THE SATO-TATE CONJECTURE FOR HILBERT MODULAR FORMS

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1 THE SATO-TATE CONJECTURE FOR HILBERT MODULAR FORMS THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY Abstract. We proe the Sato-Tate conjecture for Hilbert modular forms. More precisely, we proe the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of GL 2 (A F ), F a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques deeloped by Taylor et. al., but makes use of automorphy lifting theorems oer ramified fields, together with a topological argument with local deformation rings. In particular, we gie a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary n-dimensional Galois representations. Contents 1. Introduction 1 2. Notation 5 3. An automorphy lifting theorem 5 4. A character building exercise Twisting and untwisting Potential automorphy in weight Hilbert modular forms 47 References Introduction 1.1. In this paper we proe the Sato-Tate conjecture for Hilbert modular forms. More precisely, we proe the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of GL 2 (A F ), F a totally real field, which are not of CM type. Seeral special cases of this result were proed in the last few years. The paper [HSBT06] proes the result for elliptic cures oer totally real fields which hae potentially multiplicatie reduction at some place, and it is straightforward to extend this result to the case of cuspidal automorphic representations of weight 0 (i.e. those corresponding to Hilbert modular forms of parallel weight 2) which are a twist of the Steinberg representation at some finite place. The case of modular forms (oer Q) of weight 3 whose corresponding automorphic representations are a twist of the Steinberg representation at some finite place was treated in [Gee09], ia an argument that depends on the existence of infinitely many ordinary places. The 2000 Mathematics Subject Classification. 11F33. The second author was partially supported by NSF grant DMS

2 2 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY case of modular forms (again oer Q) was proed in [BLGHT09] (with no assumption on the existence of a Steinberg place). The main new features of the arguments of [BLGHT09] were the use of an idea of Harris ([Har07]) to ensure that potential automorphy need only be proed in weight 0, together with a new potential automorphy theorem for n-dimensional Galois representations which are symmetric powers of those attached to non-ordinary modular forms. Recent deelopments in the theory of the trace formula remoe the need for an assumption of the existence of a Steinberg place in both this theorem and in the case of elliptic cures oer totally real fields. In summary, the Sato-Tate conjecture has been proed for modular forms and for elliptic cures oer totally real fields, but is not known in any non-triial case for Hilbert modular forms not of parallel weight 2 oer any field other than Q. It seems to be hard to extend the arguments of either [Gee09] or [BLGHT09] to the general case; in the former case one has no way of establishing the existence of infinitely many ordinary places (although it is conjectured that the set of such places should be of density one), and in the latter case one has no control oer the mixture of ordinary and supersingular places oer any rational prime. In this paper, we adopt a new approach: we combine the approach of [Gee09], which is based on taking congruences to representations of GL 2 (A F ) of weight 0, with the twisting argument of [Har07] (or rather the ersion of this argument used in [BLGHT09]). These techniques do not in themseles suffice to proe the result, as one has to proe a automorphy lifting theorem for non-ordinary representations oer a ramified base field. No such theorems are known for representations of dimension greater than two. The chief innoation of this paper is a new technique for proing such results. Our new automorphy lifting theorem uses the usual Taylor-Wiles-Kisin patching techniques, but rather than identifying an entire deformation ring with a Hecke algebra, we proe that certain global Galois representations, whose restrictions to decomposition groups lie on certain components of the local lifting rings, are automorphic. That this is the natural output of the Taylor-Wiles-Kisin method is at least implicit in the work of Kisin, cf. section 2.3 of [Kis07a]. One has to be somewhat careful in making this precise, because it is necessary to use fixed lattices in the global Galois representations one considers, and to work with lifting rings rather than deformation rings. In particular, it is not clear that the set of irreducible components of a local lifting ring containing a particular O K -alued point, K a finite extension of Q l, is determined by the equialence class of the corresponding K-representation. This necessitates a good deal of care to work with O K -liftings throughout the paper. Effectiely (modulo the remarks about lattices in the preious paragraph) the automorphy lifting theorem that we proe tells us that if we are gien two congruent n-dimensional l-adic regular crystalline essentially self dual representations of G F (the absolute Galois group of a totally real field F ) with the same l-adic Hodge types, with the same ramification properties, and satisfying a standard assumption on the size of the mod l image, then if one of them is automorphic, so is the other. By the same ramification properties, we mean that they are ramified at the same set of places, and that the points determined by the two representations on the corresponding local lifting rings lie on the same components. For example, we require that the two representations hae unipotent ramification at exactly the

3 SATO-TATE 3 same set of places; we do not know how to adapt Taylor s techniques for aoiding Ihara s lemma ([Tay08]) to this more general setting. The local deformation rings for places not diiding l are reasonably well-understood, so that it is possible to erify that this condition holds at such places in concrete examples. On the other hand, the components of the crystalline deformation rings of fixed weight are not at all understood if l ramifies in F, unless n = 2 and the representations are Barsotti-Tate, when there are at most two components, corresponding to ordinary and non-ordinary representations. This might appear to preent us from being able to apply our theorem to any representations at all. We get around this problem by making use of the few cases where the components are known. Specifically, we use the cases where n = 2 and either the representations are Barsotti-Tate; or F is unramified in l, and the representations are crystalline of low weight. To explain how we are able to bootstrap from these two cases, we now explain the main argument. We begin with a regular algebraic cuspidal representation π of GL 2 (A F ), assumed not to be of CM type. By a standard analytic argument, it suffices to proe that for each n 1 the (n 1)-st symmetric power of π is potentially automorphic, in the sense that there is a finite Galois extension F /F of totally real fields and an automorphic representation π n of GL n (A F ) whose L-function is equal to that of the base change to F of the (n 1)-st symmetric power L-function of π. Equialently, if we fix a prime l, then it suffices to proe that the (n 1)-st symmetric power of an l-adic Galois representation corresponding to π is potentially automorphic, i.e. that its restriction to G F is automorphic. This is what we proe. We choose l to be large and to split completely in F, and such that π is unramified at all places of F lying oer l. We begin by making a preliminary solable base change to a totally real field F /F, such that the base change π F of π to F is either unramified or an unramified twist of the Steinberg representation at each finite place of F. We then choose an automorphic representation π of GL 2 (A F ) of weight 0 which is congruent to π, which for any place l is unramified (respectiely an unramified twist of the Steinberg representation) if and only if π is, and which is a principal series representation (possibly ramified) or a supercuspidal representation for all l. Furthermore we choose π so that for places l, π is ordinary if and only if π is ordinary. We now proe that the (n 1)-st symmetric power of π is potentially automorphic oer some finite Galois extension F of F. This is straightforward, although it is not quite in the literature. This is the only place that we need to make use of a potential automorphy theorem for an n-dimensional Galois representation, and the theorems of [HSBT06] (or rather the ersions of them which are now aailable thanks to improements in our knowledge of the trace formula, which remoe the need for discrete series hypotheses) would suffice, but for the conenience of the reader we use a theorem from [BLGHT09] (which, for instance, already include the improements made possible by our enhanced understanding of the trace formula) instead. This also allows us to aoid haing to make an argument with Rankin-Selberg conolutions as in [HSBT06]. We note that the theorem we use from [BLGHT09] is for ordinary representations, rather than the far more technical result for supersingular representations that is also proed in [BLGHT09].

4 4 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY We now wish to deduce the potential automorphy of the (n 1)-st symmetric power of π, or rather the automorphy of the corresponding l-adic Galois representation r : G F GL n (Q l ), from the automorphy of the l-adic Galois representation r : G F GL n (Q l ) corresponding to the (n 1)-st symmetric power of π. We cannot directly apply our automorphy lifting theorem, because the Hodge-Tate weights of r and r are different. Instead, we employ an argument of Harris ([Har07]), and tensor both r and r with representations obtained by the automorphic induction of algebraic characters of a certain CM field. The choice of the field and the characters is somewhat delicate, in order to presere arious technical assumptions for the automorphy lifting theorem, in particular the assumption of big residual image. The two characters are chosen so that the resulting Galois representations r and r are potentially crystalline with the same Hodge-Tate weights. The representation r is automorphic, by standard results on automorphic induction. We then apply our automorphy lifting theorem to deduce the automorphy of r. The automorphy of r then follows by an argument as in [Har07] (although we employ a ersion of this which is ery similar to that used in [BLGHT09]). In order to apply our automorphy lifting theorem, we need to check the local hypotheses. At places not diiding l, these essentially follow from the construction of π, together with a path-connectedness argument, and a check (using the Ramanujan conjecture) that a certain point on a local lifting ring is smooth. At the places diiding l the argument is rather more inoled. At the places where π is ordinary the hypothesis can be erified (after a suitable base change) using the results of [Ger09]. At the non-ordinary places we proceed more indirectly. For each non-ordinary l we choose two local 2-dimensional l-adic representations ρ and ρ of G F which are induced from characters of quadratic extensions. The representations ρ and ρ are chosen to be congruent to the local Galois representations attached to π, π respectiely, with ρ crystalline of the same Hodge-Tate weights as the local representation attached to π, and ρ non-ordinary and potentially Barsotti-Tate. Then ρ is on the same component of the local crystalline lifting ring as the local representation attached to π, and a similar statement is true for ρ and π after a base change to make it crystalline (using the knowledge of the components of Barsotti-Tate lifting rings mentioned aboe). Since the image of an irreducible component under a continuous map is irreducible, a straightforward argument shows that we need only check that the Galois representations corresponding to the (n 1)-st symmetric powers of ρ and ρ, when tensored with the Galois representations obtained from the characters induced from the CM field, lie on a common component of a crystalline deformation ring (possibly after a base change). We ensure this by choosing our characters so that the two Galois representations are both direct sums of unramified twists of the same crystalline characters, and making a path-connectedness argument. We should note that we hae suppressed some technical details in the aboe outline of our argument; we need to take considerable care to ensure that the hypotheses relating to residual Galois representations haing big image are satisfied. In addition, as mentioned aboe, rather than working with Galois representations alued in fields it is essential to work with fixed lattices throughout. We now outline the structure of the paper. In section 2 we recall some basic notation and definitions from preious papers on automorphy lifting theorems. The automorphy lifting theorem is proed in section 3, together with some results on

5 SATO-TATE 5 the behaiour of local lifting rings under conjugation and functorialities. The most technical section of the paper is section 4, where we construct the characters of CM fields that we need in the main argument. In section 5 we recall arious standard results on base change and automorphic induction, and gie an exposition of Harris s trick in the leel of generality we require. In section 6 we proe a potential automorphy theorem in weight 0; the precise result we require is not in the literature, and while it is presumably clear to the experts how to proe it, we proide the details. Finally, in section 7 we carry out the strategy described aboe, and deduce the Sato-Tate conjecture. We would like to thank Richard Taylor for some helpful discussions related to the content of this paper. We would also like to thank Florian Herzig and Sug Woo Shin for their helpful comments on an earlier draft. 2. Notation If M is a field, we let denote its absolute Galois group. We write all matrix transposes on the left; so t A is the transpose of A. Let ɛ denote the l-adic or mod l cyclotomic character. If M is a finite extension of Q p for some p, we write I M for the inertia subgroup of. If R is a local ring we write m R for the maximal ideal of R. We fix an algebraic closure Q of Q, and regard all algebraic extensions of Q as subfields of Q. For each prime p we fix an algebraic closure Q p of Q p, and we fix an embedding Q Q p. In this way, if is a finite place of a number field F, we hae a homomorphism G F G F. We will use some of the notation and definitions of [CHT08] without comment. In particular, we will use the notions of RACSDC and RAESDC automorphic representations, for which see sections 4.2 and 4.3 of [CHT08]. We will also use the notion of a RAECSDC automorphic representation, for which see section 1 of [BLGHT09]. If π is a RAESDC automorphic representation of GL n (A F ), F a totally real field, and ι : Q l C, then we let rl,ι (π) : G F GL n (Q l ) denote the corresponding Galois representation. Similarly, if π is a RAECSDC or RACSDC automorphic representation of GL n (A F ), F a CM field (in this paper, all CM fields are totally imaginary), and ι : Q l C, then we let rl,ι (π) : G F GL n (Q l ) denote the corresponding Galois representation. For the properties of r l,ι (π), see Theorems 1.1 and 1.2 of [BLGHT09]. If K is a finite extension of Q p for some p, we will let rec K be the local Langlands correspondence of [HT01], so that if π is an irreducible complex admissible representation of GL n (K), then rec K (π) is a Weil-Deligne representation of the Weil group W K. If K is an archimedean local field, we write rec K for the local Langlands correspondence of [Lan89]. We will write rec for rec K when the choice of K is clear. 3. An automorphy lifting theorem 3.1. The group G n. Let n be a positie integer, and let G n be the group scheme oer Z which is the semidirect product of GL n GL 1 by the group {1, j}, which acts on GL n GL 1 by j(g, µ)j 1 = (µ t g 1, µ).

6 6 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY There is a homomorphism ν : G n GL 1 sending (g, µ) to µ and j to 1. Write g 0 n for the trace zero subspace of the Lie algebra of GL n, regarded as a Lie subalgebra of the Lie algebra of G n. Suppose that F is an imaginary CM field with totally real subfield F +. If R is a ring and r : G F + G n (R) is a homomorphism with r 1 (GL n (R) GL 1 (R)) = G F, we will make a slight abuse of notation and write r GF (respectiely r GFw for w a place of F ) to mean r GF (respectiely r GFw ) composed with the projection GL n (R) GL 1 (R) GL n (R) l-adic automorphic forms on unitary groups. Let F + denote a totally real number field and n a positie integer. Let F/F + be a totally imaginary quadratic extension of F + and let c denote the non-triial element of Gal(F/F + ). Suppose that the extension F/F + is unramified at all finite places. Assume that n[f + : Q] is diisible by 4. Under this assumption, we can find a reductie algebraic group G oer F + with the following properties: G is an outer form of GL n with G /F = GLn/F ; for eery finite place of F +, G is quasi-split at ; for eery infinite place of F +, G(F + ) = U n (R). We can and do fix a model for G oer the ring of integers O F + of F + as in section 2.1 of [Ger09]. For each place of F + which splits as ww c in F there is a natural isomorphism ι w : G(F + ) GL n (F w ) which restricts to an isomorphism between G(O F + ) and GL n (O Fw ). If is a place of F + split oer F and w is a place of F diiding, then we let Iw(w) denote the subgroup of GL n (O Fw ) consisting of matrices which reduce to an upper triangular matrix modulo w; U 0 (w) denote the subgroup of GL n (O Fw ) consisting of matrices whose last row is congruent to (0,..., 0, ) modulo w; U 1 (w) denote the subgroup of U 0 (w) consisting of matrices whose last row is congruent to (0,..., 0, 1) modulo w. Let l > n be a prime number with the property that eery place of F + diiding l splits in F. Let S l denote the set of places of F + diiding l. Let K be an algebraic extension of Q l in Q l such that eery embedding F Q l has image contained in K. Let O denote the ring of integers in K and k the residue field. Let S l denote the set of places of F + diiding l and for each S l, let ṽ be a place of F oer. Let W be an O-module with an action of G(O F +,l). Let V G(A F ) be a + compact open subgroup with l G(O F +,l) for all V, where l denotes the projection of to G(F + l ). We let S(V, W ) denote the space of l-adic automorphic forms on G of weight W and leel V, that is, the space of functions f : G(F + )\G(A F +) W with f(g) = 1 l f(g) for all V. Let Ĩl denote the set of embeddings F K giing rise to one of the places ṽ. Let (Z n +) e I l denote the set of λ (Z n ) e I l with λ τ,1 λ τ,2... λ τ,n for all embeddings τ Ĩl. To each λ (Z n +) e I l we associate a finite free O-module M λ with a continuous action of G(O F +,l) as in Definition of [Ger09]. The representation M λ is the tensor product oer τ Ĩl of the irreducible algebraic representations of

7 SATO-TATE 7 GL n of highest weights gien by the λ τ. We write S λ (V, O) instead of S(V, M λ ) and similarly for any O-module A, we write S λ (V, A) for S(V, M λ O A). Assume from now on that K is a finite extension of Q l. Let l denote the product of all places in S l. Let R and S a denote finite sets of finite places of F + disjoint from each other and from S l and consisting only of places which split in F. Assume that each S a is unramified oer a rational prime p with [F (ζ p ) : F ] > n. Let T = S l R Sa. For each T fix a place ṽ of F diiding, extending the choice of ṽ for S l. Let U = U be a compact open subgroup of G(A F + ) such that U = G(O F + ) if R S a splits in F ; U = ι 1 e ker(gl n (O Fe ) GL n (k(ṽ))) if S a ; U is a hyperspecial maximal compact subgroup of G(F + ) if is inert in F. We note that if S a is non-empty then U is sufficiently small (which means that its projection to G(F + ) for some place F + contains no finite order elements other than the identity). For any O-algebra A, the space S λ (U, A) is acted upon by the Hecke operators T (j) w := ι 1 w ([ GL n (O Fw ) ( ) ϖw 1 j 0 GL 0 1 n (O Fw ) n j for w a place of F, split oer F + and not lying oer T, j = 1,..., n and ϖ w a uniformizer in O Fw. We let T T λ (U, A) be the A-subalgebra of End A(S λ (U, A)) generated by these operators and the operators (T w (n) ) 1. To any maximal ideal m of T T λ (U, O) one can associate a continuous representation r m : G F GL n (T T λ (U, O)/m) characterised by the following properties: (1) r c m = r mɛ 1 n ; (2) r m is unramified outside T. If T is a place of F + which splits as ww c in F and Frob w is the geometric Frobenius element of G Fw /I Fw, then r m (Frob w ) has characteristic polynomial X n ( 1) j (Nw) j(j 1)/2 T w (j) X n j ( 1) n (Nw) n(n 1)/2 T w (n). The maximal ideal m is said to be non-eisenstein if r m is absolutely irreducible. In this case, r m can be extended to a homomorphism r m : G F + G n (T T λ (U, O)/m) (in the sense that r m GF = ( r m, ɛ 1 n )) with r m 1 ((GL n GL 1 )(T T λ (U, O)/m)) = G F. Also, any such extension has a continuous lifting with the following properties: r m : G F + G n (T T λ (U, O) m ) (0) rm 1 ((GL n GL 1 )(T T λ (U, O) m)) = G F. (1) ν r m = ɛ 1 n δ µm F/F where δ + F/F + is the non-triial character of Gal(F/F + ) and µ m Z/2Z. (2) r m is unramified outside T. If T is a place of F + which splits as ww c in F and Frob w is the geometric Frobenius element of G Fw /I Fw, then r m (Frob w ) has characteristic polynomial X n ( 1) j (Nw) j(j 1)/2 T w (j) X n j ( 1) n (Nw) n(n 1)/2 T w (n). ])

8 8 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY (3) If S l, and ζ : T T λ (U, O) Q l is a homomorphism of O-algebras, then ζ r m GFe is crystalline of l-adic Hodge type λe (in the sense of Definition below) Local deformation rings. Let l be a prime number and K be a finite extension of Q l with residue field k and ring of integers O, and write m O for the maximal ideal of O. Let C O be the category of complete local Noetherian O-algebras with residue field isomorphic to k ia the structural homomorphism. As in section 3 of [BLGHT09], we consider an object R of C O to be geometrically integral if for all finite extensions K /K, the algebra R O O K is an integral domain. Let M be a finite extension of Q p for some prime p possibly equal to l and let ρ : GL n (k) be a continuous homomorphism. Then the functor from C O to Sets which takes A C O to the set of continuous liftings ρ : GL n (A) of ρ is represented by a complete local Noetherian O-algebra Rρ. We call this ring the uniersal O-lifting ring of ρ. We write ρ : GL n (Rρ ) for the uniersal lifting. The following definitions will proe to be useful later. Definition Suppose ρ : GL 2 (O) is a representation. We can think of this as putting a -action on the ector space K 2 (=V, say), in a way that stabilizes the lattice gien by the standard basis {e 0, e 1 }, where e 0 = 1, 0, e 1 = 0, 1. Considering Sym n 1 ρ as a quotient of V (n 1), we hae an ordered basis {g 0,..., g n 1 } of Sym n 1 V, where g i is the image of e (n 1 i) 0 e i 1. We call this the O-basis of Sym n 1 V inherited from our original basis in ρ. Definition Suppose ρ : GL n (O), ρ : GL n (O) are representations, which we think of as putting -actions on the ector spaces V ρ = K n, V ρ = K n in a way that stabilizes the standard basis of each. In this situation we hae an ordered O-basis on V ρ O V ρ gien by the ectors e j f k, ordered lexicographically, where the e j are the standard O-basis in V ρ and the f k are the standard basis in V ρ. We call this the O-basis of ρ O ρ inherited from our original bases Local deformations (p = l case). Suppose that p = l. In this section we will define an equialence relation on crystalline lifts of ρ. For this, we need to consider certain quotients of Rρ. Assume that K contains the image of eery embedding M K. Definition Let (Z n +) Hom(M,K) denote the subset of (Z n ) Hom(M,K) consisting of elements λ which satisfy for eery embedding τ. λ τ,1 λ τ,2... λ τ,n Let λ be an element of (Z n +) Hom(M,K). We associate to λ an l-adic Hodge type λ in the sense of section 2.6 of [Kis08] as follows. Let D K denote the ector space K n. Let D K,M = D K Ql M. For each embedding τ : M K, we let D K,τ = D K,M K M,1 τ K so that D K,M = τ D K,τ. For each τ choose a decreasing filtration Fil i D K,τ of D K,τ so that dim K gr i D K,τ = 0 unless i = (j 1) + λ τ,n j+1 for some j = 1,..., n in which case dim K gr i D K,τ = 1. We define a decreasing

9 SATO-TATE 9 filtration of D K,M by K Ql M-submodules by setting Fil i D K,M = τ Fil i D K,τ. Let λ = {D K, Fil i D K,M }. We now recall some results of Kisin. Let λ be an element of (Z n +) Hom(M,K) and let λ be the associated l-adic Hodge type. Definition If B is a finite K-algebra and V B is a free B-module of rank n with a continuous action of that makes V B into a de Rham representation, then we say that V B is of l-adic Hodge type λ if for each i there is an isomorphism of B Ql M-modules gr i (V B Ql B dr ) GM (gr i D K,M ) K B. Corollary of [Kis08] implies that there is a unique l-torsion free quotient R λ,cr ρ of Rρ with the property that for any finite K-algebra B, a homomorphism of O-algebras ζ : Rρ B factors through R λ,cr ρ if and only if ζ ρ is crystalline of l-adic Hodge type λ. Moreoer, Theorem of [Kis08] implies that Spec R λ,cr ρ [1/l] is formally smooth oer K and equidimensional of dimension n n(n 1)[M : Q l]. By Lemma of [Ger09] there is a quotient R λ,cr ρ of R λ,cr ρ corresponding to a union of irreducible components such that for any finite extension E of K, a homomorphism of O-algebras ζ : R λ,cr ρ E factors through R λ,cr ρ if and only if ζ ρ is crystalline and ordinary of weight λ. We now introduce an equialence relation on continuous representations GL n (O) lifting ρ. Definition Suppose that ρ 1, ρ 2 : GL n (O) are two continuous lifts of ρ. Then we say that ρ 1 ρ 2 if the following hold. (1) There is a λ (Z n +) Hom(M,K) such that ρ 1 and ρ 2 both correspond to points of R λ,cr ρ (that is, ρ 1 O K and ρ 2 O K are both crystalline of l-adic Hodge type λ ). (2) For eery minimal prime ideal of R λ,cr ρ, the quotient R λ,cr ρ / is geometrically integral. (3) ρ 1 and ρ 2 gie rise to closed points on a common irreducible component of Spec R λ,cr ρ [1/l]. In (3) aboe, note that the irreducible component is uniquely determined by either of ρ 1, ρ 2 because Spec R λ,cr ρ [1/l] is formally smooth. Note also that we can always ensure that (2) holds by replacing O with the ring of integers in a finite extension of K. Suppose that ρ 1 ρ 2 as aboe and let M /M be a finite extension. Assume that K contains the image of eery embedding M K. Then we claim that ρ 1 GM ρ 2 GM. Indeed, let λ be such that ρ 1 and ρ 2 hae l-adic Hodge type λ. Define λ (Z n +) Hom(M,K) by λ τ = λ τ M for all τ : M K. Then restriction to gies rise to an O-algebra homomorphism Rρ GM R λ,cr ρ which factors through R λ,cr ρ GM (using the fact that R λ,cr ρ is reduced and l-torsion free). The result now follows from the formal smoothness of Spec R λ,cr ρ GM [1/l], which implies

10 10 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY that the image of any irreducible component of Spec R λ,cr ρ [1/l] is contained in a unique irreducible component of Spec R λ,cr ρ GM [1/l]. In a similar ein, it follows that if n = 2 and ρ 1 ρ 2, then Sym k 1 ρ 1 Sym k 1 ρ 2 for all k 1, where we take the O-basis on the Sym k 1 ρ i inherited from the bases we hae on the ρ i, in the sense of Definition We will make one final ariation on this theme. Suppose ρ : GL m (O) is crystalline of l-adic Hodge type λ for some m and some λ (Z m + ) Hom(M,K), and ρ 1 ρ 2 are as aboe. (Note n need no longer be 2.) Then ρ 1 O ρ ρ 2 O ρ, where we take as O-basis on the ρ i O ρ the inherited bases in the sense of Definition Lemma Let ρ : GL n (k) be a continuous homomorphism. Suppose ρ 1, ρ 2 : GL n (O) are two lifts of ρ with ρ 1 ρ 2. If O denotes the ring of integers in a finite extension of K with residue field k, then ρ 1 ρ 2, regarded as lifts of ρ k k to GL n (O ). Proof. Let λ (Z n +) Hom(M,K) be such that ρ 1 and ρ 2 hae l-adic Hodge type λ. Let R = R λ,cr ρ and R = R O O. We need to show that ρ 1 and ρ 2 gie rise to closed points of Spec R [1/l] lying on a common component. Note that if C is an irreducible component of Spec R [1/l], then the image of C in Spec R[1/l] is an irreducible component. Indeed, the image of C in Spec R[1/l] is irreducible and closed (as R R is finite). If x is a closed point of Spec R [1/l] lying in C with image x in Spec R[1/l], then the completed local rings of Spec R [1/l] and Spec R[1/l] at x and x respectiely are isomorphic. We deduce that the image of C has the same dimension as C and hence is an irreducible component. Now, let x 1 and x 2 denote the closed points of Spec R[1/l] corresponding to ρ 1 and ρ 2 and let C denote the irreducible component of Spec R[1/l] containing x 1 and x 2. Then we claim that the preimage of C in Spec R [1/l] is irreducible. Indeed, suppose there are two distinct irreducible components C and C of Spec R [1/l] mapping to C. Then there are points x 1 and x 1 of C and C respectiely mapping to x 1. Howeer, the preimage of x 1 in Spec R [1/l] consists of a single point (let m denote the maximal ideal of R[1/l] corresponding to x 1. Then the fibre oer x 1 is gien by the spectrum of (R[1/l]/m) O O = K O O = K.) Thus x 1 = x 1 lies in the intersection of C and C, contradicting the formal smoothness of Spec R [1/l] Local deformations (p l case). Suppose now that p l. By Theorem of [Gee06a], Spec R ρ [1/l] is equidimensional of dimension n2. Definition Let ρ 1, ρ 2 : GL n (O) be two lifts of ρ. We say that ρ 1 O ρ 2 if the following hold. (1) For each minimal prime ideal of R, the quotient R / is geometrically irreducible. (2) ρ 1 corresponds to a closed point of Spec Rρ [1/l] which is contained in a unique irreducible component and this irreducible component also contains the closed point corresponding to ρ 2. We remark that, we can always replace O by the ring of integers in a finite extension of K so that condition (1) aboe holds. Also, condition (1) ensures that if ρ 1 O ρ 2 and if O is the ring of integers in a finite extension of K then ρ 1 O ρ 2.

11 SATO-TATE Properties of O and. Lemma is an equialence relation. Proof. This follows immediately from the definitions. Lemma Let M be a finite extension of Q p for some prime p. Let ρ : GL n (k) be a continuous homomorphism. If p l, let R = Rρ. If p = l, assume that K contains the image of each embedding M Q l and let R = R λ,cr ρ for some λ (Z n +) Hom(M,K). Let O denote the ring of integers in a finite extension of K. Let ρ and ρ be two lifts of ρ to O giing rise to closed points of Spec R[1/l]. Suppose that after conjugation by an element of ker(gl n (O ) GL n (O /m O )) they differ by an unramified twist. Then an irreducible component of Spec R[1/l] contains ρ if and only if it contains ρ. Proof. The uniersal unramified O-lifting ring of the triial character k is gien by O[[Y ]] where the uniersal lift χ sends Frob M to 1 + Y. Let R[[Y, X]] = R[[Y ]][[X ij : 1 i, j n]]. Let ρ denote the uniersal lift of ρ to R. Consider the lift (1 n + (X ij ))ρ (1 n + (X ij )) 1 χ of ρ to R[[Y, X]]. This lift gies rise to a homomorphism Rρ R[[Y, X]] which factors through R. Let α denote the resulting O-algebra homomorphism R R[[Y, X]]. Let ι : R R[[Y, X]] be the standard R-algebra structure on R[[Y, X]]. The minimal prime ideals of R[[Y, X]] and R are in natural bijection (if is a minimal prime of R then ι( ) generates a minimal prime of R[[Y, X]]). Let be a minimal prime of R. We claim that the kernel of the map β : R R[[Y, X]]/ι( ) = (R/ )[[Y, X]] induced by α is. Indeed, the R-algebra homomorphism (with R[[Y, X]] considered as an R-algebra ia ι) γ : R[[Y, X]] R which sends Y and each X ij to 0 is a section to the map β. The composition R β γ (R/ )[[Y, X]] R/ is thus the natural reduction map. In particular its kernel is. Since ker(β) ker(γ β) = and is minimal, we deduce ker(β) =. The lemma follows. Lemma Let M be a finite extension of Q l. Let ρ : GL n (k) be the triial representation, and let ρ and ρ : GL n (O) be two crystalline lifts of ρ of l-adic Hodge type λ which are GL n (O)-conjugate. Then ρ ρ. Proof. Take g GL n (O) with ρ = gρg 1. Let A = O X ij, Y /(Y det(x ij ) 1) where O X ij, Y is the m O -adic completion of O[X ij, Y ]. Let ρ A : GL n (A) be gien by XρX 1, where X is the matrix (X ij ). By Lemma of [Ger09], there is a continuous homomorphism Rρ A such that ρ A is the push-forward of the uniersal lifting ρ : GL n (Rρ ). Now, for any Q l-point of A, the corresponding specialisation of ρ A is a Q l -conjugate of ρ, and is thus crystalline of l-adic Hodge type λ, so corresponds to a Q l -point of R λ,cr ρ. Since the Q l -points of A are dense in Spec A, we conclude that the homomorphism Rρ A factors through R λ,cr ρ. Now, Spec A is irreducible, and the points x and x of Spec R λ,cr ρ corresponding to ρ and ρ respectiely are in the image of the map Spec A Spec R λ,cr ρ, because they correspond to specialising the matrix X to the matrices 1 n and g respectiely. The result follows.

12 12 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY Corollary Let M be a finite extension of Q l. Let ρ : GL n (k) be the triial representation, and let ρ, ρ : GL n (O) be two crystalline lifts of ρ which are both GL n (O)-conjugate to direct sums of unramified twists of a common set of crystalline characters. Then ρ ρ. Proof. After applying Lemma 3.4.3, we may assume that ρ = n i=1ρ i and ρ = n i=1ρ i where ρ i and ρ i are crystalline characters O which differ by an unramified twist for each i and reduce to the triial character modulo m O. It suffices to check that the corresponding points x and x of R λ,cr ρ are path-connected. As in the proof of Lemma 3.4.2, the uniersal unramified O-lifting ring of the triial character k is gien by O[[Y ]] with the uniersal lifting χ sending Frob M to 1 + Y. Taking n copies of this character, we obtain a lifting n i=1ρ i χ i of ρ to O[[Y 1,..., Y n ]], and thus a continuous map Spec O[[Y 1,..., Y n ]] Spec R λ,cr Both x and x are in the image of this map, so the result follows. The following is Lemma of [Ger09]. ρ. Lemma Suppose M is a finite extension of Q l and ρ : GL n (k) is the triial representation. If the ring R λ,cr ρ is non-zero, then it is irreducible. Lemma Let M be a finite extension of Q p for some prime p and let ρ : GL n (k) be a continuous homomorphism. Let ρ 1, ρ 2 : GL n (O) be two lifts of ρ. If p l, suppose that ρ 1 O ρ 2 and ρ 2 O ρ 1. If p = l, assume that ρ 1 ρ 2. Let χ 1, χ 2 : O be continuous characters with χ 1 = χ 2 and χ 1 IM = χ 2 IM. Suppose in addition that if p = l then χ 1 and χ 2 are crystalline. Then χ 1 ρ 1 O χ 2 ρ 2 if p l and χ 1 ρ 1 χ 2 ρ 2 if p = l. Proof. We treat the case p l, the other case being similar. Let χ = χ 1 = χ 2. Then the operation of twisting by χ 1 defines an isomorphism of the lifting problems of ρ and χρ. It therefore defines an isomorphism Rχρ Rρ. It follows that χ 1 ρ 1 O χ 1 ρ 2 and that χ 1 ρ 2 gies rise to a closed point of Spec Rχρ [1/l] lying on a unique irreducible component. Since χ 1 and χ 2 differ by a residually triial unramified twist, an easy argument shows that this component also contains χ 2 ρ 2 (c.f. the proof of Corollary 3.4.4). It follows that χ 1 ρ 1 O χ 2 ρ Global deformation rings. Let F/F + be a totally imaginary quadratic extension of a totally real field F +. Let c denote the non-triial element of Gal(F/F + ). Let k denote a finite field of characteristic l and K a finite extension of Q l, inside our fixed algebraic closure Q l, with ring of integers O and residue field k. Assume that K contains the image of eery embedding F Q l and that the prime l is odd. Assume that eery place in F + diiding l splits in F. Let S denote a finite set of finite places of F + which split in F, and assume that S contains eery place diiding l. Let S l denote the set of places of F + lying oer l. Let F (S) denote the maximal extension of F unramified away from S. Let G F +,S = Gal(F (S)/F + ) and G F,S = Gal(F (S)/F ). For each S choose a place ṽ of F lying oer and

13 SATO-TATE 13 let S denote the set of ṽ for S. For each place of F + we let c denote a choice of a complex conjugation at in G F +,S. For each place w of F we hae a G F,S -conjugacy class of homomorphisms G Fw G F,S. For S we fix a choice of homomorphism G Fe G F,S. Fix a continuous homomorphism r : G F +,S G n (k) such that G F,S = r 1 (GL n (k) GL 1 (k)) and fix a continuous character χ : G F +,S O such that ν r = χ. Assume that r GF,S is absolutely irreducible. As in Definition of [CHT08], we define a lifting of r to an object A of C O to be a continuous homomorphism r : G F +,S G n (A) lifting r and with ν r = χ; two liftings r, r of r to A to be equialent if they are conjugate by an element of ker(gl n (A) GL n (k)); a deformation of r to an object A of C O to be an equialence class of liftings. Similarly, if T S, we define a T -framed lifting of r to A to be a tuple (r, {α } T ) where r is a lifting of r and α ker(gl n (A) GL n (k)) for T ; two T -framed liftings (r, {α } T ), (r, {α } T ) to be equialent if there is an element β ker(gl n (A) GL n (k)) with r = βrβ 1 and α = βα for T ; a T -framed deformation of r to be an equialence class of T -framed liftings. For each place S, let R r GFe denote the uniersal O-lifting ring of r GFe and let R e denote a quotient of R r GFe which satisfies the following property: (*) let A be an object of C O and let ζ, ζ : R r GFe A be homomorphisms corresponding to two lifts r and r of r GFe which are conjugate by an element of ker(gl n (A) GL n (k)). Then ζ factors through R e if and only if ζ does. We consider the deformation problem S = (F/F +, S, S, O, r, χ, {R e } S ) (see sections 2.2 and 2.3 of [CHT08] for this terminology). We say that a lifting r : G F +,S G n (A) is of type S if for each place S, the homomorphism R r GFe A corresponding to r GFe factors through R e. We also define deformations of type S in the same way. Let Def S be the functor C O Sets which sends an algebra A to the set of deformations of r to A of type S. Similarly, if T S, let Def T S be the functor C O Sets which sends an algebra A to the set of T -framed liftings of r to A which are of type S. By Proposition of [CHT08] these functors are represented by objects R uni S and R T S respectiely of C O. Lemma Let M be a finite extension of Q p for some prime p. Let ρ : GL n (k) be a continuous homomorphism. If p l, let R be the maximal l-torsion free quotient of Rρ. If p = l, assume that K contains the image of each embedding M Q l and let R = R λ,cr ρ for some λ (Z n +) Hom(M,K). Then R satisfies property ( ) aboe.

14 14 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY Proof. This can be proed in exactly the same way as Lemma Automorphy lifting CM Fields. Theorem Let F be an imaginary CM field with totally real subfield F + and let c be the non-triial element of Gal(F/F + ). Let n Z 1 and let l > n be a prime. Let K Q l denote a finite extension of Q l with ring of integers O and residue field k. Assume that K contains the image of eery embedding F Q l. Let ρ : G F GL n (O) be a continuous representation and let ρ = ρ mod m O. Suppose that ρ enjoys the following properties: (1) ρ c = ρ ɛ 1 n. (2) The reduction ρ is absolutely irreducible and ρ(g F (ζl )) GL n (k) is big (see Definition 4.1.1). (3) (F ) ker ad ρ does not contain ζ l. (4) There is a continuous representation ρ : G F GL n (O), a RACSDC automorphic representation π of GL n (A F ) which is unramified aboe l and ι : Q l C such that (a) ρ O Q l = rl,ι (π) : G F GL n (Q l ). (b) ρ = ρ. (c) For all places l of F, either ρ GF and π are both unramified, or ρ GF O ρ GF. (d) For all places l, ρ GF ρ GF. Then ρ is automorphic. Proof. Choose a place 1 of F not diiding l such that 1 is unramified oer a rational prime p with [F (ζ p ) : F ] > n; 1 does not split completely in F (ζ l ); ρ and π are unramified at 1 ; ad ρ(frob 1 ) = 1. Extending O if necessary, choose an imaginary CM field L/F such that: L/F is solable; L is linearly disjoint from F ker r (ζ l ) oer F ; 4 [L + : F + ] where L + denotes the maximal totally real subfield of L; L/L + is unramified at all finite places; Eery prime of L diiding l is split oer L + and eery prime where ρ GL or π L ramifies is split oer L + (here π L denotes the base change of π to L); Eery place of L oer 1 or c 1 is split oer L +. Moreoer, 1 and c 1 split completely in L; Eery place l of F splits completely in L; If l is a place of F and at least one of ρ GF or π is ramified, then splits completely in L. Let G be an algebraic group as in section 3.2 (with F/F + /OL replaced by + L/L + ). By Théorème 5.4 and Corollaire 5.3 of [Lab09] there exists an automorphic representation Π of G(A L +) such that π L is a strong base change of Π. Let S l

15 SATO-TATE 15 denote the set of places of L + diiding l and let R denote the set of places of L + not diiding l and lying under a place of L where ρ or π L is ramified. Let S a denote the set of places of L + lying oer the restriction of 1 to F +. Let T = S l R Sa. For each place T, choose a place ṽ of L lying oer it and let T denote the set of ṽ for T. Let U = U G(A L + ) be a compact open subgroup such that U = G(O L + ) for S l and for T split in L; U is a hyperspecial maximal compact subgroup of G(L + ) for each inert in L; U is such that Π U {0} for R; U = ker(g(o L + ) G(k )) for S a. Extend K if necessary so that it contains the image of eery embedding L Q l. For each S l, let λ e be the element of (Z n +) Hom(L e,k) with the property that ρ GLe and ρ GLe hae l-adic Hodge type λe. Let Ĩl denote the set of embeddings L K giing rise to one of the places ṽ. Let λ = (λ e ) Sl regarded as an element of (Z n +) e I l in the eident way and let S λ (U, O) be the space of l-adic automorphic forms on G of weight λ introduced aboe. Let T T λ (U, O) be the O-subalgebra of End O(S λ (U, O)) ) 1 for w a place of L split oer L +, not lying oer T and j = 1,..., n. The eigenalues of the operators T w (j) on the space (ι 1 Π ) U gie rise to a homomorphism of O-algebras T T λ (U, O) Q l. Extending K if necessary, we can and do assume that this homomorphism takes alues in O. Let m denote the unique maximal ideal of T T λ (U, O) containing the kernel of this generated by the Hecke operators T (j) w, (T (n) w homomorphism. Let δ L/L + be the quadratic character of G L + corresponding to L. By Lemma of [CHT08] we can and do extend ρ and ρ to homomorphisms r, r : G L + G n (O) with r k = r k : G L + G n (k), r GL = (ρ GL, ɛ 1 n ), r GL = (ρ GL, ɛ 1 n ) and ν r = ν r = ɛ 1 n δ µ L/L + for some µ (Z/2Z). Let r = r k : G L + G n (k). For R S a, let R r GLe denote the maximal l-torsion free quotient of R r GLe. Note that R r GLe so R r GLe S := is formally smooth oer O for S a by Lemma of [CHT08] = R r GLe. Consider the deformation problem ( L/L +, T, T, O, r, ɛ 1 n δ µ L/L +, {R λ e,cr r GLe } Sl {R r GLe } R Sa ). By Lemma 3.5.1, the rings R λ e,cr r GLe for S l and R r GLe for R satisfy the property (*) of section 3.5. Let RS uni be the object representing the corresponding deformation functor. Note that r GL is GL n (k)-conjugate to r m where r m is the representation associated to the maximal ideal m of T T λ (U, O) in section 3.2. After conjugating we can and do assume that r GL = r m. Since m is non-eisenstein, we hae as aboe a continuous lift r m : G L + G n (T T λ (U, O) m ) of r. Properties (0)-(3) of r m and the fact that T T λ (U, O) m is l-torsion free and reduced imply that r m is of type S. Hence r m gies rise to an O-algebra homomorphism T T λ (U, O) m R uni S

16 16 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY (which is surjectie by property (2) of r m ). To proe the theorem it suffices to show that the homomorphism RS uni O corresponding to r factors through T T λ (U, O) m. We define ( ) ) R loc := Sl R λ e,cr r GLe ( R Sa R r GLe where all completed tensor products are taken oer O. Note that R loc is equidimensional of dimension 1 + n 2 #T + [L + : Q]n(n 1)/2 by Lemma 3.3 of [BLGHT09]. Sublemma. There are integers q, g Z 0 with R r GLe 1 + q + n 2 #T = dim R loc + g and a module M for both R := R loc [[x 1,..., x g ]] and S := O[[z 1,..., z n 2 #T, y 1,..., y q ]] such that: (1) M is finite and free oer S. (2) M /(z i, y j ) = S λ (U, O) m. (3) The action of S on M can be factored through an O-algebra homomorphism S R. (4) There is a surjection R RS uni whose kernel contains all the z i and y j which is compatible with the actions of R /(z i, y j ) and RS uni on M /(z i, y j ) = S λ (U, O) m. Moreoer, there is a lift rs uni : G L + G n (RS uni ) of r representing the uniersal deformation so that for each T, the composite R loc R RS uni arises from the lift rs uni GLe. Assuming the sublemma for now, let us finish the proof of the theorem. Since R is equidimensional of dimension dim R loc +g, it follows from (1) and (3) that the support of M in R is a union of irreducible components. (Indeed by Lemma 2.3 of [Tay08] it is enough to check that the m R -depth of M is equal to dim R = dim S. By (3) it is enough to check the same statement for the m S -depth, and this is immediate from (1)). The conjugacy class of r determines a homomorphism O so that r is ker(gl n (O) GL n (k))-conjugate to ζ rs uni. By lies on a unique irreducible component C e of ζ : RS uni the choice of L, for each R, r GLe Spec R r GLe [1/l]. By Lemma 3.4.2, C e is also the unique irreducible component of Spec R r GLe containing ζ rs uni GLe. For l, a similar argument shows that ζ rs uni GLe and r GLe lie on the same irreducible component C e of Spec R λ e,cr [1/l]. r GLe The conjugacy class of r determines a homomorphism ζ : RS uni O so that r is ker(gl n (O) GL n (k))-conjugate to ζ rs uni. By Lemma 3.4.2, the set of irreducible components of Spec R r GLe [1/l] containing ζ rs uni GLe is equal to the set of components containing r GLe. By the choice of L it follows that C e contains ζ rs uni GLe. A similar argument, using part (d) of assumption (4) of the theorem, shows that C e contains ζ rs uni GLe for l. By part 5 of Lemma 3.3 of [BLGHT09] the irreducible components C e for S l R determine an irreducible component C r of Spec R (as mentioned aboe, is formally smooth oer O for S a ). Moreoer, ζ composed with the R r GLe surjection R RS uni of part (4) of the sublemma gies rise to a closed point of Spec R [1/l] which is in the support of M and which lies in C r but does not lie in any other irreducible component of Spec R. We deduce that C r is in the support

17 SATO-TATE 17 of M. Since the closed point of Spec R [1/l] corresponding to ζ lies in C r it is also in the support of M and we are done by assertion (4) of the sublemma. Proof of sublemma. We apply the Taylor-Wiles-Kisin patching method. Let q 0 = [L + : Q]n(n 1)/2 + [L + : Q]n(1 ( 1) µ n )/2. If (Q, Q, {ψ e } Q ) is a triple where Q is a finite set of places of L + disjoint from T and consisting of places which split in L; Q consists of one place ṽ of L oer each place Q; for each Q, r GLe = ψe s e where dim ψ e = 1 and ψ e is not isomorphic to any subquotient of s e ; then for each Q, let R ψ e r GLe denote the quotient of R r GLe corresponding to lifts r : G Le GL n (A) which are ker(gl n (A) GL n (k))-conjugate to a lift of the form ψ s where ψ lifts ψ e and s is an unramified lift of s e. We then introduce the deformation problem ( ) S Q = L/L +, T Q, T Q, O, r, ɛ 1 n δ µ L/L, {R λ e,cr + r GLe } Sl {R r GLe } R Sa {R ψ e r GLe } Q. We define deformations (resp. T -framed deformations) of r of type S Q in the eident manner and let RS uni Q (resp. R T S Q ) denote the uniersal deformation ring (resp. T - framed deformation ring) of type S Q. By Proposition of [CHT08] we can and do choose an integer q q 0 and for each N Z 1 a tuple (Q N, Q N, {ψ e } QN ) as aboe with the following additional properties: #Q N = q for all N; N 1 mod l N for Q N ; the ring R T S QN can be topologically generated oer R loc by q q 0 = q [L + : Q]n(n 1)/2 [L + : Q]n(1 ( 1) µ n )/2 elements. For each N 1, let U 1 (Q N ) = U 1(Q N ) and U 0 (Q N ) = U 0(Q N ) be the compact open subgroups of G(A L ) with U + i (Q N ) = U for Q N, i = 0, 1 and U i (Q N ) = ι 1 e U i(ṽ) for Q N, i = 0, 1. Note that we hae natural maps T T Q N λ (U 1 (Q N ), O) T T Q N λ (U 0 (Q N ), O) T T Q N λ (U, O) T T λ (U, O). Thus m determines maximal ideals of the first three algebras in this sequence which we denote by m QN for the first two and m for the third. Note also that T T Q N λ (U, O) m = T T λ (U, O) m by the proof of Corollary of [CHT08]. For each Q N choose an element φ e G Le lifting geometric Frobenius and let ϖ e O Le be the uniformiser with Art Le ϖ e = φ e L ab e. Let P e (X) T T Q N λ (U 1 (Q N ), O) mqn [X] denote the characteristic polynomial of r mqn (φ e ). By Hensel s lemma, we can factor P e (X) = (X A e )Q e (X) where A e lifts ψ e (φ e ) and Q e (A e ) is a unit in T T Q N λ (U 1 (Q N ), O) mqn. For i = 0, 1 and α L e of non-negatie aluation, consider the Hecke operator ([ ( ) ]) V α := ι 1 1n 1 0 e U i (ṽ) U 0 α i (ṽ)

18 18 THOMAS BARNET-LAMB, TOBY GEE, AND DAVID GERAGHTY on S λ (U i (Q N ), O) mqn. Let G QN = U 0 (Q N )/U 1 (Q N ) and let QN denote the maximal l-power order quotient of G QN. Let a QN denote the kernel of the augmentation map O[ QN ] O. For i = 0, 1, let H i,qn = Q e (V ϖe )S λ (U i (Q N ), O) mqn. Q N and let T i,qn denote the image of T T Q N λ (U i (Q N ), O) in End O (H i,qn ). Let H = S λ (U, O) m. We claim that the following hold: (1) For each N, the map Q N Q e (V ϖe ) : H H 0,QN is an isomorphism. (2) For each N, H 1,QN is free oer O[ QN ] with H 1,QN /a QN H0,QN. (3) For each N and each Q N, there is a character with open kernel V e : L e T 1,Q N so that (a) for each α L e of non-negatie aluation, V α = V e (α) on H 1,QN ; (b) (r mqn T 1,QN ) WLe = s (Ve Art 1 L e ) with s unramified, lifting s e and (V e Art 1 L e ) lifting ψ e. To see this, note that Lemmas and of [CHT08] imply that P e (V ϖe ) = 0 on S λ (U 1 (Q N ), O) mqn. Property (1) now follows from Lemma of [CHT08] together with Lemma of [CHT08] and the fact that T T Q N λ (U, O) m = T T λ (U, O) m. Property (3) follows exactly as in the proof of part 8 of Proposition of [CHT08]. Note that H 1,QN is a T T λ (U 1(Q N ), O) mqn [G QN ]-direct summand of S λ (U 1 (Q N ), O) mqn. Moreoer, it follows from the fact that U is sufficiently small (see Lemma of [CHT08]) that S λ (U 1 (Q N ), O) mqn is finite free oer O[G QN ] with G QN -coinariants isomorphic to S λ (U 0 (Q N ), O) mqn ia the trace map tr GQN. It follows that H 1,QN has G QN -coinariants isomorphic to H 0,QN ia tr GQN. Finally, note that by (3) the action of α = (α e ) QN G QN on H 1,QN is gien by Q N V e (α e ). Since each ψ e is unramified, the action of G QN on H 1,QN must factor through QN and (2) follows. For each N, the lift r mqn T 1,QN of r is of type S QN and gies rise to a surjection RS uni QN T 1,QN. Thinking of QN as the maximal l-power quotient of Q N I Le, the determinant of any choice of uniersal deformation rs uni QN gies rise to a homomorphism QN (R uni R uni S QN R T S. Let R T S QN S QN ). We thus hae homomorphisms O[ QN ] = R uni S and R T S QN /a QN = and natural isomorphisms R uni S QN /a QN T = O[[X,i,j : T, i, j = 1,..., n]]. Choose a lift rs uni : G L + G n (RS uni ) representing the uniersal deformation. The tuple (rs uni, (1 n + X,i,j ) T ) gies rise to an isomorphism R T S R uni S O T.