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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 deformations 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 ofBöckle,[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 deformations 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 Böckle 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 presented 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 homomorphism 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 conditions: • 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 paragraphs, 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 characteristic 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 ramified. 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