Table Of ContentJean-Marc Fontaine and Yi Ouyang
Theory of p-adic Galois
Representations
Springer
Contents
0 Preliminary................................................ 1
0.1 Inverse limits and Galois theory ........................... 1
0.1.1 Inverse limits...................................... 1
0.1.2 Galois theory...................................... 3
0.2 Witt vectors and complete discrete valuation rings........... 5
0.2.1 Nonarchimedean fields and local fields. ............... 5
0.2.2 Witt vectors. ..................................... 8
0.2.3 Structure of complete discrete valuation rings ......... 13
0.2.4 Cohen rings. ...................................... 17
0.3 Galois groups of local fields ............................... 20
0.3.1 Ramification groups of finite Galois extension. ........ 20
0.3.2 Galois group of Ks/K.............................. 22
0.3.3 The functions Φ and Ψ. ............................ 23
0.3.4 Ramification groups of infinite Galois extension........ 27
0.3.5 Different and discriminant. ......................... 28
0.4 Ramification in p-adic Lie extensions....................... 30
0.4.1 Sen’s filtration Theorem. ........................... 30
0.4.2 Totally ramified Z -extensions....................... 35
p
0.5 Continuous Cohomology.................................. 38
0.5.1 Abelian cohomology................................ 38
0.5.2 Non-abelian cohomology. ........................... 41
1 ‘-adic representations of local fields: an overview .......... 45
1.1 ‘-adic Galois representations .............................. 45
1.1.1 Linear representations of topological groups. .......... 45
1.1.2 ‘-adic representations. ............................. 46
1.1.3 Representations arising from linear algebra............ 47
1.1.4 Examples of ‘-adic Galois representations. ............ 48
1.2 ‘-adic representations of finite fields........................ 51
1.2.1 ‘-adic Galois representations of finite fields............ 51
1.2.2 ‘-adic geometric representations of finite fields......... 51
ii Contents
1.3 ‘-adic representations of local fields ........................ 54
1.3.1 ‘-adic representations of local fields. ................. 54
1.3.2 An alternative description .......................... 58
1.3.3 The case of a finite residue field. .................... 60
1.3.4 Geometric ‘-adic representations of G ............... 62
K
2 p-adic Representations of fields of characteristic p ......... 65
2.1 B-representations and regular G-rings ...................... 65
2.1.1 B-representations. ................................. 65
2.1.2 Regular (F,G)-rings................................ 66
2.2 Mod p Galois representations of fields of characteristic p>0 .. 71
2.2.1 E´tale ϕ-modules over E............................ 71
2.2.2 The functor M. ................................... 73
2.2.3 The inverse functor V. ............................. 74
2.3 p-adic Galois representations of fields of characteristic p>0 .. 78
2.3.1 E´tale ϕ-modules over E............................. 78
2.3.2 The field Ecur...................................... 80
2.3.3 O and Z representations......................... 81
Edur p
2.3.4 p-adic representations. ............................. 83
2.3.5 Down to earth meaning of the equivalence of categories. 84
3 C-representations and Methods of Sen..................... 87
3.1 Krasner’s Lemma and Ax-Sen’s Lemma .................... 87
3.1.1 Krasner’s Lemma. ................................. 87
3.1.2 Ax-Sen’s Lemma................................... 88
3.2 Classification of C-representations ......................... 92
3.2.1 Almost ´etale descent. .............................. 92
3.2.2 Decompletion. .................................... 94
3.2.3 Study of C-representations.......................... 96
3.2.4 Sen’s operator Θ................................... 97
3.3 Sen’s operator Θ and the Lie algebra of ρ(G). ...............100
3.3.1 Main Theorem. ...................................100
3.3.2 Application of Sen’s filtration Theorem. ..............103
3.4 Sen’s method............................................104
3.4.1 Tate-Sen’s conditions (TS1), (TS2) and (TS3). ........105
3.4.2 Almost ´etale descent...............................106
3.4.3 Decompletion .....................................107
3.4.4 Applications to p-adic representations................108
3.5 C-admissible representations..............................110
3.5.1 Notations for the rest of the book....................110
3.5.2 K-admissible p-adic representations..................111
3.5.3 C-admissible representations. .......................113
Contents iii
4 The ring R and (ϕ,Γ)-module .............................115
4.1 The ring R .............................................115
4.1.1 The ring R(A).....................................115
4.1.2 Basic properties of the ring R. ......................117
4.1.3 The multiplicative group FrR∗. .....................119
4.2 The action of Galois groups on R..........................121
4.2.1 The action of Galois groups. ........................121
4.2.2 R(K0cyc/pOKcyc), ε and π. ..........................121
0
4.2.3 A fundamental theorem.............................123
4.3 An overview of Galois extensions...........................125
4.3.1 A summary of Galois extensions of K and E .........125
0 0
4.3.2 The field Be and its subrings.........................127
4.4 (ϕ,Γ)-modules and p-adic Galois representations ............129
4.4.1 (ϕ,Γ)-modules. ...................................129
4.4.2 The operator ψ....................................132
5 de Rham representations ..................................135
5.1 Hodge-Tate representations ...............................135
5.2 de Rham representations .................................138
5.2.1 The homomorphism θ. .............................138
5.2.2 The rings B+ and B . ...........................140
dR dR
5.2.3 The element t. ....................................143
5.2.4 Galois cohomology of B ..........................145
dR
5.2.5 de Rham representations............................146
5.2.6 A digression.......................................149
5.3 Overconvergent rings and overconvergent representations .....150
5.3.1 The rings of Overconvergent elements. ...............151
5.3.2 Overconvergent representations .....................157
5.3.3 Tate-Sen’s method for Ae(0,r] ........................157
5.3.4 The ring Be]0,r]. ...................................162
6 Semi-stable p-adic representations ........................165
6.1 The rings B and B ...................................165
cris st
6.1.1 The ring B . ....................................165
cris
6.1.2 The Frobenius map ϕ on B .......................168
cris
6.1.3 The logarithm map. ...............................168
6.1.4 The ring B . .....................................171
st
6.1.5 The operators ϕ and N on B . .....................174
st
6.2 Some properties about B . ..............................174
cris
6.2.1 Some ideals of W(R). ..............................174
6.2.2 A description of A ...............................176
cris
6.2.3 The filtration by I[r]. ..............................179
6.3 Semi-stable p-adic Galois representations ...................183
6.4 Filtered (ϕ,N)-modules ..................................185
6.4.1 Definitions. .......................................185
iv Contents
6.4.2 t (D) and t (D)..................................187
N H
6.4.3 Admissible filtered (ϕ,N)-modules. ..................191
6.5 Statement of Theorem A and Theorem B...................192
6.5.1 de Rham implies potentially semi-stable. .............192
6.5.2 Weakly admissible implies admissible.................195
7 Proof of Theorem A and Theorem B ......................197
7.1 Admissible filtered (ϕ,N)-modules of dimension 1 and 2......197
7.1.1 Hodge and Newton polygons. .......................197
7.1.2 The case when the filtration is trivial.................199
7.1.3 Tate’s twist. ......................................199
7.1.4 Admissible filtered (ϕ,N)-modules of dimension 1......200
7.1.5 Admissible filtered (ϕ,N)-modules of dimension 2......200
7.2 Proof of Proposition B1 ..................................204
7.2.1 Construction of the natural isomorphism..............204
7.2.2 Unramified representations..........................205
7.2.3 Reduction to the algebraically closed residue field case..206
7.2.4 Representations of dimension 1. .....................207
7.2.5 End of proof of Proposition B1. .....................208
7.3 Q -representations and filtered (ϕr,N)-modules.............210
pr
7.3.1 Definitions. .......................................210
7.3.2 Main properties. ..................................211
7.3.3 The Q -representation V . ........................213
pr (r)
7.4 Outline of the proof .....................................216
7.4.1 Reduction of Proposition B2 to Proposition B. ........216
7.4.2 Outline of the Proof of Propositions A and B..........218
7.5 Proof of Proposition 2A and Proposition 2B ................219
7.5.1 Proof of Proposition 2A............................219
7.5.2 Fundamental complex of D. ........................220
7.6 Reduction to the case t =0..............................223
H
7.6.1 The case for V. ...................................223
7.6.2 The case for D. ...................................224
7.7 Finish of proof ..........................................225
7.7.1 The Fundamental Lemma of Banach-Colmez space.....225
7.7.2 Application of the Fundamental Lemma. .............229
7.7.3 Recurrence of the Hodge polygon and end of proof. ....231
References.....................................................237
0
Preliminary
0.1 Inverse limits and Galois theory
0.1.1 Inverse limits.
In this subsection, we always assume that A is a category with infinite prod-
ucts. In particular, one can let A be the category of sets, of (topological)
groups, of (topological) rings, of left (topological) modules over a ring A. Re-
call that a partially ordered set I is called a directed set if for any i,j ∈ I,
there exists k ∈I such that i≤k and j ≤k.
Definition 0.1.Let I be a directed set. Let (A ) be a family of objects
i i∈I
in A. This family is called an inverse system(or a projective system) of A
over the index set I if for every pair i ≤ j ∈ I, there exists a morphism
ϕ :A →A such that the following two conditions are satisfied:
ji j i
(1) ϕ =Id;
ii
(2) For every i≤j ≤k, ϕ =ϕ ϕ .
ki ji kj
Definition 0.2.The inverselimit(or projectivelimit) of a given inverse sys-
tem A =(A ) is defined to be an object A in A
• i i∈I
n Y o
A=limA = (a )∈ A : ϕ (a )=a for every pair i≤j ,
←− i i i ji j i
i∈I i∈I
such that the natural projection ϕ :A→A , a=(a ) 7→a is a morphism
i i j j∈I i
for each i∈I.
Remark 0.3.One doesn’t need the set I to be a directed set but only to be
a partially ordered set to define an inverse system. For example, let I be a
Q
set with trivial ordering, i.e. i ≤ j if and only if i = j, then limA = A .
←− i i
i∈I i∈I
However, this condition is usually satisfied and often needed in application.
By the inverse system condition, one can see immediately ϕ = ϕ ϕ for
i ji j
every pair i≤j. Actually, A is the solution of the universal problem:
2 0 Preliminary
Proposition 0.4.Let (A ) be an inverse system in A, A be its inverse limit
i
and B be an object in A. If there exist morphisms f : B → A for all
i i
i ∈ I such that for every pair i ≤ j, f = ϕ ◦f , then there exists a unique
i ji j
morphism f :B →A such that f =ϕ ◦f.
j j
Proof. This is an easy exercise. ut
By definition, if A is the category of topological spaces, i.e., if X is a
i
topological space for every i ∈ I and ϕ ’s are continuous maps, then X =
ij
limX is a topological space equipped with a natural topology, the weakest
←− i
i∈I
topology suchthatalltheϕ ’sarecontinuous.Recallthattheproducttopology
i
Q
ofthetopologicalspace X istheweakest topology suchthattheprojection
i
i∈I
Q
pr : X → X is continuous for every j ∈ I. Thus the natural topology
j i j
i∈I
Q
of X is the topology induced as a closed subset of X with the product
i
i∈I
topology.
For example, if each X is endowed with the discrete topology, then X is
i
endowed with the topology of the inverse limit of discrete topological spaces.
In particular, if each X is a finite set endowed with discrete topology, then
i
we will get a profinite set (inverse limit of finite sets). In this case, since
Q Q
limX ⊂ X is closed, and since X , as the product space of compact
←− i i i
i∈I i∈I
spaces, is still compact, limX is compact too. In this case one can see that
←− i
limX is also totally disconnected.
←− i
If moreover, each X is a (topological) group and if the ϕ ’s are (continu-
i ij
ous)homomorphismsofgroups,thenlimX isagroupwithϕ :lim X →X
←− i i ←−j j i
a (continuous) homomorphism of groups.
If the X ’s are finite groups endowed with discrete topology, the inverse
i
limitinthiscaseisaprofinitegroup.Thusaprofinitegroupisalwayscompact
and totally disconnected. As a consequence, all open subgroups of a profinite
group are closed, and a closed subgroup is open if and only if it is of finite
index.
Example 0.5.(1) For the set of positive integers N∗, we define an ordering
n ≤ m if n | m. For the inverse system (Z/nZ)n∈N∗ of finite rings where the
transition map ϕ is the natural projection, the inverse limit is
mn
Zb = lim Z/nZ
←−
n∈N∗
.
(2) Let ‘ be a prime number, for the sub-index set {‘n :n∈N} of N∗,
Z = limZ/‘nZ
‘ ←−
n∈N
0.1 Inverse limits and Galois theory 3
istheringof‘-adicintegers.TheringZ isacompletediscretevaluationring
‘
with the maximal ideal generated by ‘, the residue field Z/‘Z = F , and the
l
fraction field
(cid:20) (cid:21) ∞
1 [
Q =Z = ‘−mZ
‘ ‘ ‘ ‘
m=0
being the field of ‘-adic numbers.
If N ≥ 1, let N = ‘r1‘r2···‘rh be its primary factorization. Then the
1 2 h
isomorphism
h
Y
Z/NZ’ Z/‘riZ
i
i=1
induces an isomorphism of topological rings
Y
Zb ’ Z‘.
‘primenumber
0.1.2 Galois theory.
Let K be a field and L be a (finite or infinite) Galois extension of K. The
Galois group Gal(L/K) is the group of the K-automorphisms of L, i.e.,
∼
Gal(L/K)={g :L→L, g(γ)=γ for all γ ∈K}.
Denote by E the set of finite Galois extensions of K contained in L and
order this set by inclusion, then for any pair E,F ∈ E, one has EF ∈ E and
S
E,F ⊂ EF, thus E is in fact a directed set and L = E. As a result, we
E∈E
can study the inverse limits of objects over this directed set. For the Galois
groups, by definition,
γ =(γ )∈ limGal(E/K) if and only if (γ )| =γ for E ⊂F ∈E.
E ←− F E E
E∈E
Galois theory tells us that the following restriction map is an isomorphism
∼
Gal(L/K)−→ limGal(E/K)
←−
E∈E
g 7−→(g| ):g| the restriction of g in E.
E E
Fromnowon,weidentifythetwogroupsthroughtheaboveisomorphism.Put
thetopologyoftheinverselimitwiththediscretetopologyoneachGal(E/K),
the group G=Gal(L/K) is then a profinite group, endowed with a compact
andtotallydisconnectedtopology,whichiscalledtheKrull topology.Wehave
Theorem 0.6(Fundamental Theorem of Galois Theory). There is a
one-one correspondence between intermediate field extensions K ⊂ K0 ⊂ L
4 0 Preliminary
andclosedsubgroupsH ofGal(L/K)givenbyK0 →Gal(L/K0)andH →LH
where LH ={x∈L|g(x)=x for all g ∈H} is the invariant field of H.
Moreover,theabovecorrespondencegivesone-onecorrespondencesbetween
finite extensions (resp. finite Galois extensions, Galois extensions) of K con-
tained in L and open subgroups (resp. open normal subgroups, closed normal
subgroups) of Gal(L/K).
Remark 0.7.(1) Given an element g and a sequence (gn)n∈N of Gal(L/K),
the sequence (gn)n∈N converges to g if and only if for all E ∈ E, there exists
n ∈N such that if n≥n , then g | =g| .
E E n E E
(2) The open normal subgroups of G are the groups Gal(L/E) for E ∈E,
and there is an exact sequence
1−→Gal(L/E)−→Gal(L/K)−→Gal(E/K)−→1.
(3) A subgroup of G is open if and only if it contains an open normal
subgroup. A subset X of G is an open set if and only if for all x ∈ X, there
exists an open normal subgroup H such that xH ∈X.
x x
(4) If H is a subgroup of Gal(L/K), then LH = LH with H being the
closure of H in Gal(L/K).
We first give an easy example:
Example 0.8.LetK beafinitefieldwithqelements,andletK beanalgebraic
closure of K with Galois group G=Gal(K/K).
For each n ∈ N, n ≥ 1, there exists a unique extension K of degree
n
n of K contained in Ks. The extension K /K is cyclic with Galois group
n
Gal(K /K)’Z/nZ=hϕ iwhereϕ =(x7→xq)isthearithmetic Frobenius
n n n
of Gal(K /K). We have the following diagram
n
∼
G −−−−→ limGal(K /K)
←− n
yo yo
Zb −−−∼−→ limZ/nZ.
←−
Thus the Galois group G’Zb is topologically generated by ϕ∈G: ϕ(x)=xq
for x ∈ Ks, i.e., with obvious conventions, any elements of G can be written
uniquely as g = ϕa with a ∈ Zb. The element ϕ is called the arithmetic
Frobenius and its inverse ϕ−1 is called the geometric Frobenius of G.
IfK =F ,thearithmeticFrobenius(x7→xp)iscalledtheabsolute Frobe-
p
nius anddenotedasσ.Moreover,foranyfieldkofcharacteristicp,wecallthe
endomorphism σ :x7→xp the absolute Frobenius of k. σ is an automorphism
if and only if k is perfect.
In the case K = Q, let Q be an algebraic closure of Q, and let GQ =
Gal(Q/Q).
