Lifting $N$-dimensional Galois representations to characteristic zero

0.34 MB·English

by  Jayanta Manoharmayum

LIFTING N-DIMENSIONAL GALOIS REPRESENTATIONS TO
CHARACTERISTIC ZERO
3
1 JAYANTAMANOHARMAYUM
0
2
Abstract. Let F be a number field, let N ≥ 3 be an integer, and let k

n be a finite field of characteristic ℓ. We show that if ρ : GF −→ GLN(k) is
a a continuous representation with image of ρ containing SLN(k) then, under
J moderate conditions atprimesdividingℓ∞,thereisacontinuous representa-

9 tion ρ : GF −→ GLN(W(k)) unramified outside finitely many primes with
2 ρ∼ρmodℓ.

]
T
N
1. Introduction
.
h
t LetF beanumberfield. Supposewearegivenacontinuousℓ-adicrepresentation
a
m ρ : GF −→ GLN(Qℓ) unramified outside finitely many places. The representation
ρ then takes values in a finite extension of Qℓ, and on reducing modulo ℓ a stable
[
lattice one gets a continuous representation ρ : GF −→ GLN(Fℓ), unique up to

1 semi-simplification. Conversely, one can ask if a given mod ℓ representation ρ :
v GF −→GLN(Fℓ)is necessarilythe reductionofanℓ-adicrepresentation. Thiswas
3 answered in the affirmative when N = 2 by Ramakrishna in [8]. In this article,
0
we generalise the method of Ramakrishna, loc. cit., to N ≥ 3 and provide an
1
answer to the finding characteristic zero lifts when the image of ρ and the residue
7
. characteristic ℓ are ‘big’.
1
Before we describe the main result of this article, we recall that if ρ : G −→
0 F
3 GLN(A) is a representation then adρ (resp. ad0ρ) is the A[GF]-module consisting
1 of N ×N matrices over A (resp. N ×N matrices over A with trace 0) with the
: action of g ∈G on a matrix M given by ρ(g)Mρ(g)−1. Let’s also recall that the
v F
i representationρ:GF −→GLN(A)aboveissaidtobetotallyevenifthe projective
X
image of the decomposition group at each infinite place of F is trivial.
r
a Main Theorem. Fix an integer N ≥3. Let k be a finite field of characteristic ℓ,
and let ρ : G −→ GL (k) be a continuous representation of the absolute Galois
F N
group of a number field F. Let W := W(k) denote the Witt ring of k, and fix a
continuouscharacterχ:G −→W× liftingthedeterminantofρ(i.e. χ (mod ℓ)=
F
detρ). Assume that:
(1) The image of ρ contains SL (k);
N
(2) ρ is not totally even;
(3) If v is a place of F lying above ℓ then H0 G ,ad0ρ(1) =(0).
Fv
Supposethatℓ>N3[F:Q]N. Therethenexistsaglob(cid:0)al deformatio(cid:1)n condition D with
determinant χ for ρ such that the universal deformation ring for type D deforma-
tions of ρ is a power series ring over W in at least N −2 variables. In particular,
2000 Mathematics Subject Classification. 11F80.
1
2 JAYANTAMANOHARMAYUM
there is a continuous representation ρ : G −→GL (W) with determinant χ sat-
F N
isfying the following properties:
• ρ (mod ℓ)∼ρ; and,
• ρ is unramified outside finitely many primes.
We can remove the local hypothesis at ℓ and say more when the number field is
QandN =3. Moreprecisely,letρ:G −→GL (k)satisfythefirsttwoconditions
Q 3
of the main theorem (so ρ is odd and its image contains SL (k)). Then ρ has a
3
lifting to GL (W(k)) whenever ℓ≥13 or ℓ=7 and the fixed field ofad0ρ does not
3
contain cos(2π/7). See Theorem 6.2.
Essentially, the claim made above is that a residual Galois representation with
bigimage(includingtheassumptionthatℓislarge)andgoodpropertiesatℓadmits
characteristic zero liftings. We follow Mazur’s development of deformation theory
as presented in [6]; a brief working recall of the main definitions is given in section
2.
The basic organisational principle underlying our approach is a beautiful result
ofB¨ockle,[1,Theorem4.2],relatingthestructureofauniversaldeformationringto
itslocal(uni)versalcomponents. ForaprecisestatementseeTheorem2.2insection
2.2. The problem thus becomes one of finding a global deformation condition with
smoothlocalcomponentsandtrivialdualSelmergroup. Itisperhapsworthnoting
here that the two requirements are not completely independent of each other (as
can be seen from the discussion in section 2.2). Ramakrishna’s great insight, in
the GL case, is to show how to reduce the size of the dual Selmer group by a
2
clever tweaking of the global deformation condition at some primes. We extend
this strategy.
There are two key ingredients in being able to make such an extension. Firstly,
we prove a cohomological result which gives conditions under which a subspace
of H1(G ,M) can be distinguished by its restriction at a prime. This provides
F
us with a collection of primes where an adjustment of the local condition can
result in a smaller dual Selmer group. Secondly, we need to produce enough local
deformationsfortherestrictionofρtoalocaldecompositiongroupataprimev ∤ℓ.
TherearecomplicationswhentheresiduecharacteristicofF isrelativelysmall(for
v
instance,whenthe residue characteristicis notbiggerthanN), andwe avoidthese
byassuming[F (ζ ):F ]≥3N. (See Theorem4.3.) The conditionℓ>N3[F:Q]N is
v ℓ v
an easy—but not an economic—bound that allows us to avoid local complications
at small primes for general N, ℓ.
Whilethe hypothesisatprimesaboveℓensuresthatwedonothavetodealwith
the more difficult problem of studying local deformations at ℓ, it does still cover
a wide range of examples. Note that the hypothesis at a prime v|ℓ is equivalent
to the assumption that the only G -equivariant homomorphism from ρ to ρ(1) is
Fv
the zero map. The exceptions can be easily classified for small N, and we do so
for the case when N = 3 and F = Q. We do not attempt to put any geometric
condition as the representations we are looking at might not even have the right
duality property (to link up with automorphic forms).
A similargeneralisationofRamakrishna’slifting technique to GL wasalsoob-
N
tained by Hamblen, [5], about the same time when an earlierversionof this article
was first prepared. Even so, we hope that this article still carries an interest. For
one, the results are different (Hamblen assumes the ground field to be Q and uses
LIFTING N-DIMENSIONAL GALOIS REPRESENTATIONS TO CHARACTERISTIC ZERO 3
different local conditions). Additionally, we hope that the study of local defor-
mations presented here, in particular the existence of smooth deformations, has
independentmerit. Althoughsomeofthe localanalysisalsoappearsin[3], there is
a difference in approach(for instance in the study of tamely ramified deformations
and also in the role of tensor product of deformations).
Notation 1.1. The ℓ-adic cyclotomic character is always denoted by ω and ω is
the mod ℓ-cyclotomic character. The term ‘prime’ on its own always indicates
a finite prime except when the context makes it clear that we are also including
infinte primes. If F is a number field, we assume we are given fixed embeddings
F ֒→ F for each prime v (including the infinite ones). If F is unramified at the
v
prime v we shall view Frob as element of G via the embedding F ֒→F . If A is
v F v
a topologicalring and ρ:G −→GL (A) is a continuous representation, we shall
F N
denotetherestrictionofρtoadecompositiongroupatv byρ .Weshallfrequently
v
use H∗(F,M) to denote H∗(G ,M). The group of unramified cohomology classes
F
at a prime is indicated by the presence of a subscript (as in H∗ ).
nr
IfkisafinitefieldthentheWittringofkwillbedenotedbyW(k)andx∈W(k)
denotes the Teichmu¨ller lift of x ∈ k. A CNL W(k)-algebra, or simply a CNL
algebra if the finite field k is clear, is shorthand for a complete, Noetherian, local
b
algebra with residue field k. If χ (resp. ρ) is a W(k) valued character (resp.
homomorphism) then we will use the same letters for their extension to a CNL
W(k)-algebra.
2. Preliminaries
We now give a brief summary of deformation theory and discuss some of the
key tools used in studying global deformation conditions. Aside from setting out
notation, we hope that the discussion in this section (following Theorem 2.2 in
particular) will make transparent the basic argument and structure of this article.
2.1. Deformation conditions in general. We begin with a brief summary of
what a deformation condition means since, for the most part, we shall be involved
in checking that the properties we specify at a local decomposition group give
a deformation condition. We shall follow §23, §26 of [6], except for some minor
adjustments.
Let Π be a profinite group satisfying the “finiteness at ℓ” property of Mazur
(§1 of [6]). For our purposes, a representationof Π is a continuous homomorphism
ρ:Π−→GL (A) whereA is atopologicalring. The underlying freeA-module on
N
which Π acts will be denoted by V(ρ). Given two representations
ρ :Π−→GL (A), ρ :Π−→GL (B)
A N B N
andamorphismf :A−→B in the relevantcategory,we saythatρ is alift ofρ
A B
if fρ =ρ .
A B
If ρ : Π −→ GL (A), ρ : Π −→ GL (A) are two representations then
1 n 2 m
Hom(V(ρ ),V(ρ )), or just simply Hom(ρ ,ρ ), is shorthand for the A[Π]-module
1 2 1 2
of A-linear maps from V(ρ ) to V(ρ ). As a representation Hom(ρ ,ρ ) can
1 2 1 2
be described as the group of m × n matrices over A with Π action given by
(g,M) −→ ρ (g)Mρ (g)−1. We shall take ρ ⊗ρ : G −→ GL (A) to mean
2 1 1 2 F mn
the representation(gotten fromV(ρ )⊗V(ρ ))expressedwith respectto the basis
1 2
v ⊗w ,...,v ⊗w ,...,v ⊗w ,...,v ⊗w wherev ,...,v andw ,...,w are
1 1 1 m n 1 n m 1 n 1 m
4 JAYANTAMANOHARMAYUM
the basesforρ andρ respectively. Note thatHom(ρ ,ρ )is naturallyisomorphic
1 2 1 2
to ρ∗⊗ρ where ρ∗ is the dual representation for ρ .
1 2 1 1
Let Rep (Π;k) denote the following category:
N
• Objects are pairs (A,ρ ) where A is a CNL W(k)-algebra and ρ :Π−→
A A
GL (A) is a representation;
N
• A morphism from (A,ρ ) to (B,ρ ) is a pair (f,M) where f :A−→B is
A B
a morphism of local rings and M ∈GL (B) satisfies fρ =Mρ M−1.
N A B
Givena representationρ:Π−→GL (k), a deformation condition D for ρ is a full
N
subcategory D⊆Rep (Π;k) satisfying the following properties:
N
(DC0) (k,ρ)∈D, and if (A,ρ )∈D then ρ∼ρ mod m .
A A A
(DC1) If(A,ρ )isanobjectinD and(f,M):(A,ρ )−→(B,ρ )isamorphism,
A A B
then (B,ρ ) is also in D.
B
(DC2) Given a cartesian diagram
A× B −−π−B−→ B
C
πA β
 α 
A −−−−→ C
y y
of Artinian CNL algebras A,B,C with β small, an object (A× B,ρ) of
C
Rep (Π;k) is in D if and only if (A,π ρ),(B,π ρ) are in D.
N A B
We say that ρ : Π −→ GL (A), or (A,ρ), is of type D if (A,ρ) is in D. If χ :
N
Π −→ W× is a character, we say that D has determinant χ if detρ = χ for any
(A,ρ)∈D. The deformation condition D is said to be smooth if for any surjection
f : A −→ B and an object (B,ρ ) of type D, there is an object (A,ρ ) in D
B A
such that fρ = ρ . It is sufficient to verify the smoothness condition for small
A B
extensionsonly. ThetangentspaceofD willbedenotedbyTD,andwillbeviewed
as a k-subspace of H1(Π,adρ) (it is a subspace of H1(Π,ad0ρ) if the determinant
is fixed).
In practice, conditions (DC0), (DC1), and the only if part of condition (DC2),
will almost always be immediate. If D is a deformation condition for ρ : Π −→
GL (k), the functor
N
D(A):={type D liftings ρ:Π−→GL (A) of ρ}/strict equivalence
N
is nearly representable. If D is smooth then the (uni)versal deformation ring is a
power series ring.
Ourobjectiveis toproduce(uni)versaldeformationringswhicharepowerseries
rings. In view of the following lemma, one can make use of extension of scalars to
produce such (uni) versal deformation rings.
Lemma 2.1. Let k ⊂ k be finite fields of characteristic ℓ, and let ρ : Π −→
0 1 0
GL (k ) be a representation. Denote by ρ : Π −→ GL (k ) the extension of
n 0 1 n 1
scalars of ρ to GL (k ).
0 n 1
Given a deformation condition D ⊆ Rep (Π;k ), let D be the full subcat-
1 n 1 0
egory of Rep (Π;k ) consisting of those objects (A,ρ) ∈ Rep (Π;k ) such that
n 0 n 0
A⊗ W(k ),ρ⊗W(k ) ∈D . Then:
W(k0) 1 1 1
(1) D is a deformation condition for ρ , and dim TD =dim TD .
(cid:0) 0 (cid:1) 0 k0 0 k1 1
(2) Let R ,R be the (uni)versal deformation rings of type D ,D . Then there
0 1 0 1
is an isomorphism R −→R ⊗ W(k ). In particular, if R is a power
1 0 W(k0) 1 1
series ring then so is R .
0
LIFTING N-DIMENSIONAL GALOIS REPRESENTATIONS TO CHARACTERISTIC ZERO 5
Proof. Checking that D is a deformation condition is straightforward. Extension
0
of scalars give a natural isomorphism between H1(Π,adρ )⊗k and H1(Π,adρ ).
0 1 1
Thus there is a subspace L ⊆ H1(Π,adρ ) such that L⊗k = TD . One then
0 1 1
checks that L has to be the tangent space for D .
0
Forthesecondpart,thereisasurjectionR −→R ⊗W(k ).Sincetheextension
1 0 1
W(k )/W(k )issmooth,thetangentspaceforR ⊗W(k )hasthesamedimension
1 0 0 1
as the tangent space for R . Hence the surjection is an isomorphism. (cid:3)
0
2.2. Global deformations. NowletF beanumberfieldandletk beafinitefield
ofcharacteristicℓ. Fixanabsolutelyirreduciblerepresentationρ:G −→GL (k)
F N
and a character χ:G −→W× such that χ (mod ℓ)=detρ.
F
Informally, a global deformation condition specifies that we consider liftings of
ρ : G −→ GL (k) with prescribed local behaviour. More precisely: Suppose
F N
we are given, for each prime v of F, a deformation condition D for ρ| with
v v
determinant χ. Furthermore, we require that the deformation condition D is
v
unramified for almost all primes v. The global deformation condition {D } with
v
determinantχforρ isthenthe fullsubcategoryofRep (G ;k)consistingofthose
N F
objects (A,ρ)∈Rep (G ;k) such that detρ=χ and (A,ρ| )∈D for all v.
N F v v
For a global deformation condition D with determinant χ for ρ, we shall denote
the localconditionatv by D (so D ={D }). We define the ramificationset Σ(D)
v v
to be the finite set consisting of those primes v of F where D is not unramified,
v
primeslyingaboveℓand∞,andprimeswhereρandχareramified. ThusDispre-
cisely a deformation condition for ρ| with prescribed local components
Gal(FΣ(D)/F)
(cf. §26 of [6]). The tangent space for D is the Selmer group
H1 F,ad0ρ =ker H1(G ,ad0ρ)−→ H1(F ,ad0ρ)/TD .
{TDv} F v v
ThedualSelme(cid:0)rgroupf(cid:1)orDis(cid:16)definedasfollows. FYoreachprimevofF th(cid:17)epairing
ad0ρ×ad0(1)ρ−→k(1) obtained by taking trace induces a perfect pairing
H1 F ,ad0ρ ×H1 F ,ad0ρ(1) −→H2(F ,k(1)).
v v v
Let TD⊥ ⊆ H1 F(cid:0),ad0ρ(1)(cid:1) be th(cid:0)e annihilator(cid:1)of TD under the above pairing.
v v v
The dual Selmer group H1 F,ad0ρ(1) is then determined by the local con-
(cid:0) {TD(cid:1)⊥}
v
ditions {TD⊥} i.e.
v (cid:0) (cid:1)
H1 F,ad0ρ(1) :=ker H1(G ,ad0ρ(1))−→ H1(F ,ad0ρ(1))/TD⊥ .
{TD⊥} F v v
v
While the(cid:0)tangent sp(cid:1)ace for (cid:16)D is a very difficult obYject to get a handle on,(cid:17)re-
markablya quantitative comparisionwith the dual Selmer groupis possible by the
following formula of Wiles (Theorem 8.6.20 in [7]):
(2.1) dimH1 F,ad0ρ −dimH1 F,ad0ρ(1)
{TDv} {TDv⊥}
(cid:0) (cid:1) = (cid:0) dimTD(cid:1) −dimH0(F ,ad0ρ) .
v v
primes v≤∞
X (cid:0) (cid:1)
We now describe a beautiful result of B¨ockle which allows one to relate the
global (uni)versal deformation ring in terms of local deformation rings. Let ρ,
χ and D be as above. For each prime v, set n := dimTD . The (uni)versal
v v
deformation ring R for type D deformations of ρ| then has a presentation
R ∼=W(k)[[T ,...v,T ]]/J .vNote that J =(0) ifGvFv6∈Σ(D).
v v,1 v,nv v v
6 JAYANTAMANOHARMAYUM
Fix a presentation
W(k)[[T ,...,T ]]/J, n:=H1 (F,ad0ρ),
1 n {TDv}
forthe(uni)versalglobaldeformationringRfortypeD deformationsofρ. Restric-
tion of the (uni)versal deformation to a decomposition group at v induces a map
R −→ R which can be then lifted to a map α : W(k)[[T ]] −→ W(k)[[T ]] of
v v v,i i
local rings. (Of course α , and even R −→R, might not be unique at all.)
v v
Theorem 2.2. (Bo¨ckle, Theorem 4.2 of [1]) With notation as in the preceding
paragraphs, the ideal J is generated by the images α J together with at most
v v
dimH1 (F,ad0ρ(1)) other elements. Thus
{TD⊥}
v
(2.2) gen(J)≤ gen(J )+dimH1 (F,ad0ρ(1))
v {TD⊥}
v
v∈XΣ(D)
wheregen(J)(resp. gen(J ))istheminimalnumberofelementsrequiredtogenerate
v
the ideal J (resp. J ).
v
Toproveourmaintheorem,wemakesurethatourglobaldeformationcondition
has smooth localconditions and trivialdual Selmer group. If we can do that, then
(2.2) ensures that the global deformation ring has trivial ideal of relations and so
is smooth. The question now is how to get to such nice global deformations.
The first stepis to constructa globaldeformationproblemD with smoothlocal
deformation conditions. By (2.2) the number of global relations is then bounded
bythe dimensionofthe dualSelmergroup. The next, andcriticalstep, is to tweak
oneofthelocalconditionsD atsomeprimesothatthenewdeformationcondition
v
has smaller dual Selmer group. We shall show that this can be done provided
(2.3) dimH1 F,ad0ρ ≥N −2+dimH1 F,ad0ρ(1) .
{TDv} {TDv⊥}
We shall prove the neces(cid:0)sary res(cid:1)ults from Galois cohomolo(cid:0)gy in Secti(cid:1)on 3.
Note that by Wiles formula (2.1), the above inequality will fail if the local de-
formationconditionsare‘small’. Toensurethisdoesn’thappen, wemakesurethat
D is smooth in dimH0(F ,ad0ρ) variables at primes not dividing ℓ. The required
v v
constructions are carried out in Section 4; the precise statement we need is pre-
sented in Theorem 4.3. Given these local conditions, the hypotheses at ℓ and ∞
allows us to ensure that (2.3) is satisfied.
3. Galois cohomology
Throughout this section, K/F is a finite Galois extension of number fields with
Galois group G:=Gal(K/F) and k be a finite extension of Fℓ.
If M is a k[G]-module and ξ ∈H1(G , M) then the restriction of ξ to G is a
F K
group homomorphism. We denote by K(ξ) the field through which this homomor-
phism factorises. Note that the extensionK(ξ)/F is Galois. For ξ ∈H1(G , M),
i F
i=1,...,n,thecompositumofK(ξ ),...,K(ξ )willbedenotedbyK(ξ ,...,ξ ).
1 n 1 n
3.1. In this subsectionM is a finite k[G]-module satisfying the followingtwo con-
ditions:
• M is a simple Fℓ[GF]-module with EndFℓ[GF](M)=k;
• H1(G, M)=0.
Lemma 3.1. Let 06=ξ ∈H1(G , M). Then:
F
LIFTING N-DIMENSIONAL GALOIS REPRESENTATIONS TO CHARACTERISTIC ZERO 7
(a) The restriction ξ :Gal(K(ξ)/K)−→M is an isomorphism of G-modules.
(b) If L is a Galois extension of F with K ⊆ M, then either K(ξ) ⊆ L or
K(ξ)∩L=K.
Proof. TheimagesofGal(K(ξ)/K)andGal(K(ξ)/(K(ξ)∩L)underξaresubspaces
of M stable under the action of G. The lemma follows as M is simple. (cid:3)
Proposition 3.2. If ψ ,ψ ,...,ψ are n linearly independent classes in the k-
1 2 n
vector space H1(G , M), then K(ψ ),K(ψ ),...,K(ψ ) are linearly disjoint over
F 1 2 n
K.
Proof. We first do the case n = 2. If K(ψ ) and K(ψ ) are not linearly disjoint
1 2
over K, then by the above lemma K(ψ )=K(ψ ). The composite
1 2
M −ψ−1−−→1 Gal(K(ψ )/K)=Gal(K(ψ )/K)−ψ−→2 M
1 2
is a G-module automorphism of M. Since k is the endomorphism ring of M, ψ
1
and ψ are linearly dependent—a contradiction.
2
Suppose now that the proposition holds for n−1. Let K(ψ ,...,ψ ) be the
1 n−1
compositumof K(ψ ),...,K(ψ ). By the inductive hypothesis, we haveidentifica-
1 n
tions
Gal(K(ψ ,...,ψ )/K) −→ Gal(K(ψ )/K)×···×Gal(K(ψ )/K)
1 n−1 1 n−1
↓
M ×···×M
of G-modules.
Let V be the k-subspace of H1(G , M) spanned by ψ ,...,ψ and let E be
F 1 n−1
the set Galois extensions E/F with K ⊆ E ⊆ K(ψ ,...,ψ ) and Gal(E/K)
1 n−1
isomorphic to M as G modules. We claim that the map P(V) −→ E given by
ψ −→ K(ψ) is a bijection. That the map is an injection follows from the case
n = 2 of the proposition. Now elements of E correspond to non-trivial G module
homomorphisms from M ×···×M to M. Now
Hom (M ×···×M, M)∼=Hom (M, M)×···×Hom (M, M)
G G G
∼=k×···×k.
It follows that |P(V)|=|K|, and this establishes the claim.
We now show that K(ψ ) and K(ψ ,...,ψ ) are linearly disjoint over K.
n 1 n−1
If not, then Lemma 3.1 implies that K(ψ ) ∈ E, and so by the claim K(ψ ) =
n n
K(a ψ +···+a ψ ) for some a ,...,a ∈k. Appealing to the case n=2 of
1 1 n−1 n−1 1 n
the proposition, we see that ψ is a linear combination of ψ ,...,ψ —which is
n 1 n−1
a contradiction. (cid:3)
3.2. Now let M be an absolutely irreducible k[G]-module with H1(G, M) = 0,
and let g ∈G be a fixed element of G which acts semi-simply on M. We denote by
Mg the kernelofmultiplicationby g−1onM. Note that wehavea decomposition
M =Mg⊕(g−1)M.
Fixalsoanon-trivialsubgroupL⊆M invariantunderG withminimaldimen-
F
sion as an Fℓ-vector space. It is then straightforward to check that L is simple,
that k contains EndFℓ[GF](L)=:k′ (say), and that M ∼=L⊗k′ k. Further, we have
Mg =Lg⊗k′ k and (g−1)M =(g−1)L⊗k′ k.
8 JAYANTAMANOHARMAYUM
Proposition 3.3. With assumptions and notations as in the previous two para-
graphs, letV beafinitedimensional k-subspaceofH1(G , M).IfdimMg ≥dimV
F
we can find a lift g˜∈G of g such that the restriction map
F
V ֒→H1(G , M)−→H1(hgi, M)
F
is injective.
e
Proof. Set n:=dimV. Since H1(GF, M)∼=H1(GF, L)⊗k′ k, we can find:
• a basis ξ ,...,ξ of V,
1 n
• mlinearlyindependentcocylesψ ,...,ψ inthek′-vectorspaceH1(G , L)
1 m F
withm≥nandsuchthatξ :=ψ + a ψ forsomea ∈k, i=1,...,n.
i i ij j ij
j>n
X
Fix a lift g′ ∈ G of g. We can identify H1(hg′i, M) with Mg. For ease of
F
notation, we set
K :=K(ψ ,j >n), and K :=K(ψ ,ψ ,j >n), i=1,...,n.
0 j i i j
By Proposition 3.2, the extensions K , i=1,...,n are linearly disjoint over K .
i 0
Foreach1≤i≤n,thecocyleξ restrictstoψ onK .Sinceψ (Gal(K /K )=L
i i 0 i i 0
andξ (xg′)=ψ (x)+ξ (g′)for anyx∈Gal(K /K ), we see thatthe k-subspaceof
i i i i 0
M generated by ψ (xg′) is M.
i
Weclaimthatwecanfindx ∈Gal(K /K ), 1≤i≤n,suchthatξ (x g′),...,ξ (x g′)
i i 0 1 1 n n
generate an n-dimensional subspace of M/(g −1)M. To see this, first pick x ∈
1
Gal(K /K )suchthatξ (x g′)isnon-trivialwhenprojectedtoM/(g−1)M.Having
1 0 1 1
foundx ∈Gal(K /K ), i=1,...,jwithj <nandsuchthatξ (x g′),...,ξ (x g′)
i i 0 1 1 j j
generateaj-dimensionalsubspaceofM/(g−1)M wecanfindanx ∈Gal(K /K )
j+1 j+1 0
with the property that ξ (x g′) does not lie in the subspace of M spanned by
j+1 j+1
ξ (x g′),...,ξ (x g′) and (g−1)M (possible as this latter subspace has dimension
1 i j j
j+dim (g−1)M <dim M).
k k
Finally,usingProposition3.2,wecanfindxintheGaloisgroupofK whichacts
0
as x on each extension K /K . Set g = xg′. Then as ξ (g),...,ξ (g) generate an
i i 0 1 n
n-dimensionalsubspaceofM/(g−1)M,weseethattheimagesofξ whenrestricted
i
to H1(hgi, M) are linearly independeent. e e (cid:3)
Theorem 3.4. Let M ,...,M be n inequivalent, absolutely irreducible k[G] mod-
1 n
ules witheH1(G, M )=0, 1≤i≤n. We assume that we are given a place v of F
i
and k-subspaces V ⊆H1(G , M ) with the following properties:
i F i
• M ⊕···⊕M is unramified at v, and that Frob acts semi-simply on each
1 n v
M ;
i
• dimV ≤dimH1 (F , M ) for i=1,...,n.
i nr v i
Under the above assumptions, we can find infinitely many places w such that:
• M ⊕···⊕M is unramified at w and the images of Frob ,Frob in G are
1 n w v
the same;
• Any cohomology class in V is unramified at w;
i
• The restriction map V1⊕···⊕Vn −→Hn1r(Fw, M1)⊕···⊕Hn1r(Fw, Mn)
is injective.
Proof. Denote by K(V ) the splitting field for V overK, and by K(V ,...,V ) the
i i 1 n
compositum of K(V ). The extensions K(V ) are linearly disjoint over K because
i i
Gal(K(V )/K) is isomorphic to a subgroup of M .
i i
LIFTING N-DIMENSIONAL GALOIS REPRESENTATIONS TO CHARACTERISTIC ZERO 9
Takeg ∈GtobeanelementwhichFrob liftsandletg′ ∈Gal(K(V ,...,V )/F)
v 1 n
be a lift of g. By Proposition 3.3, we can find x ∈ Gal(K(V )/K) such that
i i
V −→ H1(hx g′i, M ) is injective. Using disjointness of the K(V )’s, we can find
i i i i
an x∈Gal(K(V ,...,V )/K) such that x acts on K(V ) as x . By the Chebotarev
1 n i i
density theorem, we can then find a place w of F lifting xg′ and unramified in
K(V ,...,V ). It is now immediate such a w satisfies the properties asked for. (cid:3)
1 n
4. Local deformation conditions
Throughout this section k is a finite field of characteristic ℓ and p is a prime
different from ℓ. We shall look at deformation conditions for a finite extension
of Qp. In particular, our objective is to construct examples of local deformation
conditions which admit a large (uni)versaldeformationring. The precise nature of
what large should mean is the content of the following definition.
Definition 4.1. Let F be a finite extension of Qp and let ρ : GF −→GLN(k) be
a representation. We saythat a deformationD for ρ is well-behaved if D is smooth
and dim TD=dim H0(G ,adρ).
F
Example 4.2. If ρ is unramified then the class of unramified liftings is a well-
behaved deformation condition. The unrestricted deformation condition is well-
behaved if H2(G ,adρ)=(0).
F
We can now state our main result asserting the existence of well-behaveddefor-
mation conditions.
Theorem 4.3. Let F be a finite extension of Qp, let k be a finite field of char-
acteristic ℓ 6= p, and let ρ : G −→ GL (k) be a representation. Assume that
F N
all irreducible components occurring in the semi-simplification of ρ are absolutely
irreducible. If p≤ N and ρ is wildly ramified assume that [F(ζ ): F] ≥ 3N where
ℓ
ζ is an ℓ-th root of unity. Then the following hold:
ℓ
(a) There exists a well-behaved deformation condition D.
(b) Suppose χ : G −→ W× is a character lifting detρ. Assume that N,ℓ
F
are co-prime. Then liftings of type D and determinant χ is a smooth
deformation condition for ρ and the dimension of its tangent is equal to
dimH0(G ,ad0ρ).
F
To construct a well-behaved deformation condition D as claimed (and also to
outline the structure of this section), we proceed as follows:
• We would like to build up D from well-behaved deformation conditions
for some decomposition of ρ. In section 4.1 we show that a good way
of decomposing ρ is to make sure that the basic blocks have no common
irreducible components, even after taking Tate twists. (See relation 4.1.)
• The blocks can then be analysed separately. There are essentially three
cases we need to consider.
– Firstly, the case when a given residual representation is tamely ram-
ified. The deformation condition in this case is to obtained by speci-
fying a Jordan–Holder decomposition for a generator of tame inertia.
See section 4.2.
– Theresidualrepresentationisatensorproductoftwosmallerrepresen-
tations. In section 4.3 we study when we can construct the candidate
well-behaved deformation by using tensor products.
10 JAYANTAMANOHARMAYUM
– The residualrepresentationis induced, in which case we try to induce
a known well-behaved deformation condition. This is done in section
4.4
• Finally,weverifythatthe hypothesesofTheorem4.3guaranteeapplicabil-
ity of the preceding steps and complete the proof in section 4.5.
The second part of Theorem 4.3 is straightforward, and we deal with it right
away:
Proof of Theorem 4.3 (b). We need only check smoothness, and for that if suffices
to check that any deformation ρ : G −→ GL (A) of type D can be twisted
F N
to a deformation with determinant χ. If ψ : G −→ A× is a character and we
F
want χ = det(ψρ), then ψN = χdetρ−1. We can find such a character ψ because
χdetρ−1 :G −→1+m and
F A
1+m −x−→−−x→N 1+m
A A
is an isomorphism. (cid:3)
4.1. Products of deformation conditions. Let F be a finite extension of Qp.
We assume we are given representations ρ : G −→ GL (k), i = 1,...,n satist-
i F di
fying
(4.1) Hom ρ ,ρ (r) =(0)
k[GF] i j
for i 6= j,r ∈ Z, and a deformation c(cid:0)ondition(cid:1)Fi for ρi, i = 1,...,n. Set ρ :=
ρ ⊕···⊕ρ and N :=d +...+d .
1 n 1 n
We shall say that a representation ρ : G −→ GL (A) is of type F := F ⊕
F N 1
···⊕F ifρ∼ρ ⊕···⊕ρ with(A,ρ )∈F . We denotebyF the fullsubcategory
n 1 n i i
of Rep (G ;k) consisting of objects (A,ρ) with ρ of type F ⊕···⊕F . We then
N F 1 n
have the following
Theorem 4.4. F is a deformation condition for ρ. The natural map
((A,ρ )∈F )n −→(A,ρ ⊕···⊕ρ )
i i i=1 1 n
induces an isomorphism of tangent spaces
TF ∼=TF ⊕···⊕TF ,
1 n
and F is well-behaved if each F is well-behaved.
i
Theorem 4.4 is an immediate consequence of the following proposition:
Proposition 4.5. Let R be a CNL algebra, and let ρ:G −→GL (R) be a lift of
F N
ρ.Wethen have, uptostrictlyequivalence, auniquedecomposition ρ∼=ρ ⊕···⊕ρ
1 n
where ρ :G −→GL (R) is a lift of ρ .
i F di i
The proof of Proposition 4.5 relies on there being no cohomological relations
betweenliftsofρ andρ wheni6=j. Moreprecisely,weneedthefollowinglemma:
i j
Lemma 4.6. Let ∗=0, 1 or 2.
(1) If i6=j then H∗ G ,Hom(ρ ,ρ ) =(0) if i6=j. Consequently, we have
F i j
H∗(GF,Hom(ρ,ρ))∼=(cid:0) H∗(GF,Hom(ρ(cid:1)1,ρ1)⊕···⊕H∗(GF,Hom(ρn,ρn)).
(2) Let A be an Artinian CNL algebra, and let ρ : G −→ GL (A), ρ :
i F di j
G −→GL (A) be lifts of ρ ,ρ ,i6=j. Then
F dj i j
H∗(G ,Hom(ρ ,ρ ))=(0).
F i j