Math 162b Topics in Number Theory: Introduction to p-adic Hodge Theory

83 Pages·2013·1.787 MB·English

by  Andrei Jorza

MATH 162B WINTER 2012
INTRODUCTION TO p-ADIC GALOIS REPRESENTATIONS
OVERVIEW
ANDREIJORZA
Introduction
A main goal of algebraic number theory is to understand continuous Galois representations of GQ =
Gal(Q/Q)
ρ:GQ →GL(n,R)
where R is a topological ring. Often R is one of the rings C,Q ,Z ,F (or subextensions of these). Here
(cid:96) (cid:96) (cid:96)
continuity is with respect to the profinite topology on GQ and the topology of R; note that the topology on
R matters: as fields, C∼=Q , but not as topological fields.
p
Thestudyofthese“global”Galoisrepresentationsisafundamentallyhardproblem,andanoftensimpler
approach is to first study continuous “local” Galois representations of GQp = Gal(Qp/Qp) → GL(n,R)
whereGQ ⊂GQ isthedecompositiongroupatp. Onedesirestounderstand/classifylocalandglobalGalois
p
representations.
1. Examples
For simplicity we’ll only talk about Q and Q .
p
Global Galois representations.
Artin representations. These are continuous Galois representations ρ : GQ → GL(n,C), which necessar-
ily have finite image (if n = 2 this means that the image is cyclic, dihedral, tetrahedral, octahedral or
icosahedral).
The main conjecture related to these representations is that the Artin L-function
(cid:89)
L(ρ,s)= det(1−Frob p−s|ρIp)−1
p
p
has analytic continuation to C if ρ does not contain the trivial representation. For function fields proven
by Weil; meromorphic continuation follows from Tate’s thesis and Brauer’s theorem on induced characters
(implies conjecture for cyclic and dihedral); Langlands proved the tetrahedral case; Tunnell proved the
octahedral case; the icosahedral case is in progress. Should follow in general from the Langlands program.
In terms of classification, two dimensional Artin representations should correspond to weight 1 modular
forms.
Mod p representations. These are continuous Galois representations ρ:GQ →GL(n,Fp).
The main conjecture is Serre’s conjecture, which says that such ρ which are irreducible and “odd” come
from modular forms with predictable weight and level. Known for Q by Khare and Wintenberger.
(cid:96)-adic representations. These are continuous Galois representations ρ : GQ → GL(n,Q(cid:96)). If they come in
compatible systems, meaning one is given a Galois representation for each prime (cid:96) such that Frob acts
p
compatibly in ρ as (cid:96) varies, then one can attach a meaningful L-function.
(cid:96)
Themainconjecture istheglobalLanglandsconjecture. Thisconjecture hasseveralcomponents, butthe
firstdirectionisthatattachedto“algebraic”automorphicrepresentationsonecanalwaysattachcompatible
systems of Galois representations. See the first homework for n=1; the n=2 case the first part is Eichler-
Shimura (weight 2), Deligne (weight ≥ 3), Deligne-Serre (weight 1). The second direction is that one can
1
describe the compatible systems that come from automorphic representations (this description requires p-
adic Hodge theory). For n = 2 the first such result was Wiles’ proof of Fermat’s Last Theorem, later the
Taniyama-Shimura conjecture; finally the Fontaine-Mazur conjecture for n=2 mostly known by Kisin.
Local Galois representations.
Complex representations. These are continuous Galois representations ρ : GQ → GL(n,C). The local
p
Langlands conjecture states that these are in bijection with smooth representations of GL(n,Q ). Proven
p
by Harris-Taylor and Henniart (90’s), a different proof given by Scholze (’10).
(cid:96)-adic representations ((cid:96)(cid:54)=p). These are continuous Galois representations ρ:GQp →GL(n,Q(cid:96)) for (cid:96)(cid:54)=p.
Grothendieck’s (cid:96)-adic monodromy theorem implies that these are in bijection with certain Weil-Deligne
representations,whicharepairs(r,N)ofacontinuous(herethismeansopenstabilizers)Galoisrepresentation
r and a nilpotent matrix N such that r(g)N = pdNr(g) where d is the exponent of Frob in g. The
p
correspondence is
ρ(Frobnσ)=r(Frobnσ)exp(t (σ)N)
(cid:96)
where t :I →I /P ∼= (cid:89)Z →Z .
(cid:96) p p p q (cid:96)
q(cid:54)=p
Remark 1. A better way to describe N is that N :ρ→ρ(−1) is an intertwining operator.
p-adic Galois representations. These are continuous Galois representations ρ:GQp →GL(n,Qp).
Although Grothendieck’s (cid:96)-adic monodromy theorem no longer applies since (cid:96) = p, one can still attach
Weil-Deligne representations to such ρ via Fontaine’s p-adic Hodge theory. This is complicated, and one of
the focuses of this course.
2. This course
Goals. The main goals of this course are the following:
(1) Enumerate all p-adic Galois representations.
(2) Enumerate the p-adic Galois representations arising from geometry.
To motivate the second goal, let’s look at the case of Fermat’s Last Theory. The theorem follows from a
modularity lifting theorem which is proven by showing that a deformation space of Galois representations
(the ring R) is isomorphic to a deformation space of modular forms (the ring T); if one allows too many
Galoisrepresentationsasdeformationsonegetsatoolargering R, sooneneedstospecifyconstraintswhich
force these deformations to correspond to modular forms and algebraic geometry).
How to study? How to study p-adic Galois representations? Seek inspiration from algebraic geometry.
Local Galois representations arise naturally in the etale cohomology of varieties over Q ; perhaps one may
p
find a different cohomology group which is simpler to describe.
The proto-example is Torelli’s theorem for compact Riemann surfaces. If X is a compact connected
Riemann surface (equivalently a smooth projective connected curve over C) then H0(X,Z) = Z since X is
connected, H2(X,Z) = Z since X is proper, while H1(X,Z) is a Z-module (of rank twice the genus of the
curve).
The Hodge decomposition theorem states that
H1(X,Z)⊗ZC∼=H1,0(X)⊕H0,1(X)
where H1,0(X)=H0(X,Ω1 ) and H0,1 =H1(X,Ω0 ) are differential cohomology groups (the left hand side
X X
is the de Rham cohomology group H1 (X/C) while the right hand side is the Hodge cohomology group
dR
H1 (X)). So from the category of smooth projective connected curves over C we obtained an object in a
Hodge
different category whose objects are “Hodge structures”, i.e., Z-modules M together with a decomposition
V = M ⊗C = V1,0⊕V0,1. The amazing thing is that the curve X is determined by its associated Hodge
structure.
So we replaced a complicated object (the curve X) with a linear algebra datum (the Hodge structure).
Thisisthedefiningthemeofp-adicHodgetheory, whereoneseekslinearalgebradatainthehopeoffinding
an equivalence of categories; this will not be possible always.
2
The algebraic geometry story. The general example is that of a proper smooth scheme of finite type
over Q . The p-adic etale cohomology group Hn(X ,Q ) gives rise to p-adic Galois representations. One
p ´et Q p
p
seeks linear algebra data in the other cohomology groups attached to X.
Hodge-Tate. IngeneralthedeRhamcohomologygroupHn (X/Q )isnotadirectsumofHodgecohomology
dR p
groups;instead,Hn (X/Q )isaQ vectorspaceendowedwithafiltration(whichcomesfromadegenerating
dR p p
spectral sequence whose first sheet contains Hodge cohomology groups). The first linear algebra datum is
gr•Hn (X/Q ) which is a graded Q vector space.
dR p p
Whileinthecaseofsmoothprojectivecomplexcurvesthecomparisonbetweenthetwocohomologygroups
occured over C, the story is more complicated over Q since it needs to take into account the Galois action.
p
Tate conjectured in the 60’s and Faltings proved in the 80’s that
H(cid:124) ´ent(X(cid:123)Q(cid:122)p,Qp(cid:125))⊗Qp B(cid:124)(cid:123)H(cid:122)T(cid:125) ∼= g(cid:124)r•HdnR(cid:123)(cid:122)(X/Qp(cid:125))⊗QpBHT
Galoisaction gwraitdhedGvaelocitsoracstpiaocne gradedvectorspace
where B is a graded vector space with Galois action over the p-adic completion C of Q (implicit in this
HT p p
is that the Galois group acts on each graded piece).
de Rham. The Hodge-Tate linear algebra datum is convenient, since it is only a graded vector space with
the Galois acting on each graded piece, but it is much too coarse to describe the Galois representation on
the etale cohomology group. One could instead not base change to C and work directly with the de Rham
p
cohomology group Hn (X/Q ) which is a filtered vector space with Galois action (meaning that the Galois
dR p
group acts on each filtered piece, but not necessarily semisimply, so not necessarily on each graded piece).
ThecorrespondingcomparisontheoremwasconjecturedbyFontaineandprovenbyFaltings(usingalmost
math), Nizio(cid:32)l (using K-theory), Beilinson (using derived geometry) and Scholze (using perfectoid spaces)
(cid:124)H´ent(X(cid:123)Q(cid:122)p,Qp(cid:125))⊗Qp B(cid:124)(cid:123)d(cid:122)R(cid:125) ∼= H(cid:124) dnR((cid:123)X(cid:122)/Qp(cid:125)) ⊗QpBdR
Galoisaction fiwltietrhedGavleocitsoarcstpioance filteredvectorspace
Crystalline. ThedeRhamlinearalgebradatumofafilteredQ vectorspacewithGaloisactionisstillconve-
p
nientandworksgenerally,butthefunctorattachingthelinearalgebradatumtotheGaloisrepresentationis
not faithful, meaning it cannot detect maps between Galois representations. However, if the proper smooth
scheme X/Q has a proper smooth model over Z , in other words it has good reduction at p, then one has
p p
anadditionalcohomologygroup: thecrystallinecohomologygroupHn (X/Z )whichisaQ filteredvector
cris p p
space with an action not of the whole Galois group but just of Frobenius.
The relevant comparison theorem is that if X has proper smooth reduction at p then
H(cid:124) ´ent(X(cid:123)Q(cid:122)p,Qp(cid:125))⊗Qp B(cid:124)(cid:123)c(cid:122)ri(cid:125)s ∼= H(cid:124) dnR((cid:123)X(cid:122)/Qp(cid:125)) ⊗QpBcris
Galoisaction filtweriethdFveroctboernisupsace filteredvectorspace
which goes via the cristalline cohomology group Hn (X/Z ).
cris p
The remarkable fact about this functor is that it is faithful on the subcategory of crystalline representa-
tions,andthatonecancharacterizeallfilteredQ vectorspaceswithFrobeniusactionthatarisefromp-adic
p
Galois representations (these are the admissible modules).
Semistable. One may object that X having proper smooth reduction at p is too strong a condition. One
may weaken this and only require that X has a “semistable” reduction at p, which means that singularities
behave like the intersection of two lines. Such singularities are still nice enough because if one leaves the
category of schemes and goes into the category of “log schemes”, such semistable singularities become “log
smooth”. In that case the log crystalline cohomology group Hn (X/Z ) is a Q filtered vector space
log-cris p p
withanactionofFrobenius andamonodromyoperator N withthecommutationrelation Nφ=pφN. (The
monodromyN issupposedtoencodethemonodromyoperatoractingonthegenericcohomologyaroundthe
singular special fiber.)
The relevant comparison theorem, proven by Tsuji in the 90’s, is
H(cid:124) ´ent(X(cid:123)Q(cid:122)p,Qp(cid:125))⊗Qp (cid:124)B(cid:123)(cid:122)st(cid:125) ∼= H(cid:124) dnR((cid:123)X(cid:122)/Qp(cid:125)) ⊗QpBst
Galoisaction wfiitltherFerdobveenctiuosraspnadceN filteredvectorspace
3
which goes via the log cristalline cohomology group Hn (X/Z ).
log-cris p
Again,remarkably,thefunctorfromsemistablerepresentationstoQ filteredvectorspaceswithFrobenius
p
and monodromy is faithful and the image can be described.
Fieldsofnorms. Whatabouttheotherp-adicGaloisrepresentations,notcomingfromalgebraicgeometry?
Describing these representations proved to be crucial in the p-adic local Langlands program. The approach
is via fields of norms of Fontaine and Wintenberger, which in effect says that there exists a characteristic p
field E such that the Galois theory of a very ramified extension of Q is the same as the Galois theory of E.
p
Then one can prove comparison theorems in the characteristic p setting and automatically transport results
to characteristic 0. This theory relies on ramification estimates due to Tate.
The plan. Time permitting we will look at:
(1) Review local class field theory
(2) Hodge-Tate(-Sen) theory and B
HT
(3) Admissible representations
(4) De Rham representations and B
dR
(5) Crystalline representations and B
cris
(6) Semistable representations and B
st
(7) Admissibility and weak admissibility
(8) Fields of norms
(9) (φ,Γ)-modules
(10) Kisin’s S-modules and integral p-adic Hodge theory
(11) Families of p-adic Galois representations
4
p-adic Galois Representations
Math 162b Winter 2012
Lecture Notes
Andrei Jorza
Last updated: 2012-03-10
Contents
1 Local Class Field Theory 2
1.1 Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Newton polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Ramification of local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Main results of local class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 C -representations 6
p
2.1 The field C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
p
2.2 Ax-Sen-Tate and Galois invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Ramification estimates and Tate periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Sen theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Admissible Representations 26
3.1 The category of Hodge-Tate representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Regular rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Admissible representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Hodge-Tate again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 de Rham Representations 31
4.1 Witt Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Perfections and the ring R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
dR
4.4 De Rham representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Crystalline and Semistable Representations 42
5.1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
cris
5.2 The fundamental exact sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
st
5.4 Filtered modules with Frobenius and monodromy . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5 Crystalline and semistable representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.6 Bloch-Kato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.7 Ordinary representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1
Sources Theselecturenotesaremashupsofvarioussources,withsomeaddedclarificationswhereIcouldn’t
follow the argument. Evidently the material presented here is treated by these sources, and in most cases it
will be lifted without acknowledgement from the “text-books” for the convenience of exposition.
Lecture 1
2012-01-04
See overview notes.
Lecture 2
2012-01-06
1 Local Class Field Theory
In square brackets I give the section numbers from the math 160b (winter 2012) course notes.
1.1 Local fields
LetK beafieldofcharacteristic0withnonarchimedeanvaluationv :K× →R(e.g.,Q orafiniteextension
p
of Q , or an algebraic extension of Q , etc.) Write v(0)=∞.
p p
Denote by O = {x ∈ K|v(x) ≥ 0}, and m = {x ∈ K|v(x) > 0}. Then O× = kerv and let
K K K
k =O /m be the residue field. The ring O is a PID and if v is discrete, i.e., Imv ⊂R is discrete, then
K K K K
there exists a uniformizer (cid:36) such that m =((cid:36) ).
K K K
The field K has a topology given by the norm |x| = (#k )−v(x) (if k is not finite, replace it by any
K K K
real number >1). K being complete means completeness in this topology.
1.1.1 Hensel’s lemma
[Math 160b Winter 2012: §I.2]
Lemma 1.1. Let K be complete with respect to v, let P ∈ O [X] be monic and let c ∈ O such that
K K
P(c) ≡ 0 (mod m ) but P(cid:48)(c) (cid:54)≡ 0 (mod m ). Then there exists c ∈ O such that c ≡ c (mod m ) and
K K K K
P(c)=0.
Remark 1. 1. The standard application is the existence of a Teichmu¨ller homomorphism ω : k× → O×
K K
such that ω(x)≡x (mod m ).
K
2. This construction is later generalized by Witt vectors.
1.1.2 Krasner’s lemma
[Math 160b Winter 2012: Problem set 2]
Lemma 1.2. Let K be complete with respect to v, and let α,β ∈ K. If v(β −α) > v(σ(α)−α) for all
σ ∈G then α∈K(β).
K(α)/K
Remark 2. 1. The standard application is to showing that if two polynomials are sufficiently close p-
adically then they have isomorphic splitting fields.
2. This can be use to show that there are finitely many local field extensions of a certain degree.
3. Conceptually, it is the first instance where approximating in the p-adic world does not lead to loss of
information.
2
1.2 Newton polygons
[Math 160b Winter 2012: §I.3]
1.2.1 Definition
d
(cid:88)
Let K be a field with valuation v. For a polynomial f = f Xk ∈K[X] the Newton polygon NP is the
k f
k=0
lower convex hull of the points (i,v(f )) and (0,∞) and (d,∞).
i
Definition 1.3. A slope of f is a slope of a segment of NP .
f
1.2.2 Newton polygons and products
Theorem 1.4. Let K and v be as before.
1. Let f,g ∈K[X] such that all slopes of f are less than all slopes of g. Then NP is the concatenation
fg
of NP and NP .
f g
2. If (d,v(f )) is a vertex of NP where h ∈ K[X] has degree n > d > 0 then there exist polynomials
d h
f,g ∈K[X] such that h=fg and NP =NP | and NP =NP | .
f h [0,d] g h [d,n]
3. If NP is pure of slope α, i.e., it consists of a segments of slope α, then all the roots of f have valuation
f
−α.
Remark 3. 1. Used to study ramification of local fields.
2. Useful for finding uniformizers. For example, ζ −1 can be shown to be a uniformizer of Q (ζ ) by
pn p pn
analyzing the Newton polygon of its minimal polynomial.
3. Can be generalized to Newton polygons of power series, which we’ll use to study log (which then will
be used to study the fundamental exact sequence and extensions of p-adic Galois representations).
1.3 Ramification of local fields
[Math 160b Winter 2012: §II.2]
1.3.1 Ramification
If L/K/Q are finite extensions write f =[k :k ] be the inertia index and e =[v (L×):v (K×)]
p L/K L K L/K K K
be the ramification index.
Definition 1.5. Say that L/K is
• unramified if e =1;
L/K
• totally ramified if f =1;
L/K
• tamely ramified if p(cid:45)e ;
L/K
• wildly ramified if p|e .
L/K
Note that these can be made sense of even for infinite extensions.
Theorem 1.6. Let L/K/Q be finite extensions.
p
1. f e =[L:K].
L/K L/K
3
2. The field Kur =K(ω(k ×)) is the maximal unramified extension of K, and K/Kur is totally ramified
K
with Galois group I , the inertia subgroup.
K
3. The field Kt = Kur((cid:36)1/n|p (cid:45) n) is the maximal tamely ramified extension of K, and K/Kt is totally
K
wildly ramified with Galois group P , the wild inertia subgroup.
K
4. Have an exact sequence 1 → I → G → G → 1 and Frob will denote both the topological
K K kK K
generator of G ∼=G ∼=FrobZ(cid:98) and some lift to G , well-defined up to conjugation.
Kur/K kK K K
5. Writing I =G and P =G have 1→I →G →G →1. Moreover,
L/K L/L∩Kur L/K L/L∩Kt L/K L/K kL/kK
L/K is unramified if and only if I ={1} and L/K is tamely ramified if and only if P ={1}.
L/K L/K
Example1.7. K =Qp(ζp)istotallyramifiedoverQpbecausevp(ζp−1)= p−11 = [K:1Qp] soeK/Qp =[K :Qp]
so fK/Qp =1.
1.3.2 Ramification filtrations
[Math 160b Winter 2012: §III.1] The subgroups G ⊃I ⊃P of more and more complex elements
L/K L/K L/K
of the Galois group fit into a ramification filtration.
Definition 1.8. If L/K is finite for u≥−1 the lower ramification filtration groups are
G ={σ ∈G |v (σ(x)−x)≥u+1,∀x∈O }
L/K,u L/K L L
Theorem 1.9. 1. G =G .
L/K,u L/K,(cid:100)u(cid:101)
2. G =G .
L/K,−1 L/K
3. G =I .
L/K,0 L/K
4. G =P .
L/K,1 L/K
5. For u>>0 have G ={1}.
L/K,u
Definition 1.10. For L/K finite consider φ :[−1,∞)→[−1,∞) given by
L/K
(cid:90) x du
φ (x)=
L/K [G :G ]
0 L/K,0 L/K,u
which is a piece-wise linear function, of slope 1 on the interval [−1,0], and slope 1/e for x>>0.
L/K
Definition 1.11. The upper ramification filtration groups are
Gu =G
L/K L/K,φ−1 (u)
L/K
Theorem 1.12 (Herbrand). Let L/M/K be finite extensions
1. Gu =Gu /(Gu ∩G ).
M/K L/K L/K L/M
2. φ =φ ◦φ .
L/K M/K L/M
Remark 4. Theorem 1.12 allows one to make sense of Gu (but not of the lower filtration).
K
Theorem 1.13 (Hasse-Arf). If L/K is a finite abelian extension then Gu =G(cid:98)u(cid:99) , i.e., the jumps in the
L/K L/K
upper filtration are at integers. In other words, the y-coordinates of the vertices of the graph of φ are
L/K
integers.
Example 1.14. If F = Q and F = F(ζ ) then F /F is totally ramified, abelian, with Galois group
p ∞ p∞ ∞
G ∼=Z× and Gn ∼=1+pnZ .
F∞/F p F∞/F p
4
1.3.3 Different
Definition 1.15. If L/K is a finite extension then
• The inverse different is D−1 ={x∈L|Tr (xO )⊂O } is a fractional ideal of L containing O .
L/K L/K L K L
• The different is D is the inverse of D−1 , i.e., D ={x∈L|xD−1 ⊂O }.
L/K L/K L/K L/K L
Remark 5. The different measures the ramification of local field extensions.
Theorem 1.16. Let L/K be a finite extension.
(cid:90) ∞
1. v (D )= (#G −1)du.
L L/K L/K,u
−1
(cid:32) (cid:33)
(cid:90) ∞ 1
2. v (D )= 1− du.
K L/K #Gu
−1 L/K
3. If I is an ideal of L then v (Tr (I))=(cid:98)v (ID )(cid:99)
K L/K K L/K
1.4 Main results of local class field theory
1.4.1 The Weil group
Recall that by Theorem 1.6 1→I →G →G →1 where G ∼=FrobZ(cid:98) .
K K kK kK K
Z
Definition 1.17. The Weil group W is the preimage via the projection map of Frob , with the topology
K K
Z
that makes I open and Frob discrete.
K K
1.4.2 The main results
Theorem 1.18. Let K/Q be a finite extension.
p
1. There exists an injective homomorphism rec :K× (cid:44)→Gab, such that:
K K
2. K× ∼=Wab, O× ∼=Iab and for n≥1, 1+mn ∼=Gab,n.
K K K K K
3. If L/K is finite then rec (x)=rec (N (x)).
L K L/K
Remark 6. 1. This identifies the ramification filtration on Gab with the Lie filtration on K×.
K
2. This is a general phenomenon, if the Galois group is a p-adic Lie group then the upper filtration and
the Lie filtration are “the same”.
Definition 1.19. Thecyclotomiccharacterχ :G →Z× isgivenbytheconditionthatg(ζ =ζχcycl(g).
cycl K p pn pn
Alternatively, χ can be obtained by lifting I →Iab ∼=O× N−K→/Qp Z× to G .
cycl K K K p K
1.5 Galois cohomology
1.5.1 Continuous cohomology
Definition 1.20. Let G be a (pro)finite group and M a topological group with a continuous G-action. Set
H0(G,M)=MG
H1(G,M)={f :G→M continuous|f(gh)=f(g)g(f(h))}/∼
where f ∼h if for some m∈M one has h(g)=mf(g)g(m)−1.
Remark 7. If M is abelian then Hi(G,M)=RiMG is the right derived functor as usual.
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1.5.2 Inflation-restriction sequence
Theorem 1.21. Let H ⊂G be a normal subgroup of a profinite group and let M be a topological group with
G action. Then one has an “exact” sequence
1→H1(G/H,MH)→H1(G,M)→H1(H,M)G/H
where exactness is categorical.
Remark 8. If M is an abelian group this follows from the usual 5-term exact sequence obtained from the
Hochschild-Serre spectral sequence.
1.5.3 Examples
Proposition 1.22. 1. If G is procyclic generated by g then
H0(G,M)=Mg
H1(G,M)=M/(g−1)M
2. (Hilbert 90) If L/K is finite then
H1(G ,L×)=0
L/K
H1(G ,L)=0
L/K
H1(G ,GL(n,L))=0
L/K
H1(G ,M (L))=0
L/K n×n
Lecture 3
2012-01-09
2 C -representations
p
2.1 The field C
p
Definition 2.1. For a p-adic field K let C =K(cid:98). If K ⊂Q write C =C .
K p K p
Proposition 2.2. 1. C (cid:54)=Q , i.e., Q is not complete.
p p p
2. C is algebraically closed.
p
Proof. 1. See problem set 2.
Choose a roots of unity, such that a ∈ Qur, a ∈ Q (a ) and [Q (a ) : Q (a )] > n. For
n n p n−1 p n p n p n−1
∞
(cid:88)
example could take a = ζ where q (cid:54)= p is a prime. Let α = a pn ∈ C and assume that
n q(n!)2 n p
n=1
m
(cid:88)
α ∈ Q . Let m = [Q (α) : Q ] and let α = a pn. Choose a Galois extension M/Q containing
p p p m n p
n=1
α,α and a .
m m
Since [M : Qp(am−1)] ≥ [Qp(am) : Qp(am−1)] > m one may find σ1,...,σm+1 ∈ GM/Qp(am−1) such
that σ (a ) are all distinct.
i m
Clearly v (α−α )≥m+1 and thus for all i one has v (σ (α)−σ (α ))≥m+1. Also, for i(cid:54)=j we
p m p i i m
have v (σ (α )−σ (α )) = v (σ(a )−σ (a ))+m. Since a is the root of a polynomial which is
p i m j m p m j m m
separable mod p, it follows that σ (a )(cid:54)∼=σ (a ) (mod p) and so v (σ(a )−σ (a ))=0.
i m j m p m j m
6