POTENTIAL AUTOMORPHY AND CHANGE OF WEIGHT.

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1 POTENTIAL AUTOMORPHY AND CHANGE OF WEIGHT. THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR Abstract. We show that a strongy irreducibe, odd, essentiay sef-dua, reguar, weaky compatibe system of -adic representations of the absoute Gaois group of a totay rea fied is potentiay automorphic. Aong the way we prove a new automorphy ifting theorem for -adic representations where we impose a new condition at, which we ca potentia diagonaizabiity. This seems to be a more fexibe condition than has been previousy considered, and aows for substantia change of weight in our automorphy ifting resut Mathematics Subject Cassification. 11F33. The second author was partiay supported by NSF grant DMS , and the fourth author was partiay supported by NSF grant DMS

2 2 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR 1. Introduction Suppose that F and M are number fieds, that S is a finite set of primes of F and that n is a positive integer. By a weaky compatibe system of n-dimensiona -adic representations of G F defined over M and unramified outside S we sha mean a famiy of continuous semi-simpe representations r λ : G F GL n (M λ ) with the foowing properties. If v / S is a finite pace of F, then for a λ not dividing the residue characteristic of v, the representation r λ is unramified at v and the characteristic poynomia of r λ (Frob v ) ies in M[X] and is independent of λ. Each representation r λ is de Rham at a paces above the residue characteristic of λ, and in fact crystaine at any pace v S which divides the residue characteristic of λ. For each embedding τ : F M the τ-hodge-tate numbers of r λ are independent of λ. In this paper we prove the foowing theorem (see Theorem 7.2.1). Theorem A. Let {r λ } be a weaky compatibe system of n-dimensiona -adic representations of G F defined over M and unramified outside S, where for simpicity we assume that M contains the image of each embedding F M. Suppose that {r λ } satisfies the foowing properties. (1) Strong Irreducibiity r λ is irreducibe after restriction to any open subgroup of G F. (2) Reguarity For each embedding τ : F M the representation r λ has n distinct τ-hodge-tate numbers. (3) Odd Essentia Sef-Duaity F is totay rea; and either each r λ factors through a map to GSp n (M λ ) with an odd mutipier character; or each r λ factors through a map to GO n (M λ ) with an even mutipier character. Moreover in either case the mutipier characters form a weaky compatibe system. Then there is a finite, Gaois, totay rea extension over which a the r λ s become automorphic. In particuar for any embedding ı : M C the partia L-function L S (ı{r λ }, s) converges in some right haf pane and has meromorphic continuation to the whoe compex pane. This is not the first paper to prove potentia automorphy resuts for compatibe systems of -adic representations of dimension greater than 2, see for exampe [HSBT06], [BLGHT09], [BLGG09]. However previous attempts ony appied to very specific, though we known, exampes (e.g. symmetric powers of the Tate modues of eiptic curves) and one had to expoit specia properties of these exampes. We beieve this is the first genera potentia automorphy theorem in dimension greater than 2, and we are hopefu that it can be appied to many exampes. One can probaby weaken the assumption that F is totay rea to simpy assume that F is CM and one might hope to weaken the strong irreducibiity assumption, but other than that we do not see how to improve much on this theorem using current methods. As one exampe appication, suppose that K is a finite set of positive integers such that the 2 #K possibe partia sums of eements of K are a distinct. For each k K

3 POTENTIAL AUTOMORPHY 3 et f k be an eiptic moduar newform of weight k+1 without compex mutipication. Then the #K-fod tensor product of the -adic representations associated to the f k is potentiay automorphic and the #K-fod product L-function for the f k has meromorphic continuation to the whoe compex pane. (See Coroary ) The proof of theorem A foows famiiar ines. One works with r λ for one suitaby chosen λ. One finds a motive X over some finite Gaois totay rea extension F /F which reaizes the reduction r λ in its mod cohomoogy and whose mod cohomoogy is induced from a character. One tries to argue that by automorphic induction the mod cohomoogy is automorphic over F, hence by an automorphy ifting theorem the -adic cohomoogy is automorphic over F, hence tautoogicay the mod cohomoogy is automorphic over F and hence, finay, by another automorphy ifting theorem r λ is automorphic over F. To find X one uses a emma of Moret-Baiy [MB89], [GPR95] and for this one needs a famiy of motives with distinct Hodge numbers, which has arge monodromy. Griffiths transversaity tes us that this wi ony be possibe if the Hodge numbers of the motives are consecutive (e.g 0, 1, 2,..., n 1). Thus the -adic cohomoogy of X wi be automorphic of a different weight (infinitesima character) than r λ and the second potentia automorphy theorem needs to incorporate a change of weight. In addition it seems that we can in genera ony expect to find X over an extension F /F which is highy ramified at. Thus our second potentia automorphy theorem needs to work over a base which is highy ramified at. These two, reated probems were the principa difficuties we faced. The origina higher dimensiona automorphy ifting theorems (see [CHT08], [Tay08]) coud hande neither of these difficuties. In the ordinary case one of us (D.G.) proved an automorphy ifting theorem that uses Hida theory and some new oca cacuations to hande both of these probems (see [Ger09]). This has had important appications, but its appicabiity is sti severey imited because we don t know how to prove that many compatibe systems of -adic representations are ordinary infinitey often. The main innovation of this paper is a new automorphy ifting theorem that handes both these probems in significant generaity. One of our key ideas is to introduce the the notion of a crystaine representation ρ of the absoute Gaois group of a oca fied K being potentiay diagonaizabe: ρ is potentiay diagonaizabe if there is a finite extension K /K such that ρ GK ies on the same irreducibe component of the universa crystaine ifting ring of ρ GK (with fixed Hodge-Tate numbers) as a sum of characters ifting ρ GK. (We remark that this does not depend on the choice of integra mode for ρ.) Ordinary crystaine representations are potentiay diagonaizabe, as are crystaine representations in the Fontaine-Laffaie range (i.e. over an absoutey unramified base and with Hodge-Tate numbers in the range [0, 2]). Potentiay diagonaizabiity is aso preserved under restriction to the absoute Gaois group of a finite extension. In this sense they behave better than crystaine representations in the Fontaine-Laffaie range which require the ground fied to be absoutey unramified. Finay potentiay diagonaizabe representations are perfecty suited to our method of proving automorphy ifting theorems that aow for a change of weight. To prove such a theorem we empoy Harris tensor product trick (see [Har07]), which was first empoyed in connection with change of weight in [BLGG09]. However the freedom that potentia diagonaizabiity gives us to make highy ramified

4 4 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR base changes in the non-ordinary case means that this method becomes more powerfu. More precisey, suppose that r λ is potentiay diagonaizabe, and that r 0 is a potentiay diagonaizabe, automorphic ift of r λ (with possiby different Hodge- Tate numbers to r λ ). In fact making a finite soube (but ikey highy ramified above ) base change we can assume they are diagonaizabe, i.e. we can take K = K in the definition of potentia diagonaizabiity. We choose a cycic extension M/F of degree n in which each prime above spits competey, and two characters θ and θ 0 of such that θ = θ 0, the restriction of Ind G F θ to an inertia group at a prime v reaizes the diagona point on the same component of the universa crystaine ifting ring of r λ GFv as r λ GFv, and the restriction of Ind G F θ 0 to an inertia group at a prime v reaizes the diagona point on the same component of the universa crystaine ifting ring of r λ GFv as r 0 GFv. Then r 0 Ind G F θ is automorphic and has the same reduction as r λ Ind G F θ 0. Moreover the restrictions of these two representations to the decomposition group at a prime v ie on the same component of the universa crystaine ifting ring of (r λ Ind G F θ 0 ) GFv. This is enough for the usua Tayor-Wies-Kisin argument to prove that r λ Ind G F θ 0 is aso automorphic, from which we can deduce (as in [BLGHT09]) the automorphy of r λ. For exampe we prove the foowing automorphy ifting theorem which we hope might be of independent interest. Theorem B. Let F be an imaginary CM fied with maxima totay rea subfied F + and et c denote the non-trivia eement of Ga (F/F + ). Suppose that is odd and et n Z 1 with > n 2. Let r : G F GL n (Q ) be a continuous irreducibe representation and et r denote the semi-simpification of the reduction of r. Aso et µ : G F + Q be a continuous character. Suppose that r and µ enjoy the foowing properties: (1) r = r c ɛ n 1 µ. (2) µ(c v ) is independent of v. (3) r ramifies at ony finitey many primes. (4) r Gv is potentiay diagonaizabe (and so in particuar potentiay crystaine) for a v. (5) The image r(g F (ζ )) is n-big and [(F ) ker ad r (ζ ) : (F ) ker ad r ] > n. (6) There is a RAECSDC automorphic representation (π, χ) of GL n (A F ) such that (r, µ) = (r,ı (π), r,ı (µ)) and π is ı-ordinary. Then (r, µ) is automorphic. (See Theorem 5.1.1, and aso section 3.1 for the definition of any terminoogy which may be unfamiiar.)

5 POTENTIAL AUTOMORPHY 5 It seems to us to be a very interesting question to carify the ubiquity of potentia diagonaizabiity. Coud every crystaine representation be potentiay diagonaizabe? (We have no reason to beieve this, but we know of no counterexampe.) A the main innovations of this paper are in sections 2.3 and 5. The rest of the paper, whie sometimes technicay daunting, wi not surprise an expert. We now expain the structure of the paper. In section 2 we study the notion of (potentia) diagonaisabiity, and the reated definitions for -adic representations of G K, K a finite extension of Q p (p ), in some generaity. In section 3 we generaise a number of resuts from our earier papers (particuary [GG09] and [BLGG09]) to work with essentiay conjugate sef-dua representations, rather than just conjugate sef-dua representations; this is for the most part just a matter of carefuy keeping track of twists. In section 4 we prove a variant of the character buiding emma of [BLGG10] (itsef a variant on those of [BLGHT09] and [BLGG09]), adapted to the case of CM extensions of CM fieds. In section 5 we prove the automorphy ifting theorem described above. In section 6 we appy these resuts to potentia automorphy. First in section 6.1 we repeat arguments from [BLGHT09] using the Dwork famiy to prove a potentia automorphy resut in for sympectic mod representations of dimension n with mutipier ɛ 1 n. In section 6.2, again foowing [BLGHT09], we use Harris tensor product trick to extend this to orthogona representations and remove the condition on the mutipier. Finay in section 6.3 we combine this with our automorphy ifting theorem to obtain our main potentia automorphy theorem for a singe - adic representation. In section 7 we appy this theorem to compatibe famiies and deduce some consequences and exampes, incuding theorem A. This paper was written at the same time as [BLGG10] and there was considerabe cross fertiization. It woud aso have been impossibe without Harris tensor product trick and it is a peasure to acknowedge our debt to him Notation. We write a matrix transposes on the eft; so t A is the transpose of A. Let g n denote the space of n n matrices with the adjoint action of GL n and et g 0 n denote the subspace of trace zero matrices. If R is a oca ring we write m R for the maxima idea of R. If Γ is a profinite group then Γ ab wi denote its maxima abeian quotient by a cosed subgroup. If ρ : Γ GL n (Q ) is a continuous homomorphism then we wi et ρ : Γ GL n (F ) denote the semi-simpification of its reduction, which is we defined up to conjugacy. If M is a fied, we et M denote its agebraic cosure and denote its absoute Gaois group. We wi use ζ n to denote a primitive n th -root of 1. Let ɛ denote the -adic cycotomic character and ɛ its reduction moduo. We wi aso et ω : µ 1 Z denote the Teichmuer ift of ɛ. If K is a finite extension of Q p for some p, we write K nr for its maxima unramified extension; I K for the inertia subgroup of G K ; Frob K G K /I K for the geometric Frobenius; and W K for the Wei group. We wi write Art K : K WK ab for the Artin map normaized to send uniformizers to geometric Frobenius eements. We wi et rec K be the oca Langands correspondence of [HT01], so that if π is an irreducibe compex admissibe representation of GL n (K), then rec K (π) is a Wei- Deigne representation of the Wei group W K. We wi write rec for rec K when the choice of K is cear. If (r, N) is a Wei-Deigne representation of W K we wi

6 6 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR write (r, N) F ss for its Frobenius semisimpification. If ρ is a continuous representation of G K over Q with p then we wi write WD(ρ) for the corresponding Wei-Deigne representation of W K. (See for instance section 1 of [TY07].) By a Steinberg representation of GL n (K) we wi mean a representation Sp n (ψ) (in the notation of section 1.3 of [HT01]) where ψ is an unramified character of K. If ρ is a continuous de Rham representation of G K over Q p then we wi write WD(ρ) for the corresponding Wei-Deigne representation of W K, and if τ : K Q p is a continuous embedding of fieds then we wi write HT τ (ρ) for the mutiset of Hodge-Tate numbers of ρ with respect to τ. Thus HT τ (ρ) is a mutiset of dim ρ integers. In fact if W is a de Rham representation of G K over Q and if τ : K Q G then the mutiset HT τ (W ) contains i with mutipicity dim Q (W τ,k K(i)) K. Thus for exampe HT τ (ɛ ) = { 1}. If K = R or C or if K is a CM fied, we wi et c denote compex conjugation, a we defined automorphism of K. If K is a number fied and v is a pace of K then we wi et [c v ] denote the conjugacy cass in G K consisting of compex conjugations associated to v. If K/Q is a finite extension we wi write Art K = Art Kv : A K /K (K ) 0 G ab K. v If v is a finite pace of K we wi write k(v) for its residue fied and Frob v for Frob Kv. If K /K is a quadratic extension of number fieds we wi denote by δ K /K the nontrivia character of A K /K N K /KA K.

7 POTENTIAL AUTOMORPHY 7 Contents 1. Introduction Loca Prerequisites Automorphy Lifting and Changing The Leve and Weight Yet more character buiding A New Automorphy Lifting Theorem Potentia Automorphy Appications. 37 References Loca Prerequisites In this section we wi reca and generaise some notions from sections 3.3 and 3.4 of [BLGG09] (note that in particuar Lemma means that we do not have to concern ourseves with particuar choices of equivaences between isomorphic residua representations). Fix a rationa prime and an isomorphism ı : Q C. Let Γ denote a topoogicay finitey generated topoogica group. Let O denote the ring of integers of a finite extension of Q in Q. Let λ denote the maxima idea of O and et F = O/λ. If ρ : Γ GL n (F) is a continuous homomorphism we wi denote by ρ : Γ GL n (R O,ρ) the universa ifting ring of ρ to a compete noetherian oca O-agebra with residue fied F. (We impose no equivaence condition on ifts other than equaity.) We wi write R ρ Q for R O,ρ OQ. Note that if O O are two such rings of integers then R O,ρ OO = R O,ρ, and so, as the notation suggests, the ring R ρ Q does not depend on the choice of ring of integers O. Lemma Let H denote the subgroup of GL n (O) consisting of eements which reduce moduo the maxima idea to an eement that centraizes the image of ρ. Then H acts naturay on RO,ρ on the right. This action fixes each irreducibe component of Spec RO,ρ [1/]. Proof. The action of h H is via the map RO,ρ R O,ρ aong which the universa ifting ρ univ pushes forward to hρ univ h 1. Suppose that h H and that O is the ring of integers of a finite extension of the fraction fied of O in Q. Suppose that ρ : Γ GL n (O ) ifts ρ. Then hρh 1 aso ifts ρ. Let O s, t denote the agebra of power series over O with coefficients tending to zero. Set A = O s, t /(s det(t1 n + (1 t)h) 1), a compete topoogica domain with the λ -adic topoogy. We have a continuous representation ρ = (t1 n + (1 t)h)ρ(t1 n + (1 t)h) 1 : Γ GL n (A). Let A 0 denote the cosed subagebra of A generated by the the matrix entries of eements of the image of ρ and give it the subspace topoogy. Then ρ : Γ GL n (A 0 ),

8 8 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR and ρ mod λ = ρ, and ρ pushes forward to ρ (resp. hρh 1 ) under the continuous homomorphism A 0 O induced by t 1 (resp. t 0). We wi show that A 0 is a compete, noetherian oca O-agebra with residue fied F. It wi foow that there is a natura map RO,ρ A0 through which the maps RO,ρ O corresponding to ρ and hρh 1 both factor. As A 0 is a domain (being a sub-ring of A) we concude that the points corresponding to ρ and hρh 1 ie on the same irreducibe component of Spec RO,ρ [1/]. As any irreducibe component contains an O -point which ies on no other irreducibe component for some O as above (because such points are Zariski dense in Spec RO,ρ [1/], see emma 2.6 of [Tay08]), we see that the emma foows. It remains to show that A 0 is a compete, noetherian, oca O-agebra with residue fied F. Let γ 1,..., γ r denote topoogica generators of Γ. Write ρ(γ i ) = a i + b i, where a i GL n (O) ifts ρ(γ i ) and where b i M n n (λ A). Then A 0 is the cosure of the O-subagebra of A generated by the entries of the b i. As these entries are topoogicay nipotent in A we get a continuous O-agebra homomorphism O[[X ijk ]] i=1,...,r; j,k=1,...,n A which sends X ijk to the (j, k)-entry of b i. Let J denote the kerne. As O[[X ijk ]]/J is compact (and A is Hausdorff) this map is a topoogica isomorphism of O[[X ijk ]]/J with its image in A and this image is cosed. Thus the image is just A 0 and we have O[[X ijk ]]/J A 0, so that A 0 is indeed a compete, noetherian, oca O-agebra with residue fied F. Let p be second rationa prime and K/Q p be a finite extension. For i = 1, 2, et be a continuous representation p. ρ i : G K GL n (O Q ) Suppose that p. We say that ρ 1 connects to ρ 2, which we denote ρ 1 ρ 2, if and ony if the reduction ρ 1 = ρ 1 mod m Q is equivaent to the reduction ρ 2 = ρ 2 mod m Q, and ρ 1 and ρ 2 define points on the same irreducibe component of Spec (R ρ 1 Q ). We say that ρ 1 strongy connects to ρ 2, which we write ρ 1 ρ 2, if ρ 1 ρ 2 and ρ 1 ies on a unique irreducibe component of Spec (R ρ 1 Q ). We make the foowing remarks. (1) By Lemma the reations ρ 1 ρ 2 and ρ 1 ρ 2 do not depend on the equivaence chosen between the reductions ρ 1 and ρ 2, nor on the GL n (O Q )- conjugacy cass of ρ 1 or ρ 2. (2) Connects is a symmetric reationship, but strongy connects may not be. (3) Strongy connects is a transitive reationship, whereas connects may not be. (4) If ρ 1 ρ 2 and ρ 2 ρ 3 then ρ 1 ρ 3.

9 POTENTIAL AUTOMORPHY 9 (5) Write WD(ρ i ) = (r i, N i ). If ρ 1 ρ 2 then r i IK = r2 IK. If ρ 1 ρ 2 and ρ 2 ρ 1 then (r 1 IK, N 1 ) = (r 2 IK, N 2 ). (See [Cho09].) (6) If ρ 1 and ρ 2 have the same reduction and if ρ 1 is unramified, then ρ 1 ρ 2 if and ony if ρ 2 is unramified. (7) If K /K is a finite extension and ρ 1 ρ 2 then ρ 1 GK ρ 2 GK. (8) If ρ 1 ρ 2 and ρ 1 ρ 2 then ρ 1 ρ 1 ρ 2 ρ 2 and ρ 1 ρ 1 ρ 2 ρ 2 and ρ 1 ρ 2. (9) If µ : G K Q is a continuous character and if ρ 1 ρ 2 then ρ 1 ρ 2 and ρ 1 µ ρ 2 µ. (10) If µ : G K Q is a continuous unramified character with µ = 1 then ρ 1 ρ 1 µ. (11) Suppose that ρ 1 is semisimpe, then ρ 1 ρ ss 1. (This is proved in the same way as emma of [Ger09].) Lemma Keep the above notation, incuding the assumption that p. Suppose that ρ 1 ρ 2. (1) If ρ 1 defines a formay smooth point of Spec (R ρ 1 Q ) then ρ 1 ρ 2. This is true if and ony if H 0 (G K, (ad ρ 1 )(1)) = (0). (2) If ıwd(ρ 1 ) F ss = rec K (π) for some generic irreducibe smooth representation π of GL n (K) over C then ρ 1 ρ 2. Proof: For the first part note that a formay smooth point of Spec (Rρ 1 Q ) can ony ie on one irreducibe component. Forma smoothness foows from the vanishing of H 2 (G K, ad ρ 1 ), which by Tate duaity is equivaent to the vanishing of H 0 (G K, (ad ρ 1 )(1)). The scheme Spec (Rρ 1 Q ) is equidimensiona of dimension n 2 (see [Cho09], or Theorem of [Gee06]). On the other hand the tangent space at ρ 1 has dimension dim Z 1 (G K, ad ρ 1 ) = n 2 dim H 0 (G K, ad ρ 1 ) + dim H 1 (G K, ad ρ 1 ) = n 2 + dim H 0 (G K, (ad ρ 1 )(1)). Thus Spec (Rρ 1 Q ) is formay smooth at ρ 1 if and ony if H 0 (G K, (ad ρ 1 )(1)). For the second part write π = Sp s1 (π 1 ) Sp st (π t ) for some supercuspida representations π i of GL ni (K) and positive integers s i, n i with s i n i = n. (We are using the notation of [HT01].) Then ρ 1 has a fitration with graded pieces ρ 1,i satisfying ıwd(ρ 1,i ) F ss = rec K (Sp si (π i )), possiby after reordering the i s. Thus (ad ρ 1 )(1) has a fitration with graded pieces Hom (ρ 1,i, ρ 1,j (1)). If this were nonzero then π i = πj det m for some max{1, 1+s j s i } m s j. Thus (π i, s i ) and (π j, s j ) are inked contradicting the fact that π is generic (see page 36 of [HT01]). Important convention: Suppose that F is a goba fied and that r : G F GL n (Q ) is a continuous representation with irreducibe reduction r. In this case there is mode r : G F GL n (O Q ) of r, which is unique up to GL n (O Q )- conjugation. If v p is a pace of F we write r GFv ρ 2 (resp. r GFv ρ 2, resp. ρ 1 r GFv ) to mean r GFv ρ 2 (resp. r GFv ρ 2, resp. ρ 1 r GFv ) = p. Now suppose that = p and that, for each continuous embedding K Q, the image is contained in the fied of fractions of O. Let {H τ } be a coection of n eement mutisets of integers parametrized by τ Hom Q (K, Q ). Then RO,ρ has a unique quotient R O,ρ,{H τ },cris which is reduced

10 10 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR and without -torsion and such that a Q point of RO,ρ factors through R O,ρ,{H τ },cris if and ony if it corresponds to a representation ρ : G K GL n (Q ) which is crystaine and has HT τ (ρ) = H τ for a τ : K Q. We wi write Rρ,{H τ },cris Q for RO,ρ,{H τ },cris O Q. This definition is independent of the choice of O. The scheme Spec (Rρ,{H τ },cris Q ) is formay smooth. (See [Kis08].) We say that ρ 1 connects to ρ 2, which we denote ρ 1 ρ 2, if and ony if the reduction ρ 1 = ρ 1 mod m Q is equivaent to the reduction ρ 2 = ρ 2 mod m Q ; ρ 1 and ρ 2 are both crystaine; for each τ : K Q we have HT τ (ρ 1 ) = HT τ (ρ 2 ); and ρ 1 and ρ 2 define points on the same irreducibe component of the scheme Spec (Rρ 1,{HT τ (ρ Q 1)},cris ). Note the foowing: (1) By the proof of emma we see that the reation ρ 1 ρ 2 does not depend on the equivaence chosen between the reductions ρ 1 and ρ 2, nor on the GL n (O Q )-conjugacy cass of ρ 1 or ρ 2. (2) is symmetric and transitive. (3) If K /K is a finite extension and ρ 1 ρ 2 then ρ 1 GK ρ 2 GK. (4) If ρ 1 ρ 2 and ρ 1 ρ 2 then ρ 1 ρ 1 ρ 2 ρ 2 and ρ 1 ρ 1 ρ 2 ρ 2 and ρ 1 ρ 2. (5) If µ : G K Q is a continuous unramified character with µ = 1 then ρ 1 ρ 1 µ. (6) If ρ 1 is semisimpe, then ρ 1 ρ ss 1. (This is proved in the same way as emma of [Ger09].) We wi ca a crystaine representation ρ : G K GL n (O Q ) diagonaizabe if it connects to some representation χ 1 χ n with χ i : G K O. We wi ca Q a representation ρ 1 : G K GL n (O Q ) potentiay diagonaizabe if there is a finite extension K /K such that ρ 1 GK is diagonaizabe. Note that if K /K is a finite extension and ρ 1 is diagonaizabe (resp. potentiay diagonaizabe) then ρ 1 GK is diagonaizabe (resp. potentiay diagonaizabe). It seems to us an interesting, and important, question to determine which potentiay crystaine representations are potentiay diagonaizabe. As far as we know they coud a be. Lemma If ρ 1 and ρ 2 are conjugate in GL n (Q ) then ρ 1 is potentiay diagonaizabe if and ony if ρ 2 is potentiay diagonaizabe. Thus we can speak of a representation ρ : G K GL n (Q ) being potentiay diagonaizabe without needing to specify an invariant attice. Proof. If ρ 1 and ρ 2 are conjugate by an eement of GL n (O Q ) then after passing to a finite extension over which ρ 1 = ρ 2 = 1 we see that ρ 1 ρ 2. Thus we may suppose that ρ 1 = gρ 2 g 1 where g = diag(d 1,..., d n ) with d i Q satisfying d n d n 1... d 1. Choosing O Q arge enough we may assume that ρ 1 and ρ 2 are defined over O and that d 1,..., d n O. Repacing K by a finite extension we may aso assume that ρ 2 1 mod d 1 /d n, in which case we aso have ρ 1 = 1. Consider the compete topoogica domain A = O t 1, s 1, t 2, s 2,..., t n 1, s n 1 /(s 1 t 1 (d 1 /d 2 ),..., s n 1 t n 1 (d n 1 /d n )).

11 POTENTIAL AUTOMORPHY 11 Let g = diag(t 1... t n 1, t 2... t n 1,..., t n 1, 1) and et ρ = gρ 2 g 1. If j i then the (i, j) entry of ρ(σ) is t i... t j 1 times the (i, j) entry of ρ 2 (σ). If i j then the (i, j) entry of ρ(σ) is s j... s i 1 d i /d j times the (i, j) entry of ρ 2 (σ). Thus we see that ρ : G K GL n (A) is a continuous homomorphism. The speciaization under t i 1 for a i is ρ 2. The speciaization under s i 1 for a i is ρ 1. As in the proof of emma we concude that ρ 1 ρ 2, and we are done. We wi estabish some cases of (potentia) diagonaisabiity beow, but first we must reca some resuts from the theory of Fontaine and Laffaie [FL82], normaized as in section of [CHT08]. Assume that K/Q is unramified and denote its ring of integers by O K. Let MF O denote the category of finite O K Z O-modues M together with a decreasing fitration Fi i M by O K Z O-submodues which are O K -direct summands with Fi 0 M = M and Fi 1 M = {0}; and Frob 1 p 1-inear maps Φ i : Fi i M M with Φ i Fi i+1 M = Φ i+1 and i Φi Fi i M = M. Let Rep O (G K ) denote the category of finite O-modues with a continuous G K - action. There is an exact, fuy faithfu, covariant functor of O-inear categories G K : MF O Rep O (G K ). The essentia image of G K is cosed under taking sub-objects and quotients. If M is an object of MF O, then the ength of M as an O-modue is [K : Q ] times the ength of G K (M) as an O-modue. Let F denote the residue fied of O and et MF F denote the fu subcategory of MF O consisting of objects kied by the maxima idea λ of O and et Rep F (G K ) denote the category of finite F-modues with a continuous G K -action. Then G K restricts to a functor MF F Rep F (G K ). If M is an object of MF F and τ is an embedding K Q, we et FL τ (M) denote the mutiset of integers i such that gr i M OK Z O,τ 1 O {0} and i is counted with mutipicity equa to the F- dimension of this space. If M is an -torsion free object of MF O then G K (M) Z Q is crystaine and for every continuous embedding τ : K Q we have HT τ (G K (M) Z Q ) = FL τ (M O F). Moreover, if Λ is a G K -invariant attice in a crystaine representation V of G K with a its Hodge-Tate numbers in the range [0, 2] then Λ is in the image of G K. (See [FL82].) Lemma Let K/Q be unramified. Let M denote an object of MF F together with a fitration M = M 0 M 1 M n 1 M n = (0) by MF F -subobjects such that M i /M i+1 has F-rank [K : Q ] for i = 0,..., n 1. Then we can find an object M of MF O which is -torsion free together with a fitration by MF O -subobjects and an isomorphism M = M 0 M 1 M n 1 M n = (0) M O F = M under which M i O F maps isomorphicay to M i for a i.

12 12 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR Proof. M has an F basis e i,τ for i = 1,..., n and τ Hom (K, Q ) such that the residue fied k K of K acts on e i,τ via τ; M j is spanned over F by the e i,τ for i > j; and for each j there is a subset Ω j {1,..., n} Hom (K, Q ) such that Fi j M is spanned over F by the e i,τ for (i, τ) Ω j. Then we define M to be the free O-modue with basis e i,τ for i = 1,..., n and τ Hom (K, Q ). We et O K act on e i,τ via τ; we define M j to the sub O-modue generated by the e i,τ with i > j; and we define Fi j M to be the O-submodue spanned by the e i,τ for (i, τ) Ω j. We define Φ j : Fi j M M by reverse induction on j. If we have defined Φ j+1 we define Φ j as foows: If (i, τ) Ω j+1 then Φ j e i,τ = Φ j+1 e i,τ. If (i, τ) Ω j Ω j+1 then Φ j e i,τ is chosen to be an O-inear combination of the e i,τ Frob p for i i which ifts Φ j e i,τ. It foows from Nakayama s emma that M is an object of MF O, and then it is easy to verify that it has the desired properties. We can now state and prove our potentia diagonaizabiity criteria. Lemma Keep the above notation, incuding the assumption = p. Suppose that ρ 1 : G K GL n (Q ) is a crystaine representation. (1) If ρ 1 has a G K invariant fitration with one dimensiona graded pieces, in particuar if it is ordinary (see section 3.1 beow), then ρ 1 is potentiay diagonaizabe. (2) If K/Q is unramified and if a the Hodge-Tate numbers of ρ 1 ie in a range of the form [a, a+ 2] for some fixed a, then ρ 1 is potentiay diagonaizabe. Proof: After passing to a finite extension so that ρ 1 becomes trivia, the first part foows from item 6 of the first numbered ist of this section. For the second we may assume (by twisting) that a = 0. Note that every irreducibe subquotient of ρ 1 IK is trivia on wid inertia and hence one dimensiona. Choose a finite unramified extension K /K such that ρ 1 (G K ) = ρ 1 (I K ). Then ρ 1 GK has a G K invariant fitration with 1-dimensiona graded pieces. From Lemma (and the discussion just proceeding it) we see that ρ 1 GK has a crystaine ift ρ 2 with the same Hodge-Tate numbers as ρ 1 GK which aso has a G K - invariant fitration with one dimensiona graded pieces. It foows from section of [CHT08] that ρ 1 GK ρ 2. From the first part of this emma we see that ρ 2 is potentiay diagonaizabe. Hence ρ 1 is aso potentiay diagonaizabe. Important convention: Suppose that F is a goba fied and that r : G F GL n (Q ) is a continuous representation with irreducibe reduction r. In this case there is mode r : G F GL n (O Q ) of r, which is unique up to GL n (O Q )- conjugation. If v is a pace of F we write r GFv ρ 2 to mean r GFv ρ 2. We wi aso say that r is (potentiay) diagonaizabe to mean that r is.

13 POTENTIAL AUTOMORPHY Automorphy Lifting and Changing The Leve and Weight Terminoogy. Continue to fix a rationa prime and an isomorphism ı : Q C. Suppose that F is a number fied and χ : A F /F C is a continuous character. If there exists a Z Hom (F,C) such that χ (F ) : x (τx) aτ 0 τ Hom (F,C) we wi ca χ agebraic. In that case we can associate to χ a de Rham, continuous character r,ı (χ) : Ga (F /F ) Q as at the start of section 1 of [BLGHT09]. If τ : F Q reca that HT τ (r,ı (χ)) = { a ı τ }. We now reca from [CHT08] and [BLGHT09] the notions of RAESDC and RAECSDC automorphic representations. In fact, it wi be convenient for us to work with a sight variant of these definitions, where we keep track of the character which occurs in the essentia (conjugate) sef-duaity. Let F be an imaginary CM rea fied with maxima totay rea subfied F +. By a RAECSDC (reguar, agebraic, essentiay conjugate sef dua, cuspida) automorphic representation of GL n (A F ) we mean a pair (π, χ) where π is a cuspida automorphic representation of GL n (A F ) such that π has the same infinitesima character as some irreducibe agebraic representation of the restriction of scaars from F to Q of GL n, χ : A F + /(F + ) C is a continuous character such that χ v ( 1) is independent of v, and π = π c (χ N F/F + det). Now et F be a totay rea fied. By a RAESDC (reguar, agebraic, essentiay sef dua, cuspida) automorphic representation π of GL n (A F ) we mean a pair (π, χ) where π is a cuspida automorphic representation of GL n (A F ) such that π has the same infinitesima character as some irreducibe agebraic representation of the restriction of scaars from F to Q of GL n, χ : A F /(F ) C is a continuous character such that χ v ( 1) independent of v, and π = π (χ det). If F is CM or totay rea we wi write (Z n ) Hom (F,C),+ for the set of a = (a τ,i ) (Z n ) Hom (F,C) satisfying a τ,1 a τ,n. If F /F is a finite extension we define a F (Z n ) Hom (F,C),+ by (a F ) τ,i = a τ F,i. Foowing [Shi09] we ca a sighty reguar if either n is odd; or if n is even and for some τ Hom (F, C) and for some odd integer i we have a τ,i > a τ,i+1. If a (Z n ) Hom (F,C),+, et Ξ a denote the irreducibe agebraic representation of

14 14 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR Hom (F,C) GLn which is the tensor product over τ of the irreducibe representations of GL n with highest weights a τ. We wi say that a RAESDC (resp. RAECSDC) automorphic representation π of GL n (A F ) has weight a if π has the same infinitesima character as Ξ a. We reca that to a RAESDC or RAECSDC representation (π, χ) of GL n (A F ) we can attach a continuous semi-simpe representation r,ı (π) : Ga (F /F ) GL n (Q ) with the properties described in Theorem 1.1 (resp. 1.2) of [BLGHT09]. In particuar r,ı (π) = r,ı (π) ɛ n 1 r,ı (χ) (resp. r,ı (π) = r,ı (π) c ɛ n 1 r,ı (χ) GF.) We aso remark (see [Shi09]) that if the weight of π is sighty reguar then for every pace v of F we have ıwd(r,ı (π) GFv ) F ss = rec(πv det (1 n)/2 v ). We wi ca representations which arise in this way for some π automorphic. We wi et r,ı (π) denote the semisimpification of the reduction of r,ı (π). We wi ca a pair (r, µ) consisting of a Gaois representation and an agebraic Gaois character automorphic if there is a RAESDC or RAECSDC representation (π, χ) such that (r, µ) = (r,ı (π), r,ı (χ)). We wi say that (r, µ) is automorphic of eve prime to if there is a RAESDC or RAECSDC representation (π, χ) such that (r, µ) = (r,ı (π), r,ı (χ)) and with π v unramified for a v. We ca a continuous representation r : Ga (F /F ) GL n (Q ) ordinary if for a v a prime of F the foowing conditions are satisfied: there is a Ga (F v /F v )-invariant decreasing fitration Fi i v on Q n such that for i = 1,..., n the graded piece gr i vq n is one dimensiona and Ga (F v /F v ) acts on it by a character χ v,i ; and there are integers b τ,i Z for τ Hom Q (F v, Q ) and i = 1,..., n and an open subgroup U F v such that (χ v,i Art Fv ) U (α) = τ:f v Q (τα) bτ,i and b τ,1 < b τ,2 < < b τ,n for a τ. We wi say that r is ordinary of weight a where a (Z n ) Hom (F,Q ),+ is defined by a τ,i = i n b i. If v is a pace of F we aso sometimes say that r Ga (F v/f v) is ordinary of weight a Fv, where a Fv denotes the eement of (Z n ) Hom Q (F v,q ),+ obtained by restricting a from Hom (F, Q ) to Hom Q (F v, Q ). In definition of [Ger09] Geraghty defines what it means for a reguar agebraic cuspida automorphic representation π of GL n (A F ) to be ı-ordinary. For our purposes the exact definition wi not be so important, rather a that wi matter are the foowing facts. We et (π, χ) denote a RAESDC or RAECSDC automorphic representation of GL n (A F ). (1) If π is ı-ordinary then r,ı (π) is ordinary.

15 POTENTIAL AUTOMORPHY 15 (2) If ψ is an agebraic character of A F /F then π is ı-ordinary if and ony if π ψ is ı-ordinary. (This foows directy from the definition.) (3) If π has weight 0 and if π v is Steinberg for a v then π is ı-ordinary. (See Lemma of [Ger09].) (4) If r,ı (π) is ordinary and if π v is unramified for a v then π is ı-ordinary. (See Lemma of [Ger09]) We remark that it is presumaby both true and provabe that π is ı-ordinary if and ony if r,ı (π) is ordinary, but to work out the detais here woud take us too far afied. Reca the foowing definition from [BLGHT09]. (See definition 7.2 of that paper.) If m is a positive integer, we wi ca a subgroup H GL n (F ) m-big if the foowing conditions are satisfied. H has no -power order quotient. H 0 (H, g 0 n(f )) = (0). H 1 (H, g 0 n(f )) = (0). For a irreducibe F [H]-submodues W of g n (F ) we can find h H and α F with the foowing properties: the eement α is a simpe root of the characteristic poynomia of h and if β is another root then α m β m ; and π h,α W i h,α (0). (Here i h,α denotes the incusion of the α-eigenspace of h into F n and π h,α denotes the h-equivariant projection of F n onto the α-eigenspace for h.) We wi use big as a synonym for 1-big. This is consistent with the definition of big in [CHT08]. Some important exampes of m-big subgroups are given in Lemmas 7.3 and 7.4 of [BLGHT09]. We add to it the foowing trivia observation. Lemma Let be a prime and m an integer coprime to. The subgroup {1} of GL 1 (F ) is m-big Automorphy ifting theorems. In this section we present two automorphy ifting theorems which sighty generaize resuts of [BLGG09] and [Ger09]. The ony extra generaity we aow is that we consider the essentiay conjugate sef-dua case rather than the conjugate sef-dua case. We simpy reduce the resuts we want to those in the iterature by a twisting argument. The first resut is a sight generaization of Theorem of [BLGG09]. It represents the natura output of the Tayor-Wies-Kisin method. Theorem Let F be an imaginary CM fied with maxima totay rea subfied F +. Suppose is odd and et n Z 1 with > n. Let r : G F GL n (Q ) be a continuous representation and et r denote the semi-simpification of its reduction. Aso et µ : G F + Q be a continuous homomorphism. Suppose that (r, µ) enjoys the foowing properties: (1) r = r c ɛ n 1 µ GF.

16 16 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR (2) µ(c v ) is independent of v. (3) The reduction r is absoutey irreducibe and r(g F (ζ )) GL n (F ) is big and (F ) ker ad r does not contain ζ. (4) There is a RAECSDC automorphic representation (π, χ) of GL n (A F ) with the foowing properties. (r, µ) = (r,ı (π), r,ı (χ)). π is unramified above. For a paces v of F at which π or r is ramified we have r,ı (π) GFv r GFv. For a paces v of F we have Then (r, µ) is automorphic. r,ı (π) GFv r GFv. We remark that by Lemma 2.2.1, if π has sighty reguar weight, then for a v we can repace the condition by the condition r,ı (π) GFv r GFv r,ı (π) GFv r GFv. Proof. Let S denote the set of primes of F + which either ie above, or above which π or r ramifies. For each v S choose once and for a a prime ṽ of F above v. Using Lemma 1.4 of [BLGHT09] it is enough to prove the theorem after repacing F by a quadratic CM extension which is ineary disjoint from F ker r (ζ ) over F and in which a the primes above S spit competey. Thus we may suppose that a primes in S spit in F. As π is unramified above, so is χ. By Lemma of [CHT08] we can find an agebraic character ψ : A F /F C which is unramified above and satisfies ψ N F/F + = χ N F/F +. Repacing (π, χ) by (π (ψ det), 1) and (r, µ) by (r r,ı (ψ), µ r,ı (χ) 1 ) we reduce the theorem to the case that χ = 1 and µ = 1, so that in particuar π is RACSDC in the terminoogy of [CHT08]. Note that by assumption r is crystaine at a paces v and thus µ GF being a constituent of (r c r ɛ n 1 ) is aso crystaine. Since a paces of F + above spit in F, we see that µ itsef is crystaine. There is an integer w such that for any prime v of F + we have (µ GF + Art Fv ) v O = N w F + F +,v v /Q. (In fact, as r,ı (π) GFv and r GFv have the same Hodge-Tate numbers, we know that w = 0, but we won t use this in the rest of the proof.) By Lemma of [CHT08] we can find a continuous character θ : G F Q such that µ GF = θθ c ; the reduction θ of θ is trivia; v S then θ is unramified at ṽ c ; θ is crystaine.

17 POTENTIAL AUTOMORPHY 17 (To appy Lemma of [CHT08] we take the S of that emma to be the set of primes of F above the set S of this emma. For v S we take ψṽc to be trivia and ψṽ = µ GF +.) Again repacing F by a finite soube CM extension in which a v the primes in S spit competey and appying Lemma 1.4 of [BLGHT09], we may suppose that θ is unramified outside S. Then we have the foowing observations. (r θ) = (r θ) c ɛ n 1. π and r θ are unramified outside S. If v S and v then and r,ı (π) GFṽ r,ı (π) GFṽc (r,ı(π) θ) GFṽc (r θ) G Fṽc, = r,ı (π) c G ɛ1 n Fṽc (r θ) c G ɛ1 n Fṽc = (r θ) GFṽ. (Because r,ı (π) GFṽc r, the ifting r GFṽc,ı(π) GFṽc ies on a unique irreducibe component of Spec (Rr GFṽc Q ). As r,ı (π) (r GFṽc,ı(π) θ) we deduce that r GFṽc,ı(π) (r GFṽc,ı(π) θ) GFṽc. This justifies the first of the above equations.) If v S and v then and r,ı (π) GFṽ r,ı (π) GFṽc r G Fṽc (r θ) G Fṽc, = r,ı (π) c G ɛ1 n Fṽc (r θ) c G ɛ1 n Fṽc = (r θ) GFṽ. Now repacing (r, µ) by (r θ, 1) we reduce the present theorem to Theorem of [BLGG09]. In some cases one can combine the argument with the eve changing method of [Tay08] and some Hida theory, to obtain a theorem with weaker hypotheses at v, but more stringent hypotheses at v. For instance, the foowing theorem can be deduced from Theorem of [Ger09]. Theorem (Geraghty). Let F be an imaginary CM fied with maxima totay rea subfied F +. Suppose that is odd and et n Z 1 with > n. Let r : G F GL n (Q ) be a continuous irreducibe representation and et r denote the semi-simpification of the reduction of r. Aso et µ : G F + Q be a continuous homomorphism. Suppose that (r, µ) enjoys the foowing properties: (1) r = r c ɛ n 1 µ. (2) µ(c v ) independent of v. (3) r ramifies at ony finitey many primes. (4) r is ordinary. (5) The image r(g F (ζ )) is big and (F ) ker ad r does not contain F (ζ ). (6) There is a RAECSDC automorphic representation (π, χ) of GL n (A F ) such that (r, µ) = (r,ı (π), r,ı (χ)) and π is ı-ordinary.

18 18 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR Then (r, µ) is automorphic. Proof. The proof is exacty anaogous to, but sighty easier than, the proof of Theorem 3.2.1, except we appea to Theorem of [Ger09] instead of Theorem of [BLGG09] Changing the eve and weight. One can combine Geraghty s Theorem (and its proof) with a method of Khare and Wintenberger [KW08] to deduce eve and weight changing theorems in the ordinary case. For instance Theorem beow is a sight generaization (from the conjugate sef-dua case to the essentiay conjugate sef-dua case ) of Theorem of [GG09]. Before proving it we need a generaization of Lemma of [CHT08]. Lemma Let F be an imaginary CM fied with maxima totay rea subfied F +. Suppose that F/F + is unramified at a finite paces. Let T be a non-empty finite set of primes of F + which spit in F. Let I be a set of embeddings F C such that I Ic is the set of a embeddings F C. For τ I et m τ be an integer. Suppose that χ : A F + /(F + ) C is an agebraic character such that χ is unramified outside T and such that χ v ( 1) is independent of v. Then there is an agebraic character unramified outside T such that and ψ : A F /F C ψ N F/F + = χ N F/F +, ψ F = τ I τ mτ (cτ) w mτ for some w. Proof. We know that χ ((F + ) ) 0 = τ I τ w for some integer w. Repacing χ by χδ F/F +, if need be, we may assume that this remains true on a of (F ) +. (This is possibe as δ F/F + is unramified at a finite paces.) Define ψ : (F ) (N F/F +A F )(ÔT F ) F C by h(n F/F +g)uγ χ(n F/F +g) τ(h) mτ (cτ)(h) w mτ τ I for h F and g A F and u (ÔT F ) and γ F. (Here ÔT F denotes v T ; v O F,v.) We wi show that this is we defined and that it is continuous with respect to the usua topoogy on A F. If this is so we can then extend it to a continuous character on A F with the desired properties. Suppose that h i (N F/F +g i )u i γ i 1 as i. We must show that χ(n F/F +g i ) τ I τ(h i ) mτ (cτ)(h i ) w mτ 1

19 POTENTIAL AUTOMORPHY 19 as i. If a F then χ(n F/F +a) = τ I τ(n F/F +a) mτ (cτ)(n F/F +a) w mτ and so we may assume that h i c h i = 1 for a i. Note that for i >> 0 the eement γ i / c γ i wi be an eement of O F with norm to F + equa to 1. Thus for i >> 0 the eement γ i / c γ i ies in the finite group of roots of unity in F. As it tends to 1 v-adicay for any v T we see that γ i (F + ) for i >> 0. Thus h 2 i = h i/ c h i 1 and u i / c u i 1 as i. Write h i = s i h i where s i {±1} Hom (F +,R) and where h i F satisfies h i 1 as i. For a prime v T define w i,v to be (u i,v + c u i,v )/2 = u i,v (1 + c u i,v /u i,v )/2 if this ies in O F +,v and 1 otherwise. Note that w i,v/u i,v 1 as i. If we set w i = (w i,v ) (ÔT F ) then we concude that w + i /u i 1 as i. Thus we may suppose that h i = s i {±1} Hom (F +,R) and that u i = w i (ÔT F ). In this case + h i, N F/F +g i, u i and γ i a ie in A F and + χ(n F/F +g i ) τ I as i as desired. τ(h i ) mτ (cτ)(h i ) w mτ = χ(h i (N F/F +g i )u i γ i ) 1 Theorem Let F be an imaginary CM fied with maxima totay rea subfied F +. Suppose that F/F + is unramified at a finite paces. Let n Z >1 be an integer with > n. Assume that the extension F/F + is spit at a paces dividing. Let S be a finite set of finite paces of F + which do not divide and which spit in F. For each v S choose a prime ṽ of F above v. Suppose that r : G F GL n (F ) is an irreducibe representation and that µ : G F + F is a continuous character which satisfy the foowing assumptions. (1) r = r c ɛ n 1 µ GF. (2) µ(c v ) is independent of v. (3) There is a RAECSDC automorphic representation (π, χ) of GL n (A F ) such that (r, µ) = (r,ı (π), r,ı (χ)), and π is ı-ordinary. Suppose π has weight b (Z n ) Hom (F,C),+ and set w = b τ,i + b cτ,n+1 i, which is independent of τ and i. (4) r ony ramifies at paces above S and paces above. (5) The image r(g F (ζ )) is big and F ker ad r does not contain ζ. For v S et C v denote an irreducibe component of Spec (R r GFṽ Q ). Finay suppose that a (Z n ) Hom (F,Q ),+ satisfies for v the restriction r GFv has an ordinary, crystaine ift of weight a Fv ; and a τ,i + a τ c,n+1 i = w for a τ and i. Then there is a RAECSDC automorphic representation (π, χ ) of GL n (A F ) such that

20 20 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR (r, µ) = (r,ı (π ), r,ı (χ )); π v is unramified for a v S. r,ı (π ) GFṽ defines a point on C v for a v S; r,ı (π ) is ordinary and crystaine; π has weight a. We remark that the assumption that F/F + is unramified at a finite paces is amost certainy not needed, but it woud take us too far afied to remove it in this paper. Proof. Note that χ ((F + ) ) 0 = N w F + /Q. The oca component π v of π is unramified at a primes outside S and the primes above, and hence the character χ N F/F + is unramified outside S and the primes above. As F/F + is unramified at a finite paces we concude that χ is unramified outside S and the primes above. By Lemma above we concude that there is an agebraic character ψ : A F /F C, which is unramified outside S and the primes above, satisfies ψ N F/F + = χ N F/F +, and ψ F = τ Hom (F,C) τ mτ, where m τ + m cτ = w for a τ. Now we simpy need to appy Theorem of [GG09] to ρ = r r,ı (ψ), and λ, where λ τ,i = a τ,i m τ. (See point 9 of the first numbered ist in section 2.2.) We remark that Theorem of [GG09] does not record the concusion that the ift is unramified away from S and, but the proof of that theorem gives this extra information. 4. Yet more character buiding. We prove an anaog of the character buiding Lemma of [BLGG10] in the case of a CM base fied. Those who are prepared to take such an extension on faith need ony famiiarize themseves with the statement of Lemma beow and can then move on to the next section. The arguments are actuay somewhat simper than those in [BLGG10] A bigness criterion. We begin with an anaog of Lemma of [BLGG09] for the case of an extension of CM fieds. Lemma Suppose that F is a CM fied, is a rationa prime, m is a positive integer not divisibe by, n is a positive integer with > 2n 2, r : G F GL n (F ) is a continuous mod Gaois representation, and M is a cycic CM extension of F of degree m such that: M is ineary disjoint from F ker r (ζ ) over F, and every prime v of F above is unramified in M. Suppose aso that θ : F is a continuous character. (1) Suppose [F ker ad r (ζ ) : F ker ad r ] > m. Then the fixed fied of the kerne of the representation ad ( r Ind G F θ ) does not contain ζ.

21 POTENTIAL AUTOMORPHY 21 (2) Suppose that r has GF (ζ ) m -big image and that there is a prime Q of M ying above a prime q of F ying in turn above a rationa prime q, such that: r is unramified at a primes above q, q, q spits competey in M, q is unramified in F ker r (ζ ), q 1 > 2n, and q #θ (I Q ), but θ is unramified at a primes above q except Q. Then (r Ind G F θ ) GF (ζ has big image. ) Proof. Much of the proof is competey unchanged from Lemma of [BLGG09]. Indeed, the ony part which changes is the statement and proof of the subemma, and the decomposition of ad r GF (ζ and its use, which depends on the subemma. ) We wi use, mosty without comment, the notation of Lemma of [BLGG09]. In particuar r = Ind G F G θ M and r = r r. The modified subemma is as foows. Subemma. Suppose θ / θ τ k = θ τ j / θ τ i (or equivaenty θ θ τ i = θ τ j θ τ k ); then either k = i and j = 0, or k = 0 and i = j. The converse aso hods. Proof. For the first part, we consider the action of inertia above q on each side of i θ θ τ = θ τ j θ τ k. The eft hand side is unramified outside Q and Q τ i ; the right hand side is unramified outside Q τ j and Q τ k ; it foows immediatey that {0, i} = {j, k}. For the converse, the fact that θ / θ τ k = θ / θ τ k is trivia. The decomposition of ad r becomes: GF (ζ ) s s m 1 (4.1.1) V j (χ) mj (V j W i ) mj j=0 χ Hom (Ga (M(ζ )/F (ζ )),F ) j=0 i=1 It is easy to see that: Each V j (χ) is irreducibe. We have V j (χ) V j (χ ) uness χ = χ and j = j. This is cear, because M is ineary disjoint from F ker r (ζ ) over F. G Each V j W i F (ζ ) = Ind G (V M(ζ j θ / θ τ i ) is irreducibe; moreover, we have ) Ind G F (ζ ) G (V M(ζ j θ / θ τ i ) G F (ζ ) ) = Ind G (V M(ζ j ) θ i / θ τ ) uness j = j and i = i. These both foow from the subemma and examining the action of inertia above q. Finay, V j (χ) G F (ζ ) = Ind G (V M(ζ j θ i / θ τ ) for a χ, i, j, j. This again foows ) from a consideration of the action of inertia above q. The verification of the cohomoogy-reated part of the definition of big image then proceeds as before. A that sti remains is to show the non-group-cohomoogy-reated part of the definition of big image. The anaysis for a copy V j (χ) ad r wi proceed exacty as before, and we must just treat submodues of ad r GF (ζ which are isomorphic )