, ΓK0 )-stable W [[π]][1/p]-lattice N(V ) in its associated (φ, ΓK0 )-module
D(V ) over E such that V 7→ N(V ) gives an equivalence of categories. Moreover, there also exists an equivalence between the category of GK0 -stable
Zp -lattices in V and a category of (φ, ΓK0 )-stable W [[π]]-lattices in N(V ).
Wach modules are typically used for a study of the reduction modulo pof crystalline GK0 -representations with larger range of Hodge-Tate weights[Berg12, BLZ04, CD11, Dou10]. Another interesting feature of Wach modules is a relation to Iwasawa theory [Ben00, BB08, Berg03, LLZ10, LLZ11,
LeZ12, LoZ13].

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

19

The theory of Wach modules generalizes to a case where K is absolutely
ramiﬁed [KR09] by using the Lubin-Tate extension for K instead of the cyclotomic extension K0 (ζp∞ ), to give a classiﬁcation of GK -stable OF -lattices
in a class of crystalline GK -representations with coeﬃcients in a ﬁnite extension F of Qp which is called F -analytic. Such a Lubin-Tate version of(φ, ΓK0 )-modules has also been investigated in an attempt to generalize thep-adic Langlands correspondence for GL2 (Qp ) to p-adic local ﬁelds otherthan Qp [Berg13+, FX13].
3.3. Breuil modules. On the other hand, for any integer r satisfying
0 ≤ r ≤ p − 1, let us think of generalizing the Fontaine-Laﬀaille theory
to semi-stable representations, by naı̈vely adding a monodromy operator
fd,[0,r]N to Fontaine-Laﬀaille modules M ∈ MFW. Then dividing the usual
i+1
i+1
relation N φ = pφN by pyields N φ= φi N for any i, which forces
us to impose the Griﬃths transversality N (M i+1 ) ⊆ M i on M , and thus
N (Di+1 ) ⊆ Di on the ﬁltered (φ, N )-module D = M ⊗W K0 . However,admissible ﬁltered (φ, N )-modules over K do not satisfy it in general. Thisdiﬃculty was resolved by Breuil [Bre97], who proved that the category ofﬁltered (φ, N )-modules over K is equivalent to a category of ﬁltered (φ, N )modules over a ring SK0 satisfying the Griﬃths transversality. Here the ringSK0 is deﬁned as follows: Let πK be a uniformizer of K and E(u) ∈ W [u]the minimal polynomial of πK over W . Then we put
E(u)n| n ∈ Z≥0 ]∧ , SK0 = S ⊗W K0 ,n!
where ∧ means the p-adic completion. The ring S has a natural divided
power ﬁltration Fili S for i ≥ 0, a σ-semilinear Frobenius endomorphism φsending u to up , a derivation N satisfying N (u) = −1, and a φ-semilinear
map φi : Fili S → S for 0 ≤ i ≤ p − 1, which deﬁne a similar structure
on SK0 . For any ﬁltered (φ, N )-module D over K, the equivalence is given
by D 7→ D = D ⊗K0 SK0 with the induced φ, N and a ﬁltration deﬁned
using the isomorphism D/E(u)D ≃ DK . Note that the ring S/pn S had also
appeared in a work of Kato [Kat94].
Using this construction, Breuil generalized the Fontaine-Laﬀaille theory to
the case of semi-stable representations with Hodge-Tate weights in [0, r]. Let
r+1
0 = D
D be a ﬁltered (φ, N )-module over K satisfying DK= 0.
K and DK
His idea is to consider S-lattices in D = D ⊗K0 SK0 and their subquotients,
instead of W -lattices in D itself. In other words, we consider a category ofﬁltered S-modules M with a Frobenius structure and a derivation, which
are called Breuil modules. It turned out that we have to keep only the lastﬁltration Filr M in the deﬁnition of ﬁltered S-modules, to make a variant of
Proposition 2.5 hold. Then Theorem 2.9 and Theorem 2.11 were generalized
to this semi-stable variant by Breuil [Bre98a, Bre99] (when e = 1, r < p − 1)
and Caruso [Car06] (when er < p−1). Theorem 2.14 and Theorem 2.16 werealso generalized under these assumptions [Bre98a, Bre98b, Car06, Car08],
S = W [u][20SHIN HATTORI

which yield the aforementioned conjecture of Serre on tame characters appearing in torsion etale cohomology, even for the case of semi-stable reduction. Note that this control on tame characters was applied to show
the non-existence or a rareness of global objects having good properties everywhere [Oze11, Oze13b, OT14]. Moreover, a classiﬁcation of ﬁnite ﬂatgroup schemes and p-divisible groups over OK via Breuil modules was also
obtained by Breuil himself [Bre00] for p ≥ 3.
Breuil modules have also been applied to modularity theorems [Sav04,
Sav05], including the complete proof of the Shimura-Taniyama conjecture[BCDT01], to a study of the reduction modulo p of semi-stable representations [CS09, CS10] and to that of generalized Serre conjectures [EGH13,
Gee11, GLS12].

3.4. Breuil-Kisin classiﬁcation of ﬁnite ﬂat group schemes. Anotherbreakthrough on integral p-adic Hodge theory was also brought by Breuil.
In search of a classiﬁcation of ﬁnite ﬂat group schemes over OK which isvalid for p = 2, he proposed the idea of a classiﬁcation via φ-modules overW [[u]] and prove such a classiﬁcation for the case where group schemes are
killed by p ≥ 3 [Bre98+, Bre02].
One of the advantages of this theory is that we can control the GK module G(K̄) associated with a ﬁnite ﬂat group scheme G over OK very
easily. In fact, choose a system of p-powerroots (πn )n≥0 of πK satisfying∪pπ0 = πK , πn+1
= πn and put K∞ = n K(πn ). This is an arithmeticallyproﬁnite non-Galois extension over K and thus, via the theory of ﬁelds ofnorms [Win83], the category of GK∞ -modules is equivalent to a category of
φ-modules over OE = W [[u]][1/u]∧ . Let G be a ﬁnite ﬂat group scheme over
OK which is killed by p ≥ 3, and let M be the φ-module over OE /pOE =k((u)) associated with the GK∞ -module G(K̄)|GK∞ . Then Breuil provedthat G is classiﬁed by a φ-stable k[[u]]-lattice M in M such that the cokernelof the map 1 ⊗ φ : k[[u]] ⊗φ,k[[u]] M → M is killed by E(u), and thus we canrecover G(K̄)|GK∞ from M. We do not lose information by the restriction to
GK∞ , since he also proved that the restriction functor G(K̄) 7→ G(K̄)|GK∞from the category of GK -modules attached to ﬁnite ﬂat group schemes overOK killed by some p-power to that of GK∞ -modules is fully faithful.
Kisin established such a classiﬁcation valid for any ﬁnite ﬂat group schemes
killed by some p-power and any p-divisible groups over OK for p ≥ 3[Kis06, Kis09a], and combining it with his modiﬁed Taylor-Wiles patchingargument, he generalized modularity theorems to the case where the basenumber ﬁeld is highly ramiﬁed over p [Kis09a]. The φ-module M is often
referred to as the Breuil-Kisin module associated with G.
Breuil-Kisin’s classiﬁcation has been generalized in various ways. Firstly,
the classiﬁcation theory of ﬁnite ﬂat group schemes and p-divisible groups
over OK was completed, after a work of Kisin [Kis09b] treating the casewhere p = 2 and group schemes are unipotent, independently by Kim

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

21[Kim12], Lau [Lau10+] and Liu [Liu13]. We also have a similar classiﬁcation of ﬁnite ﬂat group schemes and p-divisible groups over regular rings[Kim13+, Lau10, VZ10], whereas the theory of displays [Zin02] provides aclassiﬁcation theory of formal p-divisible groups applicable for more generalbase rings, via much more complicated semilinear algebraic data.
The Breuil-Kisin classiﬁcation of ﬁnite ﬂat group schemes turned out
to be useful also for studying the Hecke operators at places dividing p on
the space of p-adic modular forms, via an analysis of canonical subgroups[Hat13, Hat14a, Kas13, Tia11+].
3.5. Kisin modules and (φ, Ĝ)-modules. Secondly, for any r ≥ 0, Kisin
studied φ-modules M over W [[u]] such that the cokernel of the map 1⊗φM :W [[u]]⊗φ,W [[u]] M → M is killed by E(u)r , which are often called Kisin modules, and also those endowed with an endomorphism N on (M/uM) ⊗W K0 ,which are called (φ, N )-modules over W [[u]]. He proved that the category
0 = D
of admissible ﬁltered (φ, N )-modules D satisfying DK
K can be considered as a full subcategory of the isogeny category of (φ, N )-modules overW [[u]]. This new classiﬁcation of semi-stable GK -representations was used
to prove the existence of potentially semi-stable deformation rings [Kis08],
which led to proving many cases of the Breuil-Mézard conjecture and the
Fontaine-Mazur conjecture [Kis09c] and to a construction of the p-adic local
Langlands correspondence for GL2 (Qp ) [Kis10].
Kisin’s theory also enables us to have a control on GK∞ -stable Zp -lattices
in semi-stable GK -representations. Moreover, for r ≤ p − 1, by tensoring thering S over W [[u]] along the map φ with Kisin modules, we obtain Breuil
modules without derivation, and to recover not only the GK∞ -action but
the GK -action amounts to giving it a derivation. Using these features, Liu
generalized Theorem 2.11 to semi-stable representations with Hodge-Tateweights in [0, p − 2] which is valid for any K, in terms of Breuil modules[Liu08] (the unipotent case with Hodge-Tate weights in [0, p − 1] was shownby [Gao14+]). Kisin’s theory was generalized to the case where the residue
ﬁeld k is imperfect [BT08], using a framework of p-adic Hodge theory forimperfect residue ﬁelds [Bri06].
On the other hand, since K∞ /K is not Galois, it is not so easy to study
the action of the whole Galois group GK on semi-stable GK -representations
via Kisin’s theory, unless the range of Hodge-Tate weights is small. Liuovercame this diﬃculty, by considering the Galois closure K∞ (ζp∞ )/K of
K∞ , its Galois group Ĝ = Gal(K∞ (ζp∞ )/K) and a category of φ-modules
M over W [[u]] with a Ĝ-action on a scalar extension M⊗W [[u]],φ R̂, which arecalled (φ, Ĝ)-modules [Liu07, Liu10]. He proved an anti-equivalence between
a category of (φ, Ĝ)-modules and the category of GK -stable Zp -lattices insemi-stable GK -representations generalizing Theorem 2.11, without any restriction on Hodge-Tate weights nor on the base ﬁeld K. It was also applied
to a study of generalized Serre conjectures [GLS14]. For the case of K = K0 ,the GK0 -stable Zp -lattices in crystalline GK0 -representations are classiﬁed

22

SHIN HATTORI

both by Wach modules and (φ, Ĝ)-modules, and the attached Wach module
to such a lattice can be recovered from the attached (φ, Ĝ)-module [Liu13+].
More generally, we can develop a variant of the theory of (φ, ΓK0 )-modules
using Ĝ instead of ΓK0 [Car13, Oze13a, Tav11].
Using these developments on integral p-adic Hodge theory, Theorem 2.19
was also generalized for subquotients of semi-stable representations, whichhave a worse ramiﬁcation bound [BMs02, Hat09, Car13, CL11], and it
was applied to a vanishing of Hodge numbers of algebraic varieties over Q
with everywhere good reduction except semi-stable reduction at 3 [Abr13].
Abrashkin recently found a new method to prove such a ramiﬁcation boundof GK -representations, by using a non-canonical isomorphism between twocomplete discrete valuation ﬁelds of equal characteristic [Abr12+, Abr14+].
3.6. Ramiﬁcation theory. The classical ramiﬁcation theory as reviewed
in §2.4 has been typically applied to:(i) Deﬁnitions and calculations of invariants of global objects via local terms concerning ramiﬁcation, such as the global Artin conductor and the Grothendieck-Ogg-Shafarevich formula to compute
Euler-Poincaré characteristics of l-adic sheaves over algebraic curves[Gro77].
(ii) Proving the non-existence or a rareness of global objects satisfying strong local constraints everywhere, such as Abelian varieties [Abr87, BrK01, BrK04, BrK14, Fon85, Schf03, Schf05, Schf12,
Ver13], algebraic varieties [Abr89b, Abr13, Fon93] and Galois representations [Bru99, Bru05, Jos99, Moo00, Moo03, MT03, Tat94],
generalizing the Hermite-Minkowski theorem.
(iii) The theory of ﬁelds of norms of Fontaine-Wintenberger, to give an
isomorphism between the absolute Galois group of an arithmeticallyproﬁnite extension of a complete discrete valuation ﬁeld of mixed
characteristic with perfect residue ﬁeld and that of a complete discrete valuation ﬁeld of equal characteristic [Win83].
(iv) The theory of close local ﬁelds, including a comparison of absolute Galois groups of complete discrete valuation ﬁelds of mixed
and equal characteristics with the same perfect residue ﬁeld modulo ramiﬁcation subgroups [Del84], especially to deduce the local
Langlands correspondence of reductive groups for the equal characteristic case from the mixed characteristic case [ABPS13+, Gan13+]
by combining with [Kaz86].
(v) A local version of the Grothendieck conjecture, which states thatany local ﬁeld can be recovered from its absolute Galois group andramiﬁcation subgroups [Abr00, Abr10, Moc97].
In the classical ramiﬁcation theory as in [Ser68], it is crucial that theresidue ﬁeld k is perfect and thus for any ﬁnite extension L/K, the OK algebra OL is generated by a single element. In general, for any ﬁnite Galois

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

23

extension L/K, we can deﬁne the i-th lower numbering ramiﬁcation subgroup
Gal(L/K)i = Ker(Gal(L/K) → Aut(OL /mi+1
L )).
For the case where k is perfect, we can prove, by using the monogenicityof OL , that the j-th upper numbering ramiﬁcation subgroup Gal(L/K)j =Gal(L/K)ψL/K (j) is compatible with quotients, where ψL/K is the Herbrandfunction. This enables us to take the projective limit to deﬁne the ramiﬁcation subgroup GjK of GK . However, when k is imperfect, the lower numbering ramiﬁcation subgroups are not necessarily compatible with quotientseven after renumbering.
After various attempts to deﬁne appropriate ramiﬁcation subgroups of
GK when k is imperfect, including especially a work of Kato for Abelian
extensions [Kat89], diﬀerent deﬁnitions were proposed independently byAbbes-Saito [AS02, AS03] (geometric deﬁnition via connected components
of analytic varieties), Borger [Bor04] (reducing to the case of perfect k via
the universal residual perfection), Kedlaya [Ked07a] (diﬀerential deﬁnition
via (φ, ∇)-modules over the Robba ring) and Zhukov [Abr02, Zhu03] (forhigher dimensional local ﬁelds, via induction from lower dimensional ones).
Each deﬁnition has its own advantages and applications. Abbes-Saito’s
deﬁnition was used to deﬁne characteristic cycles of etale sheaves on higher
dimensional algebraic varieties to compute their Euler-Poincaré characteristics [Sai09, Sai14+]. One of the advantages of their deﬁnition is that it
also gives a ramiﬁcation theory of ﬁnite ﬂat algebras over OK . With thisframework, Theorem 2.18 can be shown easily from the fact that multiplicative groups are minimal in the category of ﬁnite ﬂat group schemes overOK [Hat06]. This advantage was also used to prove the overconvergenceof canonical subgroups of Abelian varieties [AM04, Hat13, Hat14a, Tia10,
Tia12] and a generalization of the theory of close local ﬁelds to the case ofimperfect k [Hat14b].
Kedlaya’s deﬁnition is more ﬂexible about base extensions than AbbesSaito’s. One of its consequences is that we can show the integrality of the
Swan conductor generalizing the classical Hasse-Arf theorem much moreeasily [Ked07a], via a reduction to the case of perfect k by adding p-basis
to k and “rotate” in the enlarged space to make ramiﬁcation concentrateon the uniformizer direction. This diﬀerential version of ramiﬁcation theory
was used to prove a conjecture of Shiho [Shi02, Conjecture 3.1.8] on semistable reduction of overconvergent F -isocrystals on smooth varieties over aperfect ﬁeld of characteristic p [Ked07b, Ked08, Ked09, Ked11]. In Zhukov’s
deﬁnition, ramiﬁcation subgroups are multi-indexed, and it is suitable fora detailed study of the structure of GK . It was used for a proof of a twodimensional analogue of the Grothendieck conjecture [Abr03] and a generalization of the theory of ﬁelds of norms to higher dimensional local ﬁelds[Abr07]. Note that the theory of higher ﬁelds of norms was also obtained in24

SHIN HATTORI

diﬀerent ways for more general settings [And06, Schl06] and generalized tothe theory of perfectoid spaces [Schz12].
Among these diﬀerent deﬁnitions, the comparison between Abbes-Saito’sand Kedlaya’s has been well studied. Note that a diﬀerential interpretationof the classical ramiﬁcation theory had been investigated, for example, by[Cre00, Ked05, Mat02, Tsu98]. For the case of imperfect k, after works of
Matsuda [Mat04] and Chiarellotto-Pulita [CP09], Xiao proved that thesetwo deﬁnitions (and also Borger’s for the equal characteristic case) yield thesame Artin and Swan conductors for almost all the cases [Xia10, Xia12],from which the integrality of the conductors for Abbes-Saito’s deﬁnition
follows.
4. Sketch of proofs
In this section, we sketch the proofs of Theorem 2.9 and Theorem 2.19,
following [FL82] and [Fon93]. The key ideas for the proofs are twofold:First, by a dévissage argument, we reduce a large part of the proofs to af,[0,p−1]
study of simple objects in the category MFk̄, which we can classify
completely. Second, passing from the period ring Acrys /pAcrys to OK̄ /pOK̄
by using the ﬁrst projection pr1 : R → OK̄ /pOK̄ , we reduce a study of the∗ (M ) to a study of equations over W and their solutions in
GK0 -module Tcrys
OK̄ . The latter enables us especially to prove the aforementioned estimate of
ramiﬁcation. Note that, as we remarked before (Remark 2.7), the original
arguments in [FL82] used the subring SFL instead of Acrys , while almost
verbatim arguments are valid also for our period ring Acrys which is more
standard these days. To mediate these two period rings and also to makethe notation simpler, in this article we will mainly work on the ringA = Im(SFL /pSFL → Acrys /pAcrys ).
4.1. Cutting oﬀ higher divided powers. Put RDP = Acrys /pAcrys . We
[0,p−1]as before. Note that a divided
consider this ring as an object of MFk
power structure on this ring is induced from that of Acrys : we put the l-th
divided power γl (x) of x ∈ Fil1 RDP = Fil1 Acrys /pFil1 Acrys by
γl (x) = γl (x̂) mod pAcrys =x̂lmod pAcrysl!

with any lift x̂ of x in Fil1 Acrys . The ring RDP is also the divided powerenvelope of the surjection pr0 : R → OK̄ /pOK̄ by [BO78, Remarks 3.20 (8)].
Recall that we have deﬁned the element β = (−p, (−p)1/p , . . .) of the ring
R before. The surjection pr1 : R → OK̄ /pOK̄ induces a σ −1 -semilinear
isomorphism of rings R/β p R → OK̄ /pOK̄ sending β to (−p)1/p . Since β p =p!γp (β) = 0 in the ring RDP , we can consider the ring RDP as an (OK̄ /pOK̄ )algebra by the map OK̄ /pOK̄ ≃ R/β p R → RDP . For any divided power
ideal I of a ring and x ∈ I, we put δ(x) = (p − 1)!γp (x) and write δ l (x) =RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

25(δ ◦ · · · ◦ δ)(x) for the l-times iteration of δ. Then we can describe thestructure of the ring RDP as follows:
Lemma 4.1. The ring homomorphism(OK̄ /pOK̄ )[Y1 , Y2 , . . . , ]/(Y1p , Y2p , . . .) → RDP
sending Yl to δ l (β) is a σ-semilinear bijection, by which we identify bothsides.
Proof. Put ξ = p + [β] ∈ W (R) as before. It is enough to show a similar
W (R)-linear map
ξn| n ∈ Z≥0 ]n!
deﬁned by Yl 7→ δ l (ξ) is an isomorphism. From the deﬁnition, we see thatthe map is surjective, and isomorphic after inverting p. Thus it is enoughto show that the ring on the left-hand side is p-torsion free. Put

W (R)[Y1 , Y2 , . . .]/(ξ p −pY1 , Y1p −pY2 , . . .) → W (R)DP = W (R)[p
Bl = W (R)[Y1 , . . . , Yl ]/(ξ p − pY1 , . . . , Yl−1
− pYl ).

For any ring B without p-torsion and any element x ∈ B, if the image of x
is a regular element of B/pB, then B/xB is p-torsion free. Since the imageof ξ p is a regular element of R, we see inductively that Bl [Yl+1 ]/(Ylp − pYl+1 )is p-torsion free. Taking the inductive limit yields the claim.
□
Using this identiﬁcation, we can explicitly describe additional structuresof RDP such as GK0 -action, ﬁltration and Frobenius maps (see [FL82, 5.9],p
where they consider the subring SFL = W (R)[ ξp ] of Acrys with similar explicit description modulo p). The action of g ∈ GK0 is written as∑∑g(
al1 ,...,ln Y1l1 · · · Ynln ) =g(al1 ,...,ln )Y1l1 · · · Ynln ,l1 ,...,ln

l1 ,...,ln

where al1 ,...,ln ∈ OK̄ /pOK̄ . Indeed, from the deﬁnition of the divided power
structure on RDP , we have δ(x) = δ(x̂) mod p for any x ∈ Fil1 RDP , where x̂
is any lift of x in Fil1 Acrys . Putting g(β) = βεa with ε = (1, ζp , ζp2 , . . .) anda ∈ Zp , we haveg(δ(β)) = δ(g(ξ)) mod p = δ(ξ + [β]([ε]a − 1)) mod p
( )
p−1
∑
p l
= δ(ξ) +p−1ξ ([β]([ε]a − 1))p−l + [β]p δ([ε]a − 1) mod p
l
l=1
( )
p−1
∑
p
−1 p
= δ(β) + βp(εa − 1)p−l + β p δ(εa − 1) = δ(β),
ll=1since β p = 0 in RDP . Thus we obtain g(Y1 ) = Y1 and g(Yl ) = Yl for any l.
Moreover, the ideal Fili RDP of RDP is written as
Fili RDP = ((−p)i/p , Yl | l ≥ 1) (i ≤ p − 1),

Filp RDP = (Yl | l ≥ 1).

26

SHIN HATTORI[0,p−1]

Recall that we consider RDP as an object of MFk
by putting (RDP )i =Fili RDP for 0 ≤ i ≤ p − 1 and (RDP )p = 0. For any i ∈ {0, . . . , p − 1}, write
any element x in the i-th ﬁltration as∑x = x0 +xl Yl + (higher terms)l≥1with x0 ∈ (−p)i/p (OK̄ /pOK̄ ) and xl ∈ OK̄ /pOK̄ . Let x̂0 be any lift of x0 in
OK̄ . From the equality (1) in §2.2, we obtain
 x̂p
i
0

(i ≤ p − 2),i (1 + Y1 ) (−p)px̂0pii
φ (x) =
(1 + Y1 ) + x1
(i = p − 1, p ≥ 3),
(−p)i

p

∑
x̂

0
(1 + Y )i +xp (i = p − 1, p = 2).
(−p)i

1

l≥1

l

Remark 4.2. In particular, we have φ1 ((−p)1/p ) = 1 + Y1 in the ring RDP .
This implies that, contrary to what is stated in [Wac97, 2.1.3.4], there exists
no non-trivial ideal I of RDP stable under φp−1 such that φ1 ((−p)1/p ) ≡1 mod I. Indeed, this forces Y1 ∈ I and thus I contains φp−1 (Y1 ) = 1.
Now we put
A = (OK̄ /pOK̄ )[Y1 ]/(Y1p ) ⊆ RDP[0,p−1]and give it the induced structure as an object of MFk
Namely, we put the i-th ﬁltration Ai of A as
Ai = ((−p)i/p , Y1 ) (i ≤ p − 1),from RDP .

Ap = 0and the i-th Frobenius map of A aspp−1
 x̂0 (1 + Y1 )i∑(i ≤ p − 2),(−p)ipφi (x0 + x1 Y1 +xl Y1l ) =
x̂
p
i
0

(1 + Y1 ) + x1 (i = p − 1).
l=2
(−p)i
Then we have φi (Fili RDP ) ⊆ A for any i ≤ p − 1. Consider the quotient[0,p−1]
RDP /A as an object of MFk
with the induced structure from RDP . It
DP
p−1
DP
satisﬁes (R /A)= R /A and φp−1 = 0 on RDP /A. Then the following∗ (M )
lemma enables us to replace RDP by the simpler ring A to study Tcrys
for the case of pM = 0.
f,[0,p−1]

Lemma 4.3. For any object M ∈ MFk, we have

HomMFk (M, RDP /A) = Ext1MFk (M, RDP /A) = 0.
In particular, we have∗
Tcrys
(M ) = HomMFk (M, A),

Ext1MFk (M, RDP ) = Ext1MFk (M, A).

This follows from the lemma below:

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

27

Lemma 4.4. Let r ∈ {0, . . . , p − 1} be an integer and M an object off,[0,r]MFk. Let N be an object of MFk such that N r = N and that for any
x ∈ N , the iteration (φrN ◦ · · · ◦ φrN )(x) of any suﬃciently large times is zero.
Then we have HomMFk (M, N ) = Ext1MFk (M, N ) = 0.
Proof. Fix a k-basis e1 , . . . , eh of M with en ∈ M in \ M in +1 and write as
φiMn (en ) =h∑cm,n emm=1

with P = (cm,n ) ∈ GLh (k). Let f : M → N be a morphism of MFk and put
f (en ) = xn ∈ N in = N r = N . Then the equality φiN ◦ f = f ◦ φiM implies(x1 , . . . , xh ) = Φ(x1 , . . . , xh )P −1
with Φ = (φiN1 , . . . , φiNh ). By assumption, the l-th iteration Φl (x1 , . . . , xh ) iszero for any suﬃciently large l and thus we obtain f = 0 by recursion.
Next consider an extension in MFk
/N

0

/E

/M

/ 0.

Choose a lift ên of en to E in . We have
φiEn (ên ) =h∑cm,n êm + bn

m=1

with some bn ∈ N . Then the extension splits if and only if there exists
an ∈ N in satisfying
φiEn (ên + an ) =h∑cm,n (êm + am ),m=1

which is equivalent to
bn + φiNn (an ) =h∑cm,n am .

m=1

Thus we obtain(a1 , . . . , ah ) = (b1 , . . . , bh )P −1 + Φ(a1 , . . . , ah )P −1 .
By recursion, we see that the equation has the unique solution(a1 , . . . , ah ) =∞∑Φl (b1 , . . . , bh )(P −1 )σ (P −1 )σll−1· · · P −1 ,l=0since the right-hand side is a ﬁnite sum.
Let F be any algebraic extension of K0 in K̄. We put
AF = (OF /pOF )[Y1 ]/(Y1p ) ⊆ A = AK̄□28

SHIN HATTORI[0,p−1]and give this subring the induced structure as an object of MFk
f,[0,p−1]
A. For M ∈ MFk, we deﬁnefrom∗Tcrys,F(M ) = HomMFk (M, AF ).
∗ (M ) on which the absolute Galois group G acts
This is a subset of Tcrys
Ftrivially.
f,[0,r]∗(M ) can also
For r ∈ {0, . . . , p − 2} and M ∈ MFk, the module Tcrys,Fbe obtained by using a far simpler ring than AF . PutrbF = {x ∈ OF | vp (x) >
}.
p−1
[0,r]

We consider the quotient OF /bF as an object of MFki(OF /bF )i = {x ∈ OF | vp (x) ≥ }/bF ,p

φi (x) =by puttingx̂p
mod bF(−p)i

for any 0 ≤ i ≤ r and (OF /bF )r+1 = 0. Here x̂ denotes any lift of x in OF .
f,[0,r], the surjecLemma 4.5. For any r ∈ {0, . . . , p − 2} and any M ∈ MFktion AF → OF /bF deﬁned by Y1 7→ 0 induces an isomorphism∗
Tcrys,F(M ) → HomMFk (M, OF /bF ).

Proof. Since r < p − 1, the map φrAF kills Y1 . Moreover, for any x̂ ∈ OF
satisfying vp (x̂) = r/(p − 1) + ε with some ε > 0, we have vp (x̂p /(−p)r ) ≥r/(p − 1) + pε. Thus the ideal I = bF AF + (Y1 ) of AF , which is contained in
the r-th ﬁltration ArF , is stable under φrAF and, for any x ∈ I, the iteration(φrAF ◦ · · · ◦ φrAF )(x) of any suﬃciently large times is zero. Thus, by considering I as an object of MFk with the induced structure from AF , Lemma 4.4
yields HomMFk (M, I) = Ext1MFk (M, I) = 0. Since I = Ker(AF → OF /bF ),the long exact sequence of Hom and Ext implies the lemma.
□
For r = 0, the ideal bK̄ is equal to the maximal ideal mK̄ and we obtainthe following corollary.
f,[0,0]Corollary 4.6. For any M ∈ MFk
iﬁed.

∗ (M ) is unram, the GK0 -module Tcrys

4.2. Classiﬁcation of simple objects. Let h be a positive integer and i a
map Z/hZ → Z. We write in = i(n) for any n ∈ Z/hZ. We suppose that i isof period h. This means that h is the minimum among the positive integers
h′ satisfying in+h′ = in for any n.
For such h and i, we deﬁne an object M (h; i) of the category MFfk̄ by⊕M (h; i) =k̄enn∈Z/hZ

with a basis {en }n∈Z/hZ and
M (h; i)l =

⊕
in ≥l

k̄en ,φin (en ) = en−1 .

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

29

In particular, en ∈ M (h; i)in \ M (h; i)in +1 . When h = 1 and i(0) = j ∈ Z,
we write M (h; i) as M (1; j).
We deﬁne an Fph -action on this object by[a] ∈ EndMFk̄ (M (h; i)),−n[a](en ) = ap

enfor any a ∈ Fph . Then we can show the following proposition.
Proposition 4.7.
(i) ([FL82], Proposition 4.4) An object M ∈ MFfk̄ issimple if and only if M is isomorphic to M (h; i) for some h ∈ Z>0
and i : Z/hZ → Z of period h.
(ii) ([FL82], Lemme 4.9) Let (h, i) be a pair of a positive integer h anda map i : Z/hZ → Z of period h. Let (h′ , i′ ) be a similar pair.
Then HomMFk̄ (M (h; i), M (h′ ; i′ )) = 0 unless h = h′ and i = i′ up
to a shift. In the latter case the two objects are isomorphic, andEndMFk̄ (M (h; i)) ≃ Fph via a 7→ [a].
(iii) ([FL82], Lemme 6.3, 6.4) Ext1MFk̄ (M, M (1; p − 1)) = 0 for any M ∈f,[0,p−1]

MFk̄.

The assertions (i) and (ii) follow from a semilinear algebraic argument
whose key point is that any σ h -semilinear automorphism of a ﬁnite dimensional k̄-vector space has a non-zero ﬁxed vector. The last assertion is shownby dévissage, after explicitly constructing a split of any extension of M (h; i)by M (1; p − 1) for i : Z/hZ → {0, . . . , p − 1} as in the proof of Lemma 4.4.
f,[0,p−1]
Proposition 4.7 (iii) implies that an object M of MFk̄is in the full
f,[0,p−1]′

subcategory MFk̄
if and only if every Jordan-Hölder factor of M is
not isomorphic to M (1; p − 1). This leads to the following corollary.
Corollary 4.8 ([FL82], Théorème 6.1 (i)). Let0/ M′
f,[0,p−1]

be an exact sequence of MFW,tor
and only if so are M ′ and M ′′ .

/M

/ M ′′/0f,[0,p−1]′. Then M is an object of MFW,tor
f,[0,p−1]′

Proof. For this, it is enough to show that if M ∈ MFW,tor

f,[0,p−1]′
W (k̄) ∈ MFW (k̄),tor .

ifthen M ⊗W

Suppose that there exists a non-zero quotient M ⊗W

W (k̄) → N satisfying N = N p−1 . Then we can ﬁnd a ﬁnite extension
k ′ /k in k̄ such that this quotient is a base extension of a quotient M ⊗W
f,[0,p−1]
W (k ′ ) → N ′ in MFW (k′ ),tor satisfying N ′ = (N ′ )p−1 . Since k ′ /k is ﬁnite, thef,[0,p−1]

latter quotient can be viewed as a quotient in MFW,tor . In this category
M ⊗W W (k ′ ) is the direct sum of ﬁnite copies of M . The image N0 of acopy of M in N ′ is non-zero, and satisﬁes N0 = N0p−1 by Proposition 2.5,
which is the contradiction.
□

30

SHIN HATTORI

Next we study the Galois representations associated with the simple obf,[0,p−1]
jects of MFk̄, which is one of the key ingredients of [FL82].
Proposition 4.9 ([FL82], Théorème 5.1). Let h be a positive integer andi : Z/hZ → {0, 1, . . . , p − 1} a map of period h.
(i) Ext1MFk̄ (M (h; i), RDP ) = 0.
∗ (M (h; i)) is an F (ii) Via the Fph -action on M (h; i), the module Tcrys
phvector space of dimension one.
∗ (M (h; i)) is given by the character(iii) The IK0 -action on Tcrys
i +pi1 +···ph−1 ih−1

θh0

: IK0 → F×,
phwhere θh is the fundamental character of level h.
Proof. By Lemma 4.3, we may replace RDP with A to compute Hom andExt1 groups. First consider any extension
/A

0

/E

/ M (h; i)/0in MFk̄ . Let {en }n∈Z/hZ be the basis of M (h; i) as before, and take a lift ênof en in E in . We have φin (ên ) = ên−1 − dn−1 with some dn−1 ∈ A. Thenthe extension splits if and only if there exists un ∈ Ain such thatφin (ên + un ) = (ên−1 + un−1 )for any n ∈ Z/hZ, which is equivalent toφin (un ) − un−1 = dn−1 .
Since this equation is linear, to prove that it has a solution, we may assume
that dn = 0 except a single index n = n0 . By permuting the indices, we may
assume n0 = 0. On the other hand, a k̄-linear map M (h; i) → A deﬁned by
en 7→ un ∈ Ain gives a morphism of MFk̄ if and only if
φin (un ) − un−1 = 0
for any n ∈ Z/hZ. Hence we reduced ourselves to showing that the equation un ∈ Ain ,φin (un ) − un−1 = 0 (n ̸= 1),(2)
 in
φ (u1 ) − u0 = d
has a solution for any d ∈ A in the case of the assertion (i), and has exactly
ph solutions for d = 0 in the case of the assertion (ii).
Putun = an + bn Y1 +

p−1
∑
l=2

in
p

cn,l Y1l ,d = d0 + d1 Y 1 +p−1∑

dl Y1ll=2

with some an ∈ (−p) (OK̄ /pOK̄ ) and bn , cn,l , dl ∈ OK̄ /pOK̄ . We also put
ε(n) = 0 for n ̸= p − 1 and ε(p − 1) = 1. Then the equation (2) is equivalentRAMIFICATION OF CRYSTALLINE REPRESENTATIONS

31

to


in
 an ∈ (−p) p (OK̄ /pOK̄ ), bn , cn,l ∈ OK̄ /pOK̄ ,{p
 ânin (1 + Y1 )in + ε(in )bpn − un−1 = 0 (n ̸= 1)(−p)
d (n = 1).
From this we see that cn,l ’s are uniquely determined by an ’s, and the solutions of the equation (2) correspond bijectively with the solutions ofinp (O /pO ), b ∈ O /pO ,

a
∈
(−p)
n
n

K̄
K̄
K̄
K̄
{


 âpn0 (n ̸= 1)p+ ε(in )bn − an−1 =(−p)in(3)d0 (n = 1),
{


p

0
(n
=
̸
1)

 in ânin − bn−1 =(−p)
d1 (n = 1).
First let us assume M (h; i) ̸= M (1; p − 1), and suppose d = 0. Then we
can show ε(in )bpn = 0 for any n, by a valuation calculation [FL82, 5.11 (iii)].
Thus bn ’s are also uniquely determined by an ’s in this case. For an ’s, we canshow that any solution (an ) of the equation (3) uniquely lifts to a solution
in OK̄ of the equation of degree p
Xnp− Xn−1 = 0 for any n(−p)inover W [FL82, Lemme 5.12].
Let K0tr be the maximal tamely ramiﬁed extension of K0 in K̄. For anyρ ∈ Z(p) , we write ρ = a/b with a ∈ Z and b ∈ Z>0 satisfying p ∤ b.
Using [Bou56, §7, n◦ 4, Proposition 5], we ﬁx once and for all (−p)ρ ∈ K0tr′′
such that ((−p)ρ )b = (−p)a and (−p)ρ (−p)ρ = (−p)ρ+ρ for any such ρ, ρ′ .
∗ (M (h; i)) =Then, from the above equation, we see that the module Tcrys
DP
HomMFk̄ (M (h; i), R ) is equal to{(en 7→ λp−n(−p)ρn (1 + Y1 )in+1 ) | λ ∈ Fph }h−1+···+pinwith ρn = in+1 +pin+2, which settles the assertions (ii) and (iii) forph −1
this case. The assertion (i) can be shown by constructing a solution of theequation (3) similarly from a solution in OK̄ of an equation over W [FL82,
5.11 (i), (ii)].
The case of M (1; p − 1) needs an extra care [FL82, 5.14]. We also havean equation over W to lift solutions, while its degree is p2 and lifts are notunique. Nevertheless, we can show for the case of d = 0 that the solutions of
the lifted equation have exactly p distinct reductions modulo p, which yields
the proposition for this case.
□

Now the assertions of Theorem 2.9 except the fullness follow formally.
∗ (M )
Indeed, Lemma 2.8 reduces assertions on the module structure of Tcrys
to the case of k = k̄. Moreover, for any object N ∈ MFk and any extension0/ Acrys,∞

/E

/N

/0

32

SHIN HATTORIin MFW , the exact sequence/ Acrys /pAcrys0/ Acrys,∞×p/ Acrys,∞/0

and the snake lemma yield the extension0/ Acrys /pAcrys/ E[p]/N/0

in MFk with E[p] = Ker(p : E → E), and the induced map
Ext1MFW (N, Acrys,∞ ) → Ext1MFk (N, Acrys /pAcrys )∗is an injection. Using this, the exactness of Tcrys
in Theorem 2.9 (i) folf,[0,p−1]

lows by showing Ext1MFW (k̄) (M, Acrys,∞ ) = 0 for any M ∈ MFW (k̄),tor by∗dévissage from Proposition 4.9 (i). By combining this exactness of Tcrys
with Proposition 4.9 (ii), we obtain∗lgZp (Tcrys(M )) = lgW (M )f,[0,p−1]

for any M ∈ MFW,tor. Applying this to M [pl ] = Ker(pl : M → M ) for
f,[0,p−1]

any l shows Theorem 2.9 (ii). For any morphism f : M → N of MFW,tor∗ (f ) = 0, applying the above length equality to Im(f ) showssatisfying Tcrys∗
f = 0 and settles the faithfulness of Tcrys
in Theorem 2.9 (i). Thus Theorem∗
2.9 (iv) follows from Corollary 4.8 modulo the fullness of Tcrys(Theorem 2.9(iii)).
∗ . Now we prove Theorem 2.9 (iii). First we may
4.3. The fullness of Tcrys∗
assume k = k̄ by a Galois descent argument using the faithfulness of Tcrys[FL82, 6.2]. By Proposition 4.7 and Proposition 4.9, the natural map∗∗(N ), Tcrys(M ))HomMFW (k̄) (M, N ) → HomZp [IK0 ] (Tcrys

is an isomorphism if M and N are simple. In order for a dévissage argument
to imply the theorem, we need to show that the natural map∗∗
Ext1MFW (k̄) (M, N ) → Ext1Zp [IK ] (Tcrys(N ), Tcrys(M ))0f,[0,p−1]′

is an injection for any objects M, N ∈ MFW (k̄),tor . To prove this injectivity∗
and the fullness of Tcrys
simultaneously, by dévissage and an induction on
the length we may assume that M and N are simple. By Theorem 2.9(ii), we only have to consider the extensions killed by p. For this case,∗ (M ) and T ∗ (N ) are tamely ramiﬁed by Proposition 4.9 (iii),
since Tcrys
crys∗ (N ) by T ∗ (M ) splits
Maschke’s theorem implies that an extension of Tcrys
crys
if and only if the extension is also tamely ramiﬁed. Thus we are reduced toshowing the following lemma.
Lemma 4.10 ([FL82], 6.11). Suppose k = k̄. Let M be an object off,[0,p−1]′∗ (M ) is tamely ramiMFk̄
. Then M is semi-simple if and only if Tcrys
ﬁed.

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

33∗ (M )GF for
To prove the lemma, we need to describe the ﬁxed part Tcrys
any algebraic extension F/K0 , which is also one of the key steps for the
proof of Theorem 2.19. This is essentially done again by uniquely lifting∗ (M ) to a solution in O
each element of Tcrys
K̄ of an equation over W andusing (OK̄ )GF = OF , as follows.

Proposition 4.11 ([FL82], Proposition 6.7). Suppose k = k̄. Let M be an
f,[0,p−1]′
object of MFk̄and F/K0 an algebraic extension in K̄. Then GF acts∗
∗ (M ) = T ∗trivially on Tcrys (M ) if and only if Tcrys
crys,F (M ).
Proof. We proceed by induction on the length of the Fontaine-Laﬀaille module M . When M is simple, this follows from the explicit description of∗ (M ) given in the proof of Proposition 4.9. Suppose that we have an
Tcrys
f,[0,p−1]′

exact sequence in MFk̄
0

/N

/M

/ M (h; i)/0∗ (M ). Then G
and that GF acts trivially on Tcrys
F also acts trivially on∗∗∗
Tcrys (N ) and thus Tcrys (N ) = Tcrys,F (N ) by the induction hypothesis. Take
the basis (en )n∈Z/hZ of M (h; i) as before and a lift ên ∈ M in of en . Thenφin (ên ) = ên−1 + dn−1
∗ (M ). Sincewith some dn−1 ∈ N . Let f : M → A be an element of Tcrys∗∗
f |N ∈ Tcrys (N ) = Tcrys,F (N ), we can write asf (dn ) = αn + βn Y1 +p−1∑γn,l Y1ll=2with some αn , βn , γn,l ∈ OF /pOF . Putf (ên ) = an + bn Y1 +p−1∑cn,l Y1l ∈ Ain

l=2
in
p

with an ∈ (−p) (OK̄ /pOK̄ ) and bn , cn,l ∈ OK̄ /pOK̄ . From φin (f (ên )) =f (φin (ên )), we obtain
( )
in
âpn
cn−1,l =− γn−1,ll (−p)in
and thus cn,l ’s are uniquely determined by an ’s in a way that if an ∈ OF /pOF
for any n, then cn,l ’s are also in OF /pOF . Moreover, we see that to give∗ (M ) with f (d ) being as above is the same as to give a solutionf ∈ Tcrysn(an , bn )n∈Z/hZ of the equation
in an p∈ (−p) p (OK̄ /pOK̄ ), bn ∈ OK̄ /pOK̄ ,ân+ ε(in )bpn − an−1 = αn−1 ,(−p)inp in âni − bn−1 = βn−1 .
(−p) n34

SHIN HATTORILet α̂n and β̂n be any lifts of αn and βn in OF , respectively. Then we
can show [FL82, Lemme 6.8] that any solution (an , bn )n∈Z/hZ of the above
equation uniquely lifts to a solution (xn , yn )n∈Z/hZ in OK̄ of the equation
over OF{ Xpn+ ε(in )Ynp − Xn−1 = α̂n−1 ,(−p)inpXnin (−p)in − Yn−1 = β̂n−1 .

Since (g(an ), g(bn )) = (an , bn ) for any g ∈ GF by assumption, the uniquenessof the lifting implies (g(xn ), g(yn )) = (xn , yn ) for any g ∈ GF . Thus xn , yn ∈(OK̄ )GF = OF and an , bn ∈ OF /pOF for any n. Hence we obtain f (ên ) ∈ AF∗
and f ∈ Tcrys,F(M ).
□For any ρ ∈ Z(p) ∩ [0, 1[, there exist uniquely h ∈ Z>0 and i : Z/hZ →{0, . . . , p − 1} of period h satisfying
ρ=i1 + pi2 + · · · + ph−2 ih−1 + ph−1 i0.
ph − 1

Note that any such (h, i) except (1, p − 1) can be obtained in this way, sinceρ < 1. Putω̄ρ = (−p)ρ (1 + Y1 )i1 ∈ AiK0 tr ,0

where (−p)ρ is chosen as before.

i

Then we have φi0 (ω̄ρ ) = ω̄pρ−i0 ∈ AKh−1
tr
0

with
pρ − i0 =

i0 + pi1 + · · · + ph−2 ih−2 + ph−1 ih−1
∈ Z(p) ∩ [0, 1[.
ph − 1

Repeating this, we see that the k̄-subspace generated byω̄ρ , φi0 (ω̄ρ ), φih−1 (φi0 (ω̄ρ )), . . .
f,[0,p−1]

deﬁnes an object of MFk̄which is isomorphic to M (h; i).
Put
Ass = Spank̄ {ω̄ρ | ρ ∈ Z(p) ∩ [0, 1[} ⊆ AK0tr
and we give this subspace the induced structure from AK0tr as an object ofMFk̄ . Then we see that Ass is isomorphic to the direct sum of representatives
f,[0,p−1]′. Moreover, we canof all isomorphism classes of simple objects of MFk̄f,[0,p−1]′

show by induction on the length that M ∈ MFk̄
only if the natural injection

is semi-simple if and∗
HomMFk̄ (M, Ass ) → Tcrys(M )is bijective.
Thus we are reduced to comparing AK0tr and Ass . Consider the quotient
AK0tr /Ass ≃

p−1
⊕
l=1

(OK0tr /pOK0tr )Y1l .

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

35

Then we have AK0tr /Ass = (AK0tr /Ass )p−1 . Moreover, a simple calculation as
in [FL82, 6.10] shows that φp−1 = 0 on this quotient. By Lemma 4.4, we
obtain HomMFk̄ (M, AK0tr /Ass ) = 0 and thus
HomMFk̄ (M, Ass ) = HomMFk̄ (M, AK0tr ).
The equality shows that M is semi-simple if and only if HomMFk̄ (M, AK0tr ) =∗ (M ), which is the same as saying that T ∗ (M ) is tamely ramiﬁed by
Tcrys
crys
Proposition 4.11. This concludes the proofs of Lemma 4.10 and Theorem
2.9 (iii).
4.4. Ramiﬁcation bound. To prove Theorem 2.19, we need a lemma of
Fontaine (which was improved later by Yoshida) relating ramiﬁcation to a
ramiﬁed variant of formal smoothness, as follows. For any algebraic extension E/K in K̄ and any non-negative real number m, putam
E/K = {x ∈ OE | vK (x) ≥ m}.
For any ﬁnite Galois extension L/K, consider the property For any algebraic extension E/K, if there exists
an OK -algebra homomorphism OL → OE /am(Pm )L/K
E/K ,
then there exists a K-algebra injection L → E.
Note that we do not require the map L → E to be a lift of the given map
OL → OE /am
E/K . The property (Pm )L/K is related to uL/K by the following
lemma.
Lemma 4.12 ([Fon85], Proposition 1.5, [Yos10], Proposition 3.3).
uL/K = inf{m ∈ R≥0 | (Pm )L/K holds}.
For this, Krasner’s lemma implies that (Pm )L/K holds for any m > uL/K .
Conversely, by explicitly constructing counterexamples for (Pm )L/K , we canshow that if (Pm )L/K holds, then m > uL/K − e−1
L/K , where eL/K is the relative ramiﬁcation index of the extension L/K [Fon85]. Furthermore, we cankill the error term by using arbitrarily large tamely ramiﬁed base extensions[Yos10].
f,[0,r]Now let r < p − 1, M ∈ MFk
and L/K0 be as in Theorem 2.19. Since∗ (M ), by Lemma 2.8
the theorem concerns only about the IK0 -action on Tcrys
we may assume k = k̄. Since r < p − 1, Lemma 4.5 gives the identiﬁcation
∗
∗
Tcrys
(M ) = HomMFk̄ (M, OK̄ /bK̄ ), Tcrys,F(M ) = HomMFk̄ (M, OF /bF )for any algebraic extension F/K0 in K̄. By Lemma 4.12, it is enough tocheck the property (Pm )L/K0 for any m > 1 + r/(p − 1).
Let E/K0 be an algebraic extension in K̄, m a real number satisfyingm > 1 + r/(p − 1) and η : OL → OE /am
E/K0 a W -algebra homomorphism.

36

SHIN HATTORILet P (X) be the minimal polynomial over W of a uniformizer πL of L. We
write aseL/K0 −1∑
eL/K0
P (X) = X+
pcs X ss=0with some cs ∈ W such that c0 ∈ W × . Let x̂ be a lift of η(πL ) in OE .
Since P (η(πL )) = 0 in the ring OE /am
E/K0 , we have P (x̂) + δ = 0 with someδ ∈ OE satisfying vp (δ) ≥ m > 1 + r/(p − 1). From the Newton polygon of
the polynomial P (X) + δ, we obtain vp (x̂) = e−1
L/K0 = vp (πL ). Hence we seethat η induces an injectionη̄ : OL /bL → OE /bE
and that η̄ respects ﬁltrations. We claim that η̄ also respects φi ’s. Indeed,
note that the i-th ﬁltration (OL /bL )i is spanned over k̄ by
eL/K0{πLj | j ∈ Z, j ≥i}.
p
For such j, put pj = eL/K0 i + l. Then−1eφ (πLj ) =
i

L/K0
∑
πLpj
l
mod
b
=
π
(
L
L
(−p)i

cs πLs )i mod bL ,

∑
x̂pj
l
mod
b
=
x̂
(
E
(−p)i

δ
cs x̂s + )i mod bE .
p

s=0
eL/K0 −1φi (η̄(πLj )) =s=0

Since pδ ∈ bE , we obtain η̄(φi (πLj )) = φi (η̄(πLj )), which proves the claim.
Thus the map η̄ deﬁnes an injection of the category MFk̄ and induces aninjection∗∗
Tcrys,L(M ) → Tcrys,E(M ).
∗ (M ) = T ∗Now Proposition 4.11 shows Tcrys
crys,L (M ). Thus we also have∗∗
∗ (M ). From the deﬁniTcrys (M ) = Tcrys,E (M ) and GE acts trivially on Tcrys
tion of L, we obtain GE ⊆ GL and L ⊆ E, namely the property (Pm )L/K0
holds for any m > 1 + r/(p − 1). This concludes the proof of Theorem 2.19
for r < p − 1.
f,[0,p−1]′
For the case of r = p − 1 and M ∈ MFk, it is enough to check theproperty (Pm )L/K0 for any m > 2. Consider a W -algebra homomorphismη : OL → OE /am
E/K0 . We see in the same way as above that the induced
injection η̄ : OL /pOL → OE /pOE and Y1 7→ Y1 deﬁne an injection AL → AE
which is compatible with ﬁltrations and Frobenius structures. Thus we have∗∗an injection Tcrys,L(M ) → Tcrys,E(M ) also in this case, which yields theproperty (Pm )L/K0 and the desired ramiﬁcation bound.

Remark 4.13. As we mentioned before, [FL82] uses the subring SFL =p
W (R)[ ξp ] instead of Acrys to construct a GK0 -representation associated with

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS
f,[0,p−1]

M ∈ MFW,tor

37. We can show that these two constructions are naturally
f,[0,p−1]

isomorphic: we claim that for any M ∈ MFW,tor, the natural map∗(M )HomMFW (M, SFL,∞ ) → HomMFW (M, Acrys,∞ ) = Tcrys

is bijective andExt1MFW (M, SFL,∞ ) → Ext1MFW (M, Acrys,∞ )is injective. Indeed, by dévissage we may assume pM = 0. By [FL82,
Lemme 5.7], we have a natural identiﬁcation (OK̄ /pOK̄ )[Y1 ] ≃ SFL /pSFL
deﬁned by Y1 7→ δ(ξ) mod p. Since φp−1 (Y1p ) = 0 in this ring [FL82, 5.9],
Lemma 4.4 implies that the natural map SFL /pSFL → A induces a bijection∗
HomMFk (M, SFL /pSFL ) ≃ HomMFk (M, A) = Tcrys(M )and an injection
Ext1MFk (M, SFL /pSFL ) → Ext1MFk (M, A) = Ext1MFk (M, Acrys /pAcrys ).
The former one settles the assertion on Hom. Moreover, as in the proof of
Theorem 2.9 (i), we have a natural injection
Ext1MFW (M, SFL,∞ ) → Ext1MFk (M, SFL /pSFL )and the assertion on Ext1 follows from the commutative diagram
Ext1MFW (M, SFL,∞ ) Ext1MFW (M, Acrys,∞ ) / Ext1

/pSFL )MFk (M, S
 FL/ Ext1_

MFk (M, Acrys /pAcrys ).

References
A. Abbes and A. Mokrane: Sous-groupes canoniques et cycles évanescents padiques pour les variétés abéliennes, Publ. Math. Inst. Hautes Études Sci. 99(2004), 117–162.
[AS02]
A. Abbes and T. Saito: Ramiﬁcation of local ﬁelds with imperfect residue ﬁelds,
Amer. J. Math. 124 (2002), no. 5, 879–920.
[AS03]
A. Abbes and T. Saito: Ramiﬁcation of local ﬁelds with imperfect residue ﬁelds
II, Kazuya Kato’s ﬁftieth birthday. Doc. Math. 2003, Extra Vol., 5–72.
[Abr87] V. Abrashkin: Galois modules of group schemes of period p over the ring of
Witt vectors, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 4, 691–736, 910;
translation in Math. USSR-Izv. 31 (1988), no. 1, 1–46.
[Abr89a] V. Abrashkin: Modiﬁcation of the Fontaine-Laﬀaille functor, Izv. Akad. NaukSSSR Ser. Mat. 53 (1989), no. 3, 451–497, 671; translation in Math. USSR-Izv.
34 (1990), no. 3, 467–516.
[Abr89b] V. Abrashkin: Modular representations of the Galois group of a local ﬁeld and ageneralization of a conjecture of Shafarevich, Izv. Akad. Nauk SSSR Ser. Mat.
53 (1989), no. 6, 1135–1182, 1337; translation in Math. USSR-Izv. 35 (1990),
no. 3, 469–518.
[Abr00] V. Abrashkin: On a local analogue of the Grothendieck conjecture, Internat. J.
Math. 11 (2000), no. 2, 133–175.
[AM04]

38

SHIN HATTORI

V. Abrashkin: Ramiﬁcation theory for higher dimensional local ﬁelds, Algebraic number theory and algebraic geometry, 1–16, Contemp. Math. 300, Amer.
Math. Soc., Providence, RI, 2002.
[Abr03] V. Abrashkin: An analogue of Grothendieck’s conjecture for two-dimensional
local ﬁelds of ﬁnite characteristic, Tr. Mat. Inst. Steklova 241 (2003), Teor.
Chisel, Algebra i Algebr. Geom., 8–42; translation in Proc. Steklov Inst. Math.
2003, no. 2 (241), 2–34.
[Abr07] V. Abrashkin: An analogue of the ﬁeld-of-norms functor and of theGrothendieck conjecture, J. Algebraic Geom. 16 (2007), no. 4, 671–730.
[Abr10] V. Abrashkin: Modiﬁed proof of a local analogue of the Grothendieck conjecture,
J. Théor. Nombres Bordeaux 22 (2010), no. 1, 1–50.
[Abr12+] V. Abrashkin: A semi-stable case of the Shafarevich conjecture, to appear in
Automorphic Forms and Galois Representations, London Math. Soc. Lecture
Note Ser. 414, Cambridge Univ. Press, 2014.
[Abr13] V. Abrashkin: Projective varieties with bad semi-stable reduction at 3 only, Doc.
Math. 18 (2013), 547–619.
[Abr14+] V. Abrashkin: Ramiﬁcation estimate for Fontaine-Laﬀaille Galois modules,
arXiv:1405.3861v1 (2014).
[And06] F. Andreatta: Generalized ring of norms and generalized (φ, Γ)-modules, Ann.
Sci. École Norm. Sup. (4) 39 (2006), no. 4, 599–647.
[ABPS13+] A.-M. Aubert, P. Baum, R. Plymen and M. Solleveld: The local Langlands
correspondence for inner forms of SLn , arXiv:1305.2638v3 (2013).
[Ben00] D. Benois: On Iwasawa theory of crystalline representations, Duke Math. J.
104 (2000), no. 2, 211–267.
[BB08]
D. Benois and L. Berger: Théorie d’Iwasawa des représentations cristallines II,
Comment. Math. Helv. 83 (2008), no. 3, 603–677.
[Berg03] L. Berger: Bloch and Kato’s exponential map: three explicit formulas, Kazuya
Kato’s ﬁftieth birthday. Doc. Math. 2003, Extra Vol., 99–129.
[Berg04] L. Berger: Limites de représentations cristallines, Compos. Math. 140 (2004),
no. 6, 1473–1498.
[Berg12] L. Berger: Local constancy for the reduction mod p of 2-dimensional crystalline
representations, Bull. Lond. Math. Soc. 44 (2012), no. 3, 451–459.
[Berg13+] L. Berger: Multivariable Lubin-Tate (φ, Γ)-modules and ﬁltered φ-modules,
arXiv:1211.4431v2 (2013).
[BLZ04] L. Berger, H. Li and H. Zhu: Construction of some families of 2-dimensional
crystalline representations, Math. Ann. 329 (2004), no. 2, 365–377.
[BeK13] T. Berger and K. Klosin: On deformation rings of residually reducible Galois
representations and R = T theorems, Math. Ann. 355 (2013), no. 2, 481–518.
[Bert74] P. Berthelot: Cohomologie cristalline des schémas de caractéristique p > 0,
Lecture Notes in Math. 407, Springer-Verlag, Berlin-New York, 1974.
[BO78]
P. Berthelot and A. Ogus: Notes on crystalline cohomology, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978.
[Bor04] J. Borger: Conductors and the moduli of residual perfection, Math. Ann. 329(2004), no. 1, 1–30.
[Bou56] N. Bourbaki: Éléments de mathématique, Fasc. XX. Livre I: Théorie des ensembles. Chapitre 3: Ensembles ordonnés. Cardinaux. Nombres entiers. Secondeédition, revue et augmentée. Actualités Sci. Ind., no. 1243. Hermann, Paris,
1956.
[Bre97] C. Breuil: Représentations p-adiques semi-stables et transversalité de Griﬃths,
Math. Ann. 307 (1997), no. 2, 191–224.
[Bre98a] C. Breuil: Construction de représentations p-adiques semi-stables, Ann. Sci.
École Norm. Sup. (4) 31 (1998), no. 3, 281–327.
[Abr02]

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

39[Bre98b] C. Breuil: Cohomologie étale de p-torsion et cohomologie cristalline enréduction semi-stable, Duke Math. J. 95 (1998), no. 3, 523–620.
[Bre98+] C. Breuil: Schémas en groupes et corps des normes, unpublished, available at
http://www.math.u-psud.fr/~breuil/.
[Bre99] C. Breuil: Une remarque sur les représentations locales p-adiques et les congruences entre formes modulaires de Hilbert, Bull. Soc. Math. France 127 (1999),
no. 3, 459–472.
[Bre00] C. Breuil: Groupes p-divisibles, groupes ﬁnis et modules ﬁltrés, Ann. of Math.
(2) 152 (2000), no. 2, 489–549.
[Bre02] C. Breuil: Integral p-adic Hodge theory, Algebraic geometry 2000, Azumino(Hotaka), 51–80, Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo, 2002.
[BCDT01] C. Breuil, B. Conrad, F. Diamond and R. Taylor: On the modularity of elliptic
curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4,
843–939.
[BMs02] C. Breuil and W. Messing: Torsion étale and crystalline cohomologies, Cohomologies p-adiques et applications arithmétiques II, Asterisque 279 (2002),
81–124.
[BMz02] C. Breuil and A. Mézard: Multiplicités modulaires et représentations deGL2 (Zp ) et de Gal(Q̄p /Qp ) en l = p, Duke Math. J. 115 (2002), no. 2, 205–310.
O. Brinon: Représentations cristallines dans le cas d’un corps résiduel impar[Bri06]
fait, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 919–999.
[BC14+] O. Brinon and B. Conrad: CMI summer school notes on p-adic Hodge theory,
available at http://math.stanford.edu/~conrad/.
[BT08]
O. Brinon and F. Trihan: Représentations cristallines et F -cristaux: le cas d’uncorps résiduel imparfait, Rend. Semin. Mat. Univ. Padova 119 (2008), 141–171.
[Bru99] S. Brueggeman: The nonexistence of certain Galois extensions unramiﬁed outside 5, J. Number Theory 75 (1999), no. 1, 47–52.
[Bru05] S. Brueggeman: The nonexistence of certain nonsolvable Galois extensions of
number ﬁelds of small degree, Int. J. Number Theory 1 (2005), no. 1, 155–160.
[BrK01] A. Brumer and K. Kramer: Non-existence of certain semistable abelian varieties, Manuscripta Math. 106 (2001), no. 3, 291–304.
[BrK04] A. Brumer and K. Kramer: Semistable abelian varieties with small divisionﬁelds, Galois theory and modular forms, 13–37, Dev. Math. 11, Kluwer Acad.
Publ., Boston, MA, 2004.
[BrK14] A. Brumer and K. Kramer: Paramodular abelian varieties of odd conductor,
Trans. Amer. Math. Soc. 366 (2014), no. 5, 2463–2516.
[Car06] X. Caruso: Représentations semi-stables de torsion dans le case er < p − 1, J.
Reine Angew. Math. 594 (2006), 35–92.
[Car08] X. Caruso: Conjecture de l’inertie modérée de Serre, Invent. Math. 171 (2008),
no. 3, 629–699.
[Car13] X. Caruso: Représentations galoisiennes p-adiques et (φ, τ )-modules, Duke
Math. J. 162 (2013), no. 13, 2525–2607.
[CL11]
X. Caruso and T. Liu: Some bounds for ramiﬁcation of pn -torsion semi-stable
representations, J. Algebra 325 (2011), 70–96.
X. Caruso and D. Savitt: Polygones de Hodge, de Newton et de l’inertie modérée[CS09]des représentations semi-stables, Math. Ann. 343 (2009), no. 4, 773–789.
X. Caruso and D. Savitt: Poids de l’inertie modérée de certaines représentations[CS10]
cristallines, J. Théor. Nombres Bordeaux 22 (2010), no. 1, 79–96.
S. Chang and F. Diamond: Extensions of rank one (φ, Γ)-modules and crys[CD11]
talline representations, Compos. Math. 147 (2011), no. 2, 375–427.
B. Chiarellotto and A. Pulita: Arithmetic and diﬀerential Swan conductors[CP09]of rank one representations with ﬁnite local monodromy, Amer. J. Math. 131(2009), no. 6, 1743–1794.

40

SHIN HATTORI[CHT08] L. Clozel, M. Harris and R. Taylor: Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes Études Sci.
108 (2008), 1–181.
[Col99]P. Colmez: Représentations cristallines et représentations de hauteur ﬁnie, J.
Reine Angew. Math. 514 (1999), 119–143.
P. Colmez and J.-M. Fontaine: Construction des représentations p-adiques[CF00]
semi-stables, Invent. Math. 140 (2000), no. 1, 1–43.
[Con99] B. Conrad: Finite group schemes over bases with low ramiﬁcation, Compos.
Math. 119 (1999), no. 3, 239–320.
[CDT99] B. Conrad, F. Diamond and R. Taylor: Modularity of certain potentially
Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 2,
521–567.
[Cre00] R. Crew: Canonical extensions, irregularities, and the Swan conductor, Math.
Ann. 316 (2000), no. 1, 19–37.
P. Deligne: Les corps locaux de caractéristique p, limites de corps locaux de[Del84]
caractéristique 0, Representations of reductive groups over a local ﬁeld, 119–
157, Travaux en Cours, Hermann, Paris, 1984.
[DFG04] F. Diamond, M. Flach and L. Guo: The Tamagawa number conjecture of adjoint
motives of modular forms, Ann. Sci. École Norm. Sup. 37 (4) (2004), no. 5, 663–
727.
[Dim05] M. Dimitrov: Galois representations modulo p and cohomology of Hilbert modular varieties, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 4, 505–551.
[Dou10] G. Dousmanis: On reductions of families of crystalline Galois representations,Doc. Math. 15 (2010), 873–938.
[Eme11+] M. Emerton: Local-global compatibility in the p-adic Langlands programme for
GL2/Q , available at http://www.math.uchicago.edu/~emerton/.
[EGH13] M. Emerton, T. Gee and F. Herzig: Weight cycling and Serre-type conjectures
for unitary groups, Duke Math. J. 162 (2013), no. 9, 1649–1722.
[Fal83]
G. Faltings: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent.
Math. 73 (1983), no. 3, 349–366.
[Fon75] J.-M. Fontaine: Groupes ﬁnis commutatifs sur les vecteurs de Witt, C. R. Acad.
Sci. Paris Sér. A-B 280 (1975), Ai, A1423–A1425.
[Fon77] J.-M. Fontaine: Groupes p-divisibles sur les corps locaux, Astérisque, 47-48.
Société Mathématique de France, Paris, 1977.
[Fon83] J.-M. Fontaine:
Cohomologie de de Rham, cohomologie cristalline etreprésentations p-adiques, Algebraic geometry (Tokyo/Kyoto, 1982), 86–108,
Lecture Notes in Math. 1016, Springer, Berlin, 1983.
[Fon85] J.-M. Fontaine: Il n’y a pas de variété abélienne sur Z, Invent. Math. 81 (1985),
no. 3, 515–538.
[Fon90] J.-M. Fontaine: Représentations p-adiques des corps locaux I, The Grothendieck
Festschrift, Vol. II, 249–309, Progr. Math. 87, Birkhauser Boston, Boston, MA,
1990.
[Fon93] J.-M. Fontaine: Schémas propres et lisses sur Z, Proceedings of the Indo-French
Conference on Geometry (Bombay, 1989), 43–56, Hindustan Book Agency,
Delhi, 1993.
J.-M. Fontaine and G. Laﬀaille: Construction de représentations p-adiques,[FL82]
Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 4, 547–608.
[FMa95] J.-M. Fontaine and B. Mazur: Geometric Galois representations, Elliptic curves,modular forms & Fermat’s last theorem (Hong Kong, 1993), 41–78, Ser. Number
Theory, I, Int. Press, Cambridge, MA, 1995.

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

41

J.-M. Fontaine and W. Messing: p-adic periods and p-adic etale cohomology,
Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 179–
207, Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987.
L. Fourquaux and B. Xie: Triangulable OF -analytic (φq , Γ)-modules of rank 2,[FX13]
Algebra Number Theory 7 (2013), no. 10, 2545–2592.
[Fuj06+] K. Fujiwara: Deformation rings and Hecke algebras in the totally real case,arXiv:math/0602606v2 (2006).
[Gan13+] R. Ganapathy: The local Langlands correspondence for GSp4 over local function
ﬁelds, arXiv:1305.6088 (2013).
[Gao14+] H. Gao: Galois lattices and strongly divisible lattices in the unipotent case,arXiv:1305.2664v2 (2014).
[Gee11] T. Gee: On the weights of mod p Hilbert modular forms, Invent. Math. 184(2011), no. 1, 1–46.
[GLS12] T. Gee, T. Liu and D. Savitt: Crystalline extensions and the weight part of
Serre’s conjecture, Algebra Number Theory 6 (2012), no. 7, 1537–1559.
[GLS14] T. Gee, T. Liu and D. Savitt: The Buzzard-Diamond-Jarvis conjecture for unitary groups, J. Amer. Math. Soc. 27 (2014), no. 2, 389–435.
[Gro74] A. Grothendieck: Groupes de Barsotti-Tate et cristaux de Dieudonné, Séminairede Mathématiques Supérieures 45 (Été, 1970). Les Presses de l’Université deMontréal, Montréal, Que., 1974.
[Gro77] A. Grothendieck: Formule d’Euler-Poincaré en cohomologie étale, Cohomologie
l-adique et fonctions L, Séminaire de Géometrie Algébrique du Bois-Marie 1965–
1966 (SGA 5), Exp. X, Lecture Notes in Math. 589, Springer-Verlag, BerlinNew York, 1977.
[Hat06] S. Hattori: Ramiﬁcation of a ﬁnite ﬂat group scheme over a local ﬁeld, J. Number Theory 118 (2006), no. 2, 145–154.
[Hat09] S. Hattori: On a ramiﬁcation bound of torsion semi-stable representations overa local ﬁeld, J. Number Theory 129 (2009), no. 10, 2474–2503.
[Hat13] S. Hattori: Canonical subgroups via Breuil-Kisin modules, Math. Z. 274 (2013),
no. 3-4, 933–953.
[Hat14a] S. Hattori: Canonical subgroups via Breuil-Kisin modules for p = 2, J. Number
Theory 137 (2014), 142–159.
[Hat14b] S. Hattori: Ramiﬁcation theory and perfectoid spaces, Compos. Math. 150(2014), no. 5, 798–834.
[Hon70] T. Honda: On the theory of commutative formal groups, J. Math. Soc. Japan
22 (1970), 213–246.
[Jos99]K. Joshi: Remarks on methods of Fontaine and Faltings, Internat. Math. Res.
Notices 22 (1999), 1199–1209.
[Kas13] P. Kassaei: Modularity lifting in parallel weight one, J. Amer. Math. Soc. 26(2013), no. 1, 199–225.
[Kat89] K. Kato: Swan conductors for characters of degree one in the imperfect residue
ﬁeld case, Algebraic K-theory and algebraic number theory (Honolulu, HI,
1987), 101–131, Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 1989.
[Kat94] K. Kato: Semi-stable reduction and p-adic étale cohomology, Périodes p-adiques(Bures-sur-Yvette, 1988), Astérisque 223 (1994), 269–293.
[Kaz86] D. Kazhdan: Representations of groups over close local ﬁelds, J. Analyse Math.
47 (1986), 175–179.
[Ked05] K. Kedlaya: Local monodromy of p-adic diﬀerential equations: an overview, Int.
J. Number Theory 1 (2005), no. 1, 109–154.
[Ked07a] K. Kedlaya: Swan conductors for p-adic diﬀerential modules, I: A local construction, Algebra Number Theory 1 (2007), no. 3, 269–300.
[Ked07b] K. Kedlaya: Semistable reduction for overconvergent F -isocrystals, I: Unipotence and logarithmic extensions, Compos. Math. 143 (2007), no. 5, 1164–1212.
[FMe87]

42

SHIN HATTORIK. Kedlaya: Semistable reduction for overconvergent F -isocrystals, II: A
valuation-theoretic approach, Compos. Math. 144 (2008), no. 3, 657–672.
[Ked09] K. Kedlaya: Semistable reduction for overconvergent F -isocrystals, III: Local
semistable reduction at monomial valuations, Compos. Math. 145 (2009), no.
1, 143–172.
[Ked11] K. Kedlaya: Semistable reduction for overconvergent F -isocrystals, IV: local
semistable reduction at nonmonomial valuations, Compos. Math. 147 (2011),
no. 2, 467–523.
[Kim12] W. Kim: The classiﬁcation of p-divisible groups over 2-adic discrete valuation
rings, Math. Res. Lett. 19 (2012), no. 1, 121–141.
[Kim13+] W. Kim: The relative Breuil-Kisin classiﬁcation of p-divisible groups and ﬁnite
ﬂat group schemes, arXiv:1201.0121v4 (2013).
[Kis06]
M. Kisin: Crystalline representations and F -crystals, Algebraic geometry andnumber theory, 459–496, Progr. Math. 253, Birkhauser Boston, Boston, MA,2006.
M. Kisin: Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21[Kis08](2008), no. 2, 513–546.
[Kis09a] M. Kisin: Moduli of ﬁnite ﬂat group schemes, and modularity, Ann. of Math.
(2) 170 (2009), no. 3, 1085–1180.
[Kis09b] M. Kisin: Modularity of 2-adic Barsotti-Tate representations, Invent. Math.
178 (2009), no. 3, 587–634.
[Kis09c] M. Kisin: The Fontaine-Mazur conjecture for GL2 , J. Amer. Math. Soc. 22(2009), no. 3, 641–690.
[Kis10]M. Kisin: Deformations of GQp and GL2 (Qp ) representations, Astérisque 330(2010), 511–528.
[KR09]
M. Kisin and W. Ren: Galois representations and Lubin-Tate groups, Doc.
Math. 14 (2009), 441–461.
[Laf80]G. Laﬀaille: Groupes p-divisibles et modules ﬁltrés: le cas peu ramiﬁé, Bull.
Soc. Math. France 108 (1980), 187–206.
[Lau10] E. Lau: Frames and ﬁnite group schemes over complete regular local rings, Doc.
Math. 15 (2010), 545–569.
[Lau10+] E. Lau: A relation between Dieudonné displays and crystalline Dieudonné theory, arXiv:1006.2720v2 (2010).
[LLZ10] A. Lei, D. Loeﬄer and S. Zerbes: Wach modules and Iwasawa theory for modular forms, Asian J. Math. 14 (2010), no. 4, 475–528.
[LLZ11] A. Lei, D. Loeﬄer and S. Zerbes: Coleman maps and the p-adic regulator,
Algebra Number Theory 5 (2011), no. 8, 1095–1131.
[LeZ12] A. Lei and S. Zerbes: Signed Selmer groups over p-adic Lie extensions, J. Théor.
Nombres Bordeaux 24 (2012), no. 2, 377–403.
[Liu07]
T. Liu: Torsion p-adic Galois representations and a conjecture of Fontaine,Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 4, 633–674.
[Liu08]
T. Liu: On lattices in semi-stable representations: a proof of a conjecture of
Breuil, Compos. Math. 144 (2008), no. 1, 61–88.
T. Liu: A note on lattices in semi-stable representations, Math. Ann. 346[Liu10](2010), no. 1, 117–138.
T. Liu: The correspondence between Barsotti-Tate groups and Kisin modules[Liu13]
when p = 2, J. Théor. Nombres Bordeaux 25 (2013), no. 3, 661–676.
Compatibility of Kisin modules for diﬀerent uniformizers,[Liu13+] T. Liu:
arXiv:1302.1888v1 (2013).
[LoZ13] D. Loeﬄer and S. Zerbes: Wach modules and critical slope p-adic L-functions,
J. Reine Angew. Math. 679 (2013), 181–206.
[Mat02] S. Matsuda: Katz correspondence for quasi-unipotent overconvergent isocrystals,
Compos. Math. 134 (2002), no. 1, 1–34.
[Ked08]

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS[Mat04][Mes72][Moc97][Moo00][Moo03][MT03][Oze11][Oze13a][Oze13b][OT14][Qui73][Ram93][Sai09][Sai14+][Sav04][Sav05][Schl06][Schz12][Schf03][Schf05][Schf12][Sen72][Ser68][Ser72][Ser87]43

S. Matsuda: Conjecture on Abbes-Saito ﬁltration and Christol-Mebkhout ﬁltration, Geometric aspects of Dwork theory. Vol. I, II, 845–856, Walter de Gruyter
GmbH & Co. KG, Berlin, 2004.
W. Messing: The crystals associated to Barsotti-Tate groups: with applications
to abelian schemes, Lecture Notes in Math. 264, Springer-Verlag, Berlin-New
York, 1972.
S. Mochizuki: A version of the Grothendieck conjecture for p-adic local ﬁelds,
Internat. J. Math. 8 (1997), no. 4, 499–506.
H. Moon: Finiteness results on certain mod p Galois representations, J. Number
Theory 84 (2000), no. 1, 156–165.
H. Moon: The non-existence of certain mod p Galois representations, Bull.
Korean Math. Soc. 40 (2003), no. 3, 537–544.
H. Moon and Y. Taguchi: Reﬁnement of Tate’s discriminant bound and nonexistence theorems for mod p Galois representations, Kazuya Kato’s ﬁftiethbirthday. Doc. Math. 2003, Extra Vol., 641–654.
Y. Ozeki: Non-existence of certain Galois representations with a uniform tameinertia weight, Int. Math. Res. Not. IMRN 2011 11, 2377–2395.
Y. Ozeki: Torsion representations arising from (φ, Ĝ)-modules, J. Number Theory 133 (2013), no. 11, 3810–3861.
Y. Ozeki: Non-existence of certain CM abelian varieties with prime power torsion, Tohoku Math. J. (2) 65 (2013), no. 3, 357–371.
Y. Ozeki and Y. Taguchi: On congruences of Galois representations of number
ﬁelds, Publ. Res. Inst. Math. Sci. 50 (2014), no. 2, 287–306.
D. Quillen: Higher algebraic K-theory: I, Algebraic K-theory, I: Higher Ktheories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85–
147. Lecture Notes in Math. 341, Springer, Berlin 1973.
R. Ramakrishna: On a variation of Mazur’s deformation functor, Compos.
Math. 87 (1993), no. 3, 269–286.
T. Saito: Wild ramiﬁcation and the characteristic cycle of an l-adic sheaf, J.
Inst. Math. Jussieu 8 (2009), no. 4, 769–829.
T. Saito: Characteristic cycle and the Euler number of a constructible sheaf on
a surface, arXiv:1402.5720v1 (2014).
D. Savitt: Modularity of some potentially Barsotti-Tate Galois representations,Compos. Math. 140 (2004), no. 1, 31–63.
D. Savitt: On a conjecture of Conrad, Diamond, and Taylor, Duke Math. J.
128 (2005), no. 1, 141–197.
A. Scholl: Higher ﬁelds of norms and (φ, Γ)-modules, Doc. Math. 2006, Extra
Vol., 685–709.
P. Scholze: Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012),
245–313.
R. Schoof: Abelian varieties over cyclotomic ﬁelds with good reduction everywhere, Math. Ann. 325 (2003), no. 3, 413–448.
R. Schoof: Abelian varieties over Q with bad reduction in one prime only, Compos. Math. 141 (2005), no. 4, 847–868.
R. Schoof: Semistable abelian varieties with good reduction outside 15,
Manuscripta Math. 139 (2012), no. 1-2, 49–70.
S. Sen: Ramiﬁcation in p-adic Lie extensions, Invent. Math. 17 (1972), 44–50.
J.-P. Serre: Corps locaux, Deuxième édition. Publications de l’Université deNancago, No. VIII. Hermann, Paris, 1968.
J.-P. Serre: Propriétés galoisiennes des points d’ordre ﬁni des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331.
J.-P. Serre: Sur les représentations modulaires de degré 2 de Gal(Q̄/Q), Duke
Math. J. 54 (1987), no. 1, 179–230.

44

SHIN HATTORI

A. Shiho: Crystalline fundamental groups II—Log convergent cohomology andrigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), no. 1, 1–163.
[Tat94]
J. Tate: The non-existence of certain Galois extensions of Q unramiﬁed outside
2, Arithmetic geometry (Tempe, AZ, 1993), 153–156, Contemp. Math. 174,
Amer. Math. Soc., Providence, RI, 1994.
[Tav11] F. Tavares Ribeiro: An explicit formula for the Hilbert symbol of a formal group,
Ann. Inst. Fourier (Grenoble) 61 (2011), no. 1, 261–318.
[Tay03] R. Taylor: On icosahedral Artin representations II, Amer. J. Math. 125 (2003),
no. 3, 549–566.
[Tay06] R. Taylor: On the meromorphic continuation of degree two L-functions, Doc.
Math. 2006, Extra Vol., 729–779.
[Tay08] R. Taylor: Automorphy for some l-adic lifts of automorphic mod l Galois representations II, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 183–239.
[TW95] R. Taylor and A. Wiles: Ring-theoretic properties of certain Hecke algebras,
Ann. of Math. (2) 141 (1995), no. 3, 553–572.
Y. Tian: Canonical subgroups of Barsotti-Tate groups, Ann. of Math. (2) 172[Tia10](2010), no. 2, 955–988.
[Tia11+] Y. Tian: Classicality of overconvergent Hilbert eigenforms: case of quadratic
residue degrees, to appear in Rend. Sem. Mat. Univ. Padova.
[Tia12]
Y. Tian: An upper bound on the Abbes-Saito ﬁltration for ﬁnite ﬂat groupschemes and applications, Algebra Number Theory 6 (2012), no. 2, 231–242.
[Tsu98] N. Tsuzuki: The local index and the Swan conductor, Compos. Math. 111(1998), no. 3, 245–288.
[VZ10]
A. Vasiu and T. Zink: Breuil’s classiﬁcation of p-divisible groups over regular
local rings of arbitrary dimension, Algebraic and arithmetic structures of moduli
spaces (Sapporo 2007), 461–479, Adv. Stud. Pure Math. 58, Math. Soc. Japan,
Tokyo, 2010.
[Ver13]
H. Verhoek: Modularity of abelian varieties over Q with bad reduction in one
prime only, J. Number Theory 133 (2013), no. 9, 3065–3098.
[Wac96] N. Wach: Représentations p-adiques potentiellement cristallines, Bull. Soc.
Math. France 124 (1996), no. 3, 375–400.
[Wac97] N. Wach: Représentations cristallines de torsion, Compos. Math. 108 (1997),
no. 2, 185–240.
[Wes04] T. Weston: Unobstructed modular deformation problems, Amer. J. Math. 126(2004), no. 6, 1237–1252.
[Wil95] A. Wiles: Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2)141 (1995), no. 3, 443–551.
[Win83] J.-P. Wintenberger: Le corps des normes de certaines extensions inﬁnies decorps locaux; applications, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 1,
59–89.
[Win84] J.-P. Wintenberger: Un scindage de la ﬁltration de Hodge pour certaines variétésalgébriques sur les corps locaux, Ann. of Math. (2) 119 (1984), no. 3, 511–548.
[Xia10]L. Xiao: On ramiﬁcation ﬁltrations and p-adic diﬀerential modules, I: the equalcharacteristic case, Algebra Number Theory 4 (2010), no. 8, 969–1027.
[Xia12]
L. Xiao: On ramiﬁcation ﬁltrations and p-adic diﬀerential equations, II: mixed
characteristic case, Compos. Math. 148 (2012), no. 2, 415–463.
[Yos10] M. Yoshida: Ramiﬁcation of local ﬁelds and Fontaine’s property (Pm ), J. Math.
Sci. Univ. Tokyo 17 (2010), no. 3, 247–265.
[Zhu03] I. Zhukov: On ramiﬁcation theory in the case of an imperfect residue ﬁeld, Mat.
Sb. 194 (2003), no. 12, 3–30; translation in Sb. Math. 194 (2003), no. 11-12,
1747–1774.
T. Zink: The display of a formal p-divisible group, Cohomologies p-adiques et[Zin02]applications arithmétiques I, Astérisque 278 (2002), 127–248.
[Shi02]

RAMIFICATION OF CRYSTALLINE REPRESENTATIONS

E-mail address: shin-h@math.kyushu-u.ac.jp
Faculty of Mathematics, Kyushu University

45