Adjoint Selmer groups of automorphic Galois representations of unitary type

Let $\rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic automorphic representation of $\mathrm{GL}_n$ of unitary type. Under very mild hypotheses on $\rho$, we prove the vanishing of the (Bloch--Kato) adjoint Selmer group of $\rho$. We obtain definitive results for the adjoint Selmer groups associated to non-CM Hilbert modular forms and elliptic curves over totally real fields.


Introduction
Context. Let F be a CM number field, with maximal totally real subfield F + . Fix an algebraic closure F of F and a complex conjugation c ∈ Gal(F /F + ). We say that a cuspidal automorphic representation π of GL n (A F ) is of unitary type if it is conjugate self-dual, i.e. if it satisfies the relation π c ∼ = π ∨ . If π is conjugate selfdual and moreover regular algebraic (a condition on π ∞ ), then for any isomorphism ι : Q p → C there is an associated p-adic Galois representation r π,ι : Gal(F /F ) → GL n (Q p ), characterized up to isomorphism by compatibility with the local Langlands correspondence at each finite place of F . The conjugate self-duality of π implies the existence of an isomorphism r c π,ι ∼ = r ∨ π,ι ⊗ ǫ 1−n , where ǫ is the p-adic cyclotomic character.
This paper concerns the adjoint Bloch-Kato Selmer group of such a representation. To define it, we note that if V denotes the space on which r π,ι acts, then the conjugate self-duality of r π,ι is reflected in the existence of a perfect, symmetric, and Galois equivariant bilinear pairing ·, · : V c × V ⊗ ǫ n−1 → Q p .
The existence of this pairing allows us to extend the adjoint action of r π,ι on End(V ) to an action of Gal(F /F + ), where c ∈ Gal(F /F + ) acts by the formula c · X = −X * (and X * is the adjoint with respect to ·, · ).
We are interested in is the Bloch-Kato Selmer group for v ∤ p. General conjectures predict the vanishing of this group (see the introduction of [All16] for a detailed discussion of this in the present context). We are content here to note that this group parameterizes infinitesimal deformations of r π,ι which are at the same time conjugate self-dual and geometric, in the sense of p-adic Hodge theory.
Our results. The following is the main theorem of this paper.
Theorem A. Let F be a CM number field, and let π be a regular algebraic, cuspidal automorphic representation of GL n (A F ) of unitary type. Let p be a prime, and let ι : Q p → C be an isomorphism. Suppose that r π,ι (G F (ζ p ∞ ) ) is enormous, in the sense of Definition 2.23. Then H 1 f (F + , ad r π,ι ) = 0. For some examples of enormous subgroups, see §2.5. For example, we note that our condition is satisfied for any π such that for some finite place v of F , π v is a twist of the Steinberg representation.
We compare Theorem A with some other results in the literature that are proved using related techniques. Kisin [Kis04] proved the analogue of Theorem A for the Galois representations attached to classical holomorphic modular forms under some mild conditions on the residual representation. Allen [All16] proved a result similar to Theorem A, but assuming a stronger condition on the residual representation r π,ι , requiring in particular that it be irreducible (similar results were also obtained by Breuil-Hellmann-Schraen [BHS17]). These works use variants of the Taylor-Wiles method, which is a powerful tool for studying the deformation theory of automorphic Galois representations.
Our main motivation for this work was to prove a result valid under very weak conditions on the residual representation. In particular, we allow the case p = 2 and r π,ι trivial, which is rather far from the cases allowed by [All16]. For example, we obtain the following results for 2-dimensional representations over totally real fields.
Theorem B. Let F be a totally real number field, and let p be a prime.
(2) Let E be a non-CM elliptic curve over F , and let r p (E) : G F → GL 2 (Q p ) denote the associated p-adic representation. Then H 1 f (F, ad r p (E)) = 0. We emphasise that no additional conditions are required in either case in order to conclude the vanishing of the adjoint Selmer group.
There are three main innovations that allow us to prove a result like Theorem A. The first is a control theorem for studying the pseudodeformation theory of a representation ρ : Γ → GL n (Z p ) of a profinite group Γ. We recall that ρ has an associated pseudocharacter tr ρ, which can be defined following either Chenevier [Che14] or Lafforgue [Laf18] (the proof that these two notions are equivalent being due to Emerson [Eme18]). If the residual representation ρ is absolutely irreducible then it is known that deforming tr ρ is equivalent to deforming ρ. In general any deformation of ρ gives rise to a deformation of tr ρ, but the two notions are not equivalent.
Here we use Lafforgue's definition of pseudocharacter to show that that if ρ ⊗ Zp Q p is absolutely irreducible, then there is a reasonably strong link between deformations and pseudodeformations with coefficients in the ring Z p ⊕ ǫZ p /(p N ). Informally, deformations and pseudodeformations are "the same", up to bounded torsion which depends only on the image ρ(Γ). See Proposition 2.7 for a precise statement.
The second innovation is the formulation, by Wake and Wang-Erickson [WWE19], of functors of pseudodeformations satisfying deformation conditions (e.g. conditions arising from p-adic Hodge theory). This is an indispensable tool for making an effective comparison between pseudodeformation rings and Hecke algebras acting on classical automorphic forms.
The third innovation is related to the use of Taylor-Wiles systems in our proof. To make use of Taylor-Wiles systems in the study of automorphic forms with integral coefficients, one needs to show that if q is a Taylor-Wiles place, then the space of automorphic forms with unramified level at q is isomorphic to the space of automorphic forms with Iwahori level at q, after localization with respect to a suitable eigenvalue of the U q operator (see for example [CG18,Lemma 5.8]). One can argue along these lines only if the residual representation ρ, unramified at q by hypothesis, has the property that ρ(Frob q ) has distinct eigenvalues. This is the reason for condition (2) in the statement of [Kis04,Introduction,Theorem] and of course excludes the case where ρ is trivial. Without this "independence of q" statement, one does not have the finiteness conditions needed to carry out the Taylor-Wiles patching argument, at least as outlined in [Dia97].
In his thesis, Pan [Pan19] introduced a surprising technique to circumvent this issue. Building on Scholze's interpretation of the Taylor-Wiles patching argument using ultrafilters [Sch18], Pan constructs a huge "pre-patched module", and then shows that using suitable Hecke operators it can be cut down to a size making it suitable for use in the Taylor-Wiles argument. We have adapted his arguments to our context (in some ways more elementary, since we work with fixed weight automorphic forms, whereas [Pan19] works with completed cohomology).
Applications. Results such as Theorem A have applications to the geometry of eigenvarieties, and this is one of the main motivations for proving them (as was already the case for Kisin [Kis04]). This is because one can embed (at least locally around an irreducible point) eigenvarieties inside deformation spaces of trianguline representations. In many cases, the vanishing of H 1 f (F + , ad r π,ι ) can be used to prove that this embedding is in fact a local isomorphism.
For example, the vanishing of the adjoint Selmer group is a significant part of what it means for a p-refined Hilbert modular form to be decent, in the sense of [BH17], and therefore to admit a p-adic L-function with good interpolation properties. Another application is that one can use an understanding of the geometry of the eigenvariety to prove modularity results for Galois representations. This possibility is already suggested in Kisin's work [Kis03,(11.13)]. We will take this point of view in [NT19], where Theorem A is one of the key inputs to prove the automorphy of the symmetric power liftings of level one Hecke eigenforms (for example, Ramanujan's modular form ∆).
Organization of this paper. In Section 2 we establish our control theorem relating pseudodeformations and deformations (up to bounded torsion), and set up the Galois theoretic ingredients for the Taylor-Wiles method. In the short Section 3 we prove a simple representation-theoretic result which controls the difference between spaces of automorphic forms with hyperspecial and Iwahori level at Taylor-Wiles places. In Section 4 we carry out our variation on the Taylor-Wiles method (inspired by Pan's work) and prove a special case of Theorem A. Finally, the general case of Theorem A, together with Theorem B and some other applications are deduced in Section 5 using base change and potential automorphy.
Acknowledgements. J.T.'s work received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714405). J.N. would like to thank Carl Wang-Erickson for helpful discussions about the work [WWE19]. We are grateful to the anonymous referee and to Florian Herzig for their detailed comments on this paper.

Notation and preliminaries
If F is a field of characteristic zero, we generally fix an algebraic closure F /F and write G F for the absolute Galois group of F with respect to this choice. If F is a number field, then we will also fix embeddings F → F v extending the map F → F v for each place v of F ; this choice determines a homomorphism G Fv → G F . When v is a finite place, we will write O Fv ⊂ F v for the valuation ring, ̟ v ∈ O Fv for a fixed choice of uniformizer, Frob v ∈ G Fv for a fixed choice of Frobenius lift, k(v) = O Fv /(̟ v ) for the residue field, and q v = #k(v) for the cardinality of the residue field. When v is a real place, we write c v ∈ G Fv for complex conjugation. If S is a finite set of finite places of F then we write F S /F for the maximal subextension of F unramified outside S and G F,S = Gal(F S /F ).
If p is a prime, then we call a coefficient field a finite extension E/Q p contained inside our fixed algebraic closure Q p , and write O for the valuation ring of E, ̟ ∈ O for a fixed choice of uniformizer, and k = O/(̟) for the residue field. We write C O for the category of complete Noetherian local O-algebras with residue field k.
If A is a ring and ρ : Γ → GL n (A) is a representation, we write ad ρ for M n (A) with its adjoint Γ-action, and ad 0 ρ ⊂ ad ρ for the A[Γ]-submodule of trace 0 matrices. We will use the self-duality ad ρ × ad ρ → A, (X, Y ) → tr XY , to identify ad ρ with its dual when we e.g. define dual Selmer conditions using Tate duality (see for example §2.4).
If G is a locally profinite group and U ⊂ G is an open compact subgroup, then we write H(G, U ) for the set of compactly supported, U -biinvariant functions f : G → Z. It is a Z-algebra, where convolution is defined using the left-invariant Haar measure normalized to give U measure 1; see [NT16,§2.2]. It is free as a Z-module, with basis given by the characteristic functions [U gU ] of double cosets.
Let K be a non-archimedean characteristic 0 local field, and let Ω be an algebraically closed field of characteristic 0. We write W K ⊂ G K for the Weil group of K and I K ⊂ W K for the inertia subgroup. We use the cohomological normalisation of class field theory: it is the isomorphism Art K : K × → W ab K which sends uniformizers to geometric Frobenius elements. We use the Tate normalisation of the local Langlands correspondence for GL n : it is the bijection rec T K between isomorphism classes of irreducible, admissible Ω[GL n (K)]-modules and isomorphism classes Frobenius-semisimple Weil-Deligne representations over Ω of rank n which is normalised as in [CT14, §2.1].
If ρ : G K → GL n (Q p ) is a continuous representation (assumed to be de Rham if p equals the residue characteristic of K), then we write WD(ρ) = (r, N ) for the associated Weil-Deligne representation, and WD(ρ) F −ss for its Frobenius semisimplification.
Definition 1.1. We say that a Weil-Deligne representation (r, N ) is generic if there is no non-zero morphism (r, N ) → (r(1), N ). We say that a continuous representation ρ is generic if W D(ρ) is generic.
We note that if W D(ρ) F −ss is generic, then ρ is generic. It follows from [All16, Lemma 1.1.3] that if π is a generic irreducible admissible Q p [GL n (K)]-module and W D(ρ) F −ss = rec T K (π), then ρ is generic. If p equals the residue characteristic of K and V is the E-vector space on which ρ acts (for some E ⊂ Q p finite over Q p with ρ(G K ) ⊂ GL n (E)), we have subspaces We have H 1 f (K, End(V )) = H 1 g (K, End(V )) if and only if ρ is generic [All16, Remark 1.2.9]. Similarly, if p does not equal the residue characteristic of K, we have a subspace H 1 ur (K, V ) = ker H 1 (K, V ) → H 1 (I K , V ) . For notational compatibility with the p-adic case we write H 1 f (K, V ) = H 1 ur (K, V ) and H 1 g (K, V ) = H 1 (K, V ). Then we again have H 1 f (K, End(V )) = H 1 g (K, End(V )) if and only if ρ is generic. Let F be a number field, and let S be a finite set of finite places of F , containing the p-adic places S p . Let r : G F,S → GL n (Q p ) be a continuous representation, with underlying E-vector space V . We have global Selmer groups We note our convention that H 1 (F S /F, * ) denotes group cohomology for the group G F,S . The group H 1 f (F, V ) does not change when S is enlarged (this is why we do not record S in the notation).
If F is a number field and π is an automorphic representation of GL n (A F ), we say that π is regular algebraic if π ∞ has the same infinitesimal character as an irreducible algebraic representation of Res F/Q GL n . We recall (cf. [BLGGT14, §2.1]) that if F is a totally real or CM number field, then a pair (π, χ) comprising an automorphic representation π of GL n (A F ) and a Hecke character χ : The sign condition of [BLGGT14] in the case that F is totally real can be suppressed, as a consequence of [Pat15, Theorem 2.1].) The automorphic representations of unitary type discussed in our introduction correspond to polarized automorphic representations (π, δ n F/F + ), where δ F/F + is the quadratic character for F/F + .
If (π, χ) is a regular algebraic, cuspidal, polarized automorphic representation, then for any isomorphism ι : Q p → C there is an associated Galois representation (we refer to [BLGGT14, Theorem 2.1.1] for its properties) If F is CM, then r π,ι extends to a homomorphism r π,ι : G F + → G n (Q p ), with multiplier character ν •r π,ι = ǫ 1−n r χ,ι (G n is the algebraic group defined in [CHT08, §2.1]; here the word 'extends' is interpreted following the convention described at the top of [CHT08, p. 8]). This defines an extension of the G F action on ad r π,ι to an action of G F + . More explicitly, if we fix a choice c ∈ G F + of complex conjugation, there is a perfect, symmetric pairing ·, · on Q n p such that r π,ι (σ)v, r π,ι (σ c )w = (ǫ 1−n r χ,ι (σ)) v, w for all σ ∈ G F , v, w ∈ Q n p and c acts on ad r π,ι = End(Q n p ) by X → −X * , where X * is the adjoint with respect to ·, · .

Pseudocharacters
In this paper we use Lafforgue's notion of pseudocharacter for a reductive group in the case of GL n (see [Laf18,§11] or [BHKT,§4]), and Chenevier's notion of group determinant [Che14]. In fact, these are equivalent, but both definitions are useful. We will prove a new result about the deformation theory of pseudocharacters (Proposition 2.7) using Lafforgue's point of view, while we follow [WWE19] in using Chenevier's definition to impose deformation conditions on pseudocharacters.
2.1. Pseudocharacters vs. determinants. We begin by recalling the relevant definitions. Let Γ be a group and fix n ≥ 1.
One can define the operations of twisting and duality on pseudocharacters in a way compatible with the usual operations on representations. For example, let i : GL n → GL n be the involution given by i(g) = t g −1 . If t is a pseudocharacter, then we define a new pseudocharacter t ∨ by the formula Before giving the definition of group determinant, we fix some notation. Let A be a ring and let A -alg be the category of commutative A-algebras. If M is an A-module, then we write h M : A -alg → Sets for the functor B → M ⊗ A B. (1) D(1) = 1.
If ρ : Γ → GL n (A) is a representation, then we can define its associated group determinant D(x) = det(ρ(x)) (where we extend ρ to a homomorphism ρ : B[Γ] → M n (B) for any A-algebra B). We omit the formulae for the dual or twist of a group determinant.
We now describe the relation between pseudocharacters and group determinants.
GLn is generated as a ring by the functions λ i (g i1 . . . g ir ) (r ∈ N, 1 ≤ i 1 , . . . , i r ≤ m), together with det(g 1 . . . g m ) −1 . The axioms defining a pseudocharacter show that we have It follows that the functions t [i] (i = 0, . . . , n) together determine t.
If D is a group determinant, then we define functions (i = 0, . . . , n) We now discuss continuity. Suppose therefore that Γ is a profinite group and A is a topological ring. The definitions are as follows. Proof. In light of Theorem 2.3, it is enough to show that if t is a pseudocharacter such that each function t [i] (i = 0, . . . , n) is continuous, then t is continuous. This is again a consequence of (2.1) and [Don92, §3.1].
2.2. Pseudocharacters vs. representations. Now let p be a prime, let E/Q p be a finite extension, and let Γ be a profinite group. Let ρ : Γ → GL n (O) be a continuous homomorphism which is absolutely irreducible over E. Let t = (t m ) m≥1 = tr ρ denote the pseudocharacter associated to ρ. We consider liftings of ρ and of t to the ring A = O ⊕ ǫE/O (with ǫ 2 = 0). Clearly if ρ ′ : Γ → GL n (A) is a lifting of ρ, in the sense that ρ ′ mod (ǫ) = ρ, then t ′ = tr ρ ′ is a lifting of t. We want to show that in fact deforming ρ in this way is not too far from deforming t.
Lemma 2.8. Let π : X → Y be a separated morphism of schemes of finite type over a base S. Let G be a separated group scheme, smooth and of finite type over S, and suppose that G acts on X in such a way that π is G-equivariant for the trivial action of G on Y . Then: (1) There is a canonical morphism Ω X/Y → Hom OS (Lie G, O X ) of coherent sheaves of O X -modules.
(2) If π is a G-torsor, then this morphism is an isomorphism.
Proof. Let e : X → G×X be the morphism x → (e, x), and let µ : G×X → X × Y X be the morphism (g, x) → (x, gx). Then µ • e is the diagonal embedding, and both e and µ • e are closed immersions. The sheaf Hom OS (Lie G, O X ) may be identified with the conormal sheaf of the morphism e (see e.g. [GP11, II, Lemme 4.11.7]) while Ω X/Y may be identified with the conormal sheaf of the morphism µ • e. The existence of the morphism therefore follows from [Sta13, Lemma 01R4]. Now suppose that π is a G-torsor. In this case µ is an isomorphism, and the statement is immediate.
Let g = Lie PGL n,O and g * = Hom(g, O), g * Xm = g * ⊗ O O Xm . We apply Lemma 2.8 to the morphisms π k : X k → Y k , with G = PGL n,O , to obtain complexes (not necessarily exact) of coherent sheaves on X k : Proof. We show that π m [1/p] is a PGL n,E -torsor above a Zariski open neighbourhood of y. The same proof shows the analogous statement for the points y(γ) and y(γ, δ), and in each case implies the statement in the lemma (since a PGL n,E -torsor is in particular smooth). Let U denote the open subset of X m [1/p] corresponding to tuples (u 1 , . . . , u m ) which generate an absolutely irreducible subgroup of GL n . Then [Ric88,Theorem 4.1] shows that U is precisely the set of stable points of X m [1/p] (for the action of PGL n,E ). In particular, π m (U ) is an open subset of Y m [1/p] and U = π −1 m (π m (U )). Each point of U has a trivial stabilizer for the PGL n,E action (Schur's lemma), so it follows from [BR85, Proposition 8.2] that π m | U : U → π m (U ) is a PGL n,E -torsor, as required.
We can use the compactness of Γ to upgrade the previous lemma to the following uniform integral statement.
Lemma 2.10. We can find an integer k 1 ≥ 0 with the following properties: (1) The cohomology of the complex i * x (⋆ m ) is annihilated by p k1 .
Proof. The first part of the lemma follows by Lemma 2.9. In fact, we can find numbers k, k(γ), and k(γ, δ) such that the requirements of each point of the lemma are satisfied if k 1 is replaced by k, k(γ), and k(γ, δ) in each case. What we must show is that we can find k 1 which exceeds k, k(γ), and k(γ, δ) for all γ, δ ∈ Γ.
To this end, let us suppose that k, k(γ), and k(γ, δ) have been chosen to each take their smallest possible values. It suffices to show that k(γ) and k(γ, δ) are locally constant as functions of γ, δ ∈ Γ. Since Γ is compact, this will imply that they are in fact bounded. This local constancy is a consequence of the second part of Lemma 2.11 below.
Lemma 2.11. Let Z be a scheme of finite type over O. If z ∈ Z(O), we write i z : Spec O → Z for the corresponding morphism.
(1) Let F be a coherent sheaf on Z such that F be a complex of coherent sheaves on Z, not necessarily exact, but such that on a Zariski open neighbourhood of z ∈ Z[1/p] is an exact sequence of locally free sheaves. Then there exists a p-adically open neighbourhood U of z in Z(O) and an integer N ≥ 0 such that for all z ′ ∈ U , p N H * (i * z ′ (⋆)) = 0. Proof. In each case we are free to replace Z by a Zariski open neighbourhood of the closed point specializing z. We can therefore assume that Z = Spec B is affine.
In the first case we can assume that F corresponds to a finite B-module M and that there is an exact sequence We may assume that F [1/p] has constant rank b − k on V z , so we get a continuous map where M a×b,k ⊂ M a×b is the locally closed subscheme of matrices of rank k (equivalently with vanishing (k + 1) × (k + 1) minors but at least one non-zero k × k minor). Note that We are therefore reduced to showing that any matrix T ∈ M a×b (O)∩M a×b,k (E) has an open neighbourhood U such that for In other words, we need to show that there is an open neighbourhood U in which the Smith normal form of T is constant. Let m be the largest valuation of a non-zero minor of T . Choosing U so that for any T ′ ∈ U , the minors of T ′ are congruent to those of T modulo ̟ m+1 , we see that the Smith normal forms of T and T ′ are indeed equal. (Note that the assumption that the E-rank is constant is necessary; otherwise we have the example of We now turn to the second part. It suffices to show that we can find an integer N ≥ 0 and a p-adically open neighbourhood U of z such that for all z ′ ∈ U , p N annihilates ker , so the result follows on applying the first part to F 2 / im(F 1 → F 2 ).
We are now in a position to prove Proposition 2.7. Recall that we write A = O⊕ǫE/O. It is helpful to first note that if X is a scheme over O and x ∈ X(O), then the fibre of Proof of Proposition 2.7. Let the integer k 1 be as in Lemma 2.10. We will show that we can take k 0 = 6k 1 . Taking the Pontryagin dual of i * x (⋆ m ) and i * x(γ) (⋆ m+1 ) gives us, for any γ ∈ Γ, a commutative diagram The first vertical arrow is the identity, while the other two vertical arrows correspond to forgetting the last entry in GL m+1 n . Both rows have cohomology annihilated by p k1 . Consequently the induced map has kernel and cokernel annihilated by p 2k1 . In particular, given an element z of the target we can define an element of the source unambiguously as follows: choose a pre-image z ′ of p 2k1 z. Then z ′′ = p 2k1 z ′ depends only on z and has image p 4k1 z. Now suppose given a pseudocharacter t ′ over A lifting t. The data of the pseudocharacter t ′ (more precisely, t ′ m ) determines an element y ′ ∈ Hom(T * y Y m , E/O). We fix a choice of x ′ ∈ Hom O (T * x X m , E/O) with image equal to p k1 y ′ . This corresponds to a tuple of elements (g ′ 1 , . . . , g ′ m ) ∈ GL n (A) m lifting the element (g 1 , . . . , g m ) = (ρ(γ 1 ), . . . , ρ(γ m )). If x ′′ is any other choice of element with image equal to p k1 y ′ , then The pseudocharacter t ′ also determines elements for any γ ∈ Γ, and the pair (p k1 x ′ , p 2k1 y ′ (γ)) lies on the right-hand side of (2.2). We may define ρ ′ (γ) uniquely as follows: it is the lift of p 4k1 (p k1 x ′ , p 2k1 y ′ (γ)) associated to the map (2.2). Then ρ ′ : Γ → GL n (A) has associated trace function tr ρ ′ = α 6k1 • t ′ , and its conjugacy class under 1 + ǫg ⊗ O E/O is independent of choices. We can verify that ρ ′ is a homomorphism (i.e. that it respects multiplication) using the fact that t ′ is a pseudocharacter, together with a diagram with rows corresponding to elements x(γδ) and x(γ, δ).

Now suppose given two liftings
of the first tuple to α k1 of the second. Passing to the top row of the commutative diagram, we see that for any γ, Increasing s, we can assume that s ≥ r, that ρ ′ (γ 1 ), . . . , ρ ′ (γ m ) lie in GL n (A s ), and that there exists an open subgroup N ⊂ Γ such that for all γ ∈ N , ρ(γ) ∈ 1 + ̟ s M n (O) and t ′ m+1 (γ 1 , . . . , γ m , γ) ≡ t ′ m+1 (γ 1 , . . . , γ m , 1) mod ̟ s A s . We observe that for γ ∈ N , there is a commutative square where the horizontal arrows are the ones already considered in (2.2) (suppressing the subscripts indicating the fibre product to save space), and the vertical ones are bijections arising from the identification between The horizontal arrows have kernels annihilated by p 2k1 . Our assumptions imply that the elements of corresponding to the images of ρ ′ (γ) and ρ ′ (1) in GL n (O/̟ s ⊕ ǫ̟ −s O/O) are identified under the above bijection. This is what we needed to prove.
2.3. Pseudocharacters -Galois deformation theory. We again fix a prime p and a finite extension E/Q p with ring of integers O and residue field k. Let C O be the category of complete Noetherian local O-algebras with residue field k. Let F be a number field, and let S be a finite set of finite places of F , containing the p-adic ones. Let ρ : G F,S → GL n (k) be a continuous representation. Let D denote the associated group determinant over k.
We write Def D,S : C O → Sets for the functor which associates to each A ∈ C O the set of continuous group determinants D of G F,S over A such that D mod m A = D. Lemma 2.13. Fix an integer q ≥ 0. Then there exists an integer g 0 = g 0 (S, D, q) such that for any set Q of finite places of F such that |Q| ≤ q, there exists a surjection O X 1 , . . . , X g0 → R D,S∪Q .
Proof. Let L/F denote the extension cut out by ρ, and let M S∪Q denote the maximal pro-p extension of L unramified outside S ∪Q. Then there exists g 1 = g 1 (S, ρ, q) such that the group Gal(M S∪Q /F ) can be topologically generated by g 1 elements (because the dimension of the space H 1 (G L,S∪Q , k) can be bounded in a way depending only on q). Moreover, any deformation of D to G We write Def Proposition 2.14. The functor Def Now suppose given a lift ρ : G F,S → GL n (O) of ρ with the following properties: Proposition 2.15. There exists a canonical homomorphism Moreover, there is a constant c ≥ 1 depending only on ρ (and not on S, [a, b], or m) such that for any m ≥ 1, the kernel and cokernel of tr m are both annihilated by p c .
In applications of the proposition we will enlarge S by adding Taylor-Wiles places. This is why it is important that the constant c is independent of the set S.
of the classifying map of D. The map tr m is the one which sends [φ] to the classifying map of the pseudocharacter tr ρ φ over A m . Note that multiplication by p k on either side of (2.4) corresponds at the level of representations (resp. determinants) to composition with α k .
Next we analyze the kernel of tr m . Suppose that tr ρ φ = tr ρ. Let k 0 be the constant of Proposition 2.7: it depends only on ρ(G F,S ) ⊂ GL n (O). Then we can find X ∈ M n (E/O) such that is killed by a uniformly bounded power of p and we see that the same is true for the kernel of tr m . Now we analyze the cokernel of tr m . This is more subtle. Let D ′ be a determinant of G F,S over A m corresponding to an element of the right-hand side of (2.4). By F,S . We may assume without loss of generality that r is surjective. According to the characterization of ker( By Proposition 2.7, there exists a homomorphism ρ φ : G F,S → GL n (A) such that the associated group determinant is α k0 • D ′ . It also follows from Proposition 2.7 that the associated cohomology class We must show that there is c ≥ 0 depending only on ρ such that for each N ≥ 1, It follows that there exists a commutative diagram of A m -algebras F,S (since this holds for finite quotients of B). This implies that the finite quotients of α 2k1 • ρ φ also satisfy the condition E F,S . We deduce that the cokernel of tr m is annihilated by p 2k0+2k1 .
2.4. Pseudocharacters -Taylor-Wiles data. We continue our discussion of the Galois deformation theory of pseudocharacters, now focusing on what happens when we impose conjugate self-duality conditions and allow additional primes of ramification. We thus fix the following notation: • p is a prime and E/Q p is a coefficient field.
• F/F + is a CM quadratic extension of a totally real field. • S is a finite set of finite places of F + containing the p-adic ones. We assume that each place of S splits in F , and fix for each v ∈ S a choice of place v of F lying above v. We set is absolutely irreducible. We set ρ = r| GF,S : G F,S → GL n (O) and write D for the group determinant of ρ and D for its reduction modulo ̟. We set Note that χ| GF v is then semistable and there exists w ∈ Z such that χǫ w has finite order. We assume moreover that a + b = w.
Then ρ determines a homomorphism R S → O, and we write q S for its kernel.
We define Selmer conditions The corresponding Selmer groups are defined by These Selmer groups are finite length O-modules. We denote their length by (1)). Taking inverse limits with respect to the projection maps W m+1 → W m and direct limits with respect to the injections W m ∼ = ̟W m+1 ⊂ W m+1 , we also define Selmer groups with characteristic 0 and divisible coefficients: Proposition 2.15 has the following consequence: Proposition 2.16. For each m ≥ 1, there is a canonical homomorphism Proof. The map (2.4) is Gal(F/F + )-equivariant for the action on the left-hand side induced by the G F + ,S action on W m and the action on the right-hand side defined as follows: the non-trivial element c ∈ Gal(F/F + ) acts on R [a,b] D,S by sending a pseudocharacter D ′ to (D ′ ) c,∨ ⊗ χ| GF,S , and this induces an action on the righthand side of (2.4). Note that this action makes sense because of our condition a + b = w. The right-hand side of (2.5) is the c-invariants in the right-hand side of (2.4). The left-hand side of (2.5) maps to the c-invariants in the left-hand side of (2.4) with bounded kernel and cokernel. This is enough.
Here is a variant that will be useful when it comes to deduce our main vanishing result.
(1) There is an isomorphism tr E,S : Proof. The first part follows from Proposition 2.16 by taking the inverse limit over m and inverting p. The main result of [Liu07] implies that H 1 , so the second part follows. The third part follows from the equality of the respective local Selmer groups in the generic case. Lemma 2.18. The natural map H 1 Proof. We consider the commutative diagram with exact rows: The central vertical map is surjective. The right vertical map is injective (by the observation preceding this lemma). So the left vertical map is surjective.
If Q is a set of finite places of F + and N is a positive integer, we say that Q is a set of Taylor-Wiles places of level N (relative to r, S) if it satisfies the following conditions: • Q ∩ S = ∅.
Q is a set of Taylor-Wiles places of level N , Q is a set consisting of a choice, for each v ∈ Q, of a place v of F lying above v, and (α v,1 , . . . , α v,n ) is a choice of ordering of the eigenvalues of ρ(Frob v ).
Lemma 2.19. Suppose that the following conditions are satisfied: (1) For each v ∈ S, ρ| GF v is generic.
Then there exists d ≥ 0 with the following property: for every N ≥ 1, every Taylor-Wiles datum (Q, Q, (α v,1 , . . . , α v,n ) v∈ Q ) of level N , and every 1 ≤ m ≤ N , we have: Proof. Fix a Taylor-Wiles datum. By the usual Greenberg-Wiles formula, we have where l v,m = l(L v,m ) and l denotes the length of an O-module. The contribution from the infinite places is m[F + : Q]n(n − 1)/2, up to a uniformly bounded error. The global terms h 0 (F + , W m ) and h 0 (F + , W m (1)) are both uniformly bounded, the first since ρ is absolutely irreducible and the second since ρ is absolutely irreducible and ρ, ρ(1) have different sets of Hodge-Tate weights.
, since we are assuming m ≤ N , and this is bounded above by nm Finally, suppose that v ∈ S p . Let R ,[a,b] v ∈ C O denote the object representing the functor of lifts of ρ| GF v whose projections to Artinian quotients are subquotients of lattices in semistable representations with all Hodge-Tate weights in the interval Corollary 2.21. Suppose that ρ satisfies the hypotheses of Lemma 2.19. Then there exists d ∈ N such that for all N ∈ N and every Taylor-Wiles datum of level N , there is a map Proof. Using Lemma 2.19 and Lemma 2.20, we see that it is enough to find d 0 , d 1 ∈ N such that for any 1 ≤ m ≤ N and any Taylor-Wiles datum of level N , we have We treat these in turn. For the first inequality, we note that Lemma 2.18 shows that the map For the second inequality, we note that the kernel of the natural map (2.8) is contained in the kernel of the map which is a subquotient of H 0 (F + , W E/O (1)) (which is finite, by the same argument showing boundedness of h 0 (F + , W m (1)) in the proof of Lemma 2.19). We see that (2.7) will hold with d 1 = h 0 (F + , W E/O (1)) provided that the map (2.8) sends H 1 W N (1)). Recalling the definition of our local conditions, this means we must show that if v ∈ S p , then the map However, this follows immediately from the definitions.
For Corollary 2.21 to be useful, we need to be able to find Taylor-Wiles data with good properties. To do this, we first introduce a useful definition. (1) For all simple E[H]-submodules V ⊂ W 0 E = ad 0 ρ ⊗ E, we can find h ∈ H with n distinct eigenvalues and α ∈ E such that α is an eigenvalue of h and tr e h,α V = 0 (where e h,α ∈ W E denotes the h-equivariant projection to the α-eigenspace).
(2) For all simple E[H]-submodules V ⊂ W E , we can find h ∈ H with n distinct eigenvalues and α ∈ E such that α is an eigenvalue of h and tr e h,α V = 0.
Proof. We note that (1) and (2) are equivalent because the scalar matrices Z E ⊂ W E give a complement to W 0 E in W E , and the condition tr e h,α Z E = 0 is satisfied for any regular semisimple element h ∈ H and eigenvalue α ∈ E.
If h ∈ GL n (O) has n distinct eigenvalues, then it acts semisimply on W E . In particular, there is a unique h-invariant direct sum decomposition The condition that there exists an eigenvalue α ∈ E of h such that tr e h,α V = 0 is equivalent to the condition that the projection of V to W h E is non-zero, or equivalently that V h = 0. This is in turn equivalent to the condition that V ⊂ (h − 1)W E . This shows that (2) and (3) are equivalent. Now we show that (3) and (4) are equivalent. For this we note that there is a GL n (O)-equivariant, inclusion-preserving bijection between the E-subspaces of W E and the divisible O-submodules of W E/O ; this sends V ⊂ W E to V + W/W and The proof in this case is complete on noting that (h − 1)W E corresponds to (h − 1)W E/O under this bijection.
Definition 2.23. We say that a subgroup H ⊂ GL n (O) is enormous if for all simple E[H]-submodules V ⊂ W E , we can find h ∈ H with n distinct eigenvalues in E and α ∈ E such that α is an eigenvalue of h and tr e h,α V = 0.
Remark 2.24. The above is a natural analogue of the definition of enormous subgroups in positive characteristic [KT17,Definition 4.10]. In contrast to the positive characteristic case, we do not need to assume any vanishing of cohomology groups for H. The necessary vanishing will follow from the purity of our Galois representations (see [ (1)), and suppose that ρ satisfies the following conditions: (1) For all but finitely many finite places v ∤ S of F , the eigenvalues of ρ(Frob v ) are algebraic numbers which have absolute value q w/2 v with respect to any complex embedding.
Then there exists d ∈ N such that for any N ∈ N we can find a Taylor-Wiles datum Proof. If Q is a set of Taylor-Wiles places then we have an exact sequence Suppose we could find σ 1 , . . . , σ q ∈ G F (ζ p ∞ ) such that (a) for each i = 1, . . . , q, ρ(σ i ) has n distinct eigenvalues in E; (b) the kernel of the map (product of restriction maps associated to the homomorphisms Z → G F + ,S , the i th such homomorphism sending 1 to σ i ) is a finite length O-module.
Then consideration of the following diagram: shows that the kernel of the map has length bounded independently of N (note that H 0 (F + , W E/O (1)) is a finite length O-module). An application of the Chebotarev density theorem would then yield the theorem.
To complete the proof, it therefore suffices to show that for any non-zero homomorphism f : E/O → H 1 (F S /F + , W E/O (1)), we can find σ ∈ G F (ζ p ∞ ) such that ρ(σ) has n distinct eigenvalues in E and Res is still non-zero (as then we can argue by induction to get σ 1 , . . . , σ q satisfying conditions (a), (b) above).
Putting everything together, we obtain: Corollary 2.27. Let q ≥ corank O H 1 (F S /F + , W E/O (1)), and suppose that ρ satisfies the following conditions: (1) For all but finitely many finite places v ∤ S of F , the eigenvalues of ρ(Frob v ) are algebraic numbers which have absolute value q w/2 v with respect to any complex embedding.
Then there exists d ∈ N such that for each N ∈ N we can find a Taylor is a quotient of (O/̟ d ) g0 , where g 0 = g 0 (S, ρ, q) is as defined in the statement of Lemma 2.13.
We need to explain how to deduce the slightly stronger result in the statement of the corollary. We first note that q S∪QN /q 2 S∪QN is a quotient of O g0 . Indeed, it is a finitely generated O-module, and there is an isomorphism , ̟), so we can apply Nakayama's lemma together with Lemma 2.13.
We may assume that N > d. In this case M = q S∪QN /(q 2 S∪QN , x 1 , . . . , x nq ) is a quotient of O g0 with the property that M/(̟ N ) is killed by ̟ d . This is only possible if M is itself killed by ̟ d , implying that M is a quotient of (O/̟ d ) g0 .

Some examples of enormous subgroups.
Let E/Q p be a coefficient field, and let H ⊂ GL n (O) be a compact subgroup.
Lemma 2.28. Suppose that for each h ∈ H, the characteristic polynomial of h has all of its roots in E.
(1) Let H ′ ⊂ H be a closed subgroup. If H ′ is enormous, then so is H.
(2) Let G ⊂ GL n denote the Zariski closure of H. If G • contains regular semisimple elements and acts absolutely irreducibly on E n , then H is enormous.
Proof. The first part is immediate from the definitions. For the second, we can assume that G = G • . Since G acts absolutely irreducibly, G(E) spans W E . Let H reg ⊂ H denote the set of regular semisimple elements. It is Zariski dense in G. Indeed, by hypothesis G reg is a non-empty Zariski open subset of G. The Zariski closure of H is contained in the union of the Zariski closure of H reg and G − G reg . This forces the Zariski closure of H reg to be equal to G. We must show that for any non-zero v ∈ W E , there exists h ∈ H reg such that tr hv = 0. If tr hv = 0 for all h ∈ H reg , then Zariski density implies that tr gv = 0 for all g ∈ G. This contradicts the fact that the elements of G(E) span W E .
Example 2.29. Let F be a totally real or CM number field, and let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A F ). Let ι : Q p → C be an isomorphism, and let ρ : G F → GL n (O) be a model of Sym n−1 r π,ι defined over O. If Sym 2 π is cuspidal, then (after possibly enlarging E) ρ(G F (ζ p ∞ ) ) is an enormous subgroup of GL n (O).
To see this, it is enough to note that the Zariski closure of the image of r π,ι contains SL 2 , and therefore that the Zariski closure of r π,ι (G F (ζ p ∞ ) ) also contains SL 2 (because passage to derived subgroup respects Zariski closure, cf. [Bor91, Ch. I, §2.1]). We can then appeal to Lemma 2.28.
We justify the claim that the Zariski closure of r π,ι (G F ) contains SL 2 . The identity component of the Zariski closure of r π,ι (G F ) is a reductive subgroup of GL 2 which contains regular semisimple elements (by [Sen73, Theorem 1]). The only possibility we need to rule out is that r π,ι (G F ) normalizes a maximal torus in GL 2 . In this case, there is a quadratic extension F ′ /F such that r π,ι | G F ′ is reducible. It's therefore enough to show that for any quadratic extension F ′ /F , r π,ι | G F ′ is irreducible. We observe that if r π,ι | G F ′ is reducible, then it's isomorphic to a sum χ 1 ⊕χ 2 of characters. Moreover, χ 1 , χ 2 are de Rham and almost everywhere unramified, so therefore can be extended to compatible systems of 1-dimensional Galois representations. It follows that r π,ι ′ | G F ′ is reducible for any other prime p ′ and isomorphism ι ′ : Q p ′ → C. In particular, Sym 2 r π,ι ′ is reducible. However, [BLGGT14, Theorem 5.5.2] implies that for a Dirichlet density 1 set of primes p ′ , the representation Sym 2 r π,ι ′ is irreducible (note that for automorphic representations of GL 3 (A F ) the asssumption of 'extremely regular weight' in loc. cit. coincides with the usual notion of regular weight).
Example 2.30. Let F be a CM field, and let π be a polarizable automorphic representation of GL n (A F ) such that for some finite place v 0 of F , π v0 is a twist of the Steinberg representation. Let ι : Q p → C be an isomorphism. Then r π,ι (G F (ζ p ∞ ) ) is enormous.
Indeed, let G denote the Zariski closure of r π,ι (G F ). Local-global compatibility at the place v 0 implies that G • contains a regular unipotent element (if v 0 |p, we argue as in [KS19, Lemma 3.2]), so in particular it acts absolutely irreducibly on E n . Then [Sch06, Proposition 4] (see also [Kat88, Classification Theorem 11.6]) shows that the derived group of G • is one of a finite list of possibilities, and that in any case it contains regular semisimple elements. We can again appeal to Lemma 2.28.

A result about Hecke algebras
Let p be a prime, let n ≥ 2, and let F v /Q l be a finite extension for some l = p. Let G = GL n (F v ), U = GL n (O Fv ), and let I ⊂ U be the standard Iwahori subgroup (i.e. the pre-image in U of the upper-triangular matrices in GL n (k(v))). Let E/Q p be a coefficient field. Our identification of O[X * (T )] with a polynomial algebra allows us to speak of polynomials as being elements of the Hecke algebra. In particular, we can think of ∆ = 1≤i<j≤n (x i − x j ) as being an element of H I , and its square ∆ 2 as being an element of the centre Z(H I ).
To simplify notation, let R = O[X * (T )] Sn , S = O[X * (T )]. Then S is a free R-module, a basis being given by the monomials x a = x a1 1 . . . x an n for a = (a 1 , . . . , a n ) ∈ Z n satisfying 0 ≤ a i ≤ i − 1 for each i = 1, . . . , n. We write B for the set of tuples a satisfying these conditions. Thus there there is a canonical (and functorial) morphism given by the formula m ⊗ s → sm. Since S is free over R, M U ⊗ R S may be identified with ⊕ a∈B M U , and the above map with (m a ) a∈B → a∈B x a · m a . The aim of this short section is to prove the following result, which will be applied in Section 4 (see Proposition 4.5). Note that ∆ n! always lies in Z(H I ). This is important for us since it means that in the global situation, ∆ n! will be in the image of the pseudodeformation ring (through which decomposition groups act via a homomorphism to the Bernstein centre).
Before proving the proposition, we establish an auxiliary result.
Lemma 3.2. Consider the n! × n! matrix P with coefficients in Z[x 1 , . . . , x n ] given by the formula P σ,a = σ(x a ) (σ ∈ S n , a ∈ B). Then there exists a unique matrix Q = (Q a,σ ) with coefficients in Z[x 1 , . . . , x n ] such that P Q = QP = ∆ n! .
Proof. It suffices to show existence, since then uniqueness follows by linear algebra over Q(x 1 , . . . , x n ). The square of the determinant of P is equal to the discriminant of the ring extension Z[e 1 , . . . , e n ] → Z[x 1 , . . . , x n ]. Using [Sta13, Tag 0C17], we see that the discriminant of this ring extension equals ∆ n! (the different ideal is generated by ∆). Therefore the determinant of P is equal to ∆ n!/2 , up to sign. The existence of the adjugate matrix implies that there is a matrix Q ′ with coefficients in Z[x 1 , . . . , x n ] such that P Q ′ = ∆ n!/2 . We then take Q = ∆ n!/2 Q ′ .
We observe that for all a ∈ B, σ, τ ∈ S n , we have σ(Q a,τ ) = Q a,στ . Indeed, this follows from the identity σ(P )σ(Q) = ∆ n! and the uniqueness of inverses. We have defined a map f : ⊕ a∈B M U → M I by the formula (m a ) a∈B → a∈B x a · m a . We define a map g : M I → ⊕ a∈B M U by the formula g(m) = (eQ a,1 m) a∈B .
We now compute f • g and g • f . We have for m ∈ M I f (g(m)) = a∈B x a eQ a,1 m = a∈B σ∈Sn x a σ(Q a,1 )σ(m).
This in turn we can rewrite as σ∈Sn a∈B P 1,a Q a,σ σ(m) = ∆ n! m.
Similarly, we have for m = (m a ) a∈B ∈ ⊕ a∈B M U : Note that S n acts trivially on M U . We can therefore rewrite the above expression as b∈B σ∈Sn This completes the proof.

Patching
In this section we prove our main technical result (Theorem 4.1).
4.1. Set-up. We suppose given the following data: • A CM number field F with maximal totally real subfield F + . We assume that F/F + is everywhere unramified and that [F + : Q] is even. • An integer n ≥ 2 and a cuspidal, polarized, regular algebraic automorphic representation (π, δ n F/F + ) of GL n (A F ) (i.e. π is of unitary type). • A prime p and an isomorphism ι : Q p → C. We assume that for each place w|p of F , π w has an Iwahori-fixed vector. • A finite set S of finite places of F + , containing the set S p of p-adic places and all places above which π is ramified. We assume that each place of S splits in F . We recall that under these conditions we define an extension of r π,ι to a homomorphism to G n , which then gives the action of G F + on ad r π,ι (see §1).
Theorem 4.1. With set-up as above, assume moreover that r π,ι (G F (ζ p ∞ ) ) is enormous. Then H 1 f (F + , ad r π,ι ) = 0. We note here that the assumptions of Lemma 2.19 hold for r π,ι by [BLGGT14, Theorem 2.1.1] (which collects together results of many people).
The proof of Theorem 4.1 will use automorphic forms on definite unitary groups. To this end, we can find the following data: • For each place v ∈ S, a choice of place v of F lying above v. We set is definite at infinity and quasi-split at each finite place of F + . • A reductive group scheme over O F + extending G.
• For each finite place v = ww c of F + which splits in F , an isomorphism We assume that the induced isomorphism ι w : G(F + v ) → GL n (F w ) is in the same inner class as the isomorphism given by inclusion G(F + v ) ⊂ GL n (F w )×GL n (F w c ), followed by projection to the first factor.
• An automorphic representation σ of G(A F + ) with the following properties: -If v|∞ is a place of F + , then the infinitesimal character of σ v respects that of π v under base change. (We recall this relation more precisely below. We can find such a G because [F + : Q] is even. The existence of σ is deduced from that of π using [Lab11, §5].) We can regard σ ∞ as an algebraic representation of the group (Res F + /Q G) C . Let I p ⊂ Hom(F, Q p ) denote the set of embeddings inducing places v ∈ S p . Then our choices determine an isomorphism Let λ = (λ τ ) τ ∈ Ip ∈ (Z n + ) Ip denote the highest weight of the algebraic representation V λ of (Res F + /Q G) Q p such that V λ ⊗ ι,Q p C ∼ = σ ∨ ∞ . We can define a highest weight ξ for (Res F/Q GL n ) Q p by letting ξ τ = λ τ and ξ τ c = −w 0 λ τ for τ ∈ I p (w 0 is the longest element in the Weyl group of GL n ). The infinitesimal character of π ∞ is the same as that of V ∨ ξ ⊗ ι,Q p C. The Hodge-Tate weights of r π,ι may be described as follows: if τ ∈ I p , then HT τ (r π,ι ) = {λ τ,1 + n − 1, λ τ,2 + n − 2, . . . , λ τ,n }.
We fix once and for all integers a ≤ b such that for all τ ∈ Hom(F, Q p ), the elements of HT τ (r π,ι ) are contained in [a, b] and a + b = n − 1.
Let E/Q p be a coefficient field containing the image of every embedding F → Q p . After possibly enlarging E, we can assume that there is a model ρ : G F,S → GL n (O) of r π,ι , which extends to a homomorphism r : G F + ,S → G n (O) such that ν • r = ǫ 1−n δ n F/F + . Let D denote the group determinant of ρ, which is then defined over k.
With these choices the pseudodeformation ring denoted R S in §2.4 is defined, as well as the prime ideal q S = ker(R S → O) determined by ρ. Moreover, for any Taylor-Wiles datum (Q, Q, (α v,1 , . . . , α v,n ) v∈Q ) we have the auxiliary ring R S∪Q . We introduce one more object here: it is the maximal quotient R S∪Q → R S∪Q,ab over which for each v ∈ Q, the restriction of the universal pseudocharacter to has a natural structure of A[U p ]-module, and the U p -invariants are S λ (U, A). It follows that S λ (U, A) has a natural structure of H(G(A ∞,p F + ), U p )-module. A standard argument (cf. [Ger19, Lemma 2.2.5]) shows that there is an isomorphism of where the sum is over automorphic representations of G( is an open compact subgroup of U and T is a finite set of places of F + containing all places such that  Proof. If we invert p then T ∅ ⊗ O Q p = µ E µ is a product of fields indexed by automorphic representations µ of G(A F + ) with µ U m = 0 and µ ∞ ∼ = σ ∞ . To prove the lemma, it suffices to show that each of the maps R D,S → E µ factors through the quotient R S : in other words, the conjugate self-duality condition and the semistability condition of (2.3). These conditions follow from local-global compatibility for the Galois representation associated to the base change of µ.

4.3.
Automorphic data associated to Taylor-Wiles data. Suppose given a set Q of Taylor-Wiles places. In this case we define open compact subgroups U 0 (Q) = v U 0 (Q) v and U 1 (Q) = v U 1 (Q) v as follows: We set ∆ Q = U 0 (Q)/U 1 (Q), which may be naturally identified with v∈Q k(v) × (p) n . We define As in Lemma 4.2, we have a surjective map R S∪Q → T Q . Note that the natural map T S∪Q λ (U, O) mQ → T ∅ is in fact an isomorphism, and so there are surjections So far we have not used any Hecke operators at places v ∈ Q. For any v ∈ Q, α ∈ F × v , and 1 ≤ i ≤ n, we let t v,i : v of the homomorphism defined just above [ACC + 18, Proposition 2.2.7] (and denoted t v,i there). That proposition shows that if π v is an irreducible admissible 1 (α), . . . , t v,n (α), and e v,i (α, π v ) ∈ Q p is the scalar by which it acts on π We define T Q 0,Q ⊂ End(S λ (U 0 (Q), O) m0,Q ) to be the subalgebra generated by T 0,Q and the elements t v,i (α) for all v ∈ Q, i = 1, . . . , n and α ∈ F × v . We define Proof. The proof is identical to that of [CHT08, Lemma 3.3.1], using that U (and hence any subgroup of U ) is sufficiently small.
]. This is a polynomial subalgebra of ⊗ v∈Q H(G(F + v ), U 0 (Q) v ) that receives an action of the group W Q = v∈Q S n . For every m ≥ 1, we have a canonical morphism of T Q -modules For each v ∈ Q, the universal pseudocharacter over R S∪Q,ab determines by restriction an n-dimensional pseudocharacter On the other hand, for each i = 1, . . . , n, there is a character α v,i : W ab F v → (T Q Q ) × given by the formula α v,i (Art F v (α)) = t v,i (α). We write α v for the pseudocharacter α v = α v,1 ⊕ · · · ⊕ α v,n .
These two families of pseudocharacters are related by the following lemma, which is a formulation of local-global compatibility at v ∈ Q. (1) The map (3) The image of the map R S∪Q,ab → T Q ֒→ T Q Q contains the Hecke operators e v,i (α) for each v ∈ Q, i = 1, · · · , n and α ∈ We caution the reader that the map R S∪Q,ab → T Q Q is not in general surjective, because of the presence of Hecke operators at Q which do not lie in the Bernstein centre.
The following proposition will be crucial for controlling our patched modules of automorphic forms. As mentioned in the introduction, this is inspired by arguments of Pan [Pan19].
there is an element f Q ∈ R S∪Q,ab such that (1) f Q kills the kernel and cokernel of η Q,m for all m ≤ N (2) The image f Q,σ of f Q under the composition of maps Proof. We set and let f Q be a pre-image of f Q in R S∪Q,ab (such a pre-image exists by Lemma 4.4). It follows from Proposition 3.1 that f Q kills the kernel and cokernel of η Q,m for all m ≤ N . If we take c = n!d then, again using Lemma 4.4, we see that the second part of the proposition is satisfied.
We give one last piece of structure. Suppose fixed an ordering Q = {v 1 , . . . , v q } and for each v ∈ Q a surjection Z p → k(v) × (p). This data determines a surjection j . The ring S WQ ∞ also has a role to play as a consequence of the following easy lemma: Lemma 4.6. The functor of deformations of the trivial pseudocharacter of Z p of dimension n is represented by O X 1 , . . . , X n Sn , with the universal characteristic polynomial χ(t) of 1 ∈ Z p given by Proof. Indeed, a residually trivial pseudocharacter of Z p of dimension n over a ring A ∈ C O is precisely a point of (GL n GL n )(A) lying over the image of the identity in (GL n GL n )(k). Now we use the identification of the adjoint quotient GL n GL n with the quotient of the diagonal maximal torus by the Weyl group. The universal deformation is given by the orbit of the matrix diag(1 + X 1 , . . . , 1 + X n ).
Consequently, there is a homomorphism S WQ ∞ → R S∪Q,ab , classifying the pullback of the tuple (γ v ) v∈Q to a tuple of n-dimensional pseudocharacters of the group Z p . There is also a homomorphism S WQ ∞ → T Q Q , classifying the pullback of (α v ) v∈Q to a tuple of n-dimensional pseudocharacters of the group Z p . This coincides with the restriction to S  • We moreover fix uniformisers ̟ v for all v ∈ Q N (for every N ). This allows us to think of the pseudocharacters γ v as pseudocharacters of k(v) × (p) × Z. Recalling that we have fixed a generator of k(v) × (p) and an ordering on Q N , for every N we have a q-tuple (γ N,1 , . . . , γ N,q ) of n-dimensional pseudocharacters of (Z p × Z) with coefficients in R N .
for each v ∈ Q N and i = 1, . . . , n. Using these actions, together with the fixed orderings on Q N , we obtain an action of the algebra on these spaces, together with an identification of A with A QN sending t (i) j to t vj ,i (̟ vj ). We have characters α . . . , n and j = 1, . . . , q. By Lemma 4.4, the pushforward of the pseudocharacter α j = tr α takes values in T N and equals the pushforward of γ N,j there.
• We can identify all the Weyl groups W QN (using our fixed orderings of Q N for each N ). We denote them all by W . There is a natural W -action on S ∞ , compatible with the maps to O[∆ N ]. The invariants S W ∞ are a regular local O-algebra, with S ∞ a finite free S W ∞ -algebra (S W ∞ is a power series algebra over the elementary symmetric polynomials in y this is the ideal of S W ∞ generated by the elementary symmetric polynomials. There is also a natural W -action on A. • We let g = nq and R ∞ = O x 1 , . . . , x g , and let q ∞ = (x 1 , . . . , x g ) ∈ Spec(R ∞ ). For each N we have a map R ∞ → R N such that q ∞ R N ⊂ q N and q N /(q 2 N , q ∞ ) is killed by a power of ̟ which is independent of N . Definition 4.8. We define Here A W QN acts on S λ (U, O/̟ m ) m via the spherical Hecke algebra action at places in Q N . We note that we naturally obtain compatible actions of A on M , M 0 and M 1 .
we equip S λ (U, O) m with an A W action (A W acts on the N factor in the product via its identification with A W QN ) and we see that we have a natural isomorphism Lemma 4.9.
(1) M 1 is a flat S ∞ -module.   It follows from Lemma 4.11 that R p acts on M 1 (this is why R p is defined the way it is). We are going to use [Pan19, Lemma 4.5.3] a few times, so we restate it here: is surjective with kernel given by elements of the form (m i ) such that for each m there exists I m ∈ F with m i ∈ M i,m for all i ∈ I m .
We have a natural map N ≥1 R N → R p which is surjective by Lemma 4.12. We also have a natural map R p → R 0 given by taking the limit over m of Lemma 4.13. The map R p → R 0 we have just defined is surjective.
Proof. We again apply Lemma 4.12: this implies that the natural map N ≥1 is surjective; on the other hand it factors through our map R p → R 0 .
From the n-dimensional pseudorepresentations (γ N,j ) N ≥1 (j = 1, . . . , q) with coefficients in N ≥1 R N , we obtain n-dimensional pseudorepresentations γ ∞,j (j = 1, . . . , q) of Z p × Z with coefficients in R p . On the other hand, M 1 has a natural structure of S ∞ ⊗ O A-module, and we have defined characters α (1) Composing γ ∞,i with the map R p → R 0 gives a pseudorepresentation which is inflated from the 'unramified quotient' Z p × Z → Z (i.e. projection to second factor).
(2) The composite of γ ∞,j with the map R p → End(M 1 ) equals the composite of α j with the map S ∞ ⊗ O A → End(M 1 ). Consequently, the map R p → End(M 1 ) is a homomorphism of S W ∞ -algebras. Proof. The first part follows from the analogous statement for each of the pseudorepresentations γ N,i (which holds because the pseudorepresentations classified by R 0 are unramified at places in Q N ). The second part follows from Lemma 4.4.
Definition 4.15. We let q p be the prime ideal in R p given by the inverse image of q 0 ⊂ R 0 .
Lemma 4.16. The image of q N ⊂ N ≥1 R N in R p is equal to q p .
Proof. Write I for the kernel of N ≥1 R N → R p and I ′ for the image of I in The ideal I is the set of elements (x N ) ∈ R N such that for each m ≥ 1 there exists It follows that I ′ is contained in the kernel of the map N ≥1 O → O. We need to show that I ′ equals the kernel. To this end, choose a tuple of elements (y N ) ∈ N ≥1 O which does lie in the kernel. Recall that there is a constant c such that the image of f QN in R N /q N = O has ̟-adic valuation ≤ c. Let I m = {N ≥ 1 | ord ̟ y N ≥ m(c + 1)}. Then I 1 ⊃ I 2 ⊃ I 2 ⊃ . . . and ∩ m≥1 I m = ∅. Moreover, each I m is in F (since (y N ) is in the kernel of the map to O).
We have (m RN f QN ) m + q N = (̟ m f m QN ) + q N , and this contains (̟ (c+1)m ) + q N . Therefore we can for each m ≥ 1 and N ∈ I m find an element x N,m ∈ (m RN f QN ) m such that x N,m + q N = y N . We define a tuple (x N ) N ≥1 ∈ N ≥1 R N by taking x N to be an arbitrary pre-image of y N if N ∈ I 1 and x N = x N,m if N ∈ I m − I m+1 .
Then (x N ) lies in I and its image in N ≥1 O equals (y N ), as required.
(1) We have an equality of ideals N ≥1 q m The possibility of proving a statement like this one is mentioned in [Pan19, Remark 4.6.10].
Proof. It suffices to prove the first part. Recall from the proof of Corollary 2.27 that there exists an integer g 0 such that for every N there exists a surjection O x 1 , . . . , x g0 → R N such that the images of the x i are in q N (since q N /q 2 N can be generated by g 0 elements). Now it suffices to prove that we have an equality . , x g0 . We conclude using the fact that for any ring R and any ideal I ⊂ R we have , and we also have I N ≥1 R = N ≥1 I when I is finitely generated.
For the statement of the next proposition, we recall that our data includes, for Proposition 4.18.
(1) The O-module is killed by ̟ c , where c is as in Corollary 2.27. (2) The natural map on completed local rings (R ∞ ) q∞ → (R p ) q p is surjective. In particular, (R p ) q p is a complete Noetherian local E-algebra with residue field E.
Proof. It follows from Corollary 2.27 that the cokernel of the map is killed by ̟ c . Applying Lemmas 4.16 and 4.17, it remains to show that the image of q ∞ /(q ∞ ) 2 in q p /(q p ) 2 is the same as the image of N ≥1 q ∞ /(q ∞ ) 2 . This is done as in the proof of [Pan19, Prop. 4.6.16]: it suffices to show that the composition of maps is surjective. Here the first map is the diagonal embedding and we regard O as a N O-algebra via the map N O → R p /q p ∼ = O. We conclude using Lemma 4.19. This shows the first part of the proposition. For the second, we see that the first part implies that each of the maps To check that lim ← −i g i is surjective, it is enough to note that the sequence (ker g i ) i≥1 satisfies the Mittag-Leffler condition (because each of these ideals has finite length, being contained in an Artinian local ring).
Lemma 4.19. Let R be a commutative ring and M a finitely generated R-module. Suppose we have a R-algebra map N ≥1 R λ → R. Then the composite map Proof. If M is finite free over R, then N ≥1 M has a ( N R)-basis given by diagonally embedded basis elements for M , and the statement is clear. In general, we write M as a quotient of a finite free R-module F . The composition of M is surjective and factors through M .
Remark 4.20. Note that we have not shown that q p is finitely generated, so we rely on the comparison with R ∞ to show that q p -adic completion (R p ) q p is q p -adically complete! Now we are going to consider the modules: Lemma 4.21.
Proof. We start with the third part: this follows immediately from Lemma 4.9, since by Proposition 4.5 the image of f in R p is not in q p . The second part also follows immediately from Lemma 4.9. It remains to show the first part. Since the inverse image of q p in S W ∞ is a W ∞ , the action of S ∞ on m 1 factors through the localisation S ∞ ⊗ S W ∞ (S W ∞ ) a W ∞ = S ∞,a∞ (note that a ∞ is the unique point of Spec(S ∞ ) in the pre-image of a W ∞ under the finite map Spec(S ∞ ) → Spec(S W ∞ )). We know from Lemma 4.9 that M 1 /a 2 ∞ is a flat S ∞ /a 2 ∞ -module, so the localisation m 1 is a flat S ∞ /a 2 ∞ -module, and hence a flat S ∞,a∞ /a 2 ∞ -module. Since m 1 /a ∞ is finite dimensional (combining the second and third parts), m 1 is finitely generated over the Artinian local ring S ∞,a∞ /a 2 ∞ . Since m 1 is a finite dimensional E-vector space, the action of the local E-algebra (R p ) q p factors through an action by (an Artinian quotient of) (R p ) q p . It follows from the third part of Lemma 4.21 that the action of (R p ) q p on m 0 ∼ = m factors through the composition of surjective maps Now we consider again our pseudorepresentations γ ∞,j (1 ≤ j ≤ q) of Z p × Z with coefficients in R p .
Proof. To show that δ j = 0, it suffices to show that for some m ≥ 1 the image of δ j under the composition Recall the constant d from Lemma 2.26. Choose m > dn(n − 1). Then it follows from Lemma 2.26 that we will be done if we can identify the image of δ j in O/̟ m with the image of the discriminant of the characteristic polynomial of a Frobenius element σ v for some v ∈ Q N . Choose m ′ so that our , where δ j,N is the image of the discriminant for the Frobenius element at the jth element of Q N . We deduce that the image of δ j in R 0 /m m ′ R0 coincides with one of these Frobenius discriminants.
The same argument shows that the image of χ j (t) mod q p splits into linear factors in O/̟ m [t] for all m ≥ 1. Indeed, for each σ v ∈ G F , the characteristic polynomial of ρ(σ v ) has all of its roots in O (this is part of the definition of an enormous subgroup of GL n (O)). Hensel's lemma implies that χ j (t) itself factors in For each j ∈ {1, . . . , q} we fix an ordering x (1) j , . . . , x (n) j of the (pairwise distinct) roots in E of the polynomial χ j (t) mod q p . For each j, we may consider the pseudorepresentation (γ ∞,j ) q p of Z p × Z with coefficients in (R p ) q p given by composing γ ∞,j with the natural map R p → (R p ) q p . This pseudorepresentation is residually multiplicity free.
Lemma 4.24. There is a unique collection of continuous characters γ (i) j : Z p ×Z → ((R p ) q p ) × (i = 1, . . . , n, j = 1, . . . , q) such that γ (i) j mod q p is the character Proof. This follows from, e.g., [BC09, Proposition 1.5.1], since in a commutative GMA we have (using the notation of loc.cit. The characters γ (i) j | Zp×0 : Z p → ((R p ) q p ) × determine an extension of the homomorphism S W ∞ → R p to a homomorphism S ∞ → (R p ) q p . This in turn naturally extends to a map from the formally smooth E-algebra (S ∞ ) a∞ and we choose a lift of this through the surjective map (see Proposition 4.18) to equip (R ∞ ) q∞ with a map from (S ∞ ) a∞ . We denote by A ′ the localization of A at the prime ideal (t (We recall that the ring A, defined at the beginning of §4.4, is a Laurent polynomial ring in elements (t Remark 4.25. The above localization is our replacement for the usual 'localization with respect to a suitable eigenvalue of the U q operator' which appears in the Taylor-Wiles method. We can only do this after patching and inverting p because we do not assume that ρ(σ v ) has distinct eigenvalues for Taylor-Wiles places v.
(1) For each i = 1, . . . , n and j = 1, . . . , q, the respective pushforwards of the characters α (2) The two structures of S ∞ -module on m ′ 1 (the standard one, and the one induced by the homomorphism S ∞ → (R p ) q p constructed above) are the same.
. . , n, j = 1, . . . , q}. By construction, the elements of X commute with each other; let T denote the Esubalgebra of End(m ′ 1 ) generated by the elements of X. Then T is an Artinian Ealgebra. The pushforwards of the characters α j are equal after pushforward to T for each j = 1, . . . , q it is enough (after the uniqueness assertion of [BC09, Proposition 1.5.1]) to show that they are equal after pushforward to each residue field of T . However, our construction shows that for each i = 1, . . . , n and j = 1, . . . , q the elements α (i) j are commuting nilpotent elements of End(m ′ 1 ) and therefore their difference α (i) j (0, 1) lies in the Jacobson radical of T . This proves the first part of the lemma. The second is an immediate consequence since the two S ∞ -module structures are determined by the two sets of characters α (i) j and γ (i) j . The third part of the lemma is equivalent to the assertion that characters γ (i) j | Zp×0 become trivial after pushforward along the map (R p ) q p → (R 0 ) q0 . Since the pseudocharacter tr γ (1) j ⊕ · · · ⊕ γ (n) j is residually multiplicity-free, the desired statement follows from uniqueness and Lemma 4.14.
The fourth part of the lemma follows from the same statement before localisation to A ′ (Lemma 4.21). We now prove the final part of the lemma. Since m ′ 1 is a direct summand of m 1 , we just need to prove that m ′ 1 is non-zero, or indeed that m ′ 0 is non-zero. For this, we note that it follows from Lemma 4.14 (compatibility of Galois and automorphic pseudocharacters) and the observation above (4.2) that the characteristic polynomial To complete the proof of this section's main theorem, we recall Brochard's freeness criterion: Let M be a non-zero A-flat B-module which is finitely generated over B. Then M is finite free over B.
Proof. We apply Brochard's criterion with A = S ∞,a∞ /a 2 ∞ , B = (R ∞ ) q∞ /a 2 ∞ , M = m ′ 1 . Note that the embedding dimension of S ∞,a∞ /a 2 ∞ is qn and, since (R ∞ ) q∞ is a power series ring over E in qn variables, the embedding dimension of (R ∞ ) q∞ /a 2 ∞ is ≤ qn. We conclude that m ′ 1 is finite free over (R ∞ ) q∞ /a 2 ∞ and therefore m ′ 0 is finite free over (R ∞ ) q∞ /a ∞ . Since the action of (R ∞ ) q∞ on m ′ 0 factors through the action of (T ∅ ) q0 , we deduce that each of the surjective maps are isomorphisms. The vanishing of the adjoint Selmer group follows from the identification of this with the reduced tangent space of (R 0 ) q0 (i.e. Proposition 2.17).
Remark 4.29. We find it convenient (or amusing) to use Brochard's freeness criterion here, although we could alternatively have worked with the (S ∞ ) a∞ -module lim ← −m ((M 1 /a m ∞ ) q p ) in place of m 1 and concluded using Auslander-Buchsbaum as in the work of Diamond and Fujiwara.

Applications
We now deduce our main theorems. We begin with a useful lemma.
Lemma 5.1. Let F be a number field, and let E/Q p be a coefficient field. Let ρ : G F → GL n (E) be a continuous representation which is unramified almost everywhere. Let S be a finite set of places of F . Then we can find a finite set T of places of F with the following property: • T ∩ S = ∅. • For any T -split finite extension F ′ /F , ρ(G F ′ (ζ p ∞ ) ) = ρ(G F (ζ p ∞ ) ).
Proof. After replacing ρ by ρ ⊕ ǫ, it is enough to show we can choose T so that ρ(G F ) = ρ(G F ′ ) if F ′ /F is T -split. Conjugate ρ so that it takes values in GL n (O), and let L ∞ /F be the extension cut out by ρ, L N /F the extension cut out by ρ N = ρ mod ̟ N . We have ρ(G F ) = lim ← −N ρ N (G F ), so it's enough to show that we can choose T so that if F ′ /F is T -split, then ρ N (G F ) = ρ N (G F ′ ) for all N ≥ 1.
To this end, we let M/F be the compositum of all of the extensions of F cut out by simple quotients of Gal(L N /F ) (for any N ≥ 1). The extension M/F is finite, because simple quotients of Gal(L N /F ) (for varying N ≥ 1) correspond to simple quotients of ρ(G F ) by closed normal subgroups. Since ρ(G F ) has a normal, closed subgroup of finite index which is a topologically finitely generated pro-p group, these quotients are finite in number.
We can therefore choose T to be any set disjoint from S and such that for each intermediate field M/M ′ /F with Gal(M ′ /F ) simple, there exists v ∈ T which does not split in M ′ .
We prove a theorem for regular algebraic, cuspidal, polarized automorphic representations. First we treat the case of a CM base field.
Proof. As in the proof of [BLGHT11, Theorem 1.2], π has a twist which is polarized with respect to δ n F/F + (i.e. of unitary type). The twist does not alter ad r π,ι , so we can assume that π is of unitary type. For any finite extension F ′ /F + , the induced map H 1 f (F + , ad r π,ι ) → H 1 f (F ′ , ad r π,ι ) is injective. It is therefore enough to find a soluble totally real extension L + /F + with the following properties: • Let L = L + F . Then r π,ι (G L(ζ p ∞ ) ) = r π,ι (G F (ζ p ∞ ) ).
• Let π L denote the base change of π (which exists and is regular algebraic, after [AC89, Ch. 3, Theorems 4.2, 5.1]). It is cuspidal, because r π,ι | GL is irreducible. Each place of L at which π L is ramified, or dividing p, is split over L + . • For every place w of L, π L,w has an Iwahori-fixed vector. To achieve this, let S be the set of places of F + dividing p or above which π is ramified, and let S F denote the set of places of F lying above a place of S. Let T F denote a set as provided by Lemma 5.1, disjoint from S F , and let T denote the set of places of F + lying below a place of T F . Then S and T are disjoint and if L + /F + is T -split, then L/F is T F -split. We can choose L + /F + to be any T -split soluble totally real extension which has the correct behaviour at the places in S, the existence of such an extension being a consequence of [CHT08, Lemma 4.1.2].
Next we treat the case of a totally real base field F . We consider a regular algebraic, cuspidal, polarized automorphic representation (π, χ) of GL n (A F ). Let ι : Q p → C be an isomorphism, and suppose that r π,ι is irreducible. Let V denote the space on which r π,ι acts. Then there is a unique G F -equivariant pairing ·, · : V × V → ǫ 1−n r χ,ι , which is symmetric if n is odd or n is even and r χ,ι (c v ) = 1 (v|∞), or antisymmetric if n is even and r χ,ι (c v ) = −1 (see [BC11] and [BLGGT14, §2.1]). We thus obtain a homomorphism r ′ π,ι : G F → GS( ·, · )(Q p ) to the general similitude group of the pairing ·, · . We write gs for the Lie algebra of this reductive group over Q p .
Theorem 5.3. Let F be a totally real number field, and let (π, χ) be a regular algebraic, cuspidal, polarized automorphic representation of GL n (A F ). Let ι : Q p → C be an isomorphism, and suppose that r π,ι (G F (ζ p ∞ ) ) is enormous. Then H 1 f (F, gs) = 0. Proof. This can be deduced from Theorem 5.2 using base change in the same way that [All16, Theorem B] is deduced from [All16, Theorem A]. We omit the details.
When n = 2, these results take a particularly simple form: Theorem 5.4. Let F be a totally real number field, and let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A F ). Let ι : Q p → C be an isomorphism. Suppose that one of the following holds: (1) π does not have CM.
(2) π has CM by an extension K/F , and K ⊂ F (ζ p ∞ ). Then H 1 f (F, ad r π,ι ) = 0. Proof. When n = 2, gs = gl 2 . Our result will follow from Theorem 5.3 if we can verify that our hypotheses imply that r π,ι (G F (ζ p ∞ ) ) is enormous. If π does not have CM, this is example 2.29.
Suppose instead that π has CM by a CM quadratic extension K/F , and K is not contained in F (ζ p ∞ ). To show that r π,ι (G F (ζ p ∞ ) ) is enormous, it is enough to show that we can find regular semisimple elements in the image of both G K(ζ p ∞ ) and G F (ζ p ∞ ) − G K(ζ p ∞ ) . Elements of the latter type exist because of our assumption that K is not contained in F (ζ p ∞ ). Now suppose for a contradiction that r π,ι ∼ = Ind GF GK χ is scalar on restriction to G K(ζ p ∞ ) . This implies that χ/χ c is trivial on G K(ζ p ∞ ) , and hence that (χ/χ c ) 2 = 1 (since c acts trivially on Gal(K(ζ p ∞ )/K)). This contradicts the fact that χ/χ c has infinite order (because its Hodge-Tate weights are all non-zero, because π is regular algebraic). This completes the proof.
We can also prove results for elliptic curves.
Theorem 5.5. Let F be a totally real number field, and let E be an elliptic curve over F . Let p be a prime, and suppose that one of the following holds: (1) E does not have CM.
(2) E has CM by a quadratic field K/Q, and K ⊂ F (ζ p ∞ ). Then H 1 f (F, ad V p (E)) = 0. Proof. If the elliptic curve E has CM, then its p-adic Galois representations are automorphic and we can appeal to Theorem 5.4. If E does not have CM, then there exists a totally real extension F ′ /F such that the p-adic Galois representations of E F ′ are automorphic (for example, by [Tay06]) and we can appeal again to the same theorem.
Combining our results with potential automorphy theorems, we can prove some more general vanishing results. Here is an example for symmetric powers of twodimensional representions.
Theorem 5.6. Let F be a CM number field, and let (π, χ) be a regular algebraic, cuspidal, polarized automorphic representation of GL 2 (A F ) such that Sym 2 π is cuspidal. Let p be a prime, and fix an isomorphism ι : Q p → C. Then for any n ≥ 1, H 1 f (F + , ad Sym n−1 r π,ι ) = 0. Proof. By [BLGGT14, Theorem 5.4.1], there exists a Galois, CM extension F ′ /F such that Sym n−1 r π,ι | G F ′ is automorphic. It suffices to show the vanishing of H 1 f ((F ′ ) + , ad Sym n−1 r π,ι ), and this follows from Theorem 5.2 once we verify that Sym n−1 r π,ι (G F ′ (ζ p ∞ ) ) is enormous. However, this follows from Example 2.29.
Finally, we give an application to vanishing results for anticyclotomic characters, as predicted by the Bloch-Kato conjecture. Over a general CM field our main theorem gives vanishing results which are not covered by known cases of the anticyclotomic main conjecture (cf. those proved in [Hid09]).
Theorem 5.7. Let F be a CM number field, and let χ : F × \A × F → C × be a unitary character of type A 0 . Let ι : Q p → C be an isomorphism, and suppose that the following conditions are satisfied: (1) χχ c = 1.
Proof. The given conditions imply that there is a character ψ : F × \A × F → C × of type A 0 such that ψ/ψ c = χ. Given this, let π denote the automorphic induction of ψ from F to F + . It is a regular algebraic, cuspidal automorphic representation of GL 2 (A F + ) and Ind G F + GF r χ,ι is a subquotient of ad r π,ι , so the desired vanishing follows from Shapiro's lemma for Bloch-Kato Selmer groups and Theorem 5.4.
Lemma 5.8. Let F be a CM field, and let χ : G F → Q/Z be a continuous character such that χχ c = 1. Then there exists a continuous character φ : G F → Q/Z such that φ/φ c = χ.
The edge morphism H 3 (F/F + , H 0 (F, Q/Z)) → H 3 (F + , Q/Z) is inflation, and is injective because the extension F/F + is CM and the map H 3 (F + , Q/Z) → v|∞ H 3 (F + v , Q/Z) is bijective. This completes the proof.