POTENTIAL AUTOMORPHY AND CHANGE OF WEIGHT.

Transcription

1 POTENTIAL AUTOMORPHY AND CHANGE OF WEIGHT. THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR Abstract. We prove an automorphy ifting theorem for -adic representations where we impose a new condition at, which we ca potentia diagonaizabiity. This resut aows for change of weight and seems to be substantiay more fexibe than previous theorems aong the same ines. We derive severa appications. For instance we show that any irreducibe, totay odd, essentiay sef-dua, reguar, weaky compatibe system of -adic representations of the absoute Gaois group of a totay rea fied is potentiay automorphic, and hence is pure and its L-function has meromorphic continuation to the whoe compex pane and satisfies the expected functiona equation Mathematics Subject Cassification. 11F33. The second author was partiay supported by NSF grant DMS , the third author was partiay supported by NSF grant DMS and the fourth author was partiay supported by NSF grants DMS and DMS and by the Oswad Veben and Simonyi Funds at the IAS. 1

2 2 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR 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 ), where λ runs over the finite paces of M, 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 5.4.1). 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) (Irreducibiity) Each r λ is irreducibe. (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 a totay odd mutipier character; or each r λ factors through a map to GO n (M λ ) with a totay even mutipier character. Moreover in either case the mutipier characters form a weaky compatibe system. Then there is a finite, Gaois, totay rea extension of F 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 [HSBT10], [BLGHT11], [BLGG11]. 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. We give an anaogous theorem when F is an imaginary CM fied. Other than this 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 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

3 POTENTIAL AUTOMORPHY 3 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 may be automorphic of a different weight (infinitesima character) than r λ and the second automorphy ifting 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 automorphy ifting 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 them. 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 notion of a potentiay 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]). Potentia 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. It seems to us to be a very interesting question to carify further 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.) The foowing gives an indication of the sort of automorphy ifting theorems we are abe to prove. (See Theorem and aso section 2.1 for the definition of any notation or terminoogy which may be unfamiiar.)

4 4 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR 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 + ). Let n denote a positive integer. Suppose that 2(n + 1) is a prime such that F does not contain a primitive th root of 1. 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) (Odd essentia conjugate-sef-duaity) r c = r µ and µ(c v ) = 1 for a v. (2) (Unramified amost everywhere) r ramifies at finitey many primes. (3) (Potentia diagonaizabiity and reguarity) r GFv is potentiay diagonaizabe (and so in particuar potentiay crystaine) for a v and for each embedding τ : F Q it has n distinct τ-hodge Tate numbers. (4) (Irreducibiity) The restriction r GF (ζ is irreducibe. ) (5) (Residua ordinary automorphy) There is a reguar agebraic, cuspida, poarized automorphic representation (π, χ) of GL n (A F ) such that (r, µ) = (r,ı (π), r,ı (χ)ɛ 1 n ) and π is ı-ordinary. Then (r, µ) is automorphic. Theorem B impies the foowing potentia automorphy theorem for a singe - adic representation, from which Theorem A can be deduced. (See Coroary and Theorem ) Theorem C. Suppose that F is a totay rea fied. Let n be a positive integer and et 2(n + 1) be a prime. Let r : G F GL n (Q ) be a continuous representation. We wi write r for the semi-simpification of the reduction of r. Suppose that the foowing conditions are satisfied. (1) (Unramified amost everywhere) r is unramified at a but finitey many primes. (2) (Odd essentia sef-duaity) Either r maps to GSp n with totay odd mutipier or it maps to GO n with totay even mutipier. (3) (Potentia diagonaizabiity and reguarity) r is potentiay diagonaizabe (and hence potentiay crystaine) at each prime v of F above and for each τ : F Q it has n distinct τ-hodge Tate numbers. (4) (Irreducibiity) r GF (ζ is irreducibe. ) Then we can find a finite Gaois totay rea extension F /F such that r GF is automorphic. Moreover r is part of a weaky compatibe system of -adic representations. (In fact, r is part of a stricty pure compatibe system in the sense of section 5.1.) This theorem has other appications besides Theorem A. For instance we mention the foowing irreducibiity resut (see Theorem 5.5.2).

5 POTENTIAL AUTOMORPHY 5 Theorem D. Suppose that F is a CM fied and that π is a reguar, agebraic, essentiay conjugate sef-dua, cuspida automorphic representation of GL n (A F ). If π has sufficienty reguar weight ( extremey reguar in the sense of section 2.1), then for in a set of rationa primes of Dirichet density 1 the n-dimensiona -adic representations associated to π are irreducibe. To prove Theorem B we empoy Harris tensor product trick (see [Har09]), which was first empoyed in connection with change of weight in [BLGG11]. However the freedom that potentia diagonaizabiity gives us to make highy ramified base changes in the non-ordinary case means that this method becomes more powerfu. More precisey, suppose that r is potentiay diagonaizabe, and that r 0 is a potentiay diagonaizabe, automorphic ift of r (with possiby different Hodge Tate numbers to r). In fact making a finite soube 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 -adic characters θ and θ 0 of G M such that θ = θ 0, the restriction of Ind G F G M θ to an inertia group at a prime v reaizes a diagona point on the same component of the universa crystaine ifting ring of r GFv as r GFv, and the restriction of Ind G F G M θ 0 to an inertia group at a prime v reaizes a diagona point on the same component of the universa crystaine ifting ring of r GFv as r 0 GFv. Then r 0 Ind G F G M θ is automorphic and has the same reduction as r Ind G F G M 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 Ind G F G M θ 0 ) GFv. This is enough for the usua Tayor Wies Kisin argument to prove that r Ind G F G M θ 0 is aso automorphic, from which we can deduce (as in [BLGHT11]) the automorphy of r. Things are a itte more compicated than this because it seems to be hard to combine this with the eve changing argument in [Tay08]. In addition a direct argument imposes minor, but unwanted, conditions on the Hodge Tate numbers of r 0 and r. So instead of going directy from the automorphy of r 0 to that of r we create two ordinary ifts r 1 and r 2 of r (at east after a base change) where r 1 has the same oca behavior away from as r 0 ; r 2 has the same oca behavior away from as r; and where the Hodge Tate numbers of r 1 and r 2 are chosen suitaby. Our new arguments aow us to deduce the automorphy of r 1 from that of r 0. D.G. s resuts in the ordinary case [Ger09] aow us to deduce the automorphy of r 2 from that of r 1. Finay appying our new argument again aows us to deduce the automorphy of r from the automorphy of r 2. To construct r 1 and r 2 we use the method of Khare and Wintenberger [KW09] based on potentia automorphy (in the ordinary case, where it is aready avaiabe: see for exampe [BLGHT11]). Aong the way we aso prove a genera theorem about the existence of -adic ifts of a given mod Gaois representation with prescribed oca behavior (see Theorem 4.3.1). We deduce a rather genera theorem about change of weight and eve (see Theorem 4.4.1) of which a very particuar instance is the foowing. Theorem E. Let n be a positive integer and et > 2(n + 1) be a prime. Fix ı : Q C. Let F be a CM fied such that a primes of F above are unramified

6 6 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR over Q and spit over the maxima totay rea subfied of F. Let π be a reguar, agebraic, poarizabe, cuspida automorphic representation of GL n (A F ) satisfying the foowing conditions: π is unramified above ; π has weight (a τ,i ) τ:f C, i=1,...,n with n 1 a τ,1 a τ,2 a τ,n 0 for a τ; and the restriction to G F (ζ ) of the mod Gaois representation r,ı (π) associated to π and ı is irreducibe. Note that in this case a τ,i + a τc,n+1 i = w is independent of τ and i. Suppose that we are given a second weight (a τ,i ) τ:f C, i=1,...,n with n 1 a τ,1 a τ,2 a τ,n 0 for a τ, such that a τ,i + a τc,n+1 i = w is aso independent of τ and i, and for a paces v of F the restriction r,ı (π) GFv has a ift which is crystaine with τ-hodge Tate numbers {a ıτ,i + n i}. Then there is a second reguar, agebraic, poarizabe, cuspida automorphic representation π of GL n (A F ) giving rise (via ı) to the same mod Gaois representation (i.e. congruent to π mod ) such that π is aso unramified above and π has weight (a τ,i ). We remark that combining the resuts of this paper with work of Caraiani, one can deduce fu oca goba compatibiity of the -adic representations associated to reguar agebraic, essentiay conjugate sef-dua, cuspida automorphic representations of GL n over a CM or totay rea fied. (See [BLGGT11] and [Car12b].) We aso remark that Stefan Patrikis and one of us (R.T.) recenty combined the methods and resuts of this paper with one further idea, originating in [Pat12], and obtained variants of theorems A and D which are perhaps more usefu in practice. (See [PT12].) More specificay they proved a version of theorem A where the irreducibiity assumption is repaced by a purity assumption. This is usefu because for many compatibe systems arising from geometry purity is known by Deigne s theorem, but irreducibiity can be hard to check. In particuar one can deduce the meromorphic continuation and functiona equation of the L-function of any reguar, pure, sef-dua motive over a totay rea fied. They aso prove a version of theorem D in which the hypothesis that π is extremey reguar is weakened to reguar, but the concusion is aso weakened to give irreducibiity ony above a set of rationa primes of positive Dirichet density. We now expain the structure of the paper. In section 1 we coect some resuts about the deformation theory of Gaois representations. These are mosty now fairy standard resuts but we reca them to fix notations and in some cases to make sight improvements. The main exception is the introduction of potentia diagonaizabiity in section 1.4, which is new and of key importance for us. In section 2 we fix some notations and we reca the existing automorphy ifting theorems (or sight generaizations of them). Very itte in this section is nove. Between the writing of the first and second versions of this paper, Jack Thorne [Tho12] has found improved versions of these theorems which aow one to remove the troubesome bigness conditions from [CHT08] and the papers that foowed it. Moreover Ana Caraiani [Car12a], [Car12b] has proved oca-goba compatibiity in a p cases, as we as proving temperedness of a reguar agebraic, poarizabe, cuspida automorphic representations, and the purity of a the Wei-Deigne representations associated to

7 POTENTIAL AUTOMORPHY 7 the -adic representation associated to automorphic representations incuding the = p case. We have taken advantage of Caraiani s and Thorne s works to optimize our own resuts. In section 3 we make use of the automorphy ifting theorems from section 2 and the Dwork famiy to prove a potentia automorphy theorem (in the ordinary case) and a theorem about ifting mod Gaois representations (again in the ordinary case). These arguments foow those of [BLGHT11] and wi not surprise an expert. In section 4 we prove our main new theorems. Section 4.1 contains our main new argument. In section 4.2 we combine this with the resuts of sections 2 and 3 to obtain our optima automorphy ifting theorem. In section 4.3 we use the same ideas to deduce an improved resut about the existence of -adic ifts of mod Gaois representations with specified oca behavior. Combining the resuts of sections 4.2 and 4.3 we deduce in section 4.4 a genera theorem about change of weight and eve for mod automorphic forms on GL n. Then in section 4.5 we use the automorphy ifting theorem of section 4.2 and our potentia automorphy theorem from section 3.3 to deduce our main new potentia automorphy resut for a singe -adic representation. In section 5 we turn to appications of our main resuts. In section 5.1 we reca definitions connected to compatibe systems of -adic representations. In sections 5.2 and 5.3 we prove some group theoretic emmas about the images of compatibe systems of -adic representations. Then in section 5.4 we deduce from the potentia automorphy theorem of section 4.5 our main theorem a potentia automorphy theorem for compatibe systems of -adic representations. Finay in section 5.5 we give further appications of our main resuts appications to fitting an -adic representation into a compatibe system and to the irreducibiity of some -adic representations associated to cusp forms on GL(n). In the appendix we record some misceaneous resuts which we use esewhere in the paper. Some of these are resuts we suspect are we known, but for which we coudn t find a reference. In these cases we give a proof. Others are resuts for which we know a reference, but which we hope it may assist the reader to reca here. We woud ike to thank the anonymous referees for a thorough and inteigent reading of our paper, and for the numerous hepfu suggestions they made to improve the exposition. We woud aso ike to thank Kevin Buzzard, Forian Herzig, Wansu Kim and James Newton for their comments on an earier draft of the paper. We are gratefu to Ana Caraiani and Jack Thorne for sharing with us eary drafts of their papers [Car12a], [Car12b] and [Tho12]; and to Brian Conrad and Jiu-Kang Yu for answering our questions about commutative agebra and hyperspecia maxima compact subgroups, respectivey. This paper was written at the same time as [BLGG12] and there was considerabe cross fertiization. Our paper woud 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 s n denote the subspace of trace zero matrices. We wi write O(n) (resp. U(n)) for the group of matrices g GL n (R) (resp. GL n (C)) such that t g c g = 1 n. If R is a oca ring we write m R for the maxima idea of R.

8 8 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR If is an abeian group we wi et tor denote its maxima torsion subgroup and tf its maxima torsion free quotient. 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 semisimpification of its reduction, which is we defined up to conjugacy. If M is a fied, we et M denote an agebraic cosure of M and G M the absoute Gaois group Ga (M/M). 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 : G M µ 1 Z denote the Teichmuer ift of ɛ. If N/M is a separabe quadratic extension we wi et δ N/M denote the non-trivia character of Ga (N/M). If Γ is a profinite group and M is a topoogica abeian group with a continuous action of Γ, then by H i (Γ, M) we sha mean the continuous cohomoogy. We wi write Q r for the unique unramified extension of Q of degree r and Z r for its ring of integers. We wi write Q nr for the maxima unramified extension of Q and Z nr for its ring of integers. We wi aso write Ẑnr for the -adic competion of Z nr nr and Q for its fied of fractions. 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. If K /K is a Gaois extension we wi write I K /K for the inertia subgroup of Ga (K /K). 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 Frobenius semi-simpe 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 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 π i is an irreducibe smooth representation of GL ni (K) for i = 1, 2 we wi write π 1 π 2 for the irreducibe smooth representation of GL n1+n 2 (K) with rec(π 1 π 2 ) = rec(π 1 ) rec(π 2 ). If K /K is a finite extension and if π is an irreducibe smooth representation of GL n (K) we wi write BC K /K(π) for the base change of π to K which is characterized by rec K (BC K /K(π)) = rec K (π) WK. If ρ is a continuous de Rham representation of G K over Q then we wi write WD(ρ) for the corresponding Wei Deigne representation of W K, and if τ : K Q 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}. We wi et c denote compex conjugation on C. We wi write Art R (resp. Art C ) for the unique continuous surjection R Ga (C/R)

9 POTENTIAL AUTOMORPHY 9 (resp. C Ga (C/C)). We wi write rec C (resp. rec R ), or simpy rec, for the oca Langands correspondence from irreducibe admissibe (Lie GL n (R) R C, O(n))- modues (resp. (Lie GL n (C) R C, U(n))-modues) to continuous, semi-simpe n- dimensiona representations of the Wei group W R (resp. W C ). (See [Lan89].) If π i is an irreducibe admissibe (Lie GL ni (R) R C, O(n i ))-modue (resp. (Lie GL ni (C) R C, U(n i ))-modue) for i = 1,..., r and if n = n n r, then we define an irreducibe admissibe (Lie GL n (R) R C, O(n))-modue (resp. (Lie GL n (C) R C, U(n))- modue) π 1 π r by rec(π 1 π r ) = rec(π 1 ) rec(π r ). If π is an irreducibe admissibe (Lie GL n (R) R C, O(n))-modue then we define BC C/R (π) to be the irreducibe admissibe (Lie GL n (C) R C, U(n))-modue defined by rec C (BC C/R (π)) = rec R (π) WC. We wi write for the continuous homomorphism = v v : A /Q R >0, where each v has its usua normaization, i.e. p p = 1/p. Now suppose that K/Q is a finite extension. We wi write K (or simpy ) for N K/Q. We wi aso write Art K = v Art Kv : A K /K (K ) 0 G ab K. If v is a finite pace of K we wi write k(v) for its residue fied, q v for #k(v), and Frob v for Frob Kv. If v is a rea 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 /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. (We hope that this wi cause no confusion with the Gaois character δ K /K. One equas the composition of the other with the Artin map for K.) If K /K is a soube, finite Gaois extension and if π is a cuspida automorphic representation of GL n (A K ) we wi write BC K /K(π) for its base change to K, an (isobaric) automorphic representation of GL n (A K ) satisfying BC K /K(π) v = BC K v /K v K (π v K ) for a paces v of K. If π i is an automorphic representation of GL ni (A K ) for i = 1, 2 we wi write π 1 π 2 for the automorphic representation of GL n1+n 2 (A K ) satisfying (π 1 π 2 ) v = π 1,v π 2,v for a paces v of K. We wi ca a number fied K a CM fied if it has an automorphism c such that for a embeddings i : K C one has c i = i c. In this case either K is totay rea, or a totay imaginary quadratic extension of a totay rea fied. In either case we wi et K + denote the maxima totay rea subfied of K. Suppose that K is a number fied and χ : A K /K C

10 10 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR is a continuous character. If there exists a Z Hom (K,C) such that χ (K ) : x (τx) aτ 0 τ Hom (K,C) we wi ca χ agebraic. In this case we can attach to χ and a rationa prime and an isomorphism ı : Q C, a unique continuous character such that for a v we have r,ı (χ) : G K Q ı r,ı (χ) WKv Art Kv = χ v. There is aso an integer wt(χ), the weight of χ, such that χ = wt(χ)/2 K. (See the discussion at the start of Section A.2 for more detais.) If F is a totay rea fied we ca a continuous character χ : A K /K C totay odd if χ v ( 1) = 1 for a v. Simiary we ca a continuous character totay odd if µ(c v ) = 1 for a v. µ : G K Q

11 POTENTIAL AUTOMORPHY 11 Contents Introduction Deformations of Gaois Representations Automorphy Lifting Theorems Potentia Automorphy The main theorems Compatibe systems. 62 Appendix A. 83 References The group G n. 1. Deformations of Gaois Representations. We et G n denote the semi-direct product of G 0 n = GL n GL 1 by the group {1, j} where j(g, a)j 1 = (a t g 1, a). We et ν : G n GL 1 be the character which sends (g, a) to a and sends j to 1. We wi aso et GSp 2n GL 2n denote the sympectic simiitude group defined by the anti-symmetric matrix J 2n = ( 0 1n 1 n 0 and we wi again et ν : GSp 2n GL 1 denote the mutipier character. Finay et GO n denote the orthogona simiitude group defined by the symmetric matrix 1 n. There is a natura homomorphism ), G n G m G n /G 0 n G m /G 0 m {±1} which sends both (j, 1) and (1, j) to 1. Let (G n G m ) + denote the kerne of this map. There is a homomorphism There is aso a homomorphism : (G n G m ) + G nm (g, a) (g, a ) (g g, aa ) j j j. I : G n GSp ( 2n ) g 0 (g, a) ( 0 a t g 1 ) 0 1n j. 1 n 0 Suppose that Γ is a group with a norma subgroup of index 2 and that γ 0 Γ. Suppose aso that A is a ring and that r : Γ G n (A) is a homomorphism with = r 1 Gn(A). 0 Write r : GL n (A) for the composition of r with projection to GL n (A). Write r(γ 0 ) = (a, (ν r)(γ 0 ))j. Then r(γ 0 δγ 1 0 )at r(δ) = (ν r)(δ)a

12 12 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR for a δ, and r(γ 0) 2 t a = (ν r)(γ 0 )a. If r : Γ GSp 2n (A) is a homomorphism with mutipier µ, then it gives rise to a homomorphism ˆr : Γ G 2n (A) which sends δ to (r(δ), µ(δ)) and γ Γ to (r(γ)j2n 1, µ(γ))j. Then = ˆr 1 G0 2n(A) and ν ˆr = µ. Simiary if r : Γ GO n (A) is a homomorphism with mutipier µ, then it gives rise to a homomorphism ˆr : Γ G n (A) which sends δ to (r(δ), µ(δ)) and γ Γ to (r(γ), µ(γ))j. Then = ˆr 1 G0 2n(A) and ν ˆr equas the product of µ with the nontrivia character of Γ/. If r : Γ G n (A) (resp. r : Γ G m (A)) is a homomorphism with r 1 Gn(A) 0 = (resp. (r ) 1 Gm(A) 0 = ) then we define I(r) = I r : Γ GSp 2n (A) and r r = (r r ) : Γ G nm (A). Note that the mutipier of I(r) equas the mutipier of r and that the mutipier of r r differs from the product of the mutipiers of r and r by the non-trivia character of Γ/. If χ : A and µ : Γ A satisfy χχ γ0 = µ, and χ(γ0) 2 = µ(γ 0 ); (i.e. the composition of χ with the transfer map Γ ab ab equas the product of µ and the non-trivia character of Γ/ ), then there is a homomorphism (χ, µ) : Γ G 1 (A) δ (χ(δ), µ(δ)) γ (χ(γγ0 1 ), µ(γ))j, for a δ and γ Γ. We have ν (χ, µ) = µ. (At the referee s suggestion we incude a proof that (χ, µ) is indeed a homomorphism in an attempt to convince the reader that a the unsupported assertions of this section can be checked in an entirey eementary way. Suppose that δ 1, δ 2 and γ 1, γ 2 Γ. Then we have and and and (χ, µ)(δ 1 δ 2 ) = (χ, µ)(δ 1 )(χ, µ)(δ 2 ) (χ, µ)(δ 1 γ 2 ) = (χ(δ 1 γ 2 γ 1 0 ), µ(δ 1γ 2 ))j = (χ, µ)(δ 1 )(χ, µ)(γ 2 ) (χ, µ)(γ 1 δ 2 ) = (χ(γ 1 δ 2 γ0 1 1δ 2 ))j = (χ(γ 1 γ0 1 0δ 2 γ0 1 1δ 2 ))j = = (χ, µ)(γ 1 )j(χ(γ 0 δ 2 γ0 1 2))j (χ, µ)(γ 1 )(µ(δ 2 )χ γ0 (δ 2 ) 1, µ(δ 2 )) = (χ, µ)(γ 1 )(χ, µ)(δ 2 ) (χ, µ)(γ 1 γ 2 ) = (χ(γ 1 γ 1 0 )χ(γ 0γ 2 ), µ(γ 1 )µ(γ 2 )) = (χ, µ)(γ 1 )j(χ γ0 (γ 2 γ 0 ), µ(γ 2 )) = (χ, µ)(γ 1 )( µ(γ 2 )χ γ0 (γ 2 γ 0 ) 1, µ(γ 2 ))j = (χ, µ)(γ 1 )( χ(γ 2 γ 0 )µ(γ 0 ) 1, µ(γ 2 ))j = (χ, µ)(γ 1 )(χ, µ)(γ 2 )

13 POTENTIAL AUTOMORPHY 13 as desired.) In the case that Γ = G F + and = G F where F is an imaginary CM fied with maxima totay rea subfied F +, we ca r : G F + G n (A) (resp. GSp 2n (A), resp. GO n (A)) totay odd if the mutipier character takes every compex conjugation to 1 (resp. 1, resp. 1). Note that if r is totay odd so is I(r) (resp. ˆr, resp. ˆr ). Suppose now that A is a fied, that r : GL n (A) is absoutey irreducibe, and that µ : Γ A is a character so that r γ0 = r µ. More precisey if γ Γ there is a b γ GL n (A), unique up to scaar mutipes, such that r(γδγ 1 )b t γ r(δ) = µ(δ)b γ for a δ. Computing r(γ 2 δγ 2 ) in two ways and using the absoute irreducibiity of r, we deduce that r(γ 2 ) is a scaar mutipe of b t γ b 1 γ. (Write r(γ 2 )r(δ)r(γ 2 ) 1 = r(γ 2 δγ 2 ) = µ(δ)b t γ r(γδγ 1 ) 1 b 1 γ = (b t γ b 1 γ )r(δ)(b t γ b 1 γ ) 1 and appy Schur s emma.) Substituting δ = γ 2 in the ast dispayed equation, we then deduce that r(γ 2 ) t b γ = ±µ(γ)b γ. One can check that the sign in the above equation is independent of γ Γ, and we wi denote it sgn (r, µ). (To see this one uses the fact that one can take b δγ = r(δ)b γ for δ and γ Γ.) Then we get a homomorphism r µ : Γ G n (A) which sends δ to (r(δ), µ(δ)) and sends γ 0 to (b γ0, sgn (r, µ)µ(γ 0 ))j. In particuar if sgn (r, µ) = 1 then ν r µ = µ, whie if sgn (r, µ) = 1 then µ 1 (ν r µ ) is the non-trivia character of Γ/. Moreover r = r Abstract deformation theory. Fix a rationa prime and et O denote the ring of integers of a finite extension L of Q in Q. Let λ denote the maxima idea of O and et F = O/λ. Let Γ denote a topoogicay finitey generated profinite group and et ρ : Γ GL n (F) be a continuous homomorphism. We wi denote by ρ = ρ O : Γ GL n (R O,ρ) the universa ifting (or framed deformation ) 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 O Q. The foowing emma is presumaby we known, but as we don t know a reference, we give a proof. Lemma Suppose that O is the ring of integers of a finite extension L /L. Then the map R O,ρ R O,ρ O O coming from the universa property of R O,ρ and ρ O O is an isomorphism. In particuar, as the notation suggests, the ring R ρ Q does not depend on the choice of L.

14 14 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR Proof. Let RO,F,ρ denote the cosed subring of R O,ρ consisting of eements which reduce to an eement of F moduo the maxima idea. Then RO,F,ρ is a compete, noetherian oca O-agebra with residue fied F and ρ O is defined over R O,F,ρ. Thus the universa property of RO,ρ gives rise to a map R O,ρ R O,F,ρ under which ρ O pushes forward to ρ O. This map extends to a O -inear map RO,ρ O O RO,ρ. We caim this is an inverse to our map RO,ρ R O,ρ O O. Under the composite RO,ρ R O,ρ the representation ρ O pushes forward to itsef, and so this map must be the identity. Consider the composite R O,ρ R O,ρ R O,ρ O O. It factors through the subring (RO,ρ O O ) F RO,ρ O O consisting of eements which reduce moduo the maxima idea to an eement of F. The representation ρ O pushes forward to itsef, and so this composite must equa the canonica incusion, and we have proved the emma. The maxima ideas are dense in Spec RO,ρ [1/] (see [Gro66, ]). A prime idea of RO,ρ [1/] is maxima if and ony if the residue fied k( ) = R O,ρ [1/] / is (topoogicay isomorphic to) a finite extension of L. (For the if part note that the image of RO,ρ in k( ) is a compact O-submodue of k( ) with fied of fractions k( ). Thus RO,ρ [1/] k( ). For the ony if part see for instance Lemma 2.6 of [Tay08].) We get a continuous representation ρ : Γ GL n (k( )). The forma competion RO,ρ [1/] is the universa ifting ring for ρ, i.e. if A is an Artinian oca k( )-agebra with residue fied k( ) and if ρ : Γ GL n (A) is a continuous representation ifting ρ, then there is a unique continuous map of k( )-agebras RO,ρ [1/] A so that ρ pushes forward to ρ. (Let R denote the image of RO,ρ in A/m A. Let A 0 denote the R-subagebra of A generated by the matrix entries of the image of ρ. Then A 0 is a compete noetherian oca O-agebra with residue fied F and ρ : Γ GL n (A 0 ). The assertion foows easiy.) The map which associates a cocyce c Z 1 (Γ, ad ρ ) to the ift (1 n + cɛ)ρ of ρ defines an isomorphism between Z 1 (Γ, ad ρ ) and the tangent space Hom k( ) (RO,ρ [1/], k( )[ɛ]/ɛ 2 ). If H 2 (Γ, ad ρ ) = (0) then RO,ρ [1/] is formay smooth at (by the argument of Proposition 2 of [Maz89], which can be easiy adapted to our current situation) of dimension dim k( ) Z 1 (Γ, ad ρ ) = n 2 + dim k( ) H 1 (Γ, ad ρ ) dim k( ) H 0 (Γ, ad ρ ), where we use continuous cohomoogy. (We earned these observations from Mark Kisin.) Let H denote the subgroup of GL n (RO,ρ ) consisting of eements which reduce moduo the maxima idea to an eement that centraizes the image of ρ. If h H then there is a unique continuous homomorphism φ h : R O,ρ R O,ρ such that ρ pushes forward to h 1 ρ h. We have that φ g φ h = φ gφg(h).

15 POTENTIAL AUTOMORPHY 15 (We remark that after definition of [CHT08] this action is defined for eements of 1 n + M n (m R O,ρ ), but it is incorrecty stated that this defines an action of the group 1 n + M n (m R ) on R O,ρ O,ρ. This is not important in the rest of [CHT08].) Lemma Keep the notation of the previous paragraph. If h H then φ h fixes each irreducibe component of Spec R O,ρ [1/]. Proof. Suppose that h H and that O is the ring of integers of a finite extension of L /L in Q, with maxima idea λ. Suppose that ρ : Γ GL n (O ) ifts ρ. Then h 1 ρh aso ifts ρ. (We are using h both for an eement of GL n (RO,ρ ) and for its image in GL n (O ) under the map RO,ρ O induced by ρ.) Reca (from Lemma A.1.2) that A = O s, t /(s det(t1 n + (1 t)h) 1) with the λ -adic topoogy is a compete topoogica domain. We have a continuous representation ρ = (t1 n + (1 t)h) 1 ρ(t1 n + (1 t)h) : Γ GL n (A). Let A 0 denote the cosed subagebra of A generated by the matrix entries of eements of the image of ρ and give it the subspace topoogy. Then ρ : Γ GL n (A 0 ), and ρ mod (λ A A 0 ) = ρ, and ρ pushes forward to ρ (resp. h 1 ρh) 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 h 1 ρh both factor. As A 0 is a domain (being a sub-ring of A) we concude that the points corresponding to ρ and h 1 ρh 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/]), we see that the emma foows. (The referee remarks that it may be hepfu to think of this argument as an instance of homotopy.) 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. Now suppose that r : Γ G n (F) is a continuous homomorphism such that Γ G n /G 0 n. Let denote the kerne of Γ G n /G 0 n and suppose that r : GL n (F) is absoutey irreducibe. Then

16 16 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR there is a universa deformation r univ : Γ G n (R univ O,r ) to a compete noetherian oca O-agebra with residue fied F, where now we consider two iftings as equivaent deformations if they are conjugate. (See section 2.2 of [CHT08].) Lemma (1) Suppose that Σ Γ has finite index, that Σ is not contained in and that r Σ is absoutey irreducibe. Then the natura map RO,r univ Σ RO,r univ induced by runiv Σ makes RO,r univ a finitey generated Runiv O,r Σ - modue. (2) Suppose that s : Γ G m (O) is a continuous homomorphism such that = s 1 Gm(O) 0 and s r is absoutey irreducibe. Then the natura map RO,r s univ R univ Runiv O,r induced by r univ s makes R univ O,r a finitey generated O,r s -modue. (3) Suppose I(r) is absoutey irreducibe and that Σ is another open subgroup of index two in Γ which does not contain ker I(r). Then the natura map R univ O,Î(r) Σ R univ O,Î(r) Σ RO,r univ induced by I(r univ ) Σ makes RO,r univ -modue. a finitey generated Proof. This is essentiay an abstraction of Lemma of [BLGG12]. Write R for RO,r univ Σ resp. RO,r s univ resp. Runiv and write m for the maxima O,Î(r) Σ idea of R. We first verify that the image of Γ in G n (RO,r univ/mruniv O,r ) is finite. In the first case we use the incusions ker(r univ mod m) ker(r univ Σ mod m) = ker(r Σ ); in the second we use the incusions ker(r univ mod m) ker((r univ s) mod m) ker(s mod m) = ker(r s) ker s; and in the third case the incusions ker(r univ mod m) ker( I(r univ ) Σ mod m) = ker Î(r) Σ. Let m denote the order of the image of Γ in G n (RO,r univ/mruniv O,r ), and et γ 1,..., γ m be eements of Γ chosen so that their images in G n (RO,r univ/mruniv O,r ) exhaust the image of Γ. Let f(t ) = (T (ζ ζ n )) F[T ] (ζ 1,...,ζ n) µ m(f) n and et A denote the maxima quotient of F[X i,j ] i,j=1,...,n over which the m th - power of the matrix (X i,j ) is 1 n. If is a prime idea of A then a the roots of the characteristic poynomia of (X i,j ) over A / are m th roots of unity and hence f(tr (X i,j )) = 0 in A/ A /. Thus there is a positive integer a such that f(tr (X i,j )) a = 0 in A. Then we get a map F[T 1,..., T m ]/(f(t 1 ) a,..., f(t m ) a ) RO,r univ/mruniv O,r T i tr r univ (γ i ). By Lemma of [CHT08] (which shows that R univ O, r is topoogicay generated as a O-agebra by the tr r univ (γ i )) we see that this map has dense image. On the other hand the source has finite cardinaity. We concude that the map is surjective

17 POTENTIAL AUTOMORPHY 17 and that RO,r univ/mruniv O,r is finite over F. Hence by Nakayama s Lemma we concude that that RO,r univ is finite over R, as desired Loca theory: p. Continue to fix a rationa prime and et O denote the ring of integers of a finite extension L of Q in Q. Let λ denote the maxima idea of O and et F = O/λ. However in this section we speciaize our discussion to the case Γ = G K, where K/Q p is a finite extension and p. Thus ρ : G K GL n (F) is continuous. Write q for the order of the residue fied of K. In this case the tangent space to RO,ρ [1/] at a maxima idea has dimension n 2 + dim k( ) H 1 (G K, ad ρ ) dim k( ) H 0 (G K, ad ρ ) = n 2 + dim k( ) H 2 (G K, ad ρ ) = n 2 + dim k( ) H 0 (G K, (ad ρ )(1)), by the oca Euer characteristic formua for Q -modues and oca duaity for Q - modues. (The proof of Lemma 9.7 of [Kis03] shows that the usua Euer characteristic formua with finite coefficients impies the anaogous statement in the case where the coefficients are finite Q -modues. Theorem of [Rub00] provides a reference for oca duaity for Q -modues.) Moreover RO,ρ [1/] is formay smooth at a maxima idea if H 0 (G K, (ad ρ )(1)) = (0). We wi ca a continuous representation ρ : G K GL n (Q ) robusty smooth (resp. smooth) if H 0 (G K, (ad ρ )(1)) = (0) for a finite extensions K /K (resp. for K = K). Our next aim is to show that the set of cosed points of Spec RO,ρ [1/] which are robusty smooth is Zariski dense, which wi impy that a irreducibe components of Spec RO,ρ [1/] are genericay formay smooth of dimension n2. (The corresponding resut for smooth points can be found in the proof of Theorem of [Gee11] or in [Cho09], but our proof seems to be different even in this specia case. This case is aready sufficient to deduce that a irreducibe components of Spec RO,ρ [1/] are genericay formay smooth of dimension n2.) Define a partia order on L by a b if a equas σ(b)ζq m where σ Ga (L/L), where ζ is a root of unity and where m Z 0. We wi write a b (a equivaent to b) if a b and b a; and we wi write a b (a comparabe to b) if a b or b a. These are both equivaence reations. Further we wi write a > b for a b but a b. Choose φ W K a ift of Frob K. If V is a finite dimensiona L-vector space with an action of W K with open kerne and if a L then we define V ((a)) (resp. V (a)) to be the L-subspace of V such that V ((a)) L L (resp. V (a) L L) is the sum of the b-generaized eigenspaces of φ in V as b runs over a eements of L with a b (resp. a b). This is independent of the choice of φ. (If φ is another choice then the actions of φ m and (φ ) m on V are equa for some m Z >0.) Thus V (a) and V ((a)) are W K -invariant. We have decompositions where a runs over L /, and V = V ((a)) V = V (a) where a runs over L /. We wi say that V has type a if V = V ((a)).

18 18 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR Lemma Suppose that (r, N) is a Wei Deigne representation of W K on a finite dimensiona L-vector space V. Then we can write u s i V = i=1 j=1 where V i,j is invariant under I K ; N : V i,j Vi,j+1 uness j = s i in which case NV i,si = (0); W K V i,j V i,j i 1 i =1 j V i,j and so we get an induced action of W K on on (V i,j i 1 i =1 j V i,j )/( i 1 i =1 j V i,j ) is irre- V i,j ; the action of W K ducibe; V i,j has type a i q 1 j for some a i Q ; and if i < i then a i a i. V i,j Proof. We may suppose that V = V (b) for some b (because N must take any V (b) to itsef). We wi construct the V i,j by recursion on i. Suppose that we have constructed V i,j for i < t. Choose a t L such that t 1 V/ ((a t )) (0) i=1 and such that if a > a t then t 1 V/ i=1 j j V i,j V i,j Aso choose an irreducibe W K -submodue t 1 V t,1 V/ i=1 ((a)) = (0) j V i,j ((a t )) and choose s t minima such that N st V t,1 = (0). Lift V t,1 to an I K -submodue Vt,1 0 V ((a t )). Then N st Vt,1 0 t 1 i=1 j V i,j. For each i < t choose j i Z >st such that a t q st a i q 1 ji. (To see that j i > s t we are using the fact that for i t we have a i a t.) Then N st Vt,1 0 V i,ji. i<t Thus if v V 0 t,1 we can write N st v = i<t N st v i for unique eements v i V i,ji s t. Set V t,1 to be the set of v i<t We see that V t,1 V ((a t )) is a Q -sub-vector space ifting V t,1, which is I K - invariant and satisfies N st V t,1 = (0). Set V t,j = N j 1 V t,1. It is not hard to see that these V i,j have a the desired properties. v i.

19 POTENTIAL AUTOMORPHY 19 then We remark that if we define an increasing fitration on V by Fi i V = V i,j i i j V F ss = u gr i V. Lemma (1) Suppose ρ : G K GL n (Q ) is a continuous representation; that ı : Q C and that π is an irreducibe smooth representation of GL n (K) over C with ıwd(ρ) F ss = rec K (π). If π is generic then ρ is smooth. (2) The cosed points in Spec RO,ρ [1/] for which ρ is robusty smooth are Zariski dense. Proof. For the first 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 has a fitration with graded pieces ρ i satisfying ıwd(ρ i ) F ss = rec K (Sp si (π i )), possiby after reordering the i s. Thus (ad ρ)(1) has a fitration with graded pieces Hom (ρ i, ρ j (1)). If this had non-zero invariants, 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 assumption that π is generic (see page 36 of [HT01]). For the second part suppose that is a cosed point of Spec RO,ρ [1/]. Set O equa to the image of RO,ρ in L = RO,ρ [1/]/ and et λ denote the maxima idea of O. By Lemma we can find a decomposition (L ) n = u i=1 V i such that for each i the sub-space V i is invariant under I K ; for each i the sub-space Fi i V = i i =1 V i is invariant under G K; and for each i we have WD(gr i V ) = Sp si (W i ), where W i has some type a i. (By Sp s (W ) we mean the Wei Deigne representation of W K whose underying representation of W K is W W (1) W (s 1) and where N : W (i) W (i+1) for i = 0,..., s 2.) Choose M Z >0 so that u ((O ) n V i ) ( M 1 O ) n. i=1 Let A ker(gl n (O [[X 1,..., X u ]]) GL n (F)) be the unique eement which preserves each (V i (O ) n ) O O [[X 1,..., X u ]] and acts on it by mutipication by (1 + M X i ). Note that A commutes with ρ (I K ). Then there is a unique continuous representation i=1 ρ : G K GL n (O [[X 1,..., X u ]]) such that ρ IK = ρ IK O O [[X 1,..., X u ]] and such that for any ift φ of Frob K to W K we have ρ(φ) = ρ (φ)a. Then ρ is a ift of ρ. If x (λ ) u write ρ x for ρ mod (X 1 x 1,..., X u x u ). Note that ρ 0 = ρ. We wi show that for (Zariski) generic x that ρ x is robusty smooth and the second part of the emma wi foow. (Note that if 0 f O [[X 1,..., X u ]] then f can not vanish on a of (λ ) u, by, for instance, Lemma 3.1 of [BLGHT11].)

20 20 THOMAS BARNET-LAMB, TOBY GEE, DAVID GERAGHTY, AND RICHARD TAYLOR If y (O ) et ν y : G K /I K (O ) be the unramified character taking Frob K to y. Then (ad ρ x )(1) has a fitration with graded pieces Note that if i = j then for any finite K /K, because Hom (V i, V j (ν (1+ M x j)/q(1+ M x i))). Hom GK (V i, V i (ν 1/q )) = (0) Hom WK (W i, W i (ν 1/q j )) = (0) for j = 1,..., s j + 1 (because, in turn, W i and W i (ν 1/q j ) wi have different types). So it remains to show that for genera x we wi aso have Hom GK (V i, V j (ν (1+ M x j)/q(1+ M x i))) = (0) for a i j and a finite K /K. Let φ W K denote a Frobenius ift and et L denote the compositum of a extensions of L of degree ess than or equa to n. Then L /L is finite. It wi do to choose (x i ) (λ ) u so that if i j and if α (resp. β) is an eigenvaue of φ on V i (resp. V j ) and if ζ is a root of unity then qαζ(1+ M x i ) β(1+ M x j ). However if such an equaity were to hod then ζ L. As L contains ony finitey many roots of unity, the x i s need ony satisfy finitey many inequaities, as desired. Suppose that C is a set of irreducibe components of Spec RO,ρ [1/] and et R O,ρ,C denote the maxima quotient of RO,ρ, which is reduced, -torsion free and has Spec RO,ρ,C [1/] supported on the components in C. Aso et D C denote the set of iftings of ρ to compete oca noetherian O-agebras R with residue fied F such that the induced map RO,ρ R factors through R O,ρ,C. By Lemma above and Lemma 3.2 of [BLGHT11] we see that D C is a deformation probem in the sense of Definition of [CHT08]. If K /K is finite and Gaois we wi et RO,ρ,K nr denote the maxima quotient of RO,ρ over which ρ (I K ) = {1 n }. Lemma The ring RO,ρ,K nr [1/] is either the zero ring or is formay smooth of dimension n 2. Proof. It suffices to show that for any maxima idea of RO,ρ,K nr [1/] we have and H 2 (Ga ((K ) nr /K), ad ρ ) = (0) dim k( ) H 1 (Ga ((K ) nr /K), ad ρ ) = dim k( ) H 0 (Ga ((K ) nr /K), ad ρ ). However H i (Ga ((K ) nr /K), ad ρ ) = H i (G K /I K, (ad ρ ) I (K ) nr /K ) (ad ρ ) G K if i = 0 = ((ad ρ ) I K ) GK /I K if i = 1 (0) otherwise. The emma foows.

21 POTENTIAL AUTOMORPHY 21 Thus RO,ρ,K nr [1/] = R O,ρ,C K nr [1/] for some finite set of components C K nr of Spec RO,ρ. Let C p-nr denote the union of C K nr over a finite Gaois extensions K /K and set RO,ρ,p-nr = R O,ρ,C p-nr. A Q -point of RO,ρ factors through R O,ρ,p-nr if and ony if it is potentiay unramified. The ring RO,ρ,p-nr [1/] is either the zero ring or is formay smooth of dimension n 2. For i = 1, 2, et ρ i : G K GL n (O Q ) be a continuous representation. 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 a common 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. (5) If ρ 1 ρ 2 and H 0 (G K, (ad ρ 1 )(1)) = (0) then ρ 1 ρ 2. (6) Write WD(ρ i ) = (r i, N i ). If ρ 1 ρ 2 then r 1 IK = r2 IK. If ρ 1 ρ 2 and ρ 2 ρ 1 then (r 1 IK, N 1 ) = (r 2 IK, N 2 ). (7) If ρ 1 and ρ 2 are unramified and have the same reduction then ρ 1 ρ 2. (8) If K /K is a finite extension and ρ 1 ρ 2 then ρ 1 GK ρ 2 GK. (9) 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. (10) 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. (11) If µ : G K Q is a continuous unramified character with µ = 1 then ρ 1 ρ 1 µ. (12) Suppose that ρ 1 is semisimpe and et Fi i be an invariant decreasing fitration on ρ 1 by O Q -direct summands, then ρ 1 i gr i ρ 1. Two of these assertions (6) and (12) require proof, so we separate them out into emmas. The first was proved in [Cho09], but as this is not yet easiy avaiabe we give a proof here. Lemma (Choi). Suppose that ρ : G K GL n (F). (1) If 1 and 2 are two maxima ideas on the same connected component of Spec R ρ Q then (ρ mod 1 ) ss I K = (ρ mod 2 ) ss I K.