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Symmetric power functoriality for holomorphic modular forms

Symmetric power functoriality for holomorphic modular forms SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS by JAMES NEWTON and JACK A. THORNE ABSTRACT Let f be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting Sym f for every n ≥ 1. We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over Q. CONTENTS Introduction ........................................................ 1 1. Definite unitary groups ................................................. 9 Part I: Analytic continuation of functorial liftings .................................... 26 2. Trianguline representations and eigenvarieties .................................... 27 3. Ping pong ........................................................ 60 Part II: Raising the level . ................................................. 65 4. Raising the level – automorphic forms ........................................ 68 5. A finiteness result for Galois deformation rings .................................... 75 6. Raising the level – Galois theory ............................................ 86 7. Level 1 case ....................................................... 94 8. Higher levels ...................................................... 104 Acknowledgements ..................................................... 110 References ......................................................... 111 Introduction Context. — Let F be a number field, and let π be a cuspidal automorphic repre- sentation of GL (A ). Langlands’s functoriality principle [Lan70, Question 5] predicts n F the existence, for any algebraic representation R : GL → GL , of a functorial lift of π n N along R; more precisely, an automorphic representation R(π) of GL (A ) which may N F be characterized by the following property: for any place v of F, the Langlands param- eter of R(π) is the image, under R, of the Langlands parameter of π . The Langlands v v parameter is defined for each place v of F using the local Langlands correspondence for GL (F ) (see [Lan89, HT01, Hen00]). n v m th The simplest interesting case is when n = 2and R = Sym is the m symmetric power of the standard representation of GL . In this case the automorphy of Sym π was proved for m = 2 by Gelbart and Jacquet [GJ78]and for m = 3, 4 by Kim and Shahidi [KS02, Kim03]. More recently, Clozel and the second author have proved the automorphy of Sym π for m ≤ 8 under the assumption that π can be realised in a space of Hilbert modular forms of regular weight [CT14, CT15, CT17]; equivalently, that the number field F is totally real and the automorphic representation π is regular algebraic, in the © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2021 https://doi.org/10.1007/s10240-021-00127-3 2 JAMES NEWTON, JACK A. THORNE sense of [Clo90b]. This includes the most classical case of automorphic representations arising from holomorphic modular forms of weight k ≥ 2. We also mention the work of th Dieulefait [Die15], which shows automorphy of the 5 symmetric power for cuspidal Hecke eigenforms of level 1 and weight k ≥ 2. On the other hand, the potential automorphy (i.e. the existence of the symmetric power lifting after making some unspecified Galois base change) of all symmetric powers for automorphic representations π associated to Hilbert modular forms was obtained by Barnet-Lamb, Gee and Geraghty [BLGG11] (the case of elliptic modular forms is due to Barnet-Lamb, Geraghty, Harris and Taylor [BLGHT11]). Results of this paper. — In this paper, we prove the automorphy of all symmetric powers for cuspidal Hecke eigenforms of level 1 and weight k ≥ 2. More precisely: Theorem A. — Let π be a regular algebraic cuspidal automorphic representation of GL (A ) 2 Q of level 1 (i.e. which is everywhere unramified). Then for each integer n ≥ 2, the symmetric power lifting n−1 Sym π exists, as a regular algebraic cuspidal automorphic representation of GL (A ). n Q In fact, we establish a more general result in which ramification is allowed: Theorem B. — Let π be a regular algebraic cuspidal automorphic representation of GL (A ) 2 Q of conductor N ≥ 1, which does not have CM. Suppose that for each prime l|N, the Jacquet module of π is non-trivial; equivalently, that π is not supercuspidal. Then for each integer n ≥ 2, the symmetric l l n−1 power lifting Sym π exists, as a regular algebraic cuspidal automorphic representation of GL (A ). n Q The class of automorphic representations described by Theorem B includes all those associated to holomorphic newforms of level  (N), for some squarefree integer N ≥ 1; in particular those associated to semistable elliptic curves over Q. We can therefore offer the following corollary in more classical language: Corollary C. — Let E be a semistable elliptic curve over Q. Then, for each integer n ≥ 2,the completed symmetric power L-function (Sym E, s) as defined in e.g. [DMW09], admits an analytic continuation to the entire complex plane. We remark that the meromorphic, as opposed to analytic, continuation of the com- pleted L-function (Sym E, s) was already known, as a consequence of the potential au- tomorphy results mentioned above. Potential automorphy results were sufficient to prove the Sato–Tate conjecture, but our automorphy results make it possible to establish effective versions of Sato–Tate (we thank Ana Caraiani and Peter Sarnak for pointing this out to us). See, for example, [Tho14b] for an unconditional result and [Mur85, BK16, RT17] for results conditional on the Riemann Hypothesis for the symmetric power L-functions. In other words, there is no quadratic Hecke character χ such that π π ⊗ χ . = SYMMETRIC POWER FUNCTORIALITY 3 Strategy. — Algebraic automorphic representations of GL (A ) are conjectured to n F admit associated Galois representations [Clo90b]. When F is totally real and π is a self-dual regular algebraic automorphic representation, these Galois representations are known to exist; their Galois deformation theory is particularly well-developed; and they admit p-adic avatars, which fit into p-adic families of overconvergent automorphic forms. We make use of all of these tools. We begin by proving the following theorem: th Theorem D. — Let n ≥ 2 be an integer and suppose that the n symmetric power lifting exists for th one regular algebraic cuspidal automorphic representation of GL (A ) of level 1. Then the n symmetric 2 Q power lifting exists for every regular algebraic cuspidal automorphic representation of GL (A ) of level 1. 2 Q We sketch the proof of Theorem D, which is based on the properties of the Coleman–Mazur eigencurve E .Werecallthatif p is a prime, the eigencurve E is a p p p-adic rigid analytic space that admits a Zariski dense set of classical points correspond- ing to pairs (f ,α) where f is a cuspidal eigenform of level 1 and some weight k ≥ 2and 2 k−1 α is a root of the Hecke polynomial X − a (f )X + p . The eigencurve admits a map κ : E → W = Hom(Z , G ) to weight space with discrete fibres; the image of (f ,α) is p p m k−2 the character x → x . We first show that for a fixed n ≥ 1, the automorphy of Sym f is a property which is “constant on irreducible components of E ”. (Here we confuse f and the automor- phic representation π that it generates in order to simplify notation.) More precisely, if (f ,α) and (f ,α ) determine points on the same irreducible component of E , then the n n automorphy of Sym f is equivalent to that of Sym f . This part of the argument, which occupies §2 of this paper, does not require a restriction to cusp forms of level 1 – see Theo- rem 2.33. It is based on an infinitesimal R = T theorem on the eigenvariety associated to a definite unitary group in n variables. Kisin (for GL )[Kis03] and Bellaïche–Chenevier (for higher rank) [BC09] have observed that such theorems are often implied by the van- ishing of adjoint Bloch–Kato Selmer groups. We are able to argue in this fashion here because we have proved the necessary vanishing results in [NT20]. To exploit this geometric property, we need to understand the irreducible com- ponents of E . This is a notorious problem. However, conjectures predict that E has a p p simple structure over a suitably thin boundary annulus of a connected component of weight space W (see e.g. [LWX17, Conjecture 1.2]). We specialise to the case p = 2, in which case Buzzard–Kilford give a beautifully simple and explicit description of the geometry of E “close to the boundary of weight space” [BK05]. More precisely, E is supported above a single connected component W ⊂ W , 2 2 which we may identify with the rigid unit disc {|w| < 1}. The main theorem of [BK05] −1 ∞ is that the pre-image κ ({|8| < |w| < 1}) decomposes as a disjoint union X of i=1 rigid annuli, each of which maps isomorphically onto {|8| < |w| < 1}.Moreover, X has the following remarkable property: if (f ,α) ∈ X is a point corresponding to a classical modular form, then the p-adic valuation v (α) (otherwise known as the slope of the pair (f ,α)) equals iv (w(κ(f , α))). p 4 JAMES NEWTON, JACK A. THORNE We can now explain the second part of the proof of Theorem D, which occupies −1 §3 of the paper. Since each irreducible component of E meets κ ({|8| < |w| < 1}),it is enough to show that each X contains a point (f ,α) such that Sym f is automorphic. This property only depends on f and not on the pair (f ,α)! Moreover, the level 1 form f −1 determines two points (f ,α), (f ,β) of κ ({|8| < |w| < 1}), which lie on components X and X satisfying i + i = (k − 1)/v (w(κ(f , α))). Starting with a well-chosen initial point i p on a given annulus X , we can jump to any other X in a finite series of swaps between i i pairs (f ,α ), (f ,β ) and moves within an annulus. We call this procedure playing ping pong, and it leads to a complete proof of Theorem D. We remark that for this second step of the proof it is essential that we work with level 1 forms, since it is only in the level 1, p = 2 case that the eigencurve E admits such a simple structure (in particular, the eigencurve is supported above a single connected com- ponent of weight space and every Galois representation appearing in E admits the same residual representation, namely the trivial 2-dimensional representation of Gal(Q/Q) over F ). We note as well that it is necessary to work with classical forms which may be ramified at the prime 2 in order for their weight characters to lie in the boundary annulus of W . We have suppressed this minor detail here. Theorem D implies that to prove Theorem A, it is enough to prove the following result: Theorem E. — For each integer n ≥ 2, there is a regular algebraic cuspidal automorphic repre- n−1 sentation π of GL (A ) of level 1 such that Sym π exists. 2 Q As in the previous works of Clozel and the second author [CT14, CT15, CT17], we achieve this by combining an automorphy lifting theorem with the construction of level-raising congruences. We aim to find f and an isomorphism ι : Q → C such that (writing r : G → GL (Q ) for the p-adic Galois representation associated to f )the f ,ι Q 2 residual representation n−1 Sym r : G → GL (F ) f ,ι Q n p is automorphic; then we hope to use an automorphy lifting theorem to verify that n−1 n−1 Sym r is automorphic, and hence that Sym f is automorphic. In contrast to the f ,ι papers just cited, where we chose r to have large image but p to be small, in order to f ,ι exploit the reducibility of the symmetric power representations of GL in small charac- teristic, here we choose r to have small image, and p to be large. f ,ι More precisely, we choose f to be congruent modulo p to a theta series, so that n−1 r = Ind ψ is induced. In this case Sym r | is a sum of characters, so its residual f ,ι f ,ι G G K automorphy can be verified using the endoscopic classification for unitary groups in n variables. The wrinkle is that the automorphy lifting theorems proved in [ANT20](gen- eralizing those of [Tho15]) require the automorphic representation π of GL (A ) (say) n K verifying residual automorphy to have a local component which is a twist of the Steinberg SYMMETRIC POWER FUNCTORIALITY 5 representation. To find such a π we need to combine the endoscopic classification with the existence of level-raising congruences. In fact, we combine two different level-raising results in order to construct the desired congruences. The first of these, original to this paper, suffices to prove Theorem E in the case that n is odd. The argument is based on a generalization of the following simple observation, which suffices to prove Theorem E in the case n = 3: let q be an odd prime power, and let U (q) denote the finite group of Lie type associated to the outer form of th GL over F .Let p be a prime such that q (mod p) is a primitive 6 root of unity. Then 3 q the unique cuspidal unipotent representation of U (q) remains irreducible on reduction modulo p, and this reduction occurs as a constituent of the reduction modulo p of a generic cuspidal representation of U (q) (see Proposition 1.15). Using the theory of depth zero types, this observation has direct consequences for the existence of congruences between automorphic representations of U . Similar arguments work for general odd n, for carefully chosen global data. We leave a discussion of the (quite intricate) details to §4. The second level-raising result, proved by Anastassiades in his thesis, allows us to n−1 2n−1 pass from the existence of Sym f to the existence of Sym f . We refer to the paper [AT21] for a more detailed discussion. It remains to extend Theorem A to the ramified case, and prove Theorem B.For this we induct on the number of primes dividing the conductor, and use an argument of ‘killing ramification’ as in the proof of Serre’s conjecture [KW09]. Thus to remove a prime l from the level we need to be able to move within a family of l -adic overconvergent modular forms to a classical form of the same tame level, but now unramified at l . This explains our assumption in Theorem B that the Jacquet module of π is non-trivial for every prime l : it implies the existence of a point associated to (a twist of) π on an l -adic eigencurve for every prime l . In a sequel to this paper [NT], we prove a new kind of automorphy lifting theorem for symmetric power Galois representations. This allows us to finally prove a version of Theorem B where the hypothesis that no local component π is supercuspidal is removed. The arguments of [NT] use only fixed weight classical automorphic forms (as opposed to overconvergent automorphic forms) but do require the results of this paper (in particular, Theorem B) as a starting point. Organization of this paper. — We begin in §1 by recalling known results on the clas- sification of automorphic representations of definite unitary groups. We make particular use of the construction of L-packets of discrete series representations of p-adic unitary groups given by Mœglin [Mœg07, Mœg14], the application of Arthur’s simple trace for- mula for definite unitary groups as explicated in [Lab11], and Kaletha’s results on the normalisation of transfer factors (in the simplest case of pure inner forms) [Kal16]. In §2 we study the interaction between the existence of symmetric power liftings of degree n with the geometry of the eigenvariety associated to a definite unitary group in n variables. The basic geometric idea is described in §2.1.In§3 we combine these results 6 JAMES NEWTON, JACK A. THORNE with the explicit description of the tame level 1, p = 2 Coleman–Mazur eigencurve to complete the proof of Theorem D. We then turn to the proof of Theorem E, which rests upon two level-raising results, only the first of which is proved here. The proof of this result is in turn split into two halves; first we give in §4 an automorphic construction of level-raising congruences using types, in the manner sketched above. Then in §6 we establish level-raising congruences of a different kind using deformation theory for residually reducible representations, as developed in [Tho15, ANT20]. These two results are applied in turn to construct our desired level-raising congruences for odd n (Proposition 7.4). A key intermediate result is a finiteness result for certain Galois deformation rings, established in §5, and which may be of independent interest. We use this to control the dimension of the locus of reducible deformations. Finally, we are in a position to prove our main theorems. In §7 we combine the preceding constructions with the main theorem of [AT21] in order to prove Theorem E and therefore Theorem A.In§8, we carry out the argument of ‘killing ramification’ in order to obtain Theorem B. The main technical challenge is to manage the hypothesis of ‘n-regularity’ which appears in our analytic continuation results (see especially Theorem 2.33). To do this we prove a result (Proposition 8.3) which takes a given automorphic representation π and constructs a congruence to an n-regular one π . This may also be of independent interest. Notation. — If F is a perfect field, we generally fix an algebraic closure F/Fand write G for the absolute Galois group of F with respect to this choice. We make the convention that a soluble extension F /F is a (finite) Galois extension with soluble Galois group Gal(F /F). When the characteristic of F is not equal to p, we write : G → Z for the p-adic th cyclotomic character. We write ζ ∈ F for a fixed choice of primitive n root of unity (when this exists). If F is a number field, then we will also fix embeddings F → F extending the map F → F for each place v of F; this choice determines a homomorphism G → G . v F F When v is a finite place, we will write O ⊂ F for the valuation ring, ∈ O for a F v v F v v fixed choice of uniformizer, Frob ∈ G for a fixed choice of (geometric) Frobenius lift, v F k(v) = O /( ) for the residue field, and q = #k(v) for the cardinality of the residue F v v field. When v is a real place, we write c ∈ G for complex conjugation. If S is a finite v F set of finite places of F then we write F /F for the maximal subextension of F unramified outside S and G = Gal(F /F). F,S S If p is a prime, then we call a coefficient field a finite extension E/Q contained inside our fixed algebraic closure Q , and write O for the valuation ring of E, ∈ O for a fixed choice of uniformizer, and k = O/( ) for the residue field. If A is a local ring, we write C for the category of complete Noetherian local A-algebras with residue field A/m . We will use this category mostly with A = EorA = O. If G is a profinite group A SYMMETRIC POWER FUNCTORIALITY 7 and ρ : G → GL (Q ) is a continuous representation, then we write ρ : G → GL (F ) for n n p the associated semisimple residual representation (which is well-defined up to conjugacy). If F is a CM number field (i.e. a totally imaginary quadratic extension of a totally + + real number field), then we write F for its maximal totally real subfield, c ∈ Gal(F/F ) for the unique non-trivial element, and δ + : Gal(F/F ) →{±1} for the unique non- F/F trivial character. If S is a finite set of finite places of F , containing the places at which F/F is ramified, we set F = F and G = Gal(F /F). S F,S S We write T ⊂ B ⊂ GL for the standard diagonal maximal torus and upper- n n n triangular Borel subgroup. Let K be a non-archimedean characteristic 0 local field, and let  be an algebraically closed field of characteristic 0. If ρ : G → GL (Q ) is a con- K n tinuous representation (which is de Rham if p equals the residue characteristic of K), then we write WD(ρ) = (r, N) for the associated Weil–Deligne representation of ρ,and F−ss WD(ρ) for its Frobenius semisimplification. We use the cohomological normalisation × ab of class field theory: it is the isomorphism Art : K → W which sends uniformizers to geometric Frobenius elements. When  = C, we have the local Langlands correspon- dence rec for GL (K): a bijection between the sets of isomorphism classes of irreducible, K n admissible C[GL (K)]-modules and Frobenius-semisimple Weil–Deligne representations over C of rank n. In general, we have the Tate normalisation rec of the local Lang- lands correspondence for GL as described in [CT14, §2.1]. When  = C,wehave T (1−n)/2 rec (π) = rec (π ⊗|·| ). If G is a reductive group over K and P ⊂ G is a parabolic subgroup and π is an G(K) admissible [P(K)]-module, then we write Ind π for the usual smooth induction. If P(K) G(K) G G = C then we write i π for the normalised induction, defined as i π = Ind π ⊗ P P P(K) 1/2 δ ,where δ : P(K) → R is the character δ (x) =| det(Ad(x)| )| (and N is the P >0 P Lie N K P P P unipotent radical of P). × × If ψ : K → C is a smooth character, then we write Sp (ψ) = (r, N) for the n n −1 −1 Weil–Deligne representation on C =⊕ C · e given by r = (ψ ◦ Art ) ⊕ (ψ|·| ◦ i=1 K −1 1−n −1 Art ) ⊕ ··· ⊕ (ψ|·| ◦ Art ) and Ne = 0, Ne = e (1 ≤ i ≤ n − 1). We write St (ψ) 1 i+1 i n K K GL −1/2 GL (K) n n for the unique irreducible quotient of i (ψ ◦ det)δ = Ind ψ ◦ det. We have B B B (K) n n rec (St (ψ)) = Sp (ψ). K n × × If F is a number field and χ : F \A → C is a Hecke character of type A (equiv- alently: algebraic), then for any isomorphism ι : Q → C thereisacontinuous character r : G → Q which is de Rham at the places v|p of F and such that for each finite χ,ι F −1 place v of F, WD(r ) ◦ Art = ι χ | . Conversely, if χ : G → Q is a continuous χ,ι F F v F v p character which is de Rham and unramified at all but finitely many places, then there × × exists a Hecke character χ : F \A → C of type A such that r = χ . In this situation 0 χ,ι we abuse notation slightly by writing χ = ιχ . If F is a CM or totally real number field and π is an automorphic representation of GL (A ), we say that π is regular algebraic if π has the same infinitesimal character as n F ∞ ∗ n an irreducible algebraic representation W of (Res GL ) . We identify X (T ) with Z F/Q n C n 8 JAMES NEWTON, JACK A. THORNE n n in the usual way, and write Z ⊂ Z for the subset of weights which are B -dominant. If ∨ n Hom(F,C) W has highest weight λ = (λ ) ∈ (Z ) , then we say that π has weight λ. τ τ ∈Hom(F,C) When F is CM, the automorphic representation π is said to be conjugate self-dual c ∨ if π π . We refer to [BLGGT14, §2.1] for the more general notion of a polarizable automorphic representation. Note that if π is conjugate self-dual, then (π, δ ) is po- F/F larized and therefore π is polarizable. If π is cuspidal, regular algebraic, and polarizable, then for any isomorphism ι : Q → C there exists a continuous, semisimple representation r : G → GL (Q ) such π,ι F n p p F−ss T −1 that for each finite place v of F, WD(r | ) rec (ι π ) (see e.g. [Car14]). (When π,ι G v v F n = 1, this is compatible with our existing notation.) We use the convention that the Hodge–Tate weight of the cyclotomic character is −1. Thus if π is of weight λ,thenfor any embedding τ : F → Q the τ -Hodge–Tate weights of r are given by π,ι HT (r ) ={λ + (n − 1), λ + (n − 2),...,λ }. τ π,ι ιτ,1 ιτ,2 ιτ,n For n ≥ 1, we define a matrix ⎛ ⎞ ⎜ ⎟ −1 ⎜ ⎟ ⎜ ⎟ = . ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ n−1 (−1) If E/F is a quadratic extension of fields of characteristic 0 then we write θ = θ : −t −1 Res GL → Res GL for the involution given by the formula θ(g) =  c(g)  . E/F n E/F n n We write U ⊂ Res GL for the fixed subgroup of θ .Then U is a quasi-split unitary n E/F n n n group. The standard pinning of GL (consisting of the maximal torus of diagonal matri- ces, Borel subgroup of upper-triangular matrices, and set {E | i = 1,..., n − 1} of root i,i+1 vectors) is invariant under the action of θ and defines an F-pinning of U ,thatwecallits standard pinning. If F is a number field or a non-archimedean local field, then we also write U for the extension of U to a group scheme over O with functor of points n n F −t −1 U (R) ={g ∈ GL (R ⊗ O ) | g =  (1 ⊗ c)(g)  }. n n O E n F n When F is a number field or a local field, we identify the dual group U = GL (C) W , n n F where W acts trivially on GL (C) and an element w ∈ W − W acts by the formula E n c F E −t −1 w · g =  g  (therefore preserving the standard pinning of GL (C)). c n n Given a partition of n (i.e. a tuple (n , n ,..., n ) of natural numbers such that 1 2 k n + n + ··· + n = n), we write L for the corresponding standard Levi sub- 1 2 k (n ,...,n ) 1 k group of GL (i.e. the block diagonal subgroup GL ×· · · × GL ⊂ GL ), and P n n n n (n ,...,n ) 1 k 1 k for the corresponding standard parabolic subgroup (i.e. block upper-triangular matrices with blocks of sizes n ,..., n ). If E is a non-archimedean characteristic 0 local field and 1 k SYMMETRIC POWER FUNCTORIALITY 9 π ,...,π are admissible representations of GL (E), ... ,GL (E), respectively, then we 1 k n n 1 k GL write π × π × ··· × π = i π ⊗ ··· ⊗ π . We write π  ···  π for the irre- 1 2 k 1 k 1 k (n ,...,n ) 1 k k k ducible admissible representation of GL (E) defined by rec ( π ) = ⊕ rec (π );it n E i E i i=1 i=1 is a subquotient of π ×· · · × π . 1 k Given a tuple (n , n ,..., n ) of natural numbers such that 2(n + ··· + n ) + n = 1 2 k 1 k−1 k n, we write M for the Levi subgroup of U given by block diagonal matrices with (n ,...,n ) n 1 k blocks of size n , n ,..., n , n , n ,..., n .Then M is a standard Levi subgroup 1 2 k−1 k k−1 1 (n ,...,n ) (with respect to the diagonal maximal torus of U ), and projection to the first k blocks gives an isomorphism M = (Res GL ×···× Res GL ) × U . We write Q (n ,...,n ) E/F n E/F n n (n ,...,n ) 1 k 1 k−1 k 1 k for the parabolic subgroup given by block upper triangular matrices (with blocks of the same sizes). If F is a non-archimedean characteristic 0 local field and π ,...,π ,π are 1 k−1 k admissible representations of GL (E), ... ,GL (E),U (F), respectively, then we write n n n k−1 k π × π × ··· × π = i π ⊗· · · ⊗ π . 1 2 k 1 k (n ,...,n ) 1. Definite unitary groups In this paper we will often use the following assumptions and notation, which we call the “standard assumptions”: • F is a CM number field such that F/F is everywhere unramified. We note + + × × × this implies that [F : Q] is even (the quadratic character of (F ) \A /O + + F F cutting out F has non-trivial restriction to F for each v|∞ but is trivial on + × (−1) ∈ (F ) ). v|∞ • p is a prime. We write S for the set of p-adic places of F . • S is a finite set of finite places of F , all of which split in F. S contains S . • For each v ∈ S, we suppose fixed a factorization v = v v in F, and write S ={ v | v ∈ S}. Let n ≥ 1 be an integer. Under the above assumptions we can fix the following data: • The unitary group G = GoverF with R-points given by the formula −t (1.0.1)G(R) ={g ∈ GL (R ⊗ F) | g = (1 ⊗ c)(g) }. n F We observe that for each finite place v of F ,G is quasi-split, while for each + + place v|∞ of F ,G(F ) is compact. We use the same formula to extend G to a reductive group scheme over O (this uses that F/F is everywhere unrami- fied). The authors apologize for using the same notation G to denote both an extension of scalars of the algebraic group G and an absolute Galois group. We hope no confusion will arise. 10 JAMES NEWTON, JACK A. THORNE • The inner twist ξ : U → G , given by the formula n,F F −1 ξ(g , g ) = (g , g  ) 1 2 1 2 n with respect to the identifications −t −1 U ={(g , g ) ∈ GL × GL | g =  g  } n,F 1 2 n n 2 n 1 n and −t G ={(g , g ) ∈ GL × GL | g = g }. F 1 2 n n 2 • Aliftof ξ to a pure inner twist (ξ, u) : U → G . We recall (see e.g. [Kal11]) n,F F 1 + that by definition, this means that u ∈ Z (F , U ) is a cocycle such that for −1σ all σ ∈ G +,wehave ξ ξ = Ad(u ). When n is odd, we define u to be the F σ 1 + cocycle inflated from Z (Gal(F/F ), U (F)), defined by the formula u = 1, n 1 u = ( , ). When n is even, we choose an element ζ ∈ F with tr (ζ ) = 0 c n n F/F 1 + and define u to be the cocycle inflated from Z (Gal(F/F ), U (F)),definedby −1 the formula u = 1, u = (ζ  ,ζ  ). (In fact, we will make essential use of this 1 c n n structure only when n is odd.) × × × • We also fix a choice of continuous character μ = μ : F \A → C such that μ| × = δ + ◦ Art + and such that if v is any place of F which is inert over F , F/F F then μ| × is unramified. + 1 + If v is a finite place of F , then the image of the cocycle u in H (F , U ) is trivial (this is true by Hilbert 90 if v splits in F, and true because det u ∈ N F if v is inert F /F v v v in F, cf. [Rog90, §1.9]). Our choice of pure inner twist (ξ, u) therefore determines a + + + U (F )-conjugacy class of isomorphisms ι : G(F ) → U (F ) (choose g ∈ U (F ) such n v n n v v v v −1 c + −1 that g g = u ;then ι is the map induced on F -points by the map Ad(g) ◦ ξ : G → c v v F + c U , which descends to F ). If v splits v = ww in F, then we have an isomorphism ι : n,F + + + G(F ) → GL (F ) (composite of inclusion G(F ) ⊂ (Res + GL )(F ) and canonical n w F/F n v v v projection (Res + GL )(F ) → GL (F )). F/F n n w + + If L /F is a finite totally real extension, then we will use the following standard notation: • We set L = L F. + + • If T is a set of places of F then we write T for the set of places of L lying above T. If w ∈ T lies above v ∈ Tand v splits v = v v in F (in particular, we suppose that we have made a choice of v|v), then we will write w for the unique place of L which lies above both w and v (in which case w splits w = w w in L). We write e.g. S for the set of places of the form w (w ∈ S ). L L We note that formation of G is compatible with base change, in the sense that the group G + is thesameasthe onegiven by formula (1.0.1) relative to the quadratic extension L SYMMETRIC POWER FUNCTORIALITY 11 L/L . The same remark applies to the pure inner twist (ξ, u). When we need to compare + + + trace formulae over F and its extension L /F (a situation that arises in §4), we will use the character μ = μ ◦ N . L F L/F 1.1. Base change and descent – first cases. — In the next few sections we summarise some results from the literature concerning automorphic representations of the group G(A +). We first give some results which do not rely on an understanding of the finer properties of L-packets for p-adic unitary groups at inert places of the extension F/F . Theorem 1.2. — Let σ be an automorphic representation of G(A +). Then there exist a parti- tion n = n + ··· + n and discrete, conjugate self-dual automorphic representations 1 k π ,...,π 1 k of GL (A ),..., GL (A ), n F n F 1 k respectively, with the following properties: (1) Let π = π  ···  π . Then for each finite place w of F below which σ is unramified, π is 1 k w unramified and is the unramified base change of σ . w| + c + −1 (2) For each place v = ww of F which splits in F, π σ ◦ ι . w v (3) For each place v|∞ of F, π has the same infinitesimal character as ⊗ W ,where W is v τ :F →C τ τ ∼ ∼ the algebraic representation of GL (F ) GL (C) such that σ W | + . = = n v n v τ G(F ) Proof. — This follows from [Lab11, Corollaire 5.3]. We call π the base change of σ .If ι : Q → C is an isomorphism, we say that σ is ι-ordinary if π is ι-ordinary at all places w|p in the sense of [Ger19, Definition 5.3]. We note that this depends only on π and the weight of π (equivalently, on σ and σ ). p p ∞ Corollary 1.3. — Let ι : Q → C be an isomorphism. Then there exists a unique continuous semisimple representation r : G → GL (Q ) with the following properties: σ,ι F n (1) For each prime-to-p place w of F below which σ is unramified, r | is unramified. σ,ι G (2) For each place v ∈ S ,r | is de Rham. p σ,ι G c + F−ss T −1 (3) For each place v = ww of F which splits in F, WD(r | ) rec (σ ◦ ι ). σ,ι G v w F w Proof. — This follows from the classification of discrete automorphic representa- tions of GL (A ) [MW89], together with the known existence of Galois representations n F attached to RACSDC (regular algebraic, conjugate self-dual, cuspidal) automorphic rep- resentations of GL (A ) (cf. [AT21, Corollary 3.4]). n F i 12 JAMES NEWTON, JACK A. THORNE We remark that if r | is multiplicity free, then the base change of σ is π = σ,ι G F(ζ ) π ···π ,where each π is a cuspidal automorphic representation of GL (A ). Indeed, 1 k i n F [MW89] shows that a non-cuspidal π would contribute a direct sum of copies of a single Galois representation twisted by powers of the cyclotomic character to r , which gives a σ,ι factor with multiplicity > 1in r | . In particular, if r | is multiplicity free then σ,ι G σ,ι G F(ζ ) F(ζ ) p p π is tempered (as each π is, by the results of [Shi11, Clo13, Car12]). Theorem 1.4. — Let π be a RACSDC automorphic representation of GL (A ). Suppose that n F π is unramified outside S. Then there exists an automorphic representation σ of G(A +) with the following properties: −1 ι (U (O )) n + (1) For each finite place v ∈ S of F, σ = 0. (2) π is the base change of σ . Proof. — This follows from [Lab11, Théorème 5.4]. 1.5. Endoscopic data and normalisation of transfer factors. — To go further we need to use some ideas from the theory of endoscopy, both for the unitary group G and for the twisted group Res GL θ . We begin by describing endoscopic data for G (cf. [Lab11, F/F n §4.2], [Rog90, §4.6]). The equivalence classes of endoscopic data for G are in bijection with pairs (p, q) of integers such that p + q = n and p ≥ q ≥ 0. Define μ = 1, μ = μ. + − We identify μ with characters of the global Weil group W using Art .Thenwecan ± F F write down an extended endoscopic triple E = (H, s,η) giving rise to each equivalence class as follows: • The group H is U × U . p q • s = diag(1,..., 1, −1,..., −1) (with p occurrences of 1 and q occurrences of −1). L L • η : H → G is given by the formulae: η : (g , g )  1 → diag(g , g )  1 ∈ GL (C) = G, 1 2 1 2 n (g , g ) ∈ GL (C) × GL (C) = H; 1 2 p q (1 , 1 )  w → diag(μ q(w)1 ,μ p(w)1 ) w(w ∈ W ) p q (−1) p (−1) q F −1 (1 , 1 )  w → diag( , )  w , p q c p q c where w ∈ W − W is any fixed element. c F F As described in [Lab11, §4.5], a choice of extended endoscopic triple E determines a E + normalisation of the local transfer factor  (v a place of F )uptonon-zeroscalar. We will fix a normalisation of local transfer factors only when n is odd, using the following observations: SYMMETRIC POWER FUNCTORIALITY 13 • The quasi-split group U , with its standard pinning, has a canonical normal- isation of transfer factors. Indeed, in this case the Whittaker normalisation of transfer factors defined in [KS99, §5] is independent of the choice of additive character and coincides with the transfer factor denoted  in [LS87]. • Our choice of pure inner twist (ξ, u) : U → G defines a normalisation of the local transfer factors for G. This normalisation of local transfer factors satisfies the adelic product formula (a very special case of [Kal18, Proposition 4.4.1]). A local transfer factor having been fixed, one can define what it means for a function H ∞ + H ∞ f ∈ C (H(F )) (resp. f ∈ C (H(A +))) to be an endoscopic transfer of a function c v c ∞ + ∞ f ∈ C (G(F )) (resp. f ∈ C (G(A +))). After the work of Waldspurger, Laumon, and c v c ∞ + ∞ Ngô, any function f ∈ C (G(F )) (resp. C (G(A +))) admits an endoscopic transfer c v c (see [Lab11, Théorème 4.3] for detailed references). We next discuss base change, or in other words, endoscopy for the twisted group Res + GL  θ . We will only require the principal extended endoscopic F/F n n L L triple (U , 1 ,η),where η : U → Res + GL is defined as follows: first, identify n n n F/F n Res + GL = (GL (C) × GL (C))  W +,where W + acts through its quotient F/F n n n F F Gal(F/F ) and an element w ∈ W + − W acts by the automorphism (g , g ) → c F F 1 2 −t −t −1 −1 L L ( g  , g  ).Then η : U → Res + GL is given by the formulae: n n n F/F n 2 1 n n t −1 η : (g)  1 → diag(g, g )  1 ∈ GL (C) × GL (C); n n (1 )  w → diag(1 , 1 )  w, (w ∈ W ); n n n F −1 (1 )  w → diag( , )  w . n c n c Following [Lab11, §4.5], we fix the trivial transfer factors in this case. By [Lab11, Lemme ∞ + U 4.1], each function φ ∈ C (Res + GL (F )  θ ) admits an endoscopic transfer φ ∈ F/F n n c v ∞ + ∞ + C (U (F )), and every function in C (U (F )) arises this way. We will follow op. cit. in n n c v c v ∞ + using the following notation: if f ∈ C (U (F )) (or more generally, if U is replaced by a n n c v ∞ + product of unitary groups) then we write f ∈ C (Res + GL (F )  θ ) for any function F/F n n c v that admits f as endoscopic transfer (with respect to the principal extended endoscopic triple defined above). If E = (H, s,η) is one of the extended endoscopic triples for G as above then, H H following [Lab11, §4.7], we set M = Res + H , and write M for the twisted space on F/F F H + M associated to the non-trivial element of Gal(F/F ). Then we may canonically identify H H M = Res + GL × GL and M = (Res + GL × GL )  (θ × θ ).Wewilluse the F/F p q F/F p q p q H ∞ same notation to describe stable base change for M . In particular, if f ∈ C (H(A +)), ∞ H then we will use f ∈ C (M (A +)) to denote a function whose endoscopic transfer (with respect to the principal extended endoscopic triple for M , defined as above) with respect to the trivial transfer factors is f (cf. [Lab11, Proposition 4.9]). Having fixed the above normalisations, we can now formulate some simple propo- sitions. 14 JAMES NEWTON, JACK A. THORNE Proposition 1.6. — Let n ≥ 1 be odd, and let v be an infinite place of F . Suppose given an extended endoscopic triple E = (H, s,η) as above and a Langlands parameter ϕ : W → H such that η ◦ ϕ is the Langlands parameter of an irreducible representation σ of G(F ).Let π be H v the (necessarily tempered, θ -invariant) irreducible admissible representation of H(F ) associated to the ∞ + Langlands parameter ϕ | ,and let f ∈ C (G(F )) be a coefficient for σ . Then there is a sign H W v v F v (v, E,ϕ ) ∈{±1} such that the identity π(f ) = (v, E,ϕ )σ (f ) = (v, E,ϕ ) holds, where H H v v H the twisted trace is Whittaker normalised (cf. [Lab11, §3.6]). Proof. — Let (ϕ ) be the L-packet of discrete series representations of H(F ) associated to ϕ . According to the main result of [Clo82], thereisasign ∈{±1} such H 1 H H that π(f ) = σ (f ). According to [Kal16, Proposition 5.10], there is 1 v,H v σ ∈(ϕ ) v v,H H asign ∈{±1} such that σ (f ) = σ (f ).Wemay take (v, E,ϕ ) = 2 2 v,H v v H σ ∈(ϕ ) v v,H H 1 2 The sign in Proposition 1.6 depends on our fixed choice of pure inner twist (be- cause it depends on the normalisation of transfer factors). We make the following basic + + but important remark, which is used in the proof of Proposition 4.6:let L /F be a finite totally real extension, and let L = L F. Then G + satisfies our standard assumptions, and comes equipped with a pure inner twist by base extension. If v is an infinite place of L , then we have the identity (v, E +,ϕ | ) = (v| +, E,ϕ ). L H W F H + ∞ + Proposition 1.7. — Let n ≥ 1 be odd, let v be a finite place of F ,and let f ∈ C (G(F )). c v Suppose given an extended endoscopic triple E = (H, s,η). (1) Suppose that v is inert in F and that f is unramified (i.e. G(O )-biinvariant). Suppose given an unramified Langlands parameter ϕ : W → H and let σ , σ be the unramified irreducible H v,H v + + representations of H(F ), G(F ) associated to the parameters ϕ , η ◦ ϕ , respectively. Let π be H H v v H + the unramified irreducible representation of M (F ) associated to ϕ | . Then there are identities H W v F H H π(f ) = σ (f ) = σ (f ), where the twisted trace is normalised so that θ fixes the unramified v,H v v v v vector of π.(If π is generic, this agrees with the Whittaker normalisation of the twisted trace.) c L (2) Suppose that v = v v splits in F. Suppose given a bounded Langlands parameter ϕ : W → H + + and let σ , σ be the representations of H(F ), G(F ) associated to the parameters ϕ , η ◦ ϕ , v,H v H H v v respectively (by the local Langlands correspondence rec for general linear groups). Let π be the F v H + H irreducible representation of M (F ) associated to ϕ | . Then there is an identity π (f ) = H W v v F v σ (f ) = σ (f ), where the twisted trace is Whittaker normalised. v,H v v Proof. — It is well-known that these identities hold up to non-zero scalar, which depends on the choice of transfer factor; the point here is that, with our choices, the scalar H H disappears. In the first part, the identity π(f ) = σ (f ) is the fundamental lemma for v,H v v stable base change [Clo90a]. The identity σ (f ) = σ (f ) is the fundamental lemma v,H v v for standard endoscopy [LN08], which holds on the nose because our transfer factors are + + identified, by the isomorphism ι : G(F ) → U (F ), with those defined in [LS87]with v n v v SYMMETRIC POWER FUNCTORIALITY 15 respect to our fixed pinning of U ; this is the ‘canonical normalisation’ of [Hal93]. If π n,F is generic then a Whittaker functional is non-zero on the unramified vector, which gives the final assertion of the first part of the proposition. H H In the second part, the equality π (f ) = σ (f ) is the fundamental lemma for v v,H v v stable base change in the split case, cf. [Rog90, Proposition 4.13.2] (where the result is stated for U(3) but the proof is valid in general). The equality σ (f ) = σ (f ) holds v,H v v because σ can be expressed as the normalised induction of a character twist of σ v v,H + + + (after choosing an appropriate embedding H → G and a parabolic subgroup of G F F F v v v containing H + ) and because the correspondence f → f can in this case be taken to be F v the corresponding character twist of the constant term along H (cf. [Rog90, Lemma 4.13.1] and [Shi11, §§3.3–3.4], noting that our normalisation of transfer factors at the place v in this case agrees on the nose with the analogue of the factor written as  in loc. cit., as follows from the definition in [LS87]). 1.8. L-packets and types for p-adic unitary groups. — Let v be a place of F inert in F. In this section we follow Mœglin [Mœg07, Mœg14] in defining L-packets of tempered representations for the group G(F ) (equivalently, given our choice of pure inner twist, U (F )). We write A(GL (F )) for the set of isomorphism classes of irreducible admissible n v representations of GL (F ) over C,and A (GL (F )) for its subset of tempered rep- n v t n v + + θ resentations. We define A(U (F )) and A (U (F )) similarly. We write A (GL (F )) n t n n v v v and A (GL (F )) for the respective subsets of θ -invariant representations (so e.g. n v θ θ A (GL (F )) is the set of irreducible representations of GL (F ) such that π := π ◦ θ = n v n v π ). Using the local Langlands correspondence rec for GL (F ) (and the Jacobson– F n v Morozov theorem), we can identify A(GL (F )) with the set of GL (C)-conjugacy classes n v n of Langlands parameters, i.e. the set of GL (C)-conjugacy classes of continuous homo- morphisms ϕ : W × SL (C) → GL (C) satisfying the following conditions: F 2 n • ϕ| is semisimple; • ϕ| is algebraic. SL (C) Then A (GL (F )) is identified with the set of parameters ϕ such that the ϕ(W ) is t n v F relatively compact, and A (GL (F )) is identified with the set of conjugate self-dual pa- n v θ θ rameters. We write A (GL (F )) ⊂ A (GL (F )) for the subset of parameters ϕ which n v + n v t t + + extend to a homomorphism ϕ : W × SL (C) → U .Suchanextension,ifitex- 2 n F F v v ists, is unique up to GL (C)-conjugacy (see e.g. [GGP12, Theorem 8.1]). The existence of such an extension ϕ can be equivalently phrased as follows: fix a decomposition l j w ∨ i c n ϕ =⊕ ρ ⊕ σ ⊕ (τ ⊕ τ ) ,where: i∈I j∈J k∈K k i j k • The integers l , m , n are all non-zero. i j k • Each representation ρ , σ , τ is irreducible and no two are isomorphic. i j k w ,∨ w ,∨ c c ∼ ∼ • For each i we have ρ = ρ and for each j we have σ = σ .For each k we i j i j w ,∨ have τ  τ . k 16 JAMES NEWTON, JACK A. THORNE n−1 • For each i, ρ is conjugate self-dual of sign (−1) and for each j , σ is conjugate i j self-dual of sign (−1) , in the sense of [GGP12,p.10]. + + Then an extension ϕ exists if and only if each integer m is even. If the extension ϕ F F v v is discrete, in the sense that Cent(GL (C), im ϕ ) is finite, then l = 1for each i ∈ I n i and the sets J, K are empty. If the parameter ϕ corresponding to a representation π ∈ A (GL (F )) is discrete, then we say that π is θ -discrete. n v + Let S denote the set of equivalence classes of pairs ((n ,..., n ), (π ,...,π )), n 1 k 1 k where (n ,..., n ) is a partition of n and π ,...,π are supercuspidal representations of 1 k 1 k GL (F ), ...,GL (F ), respectively. Two such pairs are said to be equivalent if they n v n v 1 k are isomorphic after permutation of the indices {1,..., k}. Thus we may think of an element of S as a formal sum of supercuspidal representations. We recall (see e.g. [BZ77]) that to any π ∈ A(GL (F )) we may associate the supercuspidal support sc(π) ∈ S , n v n defined by the condition that π occurs as an irreducible subquotient of the representation π × π × ··· × π (notation for induction as defined at the beginning of this paper). 1 2 k Mœglin associates to any element τ ∈ A (U (F )) its extended cuspidal support t n esc(τ ) ∈ S . We do not recall the definition here but note that its definition can be re- duced to the case where τ is supercuspidal, in the following sense: suppose that τ is a subquotient of a representation π × ··· × π × τ = i π ⊗ ··· ⊗ π ⊗ τ , 1 k−1 0 1 k−1 0 (n ,...,n ) 1 k where τ is a supercuspidal representation of U (F ).Then esc(τ ) = sc(π ) +··· + 0 n 1 k v θ θ sc(π ) + esc(τ ) + sc(π ) + ··· + sc(π ). k−1 0 k−1 1 + θ Proposition 1.9. — If τ ∈ A (U (F )) then there is a unique element π ∈ A (GL (F )) t n τ n v + v t such that esc(τ ) = sc(π ). Proof. — [Mœg07, Lemme 5.4] states that there is a unique element π = π ∈ A (GL (F )) such that esc(τ ) = sc(π). We need to explain why in fact π ∈ τ n v A (GL (F )) .[Mœg07, Théorème 5.7] states that this is true when τ is square- n v + integrable. In general, we can find a Levi subgroup M ⊂ U and an irreducible (n ,...,n ) n 1 k square-integrable representation π ⊗ ··· ⊗ π ⊗ τ of M (F ) such that τ is 1 k−1 0 (n ,...,n ) 1 k v a subquotient of i π ⊗ ··· ⊗ π ⊗ τ (see [Wal03, Proposition III.4.1]). Then 1 k−1 0 (n ,...,n ) 1 k θ θ θ π = (π × π ) × (π × π ) × ··· × (π × π ) × π , so the result follows from the τ 1 2 k−1 τ 1 2 k−1 square-integrable case. According to the proposition, there is a well-defined map + θ BC : A (U (F )) → A (GL (F )) t n n v + v t defined by BC(τ ) = π (which might be called stable base change). τ SYMMETRIC POWER FUNCTORIALITY 17 Proposition 1.10. — The map BC is surjective, and it has finite fibres. Proof. — The image of BC contains the θ -discrete representations, and the fibres of BC above such representations are finite, by [Mœg07, Théorème 5.7]. The general case can again be reduced to this one. θ −1 If π ∈ A (GL (F )) , then we define (π) = BC (π). By definition, the sets n v + (π) partition A (U (F )) and therefore deserve to be called L-packets. The following t n proposition is further justification for this. Proposition 1.11. — Let π ∈ A (GL (F )) , and fix an extension π to the twisted group n v + × ∞ GL (F )  θ . Then there are constants c ∈ C such that for any f ∈ C (GL (F )  θ): n v τ n v π(f ) = c τ(f ). τ ∈(π) Proof. — When π is θ -discrete, this is the content of [Mœg07, Proposition 5.5]. In general, (π) admits the following explicit description: decompose π = π × π × π , 1 2 where π ∈ A (GL (F )) and π ∈ A (GL (F )) is θ -discrete. Then (π) is the set 1 t n v 2 n−2n v + 1 t 1 of Jordan–Hölder factors of the induced representations π × τ as τ varies over the set 1 2 2 of elements of (π ). Using the compatibility of transfer with normalised constant terms along a parabolic (see [Mor10, Lemma 6.3.4]) we thus have an identity θ ∼ U π(f ) = (π × π × π ) (f ) = c (π × τ )(f ) 1 2 τ 1 2 1 2 τ ∈(π ) 2 2 for some constants c ∈ C . To prove the proposition, it is enough to show that if τ ,τ ∈ τ 2 2 2 (π ) are non-isomorphic then the induced representations π × τ , π × τ have no 2 1 2 1 Jordan–Hölder factors in common. This follows from [Wal03, Proposition III.4.1]. We now introduce some particular representations of U . These are built out of depth zero supercuspidal representations of U . Accordingly we first introduce some cus- pidal representations of the finite group of Lie type U (k(v)): • We write τ(v) for the unique cuspidal unipotent representation of U (k(v)) (see [Lus77,§9]). • Let k /k(v) be a degree 3 extension, and define C = ker(N : Res G → Res G ). k k( v)/k k k( v)/k(v) m k /k(v) m 3 3 3 3 Then there is a unique U (k(v))-conjugacy class of embeddings C → U (as 3 3,k(v) can be proved using e.g. [DL76, Corollary 1.14]). th Let p be a prime such that q is a primitive 6 root of unity modulo p, and let θ : C(k(v)) → C be a character of order p. Then we write λ(v, θ) 18 JAMES NEWTON, JACK A. THORNE 3,k(v) for the (negative of the) Deligne–Lusztig induction −R θ.Then λ(v, θ) is a cuspidal irreducible representation of U (k(v)) (note that C is not contained in any proper k(v)-rational parabolic of U ). 3,k(v) We define C = Res G . Then the homomorphism C(k( v)) → C(k(v)), z → z/z , k k( v)/k( v) m × c is surjective, and we define a character θ : C(k( v)) → C of order p by θ(z) = θ(z/z ). There is a unique GL (k( v))-conjugacy class of embeddings C → GL , and we write 3 3,k( v) GL 3,k( v) λ( v,θ) for the Deligne–Lusztig induction R θ.Then λ( v,θ) is a cuspidal irreducible representation of GL (k( v)). We now assume that the residue characteristic of k(v) is odd. U (F ) Proposition 1.12. — (1) Let τ = c-Ind τ(v) (compact induction). Then τ is a su- v v U (O ) percuspidal irreducible admissible representation of U (F ) and BC(τ ) = St (χ)  1,where 3 v 2 χ : F → C is the unique non-trivial quadratic unramified character. U (F ) (2) Let λ (θ) = c-Ind λ(v, θ). Then λ (θ) is a supercuspidal irreducible admissible repre- v v U (O ) sentation of U (F ). × × (3) Extend λ( v,θ) to a representation of F GL (O ) by making F act trivially, and let λ (θ) = 3 F v v v v GL (F ) 3 v c-Ind λ( v,θ).Then λ (θ) is a supercuspidal irreducible admissible representation of × v F GL (O ) 3 F v v GL (F ),and BC(λ (θ)) = λ (θ). 3 v v v Proof. — If μ is a cuspidal irreducible representation of U (k(v)),then 0 3 U (F ) v + c-Ind μ is a supercuspidal, irreducible admissible representation of U (F ) (see 0 3 U (O ) v 3 + [MP96, Proposition 6.6] – we will return to this theme shortly). The essential point there- fore is to calculate the extended cuspidal support in each case, which can be done using theresultsof[LS20] (which require the assumption that k(v) has odd characteristic). Indeed §8in op. cit. explains how to compute the reducibility points Red(π) (defined in [Mœg07, §4]) of a depth 0 supercuspidal representation, at least up to unramified twist. We compute that Red(τ ) ={(1, 3/2), (χ, 1)} or {(1, 1), (χ, 3/2)} which corresponds to BC(τ ) = St (1)  χ or BC(τ ) = St (χ)  1. Since BC(τ ) ∈ A (GL (F )) ,the v 2 v 2 v 3 v + second alternative holds. For λ (θ), we deduce that Red(λ (θ)) ={(ρ, 1)},where ρ is v v a conjugate self-dual unramified twist of λ (θ). We again conclude by sign considera- tions. Corollary 1.13. — Let n = 2k + 1 be an odd integer, and consider a representation 2k−2 θ π = St (χ)  1  ( χ ) ∈ A (GL (F )) , 2 i n v + i=1 t where χ : F → C is the unique non-trivial quadratic unramified character and for each i = 1,..., 2k − 2, χ : F → C is a character such that χ | has order 2. We can assume, after i i O w ,∨ relabelling, that χ = χ (i = 1,..., k − 1), and then (π) contains each irreducible subquo- 2k−1−i tient of the induced representation χ × χ ×· · · × χ × τ . 1 2 k−1 v SYMMETRIC POWER FUNCTORIALITY 19 w ,∨ Proof. — First we explain why we can relabel the characters so that χ = χ . 2k−1−i Considering the explicit description of A (GL (F )) , we need to explain why conjugate n v + self-dual characters must appear with even multiplicity amongst the χ . Suppose χ is i 1 conjugate self-dual. We know that χ | is the non-trivial quadratic character, so there are two possibilities for χ determined by χ ( ) =−1 or 1 (this value is also the sign of 1 1 v χ ). If the sign is −1, the multiplicity of χ is one of the even exponents m . If the sign is 1 1 j +1, dimension reasons force its multiplicity to be even. The rest of the Corollary follows from the definition of (π) in terms of extended supercuspidal supports. Note that we do not claim that (π) contains only the subquotients of this induced representation – this is not true even when k = 1. To exploit Corollary 1.13 we need to introduce some results from the theory of types. We state only the results we need, continuing to assume that n = 2k + 1 is odd. Let p denote the standard parahoric subgroup of U (O ) associated to the partition v n (1, 1,..., 1, 3); in other words, the pre-image under the reduction modulo map U (O ) → U (k(v)) of Q (k(v)). Projection to the Levi factor gives a surjective n F n (1,1,...,1,3) × k−1 homomorphism p → M (k(v)) = (k( v) ) × U (k(v)). v (1,1,...,1,3) 3 × k−1 Given a cuspidal representation σ(v) of M (k(v)) (k( v) ) × U (k(v)), (1,1,...,1,3) 3 the pair (p , σ (v)) defines a depth zero unrefined minimal K-type in the sense of × k−1 [MP96]. In this case we write E(σ ) for the set of irreducible representations of (F ) × + k−1 + + + U (O ) ⊂ M (F ) whose restriction to M (O ) = (O ) × U (O ) 3 (1,1,...,1,3) (1,1,...,1,3) 3 F v F F F v v v v is isomorphic to (the inflation of) σ(v). We have the following result. Proposition 1.14. — Let σ(v) be a cuspidal irreducible representation of M (k(v)). (1,1,...,1,3) Then: M (F ) (1,1,...,1,3) v (1) For any σ ∈ E(σ (v)), the compact induction c-Ind σ (v) is irreducible and su- k−1 (F ) ×U (O + ) percuspidal. (2) Let π be an irreducible admissible representation of U (F ). Then π | contains σ(v) if and only n p v v if π is a subquotient of an induced representation M (F ) U (1,1,...,1,3) n v i c-Ind σ (v) Q k−1 (1,1,...,1,3) (F ) ×U (O ) 3 + for some σ ∈ E(σ (v)). Proof. — See [MP96, Proposition 6.6] and [MP96, Theorem 6.11]. We now describe explicitly the two types that we need. Recall that we are assuming that the characteristic of k(v) is odd. Let ω( v) : k( v) →{±1} denote the unique non- trivial quadratic character of k( v) . • The representation τ(v, n) of p inflated from the representation ω( v) ⊗ ··· ⊗ ω( v) ⊗ τ(v) of M (k(v)). (1,1,...,1,3) 20 JAMES NEWTON, JACK A. THORNE • The representation λ(v, θ, n) of p inflated from the representation ω( v) ⊗ ··· ⊗ ω( v) ⊗ λ(v, θ) of M (k(v)) (where θ as above is a character of C(k(v)) of order p,and (1,1,...,1,3) th we assume q mod p is a primitive 6 root of unity). These types are introduced because they are related by a congruence modulo p, because th of our assumption that q mod p is a primitive 6 root of unity: Proposition 1.15. — Fix an isomorphism Q → C and use this to view τ(v, n) and λ(v, θ, n) as representations with coefficients in Q .Then: (1) τ(v, n) is irreducible. (2) λ(v, θ, n) contains τ(v, n) as a Jordan–Hölder factor with multiplicity 1. (As usual, overline denotes semi-simplified residual representation over F .) Proof. — The modular irreducibility of cuspidal unipotent representations is a gen- eral phenomenon (see [DM18]). The proposition is a statement about representations of U (k(v)), which can be proved by explicit computation with Brauer characters; see [Gec90, Theorem 4.2] (although note that there is a typo in the proof: the right-hand side of the first displayed equation should have χ in place of χ 2 ). 1 q −q The following proposition will be a useful tool for exploiting the type (p ,λ(v,θ,n)). ∞ + We introduce an associated test function φ(v,θ, n) ∈ C (U (F )): it is the function sup- c v ported on p and inflated from the character of λ(v, θ, n) .If π is an admissible rep- resentation of U (F ),then π(φ(v, θ, n)) is (up to a positive real scalar depending on normalisation of measures) the dimension of the space Hom (λ(v, θ, n), π | ). p p v v Proposition 1.16. — Assume that the characteristic of k(v) is greater than n. Let φ = φ(v,θ, n),and let E = (H, s,η) be one of our fixed endoscopic triples for U , with H = U × U . n p q θ θ Suppose given representations π , π in A (GL (F )) , A (GL (F )) , respectively, such that p q p v + q v + t t ∼ H (π ⊗ π ) (φ ) = 0. Then sc(π ) + sc(π ) = λ (θ) + χ + ··· + χ ,where χ ,...,χ : p q p q v 1 2k−2 1 2k−2 F → C are characters such that for each i = 1,..., 2k − 2, χ | = ω( v). v O Proof. — By Proposition 1.11, there is an identity ∼ H H (π ⊗ π ) (φ ) = c c (τ ⊗ τ )(φ ) p q τ τ p q p q τ ∈(π ) p p τ ∈(π ) q q × H for some constants c , c ∈ C .Now,[KV12, Theorem 2.2.6] shows that φ can be τ τ p q k(v) −1 ⊗(k−1) taken to be a weighted sum of inflations to H(O ) of characters R (θ ⊗ω( v) ) F C v i SYMMETRIC POWER FUNCTORIALITY 21 k−1 associated to conjugacy classes of embeddings C : C × Res G → H .(Ourap- i k( v)/k(v) k(v) peal to this reference is the reason for the additional assumption on the characteristic of ∼ H k(v) in the statement of the theorem.) If (π ⊗ π ) (φ ) = 0, then there exists a sum- p q mand on the right-hand side such that τ ⊗ τ contains the inflation to H(O ) of the p q F k(v) ⊗(k−1) (irreducible) representation with character −R (θ ⊗ ω( v) ). Taking into account the compatibility between parabolic induction and Deligne–Lusztig induction, the tran- sitivity of Deligne–Lusztig induction [Lus76], and Proposition 1.14, we see that for one of the representations τ , τ (the one for the factor of even rank), the extended cuspidal p q support is a sum of characters of F , each of which is the twist of an unramified character by a ramified quadratic character; and for the other of the representations τ , τ ,the ex- p q tended cuspidal support is a sum of such characters, together with λ (θ). This completes the proof. 1.17. Types for the general linear group. — In this section we record some analogues of the results of the previous section for general linear groups. Let 2 ≤ n ≤ n be an integer. Let v be a finite place of F. We assume that the characteristic of k( v) is odd. We have already introduced the notation ω( v) for the unique non-trivial quadratic character of k( v) . We introduce a further representation of the finite group GL (k( v)) of Lie type: • Let k /k( v) be an extension of degree n , and suppose that q mod p is a prim- n 1 v th × × itive n root of unity modulo p.Let  : k → C be a character of order p. 1 n Then  is distinct from its conjugates by Gal(k /k ), and we write λ( v, ) = n v GL n −1 1 (−1) R  for the Deligne–Lusztig induction. Then λ( v, ) is an Res G k /k( v) m irreducible representation of GL (k( v)). The notation λ( v, ) thus generalises that introduced in the previous section (where n = 3and  = θ ). Proposition 1.18. — Let π be an irreducible admissible representation of GL (F ),and let n v F /F denote an unramified extension of degree n . Then the following are equivalent: v,n v 1 (1) The restriction of π to GL (O ) contains λ( v, ). n F 1 v (2) There exists a continuous character χ : F → C such that χ | × =  and rec π v,n O v v,n −1 Ind (χ ◦ Art ). In particular, π is supercuspidal. W F F v,n v,n 1 Proof. — This follows from the results of [Hen92] (see especially §3.4 of that paper) and [MP96]. Let n = n − n . We write q ⊂ GL (O ) for the standard parahoric subgroup as- 2 1 v n F sociated to the partition (n , n ), i.e. the pre-image under the reduction modulo map 1 2 v GL (O ) → GL (k( v)) of P (k( v)). We write λ( v, , n) for the irreducible repre- n F n (n ,n ) v 1 2 sentation of q inflated from the representation λ( v, ) ⊗ (ω( v) ◦ det) of L (k( v)). v (n ,n ) 1 2 22 JAMES NEWTON, JACK A. THORNE We write r ⊂ q for the standard parahoric subgroup associated to the partition v v (n , 1, 1,..., 1). Then we have the following analogue of Proposition 1.14: Proposition 1.19. — Let π be an irreducible admissible representation of GL (F ). Then the n v following are equivalent: • π | contains λ( v, , n)| . r r v v • sc(π) = π +χ +···+χ ,where π satisfies the equivalent conditions of Proposition 1.18 1 1 n 1 and χ ,...,χ : F → C are characters such that for each i = 1,..., n , χ | = 1 n 2 i 2 v O ω( v). Proof. — This once again follows from the results of [MP96]. The pair (q , λ( v, , n)) is not in general a type (because λ( v, , n) is not a cus- pidal representation of L (k( v)) unless n = 1). Nevertheless, we have the following (n ,n ) 2 1 2 proposition: Proposition 1.20. — Let π be an irreducible admissible representation of GL (F ). Then the n v following are equivalent: (1) The restriction of π to q contains λ( v, , n). (2) There exist irreducible admissible representations π of GL (F ) (i = 1, 2) such that π = π i n v 1 π , the restriction of π to GL (O ) contains λ( v, ), and the restriction of π to GL (O ) 2 1 n F 2 n F 1 v 2 v contains ω( v) ◦ det. We note that in the situation of the proposition, π is the twist of an unramified representation by a quadratic ramified character. Proof. — Let P = P ,L = L ,and let N denote the unipotent radical of P. (n ,n ) (n ,n ) P 1 2 1 2 Abbreviate λ = λ( v, , n) and λ = λ| .If π is an irreducible admissible repre- N L(O ) P F sentation of GL (F ) then we define π = Hom (λ, π | ). We first show that for any n v q q v v λ P admissible representation π of GL (F ), the natural projection π → π (restriction of n v projection to unnormalised Jacquet module) is an isomorphism. Indeed, it is surjective by [Vig98, II.10.1, 1)]. To show that it is injective, let μ = λ| and let R = P , r (n ,1,1,...,1) v 1 N the unipotent radical of R. Then the pair (r , μ) is a depth zero unrefined minimal R v K-type in the sense of [MP96]. We now have a commutative diagram λ P π π μ R π π , R SYMMETRIC POWER FUNCTORIALITY 23 where the left vertical arrow is the natural inclusion and the right vertical arrow is the natural projection to co-invariants. The lower horizontal arrow is an isomorphism, by [Mor99, Lemma 3.6]. We conclude that the top horizontal arrow is injective, and there- fore an isomorphism. Suppose now that π is an irreducible admissible representation of GL (F ) and n v λ μ that π = 0. Then π = 0, so by Proposition 1.19, π is an irreducible subquotient of an induced representation π = π × χ × ··· × χ , where the inducing data is as in the 1 1 n statement of that proposition. Computation of the Jacquet module (using the geometric lemma [BZ77, Lemma 2.12]) shows that (π ) has dimension 1; therefore π must be isomorphic to the unique irreducible subquotient of π which contains λ. This is π × π , 1 2 where π is the unique irreducible subquotient of χ ×··· × χ such that π | 2 1 n 2 GL (O ) 2 n F 2 v contains ω( v) ◦ det (note that π × π is irreducible, by [Zel80, Proposition 8.5]). 1 2 Suppose instead that π = π  π = π × π ,with π , π as in the statement of the 1 2 1 2 1 2 P λ proposition. Then the geometric lemma shows that π = 0, hence π = 0. We now introduce the local lifting ring associated to the inertial type which is the analogue, on the Galois side, of the pair (q , λ( v, , n)) introduced above. We recall that th k /k( v) is an extension of degree n , q mod p is a primitive n root of unity modulo p, n 1 v 1 1 × × and  : k → C is a character of order p.Let ι : Q → C be an isomorphism, so that n p −1 × ι  : k → Q is a character with trivial reduction modulo p. Fix a coefficient field 1 p E and suppose given a representation ρ : G → GL (k) of the form ρ = σ ⊕ σ , F n v,1 v,2 v v v where: • Let F /F be the unramified extension of degree n and residue field k .Then v,n v 1 n 1 1 there is an unramified character ψ : G → k and an isomorphism σ = F v,1 v v,n Ind ψ . G v v,n −1 • σ | ⊗ω( v) ◦ Art is trivial. (In other words, σ is the twist of an unramified v,2 I v,2 F F v v representation by a ramified quadratic character.) We recall that C denotes the category of complete Noetherian local O-algebras with residue field O/ = k. Lemma 1.21. — Let R ∈ C and let ρ : G → GL (R) be a continuous lift of ρ (i.e. a O v F n v v continuous homomorphism such that ρ mod m = ρ ). Then there are continuous lifts σ : G → v R v,i F v v GL (R) of σ (i = 1, 2) with the property that σ ⊕ σ is 1 + M (m )-conjugate to ρ . n v,i v,1 v,2 n R v Moreover, each σ is itself unique up to 1 + M (m )-conjugacy. v,i n R Proof. — The splitting exists and is unique because the groups i i H (F , Hom(σ , σ )) and H (F , Hom(σ , σ )) vanish for i = 0, 1. Compare v v,1 v,2 v v,2 v,1 [Sho18, Lemma 2.3].  24 JAMES NEWTON, JACK A. THORNE Let R ∈ C denote the universal lifting ring, i.e. the representing object of the functor of all continuous lifts of ρ . We write R( v, , ρ ) for the quotient of v v R associated by [Sho18, Definition 3.5] to the inertial type τ : I → GL (Q ), v F n v v p i−1 1 −1 q −1 −1 ⊕n τ =⊕ (ι  ◦ Art ) ⊕ (ω( v) ◦ Art ) . We record the following properties of i=1 F F v v R( v, , ρ ). Proposition 1.22. — (1) The ring R( v, , ρ ) is reduced, p-torsion-free, and is supported on a union of irreducible components of R . In particular, Spec R( v, , ρ ) is O-flat and equidi- mensional of dimension 1 + n . (2) Let x : R → Q be a homomorphism, and let ρ : G → GL (Q ) be the pushforward of x F n v p v p the universal lifting, with its associated direct sum decomposition ρ = σ ⊕ σ . Then x factors x x,1 x,2 through R( v, , ρ ) if and only if there is an isomorphism σ = Ind ψ for a character v x,1 x v,n −1 −1 ψ : G → Q such that ψ | ◦ Art = ι  and σ | ⊗ ω( v) ◦ Art is trivial. x F x I F x,2 I v,n F v,n F F 1 v,n 1 v v (3) Let σ : G → GL (R( v, , ρ )) be the representation associated to the universal lifting by v,1 F n v v 1 Lemma 1.21. There exists α ∈ R( v, , ρ )/( ) such that for any Frobenius lift φ ∈ G , v v F v v 1 n det(X − σ (φ )) ≡ (X − α ) mod . v,1 v (4) Let L /F be a finite extension such that τ | is trivial, and let R denote the universal lifting ring v v v I L L of ρ| . Then the natural morphism R → R( v, , ρ ) (classifying restriction of the universal L v v v ur lifting to G ) factors over the quotient R → R that classifies unramified liftings of ρ | . L G v L L v L v v Proof. — The first two properties follow from [Sho18, Proposition 3.6]. For the n n 1 1 third, let φ be a Frobenius lift. We note that det(X−σ (φ )) = X +(−1) det σ (φ ). v v,1 v v,1 v Indeed, this can be checked at Q -points, at which σ is irreducible, induced from v,1 −1 −1 acharacter of G which extends ι  ◦ Art . Reducing modulo and apply- v,n F v,n ing Hensel’s lemma, we find that there is an element α ∈ R( v, , ρ )/( ) such that i−1 n 1   n 1 det(X − σ (φ )) ≡ (X − q α ) mod .If α = (α ) then det(X − σ (φ )) = v,1 v v v,1 i=1 v v v v (X − α ) . For the fourth part of the lemma, we need to show that the universal lifting is unramified on restriction to G . Since R( v, , ρ ) is reduced, it suffices to check this at each geometric generic point. At such a point σ is irreducible, induced from a charac- v,1 ter of G , while σ is a quadratic ramified twist of an unramified representation. The F v,2 v,n result follows. 1.23. Algebraic modular forms. — Finally, we define notation for algebraic modular forms on the group G. Retaining our standard assumptions, fix a coefficient field E ⊂ Q containing the image of each embedding F → Q , with ring of integers O,and let I n I denote the set of embeddings τ : F → E inducing a place of S .Given λ = (λ ) ∈ (Z ) , p τ τ we write V for the E[ GL (F )]-module denoted W in [Ger19, Definition 2.3]; λ n v λ v∈S it is the restriction to GL (F ) of a tensor product of highest weight representations of n v GL (E). We write V ⊂ V for the O[ GL (O )]-submodule denoted M in loc. n λ λ n F λ v∈S v cit.;itisan O-lattice. SYMMETRIC POWER FUNCTORIALITY 25 In this paper we will only consider algebraic modular forms with respect to open compact subgroups U ⊂ G(A ) which decompose as a product U = U ,and such −1 that for each v ∈ S ,U ⊂ ι GL (O ). Given such a subgroup, together with a finite p v n F v v set  of finite places of F and a smooth O[U ]-module M, finite as O-module, we + ∞ define S (U, M) to be the set of functions f : G(F )\G(A ) → V ⊗ Msuch that for λ λ O each u ∈ Uand g ∈ G(A ), u · f (gu) = f (g).(Here U actson V ⊗ M via projection to λ O U × U .) If λ = 0, we drop it from the notation and simply write S(U, M). We recall the definition of some useful open compact subgroups and Hecke oper- ators (see [Ger19, §2.3] for more details): + c • For any place v of F which splits v = ww in F, the maximal compact subgroup −1 GL (O ).If v ∈  ∪ S ,U = ι GL (O ),and 1 ≤ j ≤ n, then the unramified n F p v n F w w Hecke operator T given by the double coset operator Id 0 w j j −1 v T = ι GL (O ) GL (O ) × U n F n F w w w w 0Id n−j acts on S (U, M). + c • For any place v of F which splits v = ww in F, the Iwahori subgroup Iw ⊂ GL (O ) of matrices which are upper-triangular modulo . n F w • For any place v ∈ S and c ≥ b ≥ 0with c ≥ 1, the subgroup Iw (b, c) ⊂ p v GL (O ) of matrices which are upper-triangular and unipotent upper- n F v v −1 triangular modulo .If U = ι Iw (b, c) for each v ∈ S and 1 ≤ j ≤ n, v v p v v then the re-normalised Hecke operator U of [Ger19, Definition 2.8] acts v,λ on S (U, M). (This Hecke operator depends on our choice of uniformizer . λ v However, the ordinary part of S (U, M), defined below using these operators, is independent of choices.) + c • For any place v of F which splits v = ww in F, the principal congruence subgroup K (1) = ker(GL (O ) → GL (k( v))). v n F n −1 When U = ι Iw (b, c) for each v ∈ S , there is a canonical direct sum decomposition v v p n−ord ord ord S (U, M) = S (U, M) ⊕ S (U, M) with the property that S (U, M) is the largest λ λ λ submodule of S (U, M) where each operator U (v ∈ S , j = 1,..., n)actsinvertible λ p v,λ ([Ger19, Definition 2.13]). We recall some basic results about the spaces S (U, M). We say that U is suffi- ∞ + −1 ciently small if for g ∈ G(A ),the group G(F ) ∩ gUg is trivial. We have the following simple lemma (cf. [Ger19, p. 1351]): Lemma 1.24. — Suppose that U is sufficiently small and that M is O-flat. Then for any c c c ≥ 1,the naturalmap S (U, M) ⊗ O/ → S (U, M/( )) is an isomorphism. λ O λ After fixing an isomorphism ι : Q → C, we can describe the spaces S (U, M) in classical terms ([Ger19, Lemma 2.5]): 26 JAMES NEWTON, JACK A. THORNE Lemma 1.25. — Let ι : Q → C be an isomorphism. Then there is an isomorphism ∨ ∞ S (U, M) ⊗ C = ⊕ m(σ ) Hom ((M ⊗ C) ,σ ) λ O,ι σ U O,ι respecting the action of Hecke operators at finite places away from  ∪ S , where the sum runs over automorphic representations σ of G(A +) such that for each embedding τ : F → C inducing a place v + + of F , σ is the restriction to G(F ) of the dual of the irreducible algebraic representation of GL (C) v n of highest weight λ −1 . ι τ Part I: Analytic continuation of functorial liftings The first part of this paper (§§2 – 3) is devoted to the proof of Theorem D from th the introduction, which shows that the automorphy of the n symmetric power for one th cuspidal Hecke eigenform of level 1 implies the automorphy of the n symmetric power for all cuspidal Hecke eigenforms of level 1. As described in the introduction, the proof has two main ingredients. The first, which is the main result of §2, is that automorphy of symmetric powers can be prop- agated along irreducible components of the Coleman–Mazur eigencurve. The second ingredient, which is explained in §3, uses the main result of [BK05] and has already been sketched in the introduction. Here we make some further introductory remarks on §2.Bymakingasuitable(in particular, soluble) base change to a CM field, we translate ourselves to the setting of definite unitary groups. We start from a classical point z of an eigenvariety for a rank th 2 unitary group, E , such that the n symmetric power of the associated Galois repre- sentation is known to be automorphic. We use Emerton’s construction of eigenvarieties (involving his locally analytic Jacquet functor), and our point of view on eigenvarieties and Galois representations is particularly influenced by those of [BC09]and [BHS17]. Like the authors of [BHS17], we rely in an essential way on the results of [KPX14], which make it possible to spread out pointwise triangulations to global triangulations. We consider the diagram: E X × T 2 ps,2 2 Sym n+1 E X × T n+1 ps,n+1 n+1 Here, E is an eigenvariety for a rank n + 1 unitary group, X is a certain rigid space n ps,d of d -dimensional p-adic Galois pseudocharacters and T is a rigid space parameterising characters of a p-adic torus. Our eigenvarieties come equipped with maps to these char- acter varieties as part of their construction; combining this with the existence of a family SYMMETRIC POWER FUNCTORIALITY 27 of Galois pseudocharacters over the eigenvariety interpolating the global Langlands cor- respondence at classical points gives the closed immersions i appearing in the diagram. th The map Sym corresponds to taking the n symmetric power of the 2-dimensional pseu- docharacter. Our task is to show that if C is an irreducible component of E containing z ,then 2 0 Sym (i (C)) is contained in the image of i . A classicality result (Lemma 2.30) will then 2 n be used to show that for another classical point z of C , its symmetric power Sym (i (z )) 1 2 1 is actually the image of a classical point of E . n+1 To show that Sym (i (C)) is indeed contained in the image of i , we combine a 2 n simple lemma in rigid geometry (Lemma 2.2) with information coming from the local ge- ometry of a certain natural locally closed neighbourhood of Sym (i (z )) in X ×T 2 0 ps,n+1 n+1 which contains open subspaces of both E and Sym (i (C)). This subspace is essentially n+1 2 the trianguline variety, but since we work with spaces of pseudocharacters instead of rep- resentations we restrict to open neighbourhoods in which our pseudocharacters are abso- lutely irreducible and hence naturally lift to representations. Our results on the vanishing of adjoint Selmer groups [NT20] are used to compare E and the trianguline variety. n+1 We proceed in a similar way to the proof of [BC09, Corollary 7.6.11], which shows that vanishing of an adjoint Selmer group implies that i induces an isomorphism between n+1 completed local rings of the eigenvariety and the trianguline variety. 2. Trianguline representations and eigenvarieties Throughout this section, we let p be a prime and let E ⊂ Q be a coefficient field. We write C for the completion of Q .If X is a quasi-separated E-rigid space we let X (Q ) = X (E ), where the union is over finite extensions of E. We can naturally p E ⊂Q view X (Q ) as a subset of the set of closed points of the rigid space X (where base ex- tension of a quasi-separated rigid space is as defined in [BGR84, §9.3.6], see also [Con99, §3.1]). 2.1. An ‘analytic continuation’ lemma. — Suppose given a diagram of E-rigid spaces Y GX , where α is a closed immersion. We identify X with a subspace of G.Let x ∈ Y be a point such that β(x) ∈ X . −1 Lemma 2.2. — Suppose that β (X ) contains an affinoid open neighbourhood of x. Then for each irreducible component C of Y containing x, we have β(C) ⊂ X . 28 JAMES NEWTON, JACK A. THORNE −1 Proof. — We observe that β (X ) ∩C is a Zariski closed subset of C which contains −1 a non-empty affinoid open subset. This forces β (X ) ∩ C = C (apply [Con99, Lemma 2.2.3]), hence β(C) ⊂ X . 2.3. A Galois deformation space. — Let F, S, p be as in our standard assumptions (§1). We assume that E contains the image of every embedding τ : F → Q . 2.3.1. Trianguline deformations – infinitesimal geometry. — This section has been greatly influenced by works of Bellaïche and Chenevier [BC09, Che11]. We use the formalism of families of (ϕ,  )-modules, as in [KPX14]. Thus if v ∈ S and X is an F p E-rigid space, one can define the Robba ring R ;if V is a family of representations of X,F G over X, then the functor D of [KPX14, Theorem 2.2.17] associates to V the family v rig D (V) of (ϕ,  )-modules over X which is, locally on X, finite free over R . We refer F X,F rig v v to [HS16, §2] for the definitions of these objects, as well as more detailed references. If X = Sp A, where A is an E-affinoid algebra, we write R = R .If δ : F → A is a X,F A,F v v v continuous character, we have a rank one (ϕ,  )-module R (δ ) defined by [KPX14, F A,F v v v Construction 6.2.4]. We will also have cause to mention the (ϕ, )-cohomology groups H (−) which are defined in [KPX14, §2.3]. ϕ,γ Let v ∈ S ,and let ρ : G → GL (E) be a continuous representation. If δ = p v F n v n × (δ ,...,δ ) : (F ) → E is a continuous character, we call a triangulation of ρ of v,1 v,n v parameter δ an increasing filtration of D (ρ ) by direct summand (ϕ,  )-stable R - v v F E,F rig v v submodules such that the successive graded pieces are isomorphic to R (δ ), ... , E,F v,1 R (δ ). We say that ρ is trianguline of parameter δ if it admits a triangulation of E,F v,n v v parameter δ .If δ satisfies δ ( ) ∈ O for each i, then we say that δ is an ordinary v v v,i v v −1 parameter. Equivalently, δ is ordinary if δ ◦ Art extends to a continuous character of v v,i G for each i. For an ordinary parameter δ , ρ is trianguline of parameter δ if and only F v v v −1 if ρ has a filtration with successive graded pieces isomorphic to δ ◦ Art ,...,δ ◦ v v,1 v,n −1 Art . We say that the character δ is regular if for all 1 ≤ i < j ≤ n,wehave δ /δ = x v v,i v,j Hom (F ,E) Q v a a v v,τ for any a = (a ) ∈ Z , where by definition x (y) = τ(y) . Note that the v v,τ τ ≥0 τ a a − a v v v,τ characters x satisfy |x (p)| = p , so there is an affinoid cover of the rigid space Hom(F , G ) with each open containing only finitely many x . n reg We define T = Hom((F ) , G ), a smooth rigid space over E, and write T ⊂ v m v v T for the Zariski open subspace of regular characters (Zariski open by the finiteness × n observation in the preceding paragraph). We define W = Hom((O ) , G ) and write v m r : T → W for the natural restriction map. v v v Lemma 2.4. — Let ρ : G → GL (E) be a continuous representation. Then for any δ ∈ v F n v reg T (E), ρ admits at most one triangulation of parameter δ . If such a triangulation exists, then ρ is v v v strictly trianguline of parameter δ in the sense of [KPX14, Definition 6.3.1]. v SYMMETRIC POWER FUNCTORIALITY 29 Proof. — Suppose ρ admits a triangulation of parameter δ ,so D (ρ ) is v v v rig equipped with an increasing filtration Fil . Following [KPX14, Definition 6.3.1], we need 0 −1 to show that for each 0 ≤ i ≤ n the cohomology group H D (ρ )/ Fil (δ ) v i v,i+1 ϕ,γ rig is one-dimensional. It follows from [KPX14, Proposition 6.2.8] that 0 −1 H gr D (ρ ) (δ ) vanishes when i < j and is one-dimensional when i = j . v v,i ϕ,γ j rig The vanishing holds precisely because δ is regular. A dévissage completes the proof. Definition 2.5. — If δ : F → E is a continuous character (hence locally Q -analytic) we let the tuple (wt (δ)) be such that the derivative of δ is the map τ τ ∈Hom (F ,E) F → E x → −wt (δ)τ (x). τ ∈Hom (F ,E) Q v We can extend this discussion to Artinian local rings. Let C denote the category of Artinian local E-algebras with residue field E. If A ∈ C ,then R = R ⊗ A. If A,F E,F E E v v ρ : G → GL (A) is a continuous representation, then D (ρ ) is a free R -module. v F n v A,F v rig v If δ ∈ T (A), we call a triangulation of ρ of parameter δ an increasing filtration of v v v v D (ρ ) by direct summand (ϕ,  )-stable R -submodules such that the successive v F A,F rig v v graded pieces are isomorphic to R (δ ), ... , R (δ ). A,F v,1 A,F v,n v v If ρ : G → GL (E) is a continuous representation and F is a triangulation of v F n v parameter δ ∈ T (E), then we write D : C → Sets for the functor which associates v v ρ ,F ,δ v v v E to any A the set of equivalence classes of triples (ρ , F ,δ ),where: v v v • ρ : G → GL (A) is a lifting of ρ , continuous with respect to the p-adic topol- F n v v v ogy on A. • δ ∈ T (A) is a lifting of δ . v v • F is a triangulation of ρ of parameter δ which lifts F (note that there is a v v v † † † canonical isomorphism D (ρ ) ⊗ E = D (ρ ),as D commutes with base A v rig v rig rig change). Triples (ρ , F ,δ ), (ρ , F ,δ ) are said to be equivalent if there exists g ∈ 1 + M (m ) n A v v v v v v which conjugates ρ to ρ and takes F to F . v v v v We write D for the functor of equivalence classes of lifts ρ : G → GL (A). ρ F n v v Thus forgetting the triangulation determines a natural transformation D → D . ρ ,F ,δ ρ v v v v reg Proposition 2.6. — Suppose that δ ∈ T (E). Then the natural transformation D → v ρ ,F ,δ v v v v D is relatively representable, and injective on A-points for every A ∈ C .If ρ is absolutely irreducible, ρ v then both functors are pro-representable, in which case there is a surjective morphism R → R ρ ρ ,F ,δ v v v v of (pro-)representing objects. Proof. — If F = Q , this is contained in [BC09, Proposition 2.3.6] and [BC09, v p Proposition 2.3.9]. The general case is given by [Nak13, Lemma 2.35, Proposition 2.37, 30 JAMES NEWTON, JACK A. THORNE Corollary 2.38] (noting that Nakamura works with Berger’s category of B-pairs, which is equivalent to the category of (ϕ, )-modules over the Robba ring). A consequence of Proposition 2.6 is that when δ is regular, D (E[ ]) can be v ρ ,F ,δ v v v identified with a subspace of D (E[ ]) = H (F , ad ρ ). Since F is moreover uniquely ρ v v v determined by δ (Lemma 2.4), this subspace depends only on δ , when it is defined. We v v write H (F , ad ρ ) for this subspace. We observe that there is a natural transforma- v v tri,δ tion D → Spf O , which sends a triple (ρ , F ,δ ) to the character r (δ ). ρ ,F ,δ W ,r (δ ) v v v v v v v v v v v Evaluating on E[ ]-points, we obtain an E-linear map H (F , ad ρ ) → T W , v v r (δ ) v tri,δ v v where T W denotes the Zariski tangent space of W at the point r (δ ). This map r (δ ) v v v v v v appears in the statement of the following lemma: Lemma 2.7. — Let ρ : G → GL (E) be a continuous representation. Suppose that: v F n (1) ρ is de Rham. reg (2) ρ is trianguline of parameter δ ∈ T (E). v v (3) For each τ ∈ Hom (F , E), we have Q v wt (δ )<wt (δ )< ··· <wt (δ ). τ v,1 τ v,2 τ v,n We note that the labelled weights wt coincide with the labelled Hodge–Tate weights of ρ (cf. [KPX14, τ v Lemma 6.2.12]). Then the natural map 1 1 ker H (F , ad ρ) → T W → H (F , ad ρ) v r (δ ) v v v v tri,δ has image contained in 1 1 1 H (F , ad ρ) = ker(H (F , ad ρ) → H (F , ad ρ ⊗ B )). v v v Q d R g p Proof. — We must show that if (ρ , F ,δ ) ∈ D (E[ ]) is an element in the ρ ,F ,δ v v v v v v kernel of the map to T W ,then ρ is de Rham. When F = Q , this follows from r (δ ) v v p v v v [BC09, Proposition 2.3.4]; in general it follows from modifying their argument as in [HS16, Proposition 2.6] (the coefficients in this latter result are assumed to be a field, whilst we need coefficients E[ ], but the same proof works with any Artin local E-algebra as coefficient ring). We say that a triangulation of a representation ρ : G → GL (Q ) of parameter v F n v p δ is non-critical if for each τ ∈ Hom (F , E), the labelled weights are an increasing v Q v sequence of integers: wt (δ )<wt (δ )< ··· <wt (δ ). τ v,1 τ v,2 τ v,n SYMMETRIC POWER FUNCTORIALITY 31 In other words, if δ satisfies condition (3) of Lemma 2.7. We now give a criterion for a de Rham representation to have a triangulation satisfying this condition. This generalizes [HS16, Lemma 2.9], which treats the crystalline case. Lemma 2.8. — Let v ∈ S ,and let ρ : G → GL (Q ) be a de Rham representation p v F n satisfying the following conditions: (1) There exists an increasing filtration of the associated Weil–Deligne representation WD(ρ ) (by sub- Weil–Deligne representations) with associated gradeds given by characters χ ,...,χ : W → v,1 v,n F Q . (2) For each embedding τ : F → E,the τ -Hodge–Tate weights of ρ are distinct. v v (3) For each embedding τ : F → E,let k < ··· < k denote the strictly increasing sequence of v τ,1 τ,n τ -Hodge–Tate weights of ρ . Then we have for all τ ∈ Hom (F , E): v Q v v (χ (p)) < k + k p v,1 τ,2 τ ,1 τ =τ and for all i = 2,..., n − 1: i−1 v ((χ ...χ )(p)) < k + k  + k  . p v,1 v,i τ,i+1 τ ,i τ ,j τ =τ τ j=1 Then ρ is trianguline of parameter δ , where for each i = 1,..., n, δ : F → Q is defined by the v v v,i v p −k τ,i formula δ (x) = (χ ◦ Art (x)) τ(x) . In particular, the pair (ρ ,δ ) satisfies condition v,i v,i F v v (3) of Lemma 2.7. Proof. — The filtration of WD(ρ ) determines an increasing filtration 0 = M ⊂ v 0 M ⊂ M ⊂ ··· ⊂ M = D (ρ ) of D (ρ ) by sub-(ϕ, N, G )-modules (via the equiv- 1 2 n pst v pst v F alence of categories of [BS07, Proposition 4.1]). The main result of [Ber08] states that there is an equivalence of tensor categories between the category of filtered (ϕ, N, G )- modules and a certain category of (ϕ,  )-modules (restricting to the usual equivalence between weakly admissible filtered (ϕ, N, G )-modules and (ϕ,  )-modules associated F F v v to de Rham representations). We thus obtain a triangulation of the associated (ϕ,  )- module of ρ . A filtered (ϕ, N, G )-module M of rank 1 is determined up to isomorphism by its associated character χ : W → Q and the (unique) integers a such that F τ v p gr (M ⊗ Q ) = 0. The corresponding rank-1 (ϕ,  )-module is the one F F v F ⊗ Q ,τ ⊗id v v Q p p p −a associated to the character δ : F → Q given by the formula δ = x (χ ◦ Art ) (cf. v p v [KPX14, Example 6.2.6(3)]). What we therefore need to verify is that if a ∈ Z are the τ,i τ,i integers for which gr (M /M ⊗ Q ) = 0, then a = k . i i−1 F ,τ τ,i τ,i v,0 p 32 JAMES NEWTON, JACK A. THORNE This follows from hypothesis (3) of the lemma, together with the fact that D (ρ ) pst v is a weakly admissible filtered (ϕ, N, G )-module, as we now explain. We show by induc- tion on i that the jumps in the induced Hodge–de Rham filtration of M areasclaimed. For M , if these jumps are k then we have for each τ 1 τ,j ⎛ ⎞ 1 1 ⎝ ⎠ k + k  > v (χ ( )) τ,2 τ ,1 p v,1 v [F : Q ] [F : Q ] v p v,0 p τ =τ = t (M ) ≥ t (M ) = k  . N 1 H 1 τ ,j [F : Q ] v p Since the sequences k are strictly increasing, this is possible only if j = 1for each τ .In τ,i τ general, if the jumps of M areasexpectedand M /M has jumps k then we have i−1 i i−1 τ,j for each τ ⎛ ⎞ i−1 ⎝ ⎠ k + k + k > t (M ) τ,i+1 τ ,i τ ,j N i [F : Q ] v p j=1 τ =τ τ ⎛ ⎞ i−1 ⎝ ⎠ ≥ t (Fil ) = k + k . H i τ ,j  τ ,j [F : Q ] v p j=1 τ τ Once again this is possible only if j = i for each τ . Definition 2.9. — We say that a character δ ∈ T (Q ) is numerically non-critical if it satisfies v v the following conditions: (1) For each τ ∈ Hom (F , E), the labelled weights wt (δ ), ..., wt (δ ) are an increasing Q v τ v,1 τ v,n sequence of integers. (2) For each τ ∈ Hom (F , E),and foreach i = 1,..., n − 1, we have Q v v ((δ ...δ )(p)) < wt (δ ) − wt (δ ). p v,1 v,i τ v,i+1 τ v,i Following [BC09, Remark 2.4.6], we may reformulate Lemma 2.8 as follows: let ρ : G → GL (Q ) be a Hodge–Tate regular de Rham representation, and suppose v F n that WD(ρ ) is equipped with an increasing filtration such that the associated gradeds are given by characters χ ,...,χ : W → Q .Let k < ··· < k be the strictly increas- v,1 v,n F τ,1 τ,n v p ing sequences of τ -Hodge–Tate weights, and let δ ∈ T (Q ) be the character defined by −k τ,i the formula δ (x) = (χ ◦ Art (x)) τ(x) .Thenif δ is numerically non-critical, v,i v,i F v the representation ρ admits a non-critical triangulation with parameter δ . v v The most important case for us is that of 2-dimensional de Rham representations of G , and their symmetric powers. In this case the possible triangulations admit a par- p SYMMETRIC POWER FUNCTORIALITY 33 ticularly explicit description (cf. [Col08]; this description can be easily justified, including the case p = 2, using the results of [Ber08]): Example 2.10. — Let ρ : G → GL (Q ) be a de Rham representation with Hodge–Tate Q 2 ss weights k < k such that WD(ρ) = χ ⊕ χ with χ : W → Q . Assume for simplicity that 1 2 1 2 i Q χ = χ . Then we have the following possibilities: 1 2 −1 (1) If ρ is not potentially crystalline, then we can choose χ , χ so that χ = χ (|·| ◦ Art ).In 1 2 1 2 p this case ρ has a unique triangulation. It is non-critical, of parameter −k −k 1 2 δ = (x χ ◦ Art , x χ ◦ Art ). 1 Q 2 Q p p (2) If ρ is potentially crystalline and irreducible, or reducible and indecomposable, ρ has two triangula- tions. Both of these are non-critical, and their respective parameters are −k −k 1 2 δ = (x χ ◦ Art , x χ ◦ Art ) 1 Q 2 Q p p and −k −k 1 2 δ = (x χ ◦ Art , x χ ◦ Art ). 2 Q 1 Q p p (3) If ρ is decomposable, we can assume ρ = ψ ⊕ ψ where ψ has Hodge–Tate weight k and 1 2 i i WD(ψ ) = χ . In this case ρ admits two triangulations. The non-critical triangulation has pa- i i rameter −k −k 1 2 δ = (x χ ◦ Art , x χ ◦ Art ) 1 Q 2 Q p p and the critical triangulation has parameter −k −k 2 1 δ = (x χ ◦ Art , x χ ◦ Art ) 2 Q 1 Q p p (see for example [Ber17, Example 3.7] for the crystalline case). reg reg We now consider the global situation. We define T = T , T = T , v∈S v∈S v p p and W = W . We write r = r : T → W for the product of restriction maps. v v v∈S v∈S p p Let G = (GL × GL )  {±1} denote the group scheme defined in [CHT08, §2.1], ν : n n 1 G G → GL its character, and suppose given a continuous homomorphism ρ : G + → n 1 F ,S 1−n n G (E) such that ν ◦ ρ = δ . We write ad ρ for the E[G + ]-module given by n G + F ,S F/F adjoint action of G on the Lie algebra of GL . We write D : C → Sets for the functor n n ρ which associates to each A ∈ C the set of ker(GL (A) → GL (E))-conjugacy classes of n n 1−n n lifts ρ : G + → G (A) of ρ such that ν ◦ ρ = δ . F ,S n G + F/F If  ⊂ G is a subgroup, then ρ() ⊂ G (E) = GL (E) × GL (E), and we follow F,S n 1 [CHT08] in writing ρ| :  → GL (E) for the projection to the first factor. If v ∈ S, then there is a natural functor D → D , given by restriction ρ → ρ | . ρ ρ| G G F F v v 34 JAMES NEWTON, JACK A. THORNE reg Let δ = (δ ) ∈ T (E) be such that for each v ∈ S , ρ = ρ| is trianguline of v v∈S p v G p F parameter δ .Wedefine afunctor D = D × D ρ,F,δ ρ D ρ ,F ,δ ρ v v v v∈S v v∈S 1 + 1 + and H (F /F , ad ρ) = D (E[ ]) ⊂ H (F /F , ad ρ). S ρ,F,δ S tri,δ Proposition 2.11. — Keeping assumptions as above, suppose further that the following conditions are satisfied: (1) For each v ∈ S , ρ is de Rham. p v (2) For each v ∈ S, WD(ρ| ) is generic, in the sense of [All16, Definition 1.1.2]. (3) For each v ∈ S and for each τ ∈ Hom (F , E),wehave p Q v wt (δ )<wt (δ )< ··· <wt (δ ). τ v,1 τ v,2 τ v,n In other words, the triangulation of ρ with parameter δ is non-critical. v v 1 + (4) H (F , ad ρ) = 0. 1 + Then dim H (F /F , ad ρ) ≤ dim W . E S E tri,δ Proof. — Our assumption that WD(ρ| ) is generic means that for each v ∈ S , G p 1 1 1 1 H (F , ad ρ) = H (F , ad ρ) and for each v ∈ S − S ,H (F , ad ρ) = H (F , ad ρ) (see v v p v v f g f 1 + [All16, Remark 1.2.9]). Lemma 2.7 then implies that the map H (F /F , ad ρ) → tri,δ 1 + T W is injective (its kernel being contained in H (F , ad ρ) = 0). r(δ) 2.11.1. Trianguline representations – global geometry. — We fix a continuous pseu- docharacter τ : G → k of dimension n ≥ 1 which is conjugate self-dual, in the sense F,S ∨ 1−n that τ ◦ c = τ ⊗ . (We define pseudocharacters following Chenevier [Che14], where they are called determinants. For a summary of this theory, including what it means for a pseudocharacter of G to be conjugate self-dual, see [NT20,§2].) Let R denote F,S ps the universal pseudodeformation ring representing the functor of lifts of τ to conjugate self-dual pseudocharacters over objects of C (cf. [NT20, §2.19]). If v ∈ S ,let R de- O p ps,v note the pseudodeformation ring of τ | . We write X for the rigid generic fibre of G ps R ,and X for the rigid generic fibre of R . Then there is a natural morphism ps ps,v ps,v X → X = X of rigid spaces over E. We recall that to any representation ps ps,p ps,v v∈S ρ : G → GL (E) such that tr ρ = τ , and which is conjugate self-dual in the sense that F,S n c ∨ 1−n ρ ρ ⊗ , is associated a closed point tr ρ ∈ X (E). Conversely, if t ∈ X (E),then ps ps there exists a semi-simple conjugate self-dual representation ρ : G → GL (Q ) such F,S n that tr ρ = t, and this representation is unique up to isomorphism. If ρ : G → GL (E) is an absolutely irreducible representation such that ρ F,S n ∨ 1−n ρ ⊗ , then there is a homomorphism ρ : G + → G (E) such that ρ | = ρ and 1 F ,S n 1 G F,S 1−n a ν ◦ ρ = δ . The integer a ∈{0, 1} is uniquely determined by ρ,and anytwo G 1 F/F SYMMETRIC POWER FUNCTORIALITY 35 such extensions are conjugate by an element of GL (Q ).(See[CHT08, Lemma 2.1.4].) The following lemma extends this to objects of C : Lemma 2.12. — Let ρ : G → GL (E) be an absolutely irreducible representation such that F,S n c ∨ 1−n ρ ρ ⊗ , and fix an extension ρ as in the previous paragraph. Let A ∈ C . Then the following 1 E sets are in canonical bijection: (1) The set of ker(GL (A) → GL (E))-conjugacy classes of liftings ρ : G → GL (A) of ρ n n F,S n ∨ 1−n such that tr ρ ◦ c = tr(ρ ) ⊗ . (2) The set of ker(GL (A) → GL (E))-conjugacy classes of liftings ρ : G + → G (A) of ρ n n F ,S n 1 1−n such that ν ◦ ρ = δ . 1 F/F Proof. — There is an obvious map sending ρ to tr ρ | . We need to check that 1 1 F,S this is bijective (at the level of conjugacy classes). To check injectivity, let ρ ,ρ : G + → F ,S 1 1 G (A) be two such homomorphisms and suppose that ρ | , ρ | are ker(GL (A) → n G G n 1 F,S 1 F,S GL (E))-conjugate. We must show that ρ , ρ are themselves conjugate. We may suppose 1 1 that in fact ρ | = ρ | . In this case Schur’s lemma (cf. [CHT08, Lemma 2.1.8]), G G 1 F,S 1 F,S −1 applied to ρ (c) ρ (c),shows that ρ , ρ are equal. 1 1 1 1 Now suppose given ρ : G → GL (A) lifting ρ , and such that tr ρ ◦ c = tr(ρ ) ⊗ F,S n 1−n .Let J ∈ GL (E) be defined by ρ (c) = (J, −ν ◦ ρ (c))j (cf. [CHT08, Lemma n 1 G 1 c −t −1 1−n 2.1.1]), so that ρ(σ ) = Jρ(σ) J ⊗ for all σ ∈ G .Then[Che14, Example 3.4] F,S c   −t  −1 implies the existence of a matrix J ∈ GL (A) lifting J such that ρ (σ ) = J (ρ (σ )) (J ) for all σ ∈ G .By[CHT08, Lemma 2.1.1], this implies the existence of a homo- F,S morphism ρ : G + → G (A) lifting ρ and such that ρ | = ρ . This completes the F ,S n 1 G 1 1 F,S proof. v−irr There is a Zariski open subspace X ⊂ X consisting of those points at which ps,v ps,v p−irr v−irr the universal pseudocharacter is absolutely irreducible. We write X = X ps,p ps,v v∈S p−irr p−irr and X for the pre-image of X . Thus again there is a canonical morphism ps ps,p p−irr p−irr X → X . According to [Che14, §4.2], there exists an Azumaya algebra A over ps ps,p v−irr u × u X and a homomorphism ρ : G → A such that tr ρ is the universal pseudochar- ps,v v v v acter. Lemma 2.13. — Let ρ : G → GL (E) be an absolutely irreducible representation, cor- v F n v−irr responding to a point z = tr ρ ∈ X (E). Then there exists an affinoid open neighbourhood ps,v v−irr z ∈ U ⊂ X (E) and an isomorphism A| M (O ). U n U ps,v Proof. — Let U be an open affinoid neighbourhood of z.The stalk O is a U,z Henselian local ring ([FvdP04, Proposition 7.1.8]). Thus the stalk A is an Azumaya algebra over a Henselian local ring which is split over the closed point; it is therefore split, i.e. there exists an isomorphism A M (O ). After shrinking U , this extends to an z n U,z isomorphism A(U) M (O(U)), as desired. n 36 JAMES NEWTON, JACK A. THORNE p−irr Lemma 2.14. — (1) Let z ∈ X (E) be the closed point corresponding to the isomorphism ps,p class of a tuple (ρ ) of absolutely irreducible representations ρ : G → GL (E). Then there v v∈S v F n p v is a canonical isomorphism p−irr T X ⊕ H (F , ad ρ ). z ps,p v∈S v v p−irr (2) Let z ∈ X (E) be the closed point determined by a representation ρ : G + → G (E) such that ps F ,S n for each v ∈ S , ρ| is absolutely irreducible. Then there is a canonical isomorphism p G p−irr T X = H (G , ad ρ), z F ,S ps which has the property that the diagram p−irr T X H (G , ad ρ) z ps F ,S p−irr T X ⊕ H (F , ad ρ ) z ps,p v∈S v v commutes. Proof. — The first part follows from [Che14, §4.1], which states that the completed v−irr local ring of X at the E-point corresponding to an absolutely irreducible represen- ps,v tation ρ : G → GL (E) pro-represents the functor D . The second follows from this v F n ρ v v and Lemma 2.12. Proposition 2.15. — Let v ∈ S ,and let ρ : G → GL (E) be an absolutely irreducible p v F n reg v−irr reg representation which is trianguline of parameter δ ∈ T (E).Let z ∈ X × T be the closed point v ps,v v corresponding to the pair (ρ ,δ ).Then: v v v−irr reg (1) There exists an affinoid open neighbourhood U ⊂ X × T of z over which there exists a ps,v v universal representation ρ : G → GL (O(U )).Let V ⊂ U denote the set of points (ρ ,δ ) F n v v v v v v such that ρ is trianguline of parameter δ ,and let Z ⊂ U denote the Zariski closure of V . Then v v v v v V is the set of points of a Zariski open subspace of Z . v v (2) The Zariski tangent space of Z at z is contained in the subspace H (F , ad ρ ) of the Zariski v v v tri,δ v−irr reg tangent space of X × T at z. ps,v v v−irr reg Proof. — By Lemma 2.13, there is an affinoid neighbourhood U ⊂ X × T ps,v v of z over which there exists a universal representation ρ : G → GL (O(U )).Wecan F n v v v assume without loss of generality that U is connected. By [KPX14, Corollary 6.3.10], there is a reduced rigid space Z over E and a proper birational morphism f : Z → Z having the following properties: SYMMETRIC POWER FUNCTORIALITY 37 • For every point z ∈ Z , the absolutely irreducible representation ρ  is trian- f (z ) guline. ∗ u • There is an increasing filtration 0 = F ⊂ F ⊂··· ⊂ F = D (f (ρ ))) by 0 1 n rig coherent (ϕ,  )-stable R -submodules. F Z ,F v v −1 • There exists a Zariski closed subset Z ⊂ Z such that Z ∩ f (V ) =∅ and b b for each z ∈ Z − Z , the pullback of F to D (ρ  ) is a triangulation of • f (z ) rig parameter δ . f (z ) • Over Z − Z , F is in fact a filtration by local direct summand R - • Z ,F b v submodules. • The map f factors through a proper birational morphism f : Z → Z ,where Z is the normalisation of Z .Moreover, f factors as the composition of a se- v v quence of proper birational morphisms between normal rigid spaces Z = Z → Z → ··· Z = Z m m−1 1 v where each morphism Z → Z is glued, locally on the target, from analyti- i i−1 fications of birational projective schemes over Spec(A),with Sp(A) ⊂ Z an i−1 affinoid open. Note that the final point is a consequence of the construction in the proof of [KPX14, −1 Theorem 6.3.9]. The third point actually implies that Z = Z f (V ),hence Z = v v f (Z ) V . Since f is proper this shows that V is Zariski open in Z . v v v Let z ˜ ,..., z ˜ ∈ Z be the closed points of the normalisation with image in Z 1 m v v equal to z.For each 1 ≤ j ≤ m,let z be a closed point of the preimage of z ˜ in Z . We denote by z the image of z in any of the Z ,for 1 ≤ i ≤ m. We claim that the map j i O → O on completed local rings is injective; indeed, it follows from the final point Z ,z ˜ Z ,z v j j in the itemized list above that we need to show injectivity for each map O → O . Z ,z Z ,z i j i+1 j Each of these maps coincides with the map on complete local rings A → O associated x X,x with a (projective, birational) morphism of schemes X → Spec(A), where A is the ring of functions on an open affinoid neighbourhood of z ∈ Z , x ∈ Spec(A) is the maximal j i ideal given by z and x ∈ X is a closed point mapping to x. The complete local ring A j x is a domain (by normality and excellence of A )and themap A → O is injective (by x x X,x dominance of X → Spec(A)), so [GD71, Corollaire 3.9.8] gives the desired injectivity. The map O → O is the normalisation of O , so it is also injective. Putting Z ,z Z ,z v Z ,z ˜ v i=1 v i everything together, we have shown that the map O → O is injective. Z ,z Z ,z v i=1 v i After possibly extending E, we can assume that all of the points z in Z have residue field E. The existence of a global triangulation over Z − Z implies that for each i = 1,..., m, there is a classifying map R → O ,where F is the unique ρ ,F ,δ Z ,z v v v v Here we mean the relative analytification defined by Köpf [Köp74], see also [Con06, Example 2.2.11]. 38 JAMES NEWTON, JACK A. THORNE triangulation of ρ of parameter δ . This implies the existence of a commutative diagram v v O O U ,z Z ,z v v R O  , ρ ,F ,δ Z ,z v v v i=1 where the left vertical arrow is surjective (Proposition 2.6) and the top horizontal arrow is surjective. We have already noted that the right vertical arrow is injective. These facts together imply that the top horizontal arrow factors through a surjective map R → ρ ,F ,δ v v v O . This implies the desired result at the level of Zariski tangent spaces. Z ,z Remark 2.16. — Prompted by a referee, we note that the definition of ‘Zariski dense’ given in [KPX14, Definition 6.3.2] is somewhat non-standard. In this paper (and in other references we cite such as [BHS17]), a subset Z of a rigid space X is called Zariski dense if the smallest closed analytic subset of X which contains Z is X. In [KPX14, Definition 6.3.2] the stronger condition is imposed that Z is Zariski dense (in the usual sense) in each member of some admissible affinoid cover of X. When we apply [KPX14, Corollary 6.3.10] in the above proof, we have a Zariski dense subset of an affinoid, so there is no discrepancy between the definitions in this case. Eugen Hellmann has explained to us that the crucial result [KPX14, Theorem 6.3.9] does in fact hold with the weaker, standard definition of Zariski dense. Since it may be of interest, we sketch the argument. We start with X, δ,Mas in [KPX14, Theorem 6.3.9] and suppose we have a Zariski dense (in the usual sense) subset X ⊂ X satisfying the assumptions of loc. cit. We alg may assume that X is normal and connected, and will show that X can be enlarged to alg a subset which is Zariski dense in the stronger sense of [KPX14]. i ∨ i ∨ There are coherent sheaves H (M (δ)),H (M (δ)/t ) on X, which are lo- ϕ,γ ϕ,γ K K cally free over a non-empty (hence dense) Zariski open subset U ⊂ X. At points z in 0 ∨ 0 ∨ the Zariski dense subset X ∩ U, the fibre H (M (δ)) ⊗ k(z) = H (M (δ )) alg O z ϕ,γ X ϕ,γ z K K has dimension one and the map M → R (π )(δ ) dual to a non-zero element of z k(z) K z this fibre is surjective. The latter condition is equivalent to non-vanishing of the map 0 ∨ 0 ∨ H (M (δ)) ⊗ k(z) → H (M (δ)/t ) ⊗ k(z) for every p-adic embedding σ . O σ O X X ϕ,γ ϕ,γ K K These conditions hold over a Zariski open subset U ⊂ U. Since U contains X ∩ U, alg it is also Zariski dense in X. Moreover, U contains a Zariski dense subset of every affi- noid open V ⊂ X. Indeed, the intersection V ∩ U with the Zariski open and dense subset U contains an affinoid open subset of V. Repeating this step, the intersection V ∩ U also contains an affinoid open subset of V. We have shown that we obtain the desired enlargement of X by adjoining U . alg SYMMETRIC POWER FUNCTORIALITY 39 2.17. The unitary group eigenvariety. — Now let F, S, p, G = G be as in our standard assumptions (§1). We continue to assume that E contains the image of every embedding τ : F → Q . In particular, the reductive group Res + G splits over E. F /Q n Let U = U ⊂ G (A ) be an open compact subgroup such that for every n n,v n + v F + + finite place v ∈ SofF ,U is hyperspecial maximal compact subgroup of G (F ).We n,v n ∞,S S 1 n n −1 S S define T = O[T ,..., T ,(T ) ]⊂ O[U \G (A )/U ] to be the algebra generated n w w w n F n c + by the unramified Hecke operators at split places v = ww of F not lying in S. These operators were defined in §1.23. We write T ⊂ B = T N ⊂ GL for the usual maximal torus and upper triangular n n n n n Borel subgroup, and define E-rigid spaces W = Hom( T (O ), G ) n n F m v∈S and T = Hom( T (F ), G ). n n v m v∈S Restriction of characters determines a morphism r : T → W of rigid spaces. Note that n n the spaces T , W may be canonically identified with the spaces T , W of the previous n n section. We fix a choice of isomorphism ι : Q → C.If π is an automorphic represen- tation of G (A ) with π = 0, there is a corresponding semisimple Galois represen- n F tation r : G → GL (Q ) (cf. Corollary 1.3), which satisfies local-global compati- π,ι F,S n −1 ∞ U S bility at each place of F. The space ι (π ) is naturally an isotypic T -module, which therefore determines a homomorphism ψ : T → Q .Wecallan accessible re- n p finement of π a choice χ = (χ ) for each v ∈ S of a (necessarily smooth) character v v∈S p χ : T (F ) → Q which appears as a subquotient of the normalised Jacquet module v n v −1/2 −1 −1 ι r (π ) = ι (π δ ); equivalently, for which there is an embedding of π into N v v,N (F ) v n n v B GL the normalised induction i ιχ .Notethat χ ∈ T (Q ). v n B p Lemma 2.18. — Let π be an automorphic representation of G (A ),and let χ = (χ ) n v v∈S F p T −1 be an accessible refinement of π.Thenfor each v ∈ S , there is an increasing filtration of rec (ι π ) p v by sub-Weil–Deligne representations with graded pieces (1−n)/2 (1−n)/2 χ |·| ◦ Art ,...,χ |·| ◦ Art . v,1 F v,n F v v Proof. — Since π admits an accessible refinement, it is a subquotient of a principal k (n −1)/2 series representation. Suppose that rec (π ) =⊕ Sp (ψ |·| ) for some charac- F v i v i=1 n ters ψ : F → C . By the Langlands classification, π is isomorphic to a subquotient of i v v 40 JAMES NEWTON, JACK A. THORNE the normalised induction GL = i St (ψ ) ⊗ ··· ⊗ St (ψ ), n 1 n k P 1 k where P ⊂ GL is the standard parabolic subgroup corresponding to the partition n = n + n + ··· + n . It will therefore suffice to show the stronger statement that if α = 1 2 k α ⊗ ··· ⊗ α is a subquotient of the normalised Jacquet module of , then there is an 1 n k (n −1)/2 increasing filtration of ⊕ Sp (ψ |·| ) by sub-Weil–Deligne representations with i=1 n (n −1)/2 graded pieces given by α ◦ Art ,...,α ◦ Art . We recall that each Sp (ψ |·| ) 1 F n F i v v comes with a standard basis e , e ,..., e . We concatenate and relabel these bases so that 1 2 n k (n −1)/2 e , e ,..., e is a basis for ⊕ Sp (ψ |·| ) with e j−1 ,... e j the standard basis 1 2 n i i=1 n i 1+ n n i i i=1 i=1 (n −1)/2 for Sp (ψ |·| ). We first treat the case k = 1, n = n. After twisting we can assume that ψ = 1. Then (n−1)/2 (1−n)/2 the normalised Jacquet module of St equals |·| ⊗· · · ⊗|·| , while there is a (n−1)/2 unique invariant flag of Sp (|·| ) given by Fil = span(e ,..., e ) (i = 1,..., n) which i 1 i has the desired graded pieces. Now we return to the general case. Using [Zel80, Theorem 1.2], we see that the irreducible subquotients of the normalised Jacquet module of  are precisely the charac- −1 −1 ters β ⊗ ··· ⊗ β ,where w ∈ S is any permutation which is increasing on each w (1) w (n) of the sets {1,..., n }, {n + 1,..., n + n }, ... , {n +· · · + n + 1,..., n + ··· + n }, 1 1 1 2 1 k−1 1 k (n −1)/2 (1−n )/2 i i and (β ,...,β ) is the concentration of the tuples (ψ |·| ,...,ψ |·| ) for 1 n i i i = 1,..., k. We see that the increasing filtration of k (n −1)/2 ⊕ Sp (ψ |·| ) i=1 n given by Fil = span(e −1 ,..., e −1 ) is a filtration by sub-Weil–Deligne representations j w (1) w (j) which has the desired property. This completes the proof. If χ is an accessible refinement of π,thenwewrite ν(π, χ) ∈ T (Q ) for the char- acter −1/2 −1 (2.18.1) ν(π, χ) = κ(π) · (χ ι δ ) , v v∈S B p where κ(π) ∈ T (E) is the (B -dominant) Q -algebraic character which is the highest n n p −1 weight of ι π .If κ(π) = (κ ≥ κ ≥ ··· ≥ κ ) then the labelled Hodge– ∞ v τ,1 τ,2 τ,n τ :F →Q Tate weights of r | , in increasing order, are (−κ < −κ + 1 < ··· < −κ + n − π,ι G τ,1 τ,2 τ,n 1) . τ :F →Q We write j : T → T forthe mapdefinedbythe formula n n n −1 1−n j (ν) = ν · (1, ◦ Art ,..., ◦ Art ). n v v F F v v The reason for introducing this map is that if π is an automorphic representation of (1−n)/2 G (A +) and χ is an accessible refinement, then the parameter δ associated to χ|·| n F SYMMETRIC POWER FUNCTORIALITY 41 by the formula of Lemma 2.8 satisfies j (ν(π, χ)) = δ. We call the accessible refinement χ numerically non-critical or ordinary if δ is. Note that this property depends on the pair (π, χ) and not just on χ . 2.18.1. Emerton’s eigenvariety construction. — We now describe the construction, fol- lowing Emerton [Eme06b], of the (tame level U ) eigenvariety for G . We use Emerton’s n n construction because we do not want to restrict to considering π with Iwahori-fixed vec- tors at places in S (as is done, for example, in [BC09]) and it seems to us that Emerton’s representation-theoretic viewpoint is the most transparent way to handle this level of generality. We recall the set-up of §1.23, so for each dominant weight λ we have a module n S S (U , O/ ) of algebraic modular forms, which has a natural action of T . When λ is λ n trivial we omit it from the notation. We define p p s S(U , O) := lim lim S(U U , O/ ) n n ← − − → s U and p p S(U , E) := S(U , O) ⊗ E, n n p p so S(U , E) is an E-Banach space (with unit ball S(U , O)), equipped with an admissible n n continuous representation of G (F ). For dominant weights λ, we can consider the space ∨ p V −alg + S of locally V -algebraic vectors S(U , E) .Wehavea (G (F ) × T )-equivariant iso- λ n p n morphism p ∨ p V −alg lim S (U U , O) ⊗ V S(U , E) λ p O n λ n − → (see [Eme06b, Corollary 2.2.25]). We can also consider the space of locally Q -analytic p an vectors S(U , E) , and apply Emerton’s locally analytic Jacquet functor J to this locally n n analytic representation of G (F ). We thereby obtain an essentially admissible locally p an analytic representation J S(U , E) of T (F ), and by duality a coherent sheaf B n v n n v∈S M on T , equipped with an action of T .Wedenoteby A ⊂ End(M ) the coherent n n n n O -algebra subsheaf generated by T . Now we can define the eigenvariety, an E-rigid space, as a relative rigid analytic spectrum E := Sp A − → T n n n equipped with the canonical finite morphism ν . 42 JAMES NEWTON, JACK A. THORNE −1 We define another finite morphism ν : E → T by twisting ν by δ (see Remark n n 2.21). By construction, we also have a ring homomorphism ψ : T → O(E ),soweobtain a map on points: ∗ S ψ × ν : E (Q ) → Hom(T , Q ) × T (Q ). n n p n p p For E /E finite (with E ⊂ Q ), a point (ψ ,ν ) ∈ Hom(T , E ) × T (E ) is in the image of 0 0 n p n ψ × ν if and only if the eigenspace p  an J S(U , E ) [ψ ,ν δ ] B 0 0 B n n n is non-zero, or in other words if there is a non-zero T(F )-equivariant map v∈S p  an ν δ → J S(U , E ) [ψ ]. 0 B B 0 n n n We define the subset Z ⊂ E (Q ) of classical points to be those for which there is more- n n over a non-zero map to the Jacquet module of the locally algebraic vectors: p  alg ν δ → J S(U , E ) [ψ ]. 0 B B 0 n n n S × Lemma 2.19. — For any characters ψ : T → E and χ : T(F ) → E , we have n v∈S p an Hom χ, J S(U , E) [ψ ] T(F ) B v n n v∈S p an = Hom χ, J S(U , E) [ψ ] . T(F ) B v n n v∈S Proof. — This can be seen using Emerton’s canonical lift [Eme06a,Proposition p an 3.4.9], which identifies both sides of the equality with the same eigenspace in S(U , E) . Alternatively, we can use the left exactness of the Jacquet functor. In the latter argument S p an we need to use the fact that T acts on S(U , E) via a Noetherian ring and we then n n deduce that passing to an eigenspace for a (finitely generated) ideal in this ring commutes with the Jacquet functor. We now relate the classical points Z to refined automorphic representations. Let ∞ U A denote the set of automorphic representations π of G (A ) such that (π ) = 0, n n F let RA denote the set of pairs (π, χ) where π ∈ A and χ is an accessible refine- n n ment of π,and let Z ⊂ Hom(T , Q ) × T (Q ) denote the set of points of the form n n n p p (ψ ,ν(π,χ)),where (π, χ) ∈ RA . We note in particular the existence of the surjective π n map γ : RA → Z . n n n Lemma 2.20. — The map ψ × ν restricts to a bijection Z → Z . n n SYMMETRIC POWER FUNCTORIALITY 43 Proof. — If z ∈ Z ∩ E (E ) is a classical point defined over E , it follows from the n n above discussion on locally algebraic vectors that z arises from a non-zero map p  ∗ ∨ ν(z)δ → J lim S (U U , E )[ψ (z)]⊗  V B B λ p E n n n λ − → for some dominant weight λ. It follows from [Eme06a, Prop. 4.3.6] that such maps cor- respond bijectively with non-zero maps ∨ −1 p  ∗ ν(z)δ (λ ) → J lim S (U U , E )[ψ (z)] , B B λ p n n n − → ∨ ∨ where λ is the highest weight of V . By Lemma 1.25,wehave (π, χ) ∈ RA where π has highest weight ιλ , ψ = n ∞ π −1/2 ∗ ∨ ψ (z) and ν(z) = λ χδ = ν(π, χ) . This shows that we do indeed have an induced map Z → Z , and it is easy to see that this is a bijection. n n Remark 2.21. — An accessible refinement χ is numerically non-critical if and only 1/2 if for every v ∈ S the character ν(π, χ) δ = κ(π) χ δ has non-critical slope, in the p v B v v −1/2 1/2 sense of [Eme06a, Defn. 4.4.3]. The renormalisation (replacing χ δ with χ δ )ap- v v B B n n 1/2 pears in Emerton’s eigenvariety construction because χ δ is a smooth character ap- −1 pearing in the (non-normalised) Jacquet module ι π , whilst Bellaïche–Chenevier v,N (F ) n v normalise things to be compatible with the Hecke action on Iwahori-fixed vectors (see [BC09, Prop. 6.4.3]). Our next task is to recall some well known properties of the eigenvariety E (cf. [Bre15, §7]), variants of which are established by numerous authors in slightly differ- ent contexts (e.g. [Che04, Buz07, Eme06b, Loe11]). We follow the exposition of [BHS17] which establishes these properties for the patched eigenvariety. In order to at least sketch the proofs of these properties in our context, we first introduce a ‘spectral variety’ which will turn out to be a Fredholm hypersurface over W . n−1 We fix the element z = (z ) ∈ T (F ) with z = diag( ,..., , 1), v v∈S n v v v p v∈S v and let Y be the closed subgroup of T (F ) generated by T (O ) and z. n v n F v∈S v∈S p p The rigid space Y = Hom(Y, G ) is then identified with W × G .Asin[BHS17, §3.3], m n m p an it follows from [Eme06a, Proposition 3.2.27] that J S(U , E) has dual equal to the n n space of global sections of a coherent sheaf N on W × G .Wedefine Y to be the n n m z −1 schematic support (cf. above Définition 3.6 in [BHS17]) of N (δ ). This rigid space comes equipped with a closed immersion Y "→ W × G . The twist in the definition of z n m Y is there to ensure that this closed immersion is compatible with the map ν.Indeed,the map from E given by composing ν with the restriction map to W × G factors through n n m a finite map f : E → Y , giving us a commutative diagram: n z 44 JAMES NEWTON, JACK A. THORNE E T n n Y W × G z n m Now we state our proposition summarising the key properties of the eigenvari- ety E . Proposition 2.22. — The tuple (E ,ψ,ν,Z ) has the following properties: n n (1) E is a reduced E-rigid space, equipped with a finite morphism ν : E → T . We write κ for the n n n induced map κ : E → W . n n (2) Z ⊂ E (Q ) is a Zariski dense subset which accumulates at every point of Z (in other words, each n n n point of Z admits a basis of affinoid neighbourhoods V such that V ∩ Z is Zariski dense in V), n n ∗ S and the map ψ × ν : E (Q ) → Hom(T , Q ) × T (Q ) restricts to a bijection Z → Z . n n n n p n p p S −1 (3) For any affinoid open V ⊂ T , the map T ⊗ O(V) → O(ν V) is surjective. (4) E is equidimensional of dimension equal to dim W . For any irreducible component C ⊂ E , κ(C) n n n is a Zariski open subset of W . (5) Let z ∈ E (Q ) be a point, and suppose that δ = j (ν(z)) factors as δ = δ δ ,where δ is a n n alg sm alg strictly dominant algebraic character and δ is smooth, and that δ is numerically non-critical. Then sm z ∈ Z . ≤1 (6) ψ takes values in the subring O(E ) of bounded elements. Proof. — First we note that a tuple satisfying the first three properties is unique – we will not actually use this fact, but it can be proved in the same way as [BC09,Proposition 7.2.8] (our context is slightly different, as we equip our eigenvarieties with a map to T instead of W × G ). We also note that it is not essential for our purposes to show that E n m n is reduced (this is the most delicate of the listed properties); we could instead replace E with its underlying reduced subspace. Now we summarise how to verify these properties. Property (5) follows from Emer- ton’s ‘classicality criterion’ for his Jacquet functor [Eme06a, Theorem 4.4.5] (cf. Remark 2.21). Property (3) holds by construction. Property (4) can be established as in [BHS17, §3.3] using the spectral variety Y . More precisely, (the proof of) Lemma 3.10 in this reference shows that the closed analytic subset of W × G underlying Y is a Fredholm hypersurface, and Y has an admisible n m z z cover by affinoids (U ) on which the map to W is finite and surjective with image an i∈I n open affinoid W ⊂ W . Moreover, each U is disconnected from its complement in the i n inverse image of W and (U , N ) is a finite projective O (W )-module. i n W i i n Having established the existence of a good affinoid cover of the spectral variety, −1 we set U = f (U ). Since f is a finite map, (U ) is an admissible affinoid cover of i i i∈I E . It can then be shown, as in [BHS17, Proposition 3.11], that each affinoid O (U ) is n E i n SYMMETRIC POWER FUNCTORIALITY 45 isomorphic to a O (W )-algebra of endomorphisms of the finite projective O (W )- W i W i n n module (U , M ).Wecan nowprove Property (4)asin[BHS17, Corollaire 3.12]: this i n shows that E is equidimensional of dimension equal to W , without embedded compo- n n nents, and each irreducible component maps surjectively to an irreducible component of Y . Since irreducible components of Fredholm hypersurfaces are again Fredholm hyper- surfaces, the image of such an irreducible component is Zariski open in W (cf. [BHS17, Corollaire 3.13]). Now to establish property (2), using property (5), it suffices to show that points z ∈ E (Q ) with numerically non-critical δ = j (ν(z)) accumulate at any point z with κ(z ) n n 0 0 locally algebraic (cf. [BHS17, Théorème 3.19]). Using the good affinoid cover described in the previous paragraph, we have an affinoid neighbourhood U of z which is a finite cover of an affinoid W ⊂ W . In fact, such U form a neighbourhood basis at z (cf. [Taï16, n 0 Theorem 2.1.1, Lemma 2.1.2]). The valuations v (δ (p)) are bounded as z varies in U p v,i (with δ = j (ν(z))). It follows from the description in Definition 2.9 that there is a subset −1 of W accumulating at κ(z ) such that κ () ∩ U consists entirely of points with n 0 −1 numerically non-critical δ. The subset κ () ∩ U is Zariski dense in U. Finally, to establish property (1) it remains to prove that E is reduced. Since we showed that E is without embedded components, it suffices to prove that every irre- ducible component of E contains a reduced point. Using (4) and the Zariski density of algebraic characters in W , it suffices to show that E is reduced at every point z with n n 0 κ(z ) algebraic. We use a good affinoid neighbourhood U = Sp(B) of z as in the pre- 0 0 vious paragraph, with W = κ(U) = Sp(A). The finite A-algebra B is identified with a sub-A-algebra of End (M),where M = (U, M ) is a finite projective A-module. As A n in the proof of [Che05, Proposition 3.9], it now suffices to show that for w in a Zariski dense subset of W, the Hecke algebra T and T (F ) act semisimply on the fibre n v v∈S M ⊗ k(w) — we use the fact that an endomorphism of a projective A-module which vanishes in the fibres at a Zariski dense subset of points in W necessarily vanishes. The proof of [BHS17, Corollaire 3.20] shows that we can achieve this by choosing w so that their pre-images in U have très classique associated characters δ [BHS17, Définition 3.17] (this is a condition on characters with algebraic image in W which can be guaranteed by a ‘numerical’ condition as in the proof of [BHS17, Théorème 3.19], in particular it gives a Zariski-dense and self-accumulating subset of E ). We can replace the reference + S to [CEG 16] in the proof with the well-known assertion that the Hecke algebra T acts semisimply on lim S (U U , E) for dominant λ. Finally, property (6) follows from the λ p − → S p p fact that the T -action stabilizes the unit ball S(U , O) ⊂ S(U , E). n n n The properties established in Proposition 2.22 imply the existence of a conjugate self-dual Galois pseudocharacter T : G → O(E ) with the property that for any point n F,S n z ∈ Z corresponding to a pair (π, χ),T = tr r . This is proved as in [BC09,Proposi- n n,z π,ι ≤1 tion 7.5.4] and [Che04, Proposition 7.1.1]. The key points are that O(E ) is compact ≤1 [BC09, Lemma 7.2.11] and the map O(E ) → C given by the evaluation maps n p z∈Z n 46 JAMES NEWTON, JACK A. THORNE at each z ∈ Z is a continuous injection (by Zariski density of Z and reducedness of E ). n n n Then [Che14, Example 2.32] is used to glue together the pseudocharacters tr r to form π,ι the continuous pseudocharacter T . The pseudocharacter T determines an admissible cover E = E (τ ) as a dis- n n τ n n joint union of finitely many open subspaces indexed by G -orbits of pseudocharacters τ : G → F , over which the residual pseudocharacter satisfies the condition T = τ n F,S p n,z n (cf. [Che14, Theorem 3.17]). Fix a pseudocharacter τ : G → F . Extending E if necessary, we may assume n F,S p that τ takes values in k. We recall some of the E-rigid spaces of Galois representations defined in §2.11.1, now decorated with n subscripts. Thus X is the space of conjugate ps,n p−irr self-dual deformations of τ , X is its Zariski open subspace of pseudocharacters which n ps,n irr are irreducible at the p-adic places. We also have the subspace X of pseudocharacters ps,n which are (globally) irreducible. The existence of T determines a morphism λ : E(τ ) → n n X , and the morphism i = λ × (j ◦ ν) : E (τ ) → X × T is a closed immersion, by ps,n n n n ps,n n point (3) in the list of defining properties of E . Now assume that n ≥ 3, and let τ : G → F be a conjugate self-dual pseu- 2 F,S p n−1 docharacter of dimension 2. Let τ = Sym τ ;then τ is a conjugate self-dual pseu- n 2 n docharacter of G of dimension n. Taking symmetric powers of pseudocharacters de- F,S termines a morphism σ : X → X . On the other hand, we can define a map n,g ps,2 ps,n σ : T → T by the formula n,p 2 n n−1 n−2 n−1 ((δ ,δ )) → ((δ ,δ δ ,...,δ )) . v,1 v,2 v∈S v,2 v∈S p v,1 v,1 v,2 p We write σ = σ × σ : X × T → X × T for the product of these two morphisms. n n,g n,p ps,2 2 ps,n n We have constructed a diagram σ ◦i i n 2 n E (τ ) X × T E (τ ). 2 2 ps,n n n n Compare Lemma 2.2. Definition 2.23. — Let π be an automorphic representation of G (A +),let χ = (χ ) be 2 F v v∈S an accessible refinement of π,and let n ≥ 2. We say that χ is n-regular if for each v ∈ S the character χ = χ ⊗ χ satisfies (χ /χ ) = 1 for 1 ≤ i ≤ n − 1. v v,1 v,2 v,1 v,2 Theorem 2.24. — Let (π ,χ ) ∈ RA satisfy tr r = τ ,and let z = γ (π ,χ ) ∈ 2 2 2 π ,ι 2 2 2 2 2 E (τ )(Q ). Suppose that: 2 2 (1) The refinement χ is numerically non-critical and n-regular. (2) There exists (π ,χ ) ∈ RA such that (σ ◦ i )(z ) = i (z ),where z = γ (π ,χ ). n n n n 2 2 n n n n n n (3) For each v ∈ S , the Zariski closure of r (G ) (in GL /Q ) contains SL . p π ,ι F 2 2 2 v Then each irreducible component C of E (τ ) containing z satisfies (σ ◦ i )(C) ⊂ i (E (τ ) ). 2 2 C 2 n 2 n n n C p p Proof. — Extending E (the field over which E is defined) if necessary, we may as- sume that z ∈ E (τ )(E) and r takes values in GL (E).By[Con99, Theorem 3.4.2] 2 2 2 π ,ι 2 2 SYMMETRIC POWER FUNCTORIALITY 47 (which says that an irreducible component of E (τ ) is contained in the base change of 2 2 C an irreducible component of E (τ )), it suffices to show that each irreducible component 2 2 C of E (τ ) containing z satisfies (σ ◦ i )(C) ⊂ i (E (τ )). By Lemma 2.2,itisenoughto 2 2 2 n 2 n n n −1 show that (σ ◦ i ) (i (E (τ ))) contains an affinoid open neighbourhood of z .Toprove n 2 n n n 2 this, we will use a number of the results established so far. Since χ is a numerically non-critical refinement, the parameter δ of the associ- 2 2 ated triangulation is non-critical, in the sense that for each τ ∈ Hom(F, E), the sequence of τ -weights of δ is strictly increasing (Lemma 2.8). Passing to symmetric powers, we see that r is trianguline of parameter δ = σ (δ ),and that δ is non-critical (although it is π ,ι n n,p 2 n not necessarily numerically non-critical). reg The n-regularity of χ implies that δ ∈ T (E). We are going to apply Propo- 2 n 1 + sition 2.11 to conclude that dim H (F /F , ad r ) ≤ dim W = dim E (τ ).Note E S π ,ι n n n tri,δ that if v ∈ SthenWD(r | ) is generic, because it is pure: the base change of π π ,ι G n n F to GL (A ) exists (for example, by [Lab11, Corollaire 5.3]) and is cuspidal (because n F r is irreducible), so we can appeal to the main theorem of [Car12] (which estab- π ,ι lishes the general case; under various additional hypotheses, purity was established in [HT01, TY07, Shi11, Clo13]). If v ∈ S , let us write f : X × T → X × T for p v ps,n n ps,n,v v the natural restriction map. By Proposition 2.15 and Lemma 2.14,wecan findfor each v ∈ S an affinoid open neighbourhood U ⊂ X × T of the point f (z ) such that the p v ps,n,v v v n following properties hold: v−irr reg u • In fact, U ⊂ X × T and there exists a universal representation ρ : G → v F ps,n,v v v v GL (O(U )) over U . n v v • Let Z ⊂ U denote the Zariski closure of the set V ⊂ U of points correspond- v v v v ing to pairs (ρ ,δ ) such that ρ is trianguline of parameter δ .Thenthe Zariski v v v v tangent space of Z at f (z ) is contained in H (F , ad r | ). v v n v π ,ι G tri,δ n F We can then find an affinoid open neighbourhood U ⊂ X × T of the point z such ps,n n n that the following properties hold: −1 u • U ⊂∩ f (U ) and there exists a universal representation ρ : G → v∈S v F,S p v GL (O(U)) over U . −1 • Let Z = U ∩ (∩ f (Z )).Then Z is a closed analytic subset of U and the v∈S v 1 + Zariski tangent space of Z at the point z is contained in H (F /F , ad r ). n S π ,ι tri,δ 1 + By the main theorem of [NT20], we have H (F , ad r ) = 0, so Proposition π ,ι 2.11 implies that the Zariski tangent space of Z at point z has dimension at most dim W . Let U = E (τ ) ∩ U.Then U is an affinoid open neighbourhood of z in E (τ ).We n n n n n note that if z = γ (π ,χ ) ∈ U ,where χ is a numerically non-critical refinement, then n n n n z ∈ Z (by Lemma 2.8, and the definition of Z ). Such points accumulate at z , implying that every irreducible component of U containing the point z is contained in Z.In particular, Z contains an affinoid open neighbourhood of z in U ,sowehave dim Z ≥ n 48 JAMES NEWTON, JACK A. THORNE dim E (τ ) = dim W . It follows that dim Z = dim W ,that O is a regular local ring, n n n n Z,z and that Z is smooth at the point z . Consequently, U and Z are locally isomorphic at z , E (τ ) is smooth at the point z ,and E (τ ) has a unique irreducible component n n n n n n passing through z . Applying Lemma 2.2, we can also deduce that the unique irreducible component Z of Z containing z is contained in U . −1 Now let U = (σ ◦ i ) (U),and let g = (σ ◦ i )| : U → U.Then U is an n 2 n 2 U −1 admissible open of E (τ ),and g (Z) ⊂ U is a non-empty closed analytic subset. Let 2 2 (π ,χ ) ∈ RA be a pair such that χ satisfies the analogue of property (1) in the theorem. 2 2 2 Arguing again as in the second paragraph of the proof, we see that if the point (σ ◦ i )(γ (π ,χ )) lies in U , then it in fact lies in Z . Since such points accumulate at z , 2 2 2 2 2 −1 we see that g (Z ) contains each irreducible component of U which passes through z (and hence contains an affinoid open neighbourhood of z ). Since Z ⊂ U we deduce −1 that (σ ◦ i ) (E (τ )) contains an affinoid open neighbourhood of z . This completes n 2 n n 2 the proof. Remark 2.25. — Note that assumption (3) on the image of the local Galois rep- resentation ensures that all symmetric powers remain locally irreducible. We need this to apply the results of §2.11.1. The authors expect that, with some effort, this material could be adjusted to allow locally reducible (but globally irreducible) families of Galois representations. We also prove a version of this result in the ordinary case. We first note a well- known consequence of Hida theory: Lemma 2.26. — The Zariski closure of the classical points with ordinary refinements ord E (τ ) ⊂ E (τ ) is a union of connected components of E (τ ) which are finite over W ,and every n n n n n n n ord ord classicalpointof E (τ ) has an ordinary refinement. All points of E (τ ) with dominant locally n n n n algebraic image in W are classical. ord Proof. — We can identify E (τ ) with the generic fibre of the formal spectrum of n n S,ord ∞ Hida–Hecke algebra (a localization of the ring denoted by T (U (p ), O) in [Ger19, §2]), since this is naturally a Zariski closed subspace of X × T in which the classical ps,n n points with ordinary refinements are Zariski dense. We deduce from Hida theory that ord E (τ ) is finite over W and equidimensional of dimension dim W .Moreover, themap n n n n ord ◦ ν : E (τ ) → T factors through the open subspace T ⊂ T classifying unitary charac- n n n n ters of T (F ). n v v∈S On the other hand, we claim that every point of E (τ ) × T is contained in n n ν,T n n ord ord E (τ ) . Assuming this, these (reduced) subspaces of E (τ ) are equal and E (τ ) is n n n n n n an open and closed subspace of E (τ ). The final part of the lemma follows from the n n classicality theorem in Hida theory [Ger19, Lemma 2.25]. ◦ ord It remains to show the claimed inclusion of E (τ ) × T in E (τ ) . Suppose n n ν,T n n n n z is an E-point of E (τ ) × T (extending scalars deals with the general case). The n n ν,T n n SYMMETRIC POWER FUNCTORIALITY 49 p an ∗ character ν(z)δ then appears in the eigenspace J S(U , E) [ψ (z)]. This charac- B B n n n p an ∗ ter therefore also appears in J S(U , E) [ψ (z)] , by Lemma 2.19. Applying [Sor17, n n p ∗ Corollary 6.4] to S(U , O)[ψ (z)] (note that Sorensen’s Jacquet modules are twisted by −1 δ compared to ours), we deduce that the unitary character ν(z) appears in the ordinary p ∗ ord part Ord S(U , O)[ψ (z)]. This shows that z is a point of E (τ ) . B n n Theorem 2.27. — Let (π ,χ ) ∈ RA satisfy tr r = τ ,and let z = γ (π ,χ ) ∈ 2 2 2 π ,ι 2 2 2 2 2 E (τ )(Q ). Suppose that: 2 2 (1) The refinement χ is ordinary. (2) There exists (π ,χ ) ∈ RA such that (σ ◦ i )(z ) = i (z ),where z = γ (π ,χ ). n n n n 2 2 n n n n n n (3) The Zariski closure of r (G ) contains SL . π ,ι F 2 Then each irreducible component C of E (τ ) containing z satisfies (σ ◦ i )(C) ⊂ i (E (τ ) ). 2 2 C 2 n 2 n n n C p p Proof. — Extending E if necessary, we may assume that z ∈ E (τ )(E) and r 2 2 2 π ,ι HT−reg takes values in GL (E).Wedenoteby T ⊂ T the Zariski open subset where for each 2 n v ∈ S and τ ∈ Hom (F , E) the labelled weights wt (δ ) are distinct for i = 1,..., n. p Q v τ n,v,i By Lemma 2.12 and (a global variant of) Lemma 2.13, there is an open affinoid neighbourhood irr reg z = (tr r ,δ ) ∈ U ⊂ X × T n π ,ι n n ps,n n and a universal representation ρ : G → GL (O(U)) such that the induced representa- F,S n tion (ρ )ˆ : G → GL (O(U)ˆ ) with coefficients in the completed local ring at z extends z F n z n n n u u 1−n n to a homomorphism (ρ )ˆ : G + → G (O(U)ˆ ) with ν ◦ (ρ )ˆ = δ . z F ,S n z G z + n n n n F/F Since χ is ordinary, the parameter δ = σ (δ ) is ordinary. We denote by 2 n n,p 2 FL (O ) − → U the rigid space (equipped with a proper map to U ) classifying S -tuples p p (F ) of full flags in O . We consider the closed subspace v v∈S p U ord n Z ⊂ FL (O ) whose points z correspond to flags F which are G -stable (under the ρ -action) for each v F i −1 v ∈ S and the action of G on gr (F ) is given by δ ◦ Art where δ is the parameter p F v z,v,i z HT−reg ord of α(z). Since our parameters lie in T , Z → U is a closed immersion (it is a proper monomorphism). u ord Using the existence of (ρ )ˆ , we can view the tangent space T Z as a sub- z z n n space of H (G , ad r ). By a similar argument to Proposition 2.11, it follows from e.g. F ,S π ,ι [Ger19, Lemma 3.9] (which gives the analogue of Lemma 2.7 in the ordinary case) and ord the main theorem of [NT20] that the map T Z → T W is injective. On the other z r(δ ) n n n ord ord hand, E (τ ) ∩ U , a subspace of Z containing z , is equidimensional of dimension n n n ord ord dim W . We deduce that Z is smooth at z ,and that E (τ ) is locally isomorphic to Z n n n n at z . We complete the proof in the same way as Theorem 2.24. n 50 JAMES NEWTON, JACK A. THORNE We restate Theorem 2.24 and Theorem 2.27 in a way that does not make explicit reference to E . Corollary 2.28. — Let (π ,χ ), (π ,χ ) ∈ RA ,and let z , z ∈ E (Q ) be the corre- 2 2 2 2 2 2 2 2 p sponding points of the eigenvariety. Suppose that one of the following two sets of conditions are satisfied: (1) The refinement χ is numerically non-critical and n-regular. (2) For each v ∈ S , every triangulation of r | is non-critical. The refinement χ is n-regular. p π ,ι G F 2 (3) For each v ∈ S , the Zariski closures of the images of r | and r | contain SL . p π ,ι G π ,ι G 2 2 F F v 2 v (4) There exists an automorphic representation π of G (A +) such that n n F n−1 Sym r r . π ,ι π ,ι 2 n (5) The points z ,z lie on a common irreducible component of E ; 2 2,C 2 p or ord (1 ) The refinement χ is ordinary. ord (2 ) The Zariski closure of the image of r | contains SL . π ,ι G 2 2 F ord (3 ) There exists an automorphic representation π of G (A ) such that n n F n−1 Sym r = r . π ,ι π ,ι 2 n ord (4 ) The points z ,z lie on a common irreducible component of E (this implies that the refinement 2 2,C 2 p χ is also ordinary, by Lemma 2.26). Then there exists an automorphic representation π of G (A ) such that n F n−1 Sym r r  . π ,ι π ,ι ∞ ∞ U Proof. — Choose U ⊂ G (A ) so that (π ) = 0and take τ = tr r .Then n n 2 π ,ι F n 2 n−1 n−2 n−1 (σ ◦ i )(z ) ∈ i (E (τ )(Q )). We claim that setting χ = χ ⊗χ χ ⊗· · · ⊗χ n 2 2 n n n n,v 2,v,2 p 2,v,1 2,v,1 2,v,2 for v ∈ S defines an accessible refinement χ of π .Fix v ∈ S . To temporarily simplify p n n p notation, we write χ = χ ⊗ χ for χ . The representation π is isomorphic to either 1 2 2,v 2,v GL −1/2 2 St (ιχ |·| ) or to an irreducible parabolic induction i ιχ . In the first case, 2 1 T −1 n−1 T −1 n−1 (1−n)/2 ∼ ∼ rec (ι π ) Sym rec (ι π ) Sp (χ |·| ) + = + = n,v 2,v n 1 F F v v and χ is the unique accessible refinement of π . In the second case, n,v n,v n−1 T −1 n−1 T −1 n−1−i i (1−n)/2 −1 ∼ ∼ rec +(ι π ) = Sym rec +(ι π ) = χ χ |·| ◦ Art . n,v 2,v + 1 2 F F F v v i=0 By [Con99, Theorem 3.4.2], this assumption is equivalent to requiring that there is a finite extension of coefficient fields E /Esuch that z , z lie on a common geometrically irreducible component of E  . 2 2,E 2 SYMMETRIC POWER FUNCTORIALITY 51 Note that π is generic (the base change of π to GL (A ) is cuspidal, since r is irre- n,v n n F π ,ι ducible). We can now use the characterisation of generic representations in the Bernstein– Zelevinsky classification [Zel80, Theorem 9.7] and the compatibility with local Lang- lands [Rod82, §4.4]. It follows that no pair of characters in the above direct sum decom- GL position have ratio equal to the norm character, so the parabolic induction i ιχ is n,v irreducible and isomorphic to π . In particular, χ is an accessible refinement of π . n,v n,v n,v Taking the above discussion into account, it is straightforward to see that (σ ◦ i )(z ) is associated to the pair (π ,χ ) ∈ RA . Thus the hypotheses of Theorem 2.24 or 2 2 n n n 2.27 are satisfied, and for any (π ,χ ) as in the statement of the corollary there exists a 2 2 n−1 point z ∈ E (τ )(Q ) such that T = tr Sym r . It remains to show that z ∈ Z ,or n n n,z π ,ι n n p n in other words that z is associated to a classical automorphic representation. In the ordinary case, this follows from Lemma 2.26. In the remaining case, Lemma n−1 2.29 shows that for each v ∈ S , every triangulation of Sym r | is non-critical. It p π ,ι G 2 v follows from Lemma 2.30 that z ∈ Z . Lemma 2.29. — Let v ∈ S ,and let ρ : G → GL (Q ) be a continuous, regular de Rham p v F 2 v p representation such that WD(ρ ) has two distinct characters χ , χ as Jordan–Hölder factors, which v 1 2 satisfy (χ /χ ) = 1 for each i = 1,..., n − 1. Suppose moreover that every triangulation of ρ is 1 2 v n−1 non-critical. Then every triangulation of Sym ρ is non-critical. Proof. — We begin by describing the data of the triangulation of ρ in a bit more detail. Let K = F and let L/K be a Galois extension over which ρ becomes semi-stable. v v Let L be the maximal unramified extension of L/Q . After enlarging E, we can assume 0 p that every embedding of L in Q lands in E, and that ρ is defined over E. The filtered (ϕ, N, Gal(L/K))-module D associated to ρ consists of the following data: (1) A free L ⊗ E-module D of rank 2, equipped with a σ ⊗ 1-semilinear endomor- 0 Q phism ϕ. (2) An L ⊗ E-linear endomorphism N of D satisfying the relation Nϕ = pϕN. 0 Q (3) An L -semilinear, E-linear action of the group Gal(L/K) on D that commutes with the action of both ϕ and N. (4) A decreasing, Gal(L/K)-stable, filtration Fil D of D = D ⊗ L. • L L L For each embedding τ : L → E, we write l ⊂ D = D ⊗ E for the image of the τ τ L L⊗E,τ rank 1 step of the filtration Fil .Wecan defineanactionofthe group W on D by the • K −α(g) formula g · v = (g mod W ) ◦ ϕ ,where α(g) is the power of the absolute arithmetic Frobenius induced by g on the residue field of K. This action preserves the factors of the product decomposition D = D ,where t ranges over embeddings t : L → Eand D = D ⊗ E. Moreover, the isomorphism 0 t L ⊗E,t class of the Weil–Deligne representation D is independent of t. The data of a triangu- lation of ρ is equivalent to the data of a choice of character appearing in some (hence v 52 JAMES NEWTON, JACK A. THORNE n−1 every) D . If N is non-zero on D , then there is a unique N-stable line in Sym D . t t t n−1 Hence there is a unique triangulation of Sym ρ , induced by the unique (non-critical) triangulation of ρ , and it is also non-critical. From now on we assume that N = 0, and we proceed as indicated in [Che11, Example 3.26]. We can choose a basis e , e for D as L ⊗ E-module such that the projection of 1 2 0 Q the vectors e , e to each D is a basis of eigenvectors for the group W . 1 2 t K Having made this choice of basis, each line l is spanned by a linear combination of e , e . Our assumption that every triangulation of ρ is non-critical is equivalent to the 1 2 v requirement that l maybespannedbyavector e + a e ,where a ∈ E for all τ . Indeed, τ 1 τ 2 τ if l is spanned by e for some i, then the triangulation corresponding to the submodule τ i of D spanned by e will fail the condition required for non-criticality with respect to the embedding τ . Having made these normalisations, the condition that every triangulation of n−1 Sym ρ be non-critical is equivalent to the following statement: let I ⊂{0,..., n − 1} i |I| be a subset, and let a x ∈ E[x] be a polynomial, which is equal to (1 + a x) Q(x) i τ i∈I for some polynomial Q(x) ∈ E[x] of degree at most n − 1 −|I|,then Q(x) = 0. Polyno- mials of the latter form correspond to elements of the |I|th step of the Hodge filtration n−1 on Sym D and the statement implies that this Hodge filtration is in general position compared to the filtration induced by every triangulation. Replacing the variable x with −a x, we can assume that a =−1. As in [Che11, Example 3.26], the vanishing of the τ τ |I| successive derivatives at 1 of a x gives a non-degenerate linear system of |I| equa- i∈I tions satisfied by the a , and therefore the a are all zero. Non-degeneracy is checked i i by noticing that the determinant of the linear system is the Vandermonde determinant (i − j). i<j∈I p−irr Lemma 2.30. — Let z ∈ E (τ )(Q ) be a point with i (z) = (tr r ,δ) ∈ X (Q ) × n n n z ps,n p p reg T (Q ). Suppose that δ = j (ν(z)) = δ δ with δ algebraic and δ smooth. Suppose moreover n alg sm alg sm that, for each v ∈ S , every triangulation of r | is non-critical. Then z ∈ Z (in particular, δ is p z G n alg strictly dominant). Proof. — After extending E, we may assume that z ∈ E (τ )(E) and r takes values n n z in GL (E). We note that, since δ is locally algebraic, it follows from property (5)ofthe p−irr −1 reg eigenvariety that the subset of numerically non-critical classical points in i (X ×T ) ps,n n n accumulates at z. It follows from [KPX14, Corollary 6.3.10], applied as in Proposition p−irr −1 reg 2.15, that there is a connected affinoid neighbourhood U of z in i (X × T ),over ps,n n n which there exist representations ρ : G → GL (O(U)) for each v ∈ S with trace equal F n p v v to the restriction to G of the universal pseudocharacter and a non-empty Zariski open and dense subspace V ⊂ U such that for every z ∈ V with i (z ) = (tr r ,δ ), r is trian- n SYMMETRIC POWER FUNCTORIALITY 53 guline of parameter δ . Now we can apply [BHS17, Lemme 2.11] to deduce that δ is alg strictly dominant and r is trianguline of parameter δ. We now argue as in [BHS17, Prop. 3.28] (which is itself similar to the argument of [Che11, Prop. 4.2]). The idea of the argument is to show that failure of classicality would entail the existence of a ‘companion point’ to z, with the same associated Galois representation and a locally algebraic weight which is not strictly dominant. This would contradict [BHS17, Lemma 2.11]. Let η = ν(z)δ = η η ,with η dominant algebraic (since δ is strictly domi- B alg sm alg alg nant) and η smooth. By the construction of E and Lemma 2.19 we have a non-zero sm n space of morphisms p an ∗ 0 = Hom η, J S(U , E) [ψ (z)] . T (F ) B n v n n v∈S Now we use some of the work of Orlik–Strauch [OS15], with notation as in [Bre15, §2]. We denote by g the Q -Lie algebra of G (F ) = GL (F ) and de- n p n n v v∈S v v∈S p p note by b ⊂ g the lower triangular Borel. We define a locally analytic representation n n + n of G (F ) (see [Bre15, Thm. 2.2] for the definition of the functor F ): v∈S p B G G −1 −1 −1 n n F (ηδ ) := F U(g ) ⊗ η ,η δ . n,E sm B U(b ) alg B n n,E n B B n n −1 Note that U(g ) ⊗ η has a unique simple submodule (isomorphic to the n,E U(b ) alg n,E −1 ∨ unique simple quotient of U(g ) ⊗ η ), the algebraic representation V(η ) with n,E alg U(b ) alg n,E −1 −1 lowest (with respect to B )weight η . It follows from [Bre15, Thm. 2.2] that F (ηδ ) alg B G −1 has a locally algebraic quotient isomorphic to V(η ) ⊗ Ind η δ . alg E sm By [Bre15, Thm. 4.3], there is a non-zero space of morphisms n −1 p an ∗ 0 = Hom F (ηδ ), S(U , E) [ψ (z)] . G (F ) n v B n v∈S B n p n G −1 The Jordan–Hölder factors of F (ηδ ) can be described using [Bre15,Thm.2.2] and standard results on the Jordan–Hölder factors of Verma modules (see [Bre15, Cor. 4.6]). Suppose λ ∈ T (E) is an algebraic character. Denote by M the unique simple n λ −1 submodule of the dual Verma module U(g ) ⊗ λ . Then the Jordan–Hölder n,E U(b ) n,E G −1 factors of F (ηδ ) areall of theform JH(w, π) = F M ,π w·η alg with P a parabolic subgroup of G (F ) containing B , π a Jordan–Hölder factor n n n v∈S v −1 of the parabolic induction of η δ from B to the Levi of P ,and w an element of the sm n n We caution the reader that the version of this paper currently available on the arXiv contains a less general result than the published version, to which we appeal here. In particular, it restricts to Galois representations which are known in advance to be crystalline. 54 JAMES NEWTON, JACK A. THORNE Weyl group of (Res G ) × E, acting by the ‘dot action’ on η .Here P is maximal F /Q n Q alg n for M , in the sense of [Bre15,§2]. w·η alg p an ∗ We claim that there cannot be a non-zero morphism JH(w, π)→ S(U ,E) [ψ (z)] for w = 1. Suppose, for a contradiction, that there is such a map. It follows from [Bre16, ∗ ∗ Cor. 3.4] that we have ψ (z) = ψ (z ) (and hence an isomorphism of Galois represen- tations r = r ) for a point z ∈ E (τ )(E) with j (ν(z )) locally algebraic but not strictly z z n n n dominant (its algebraic part matches the algebraic part of j (w · η )). The argument in n alg the first paragraph of this proof, using [BHS17, Lemma 2.11], then gives a contradiction. We deduce from this that any map n −1 p an ∗ F (ηδ ) → S(U , E) [ψ (z)] B n B n −1 factors through the locally algebraic quotient V(η ) ⊗ Ind η δ . Applying [Bre15, alg E sm Thm. 4.3] again, we deduce that we have equalities p an ∗ Hom η, J S(U , E) [ψ (z)] T (F ) B n v n n v∈S −1 p an ∗ = Hom F (ηδ ), S(U , E) [ψ (z)] G (F ) B n n v n v∈S B p n −1 p alg ∗ = Hom F (ηδ ), S(U , E) [ψ (z)] G (F ) B n n v n v∈S B p n p alg ∗ = Hom η, J S(U , E) [ψ (z)] . T (F ) B n v n v∈S n In particular, our point z arises from a non-zero map p alg ∗ η → J S(U , E) [ψ (z)] . n n Applying [Eme06a, Prop. 4.3.6] and computing locally algebraic vectors as in §2.18.1 we see that such a map corresponds to a non-zero map of smooth representations −1/2 p ∗ η → J lim S (U U , E)[ψ (z)] and hence a pair (π ,ι ◦ η δ ) ∈ RA with sm B η p n sm n n n B alg n − →U corresponding classical point equal to z. We therefore have z ∈ Z . 2.31. Application to the eigencurve. — Thus far in this section we have found it conve- nient to phrase our arguments in terms of automorphic forms on unitary groups. Since our intended application will rely on particular properties of the Coleman–Mazur eigen- curve for GL , we now show how to deduce what we need for the eigencurve from what we have done so far. We first introduce the version of the eigencurve that we use. Fix an integer N ≥ 1, × × × prime to p.Let T = Hom(Q /Z × Q , G ); it is the E-rigid space parameterising char- 0 m p p p × 2 × acters χ = χ ⊗ χ of (Q ) such that χ is unramified. Let W = Hom(Z , G ), 0 0,1 0,2 0,1 0 m p p and write r : T → W for the morphism given by r (χ ⊗ χ ) = χ /χ | = 0 0 0 0 0,1 0,2 0,1 0,2 p SYMMETRIC POWER FUNCTORIALITY 55 pN −1 χ | ×.Wedenotethe map r ◦ ν : E → W by κ.Let T = O[{T , S } ] de- 0 0 0 0 l l lpN 0,2 Z 0 note the polynomial ring in unramified Hecke operators at primes not dividing Np. l 0 l 0 Here T and S are the double coset operators for the matrices and .Let l l 01 0 l U (N) = U (N) ⊂ GL (Z) = GL (Z ) be defined by 1 1 l 2 2 l l l ab U (N) = ∈ GL (Z ) : c, d − 1 ∈ NZ . 1 l 2 l l cd The eigencurve is a tuple (E ,ψ ,ν , Z ),where: 0 0 0 0 (1) E is a reduced E-rigid space, equipped with a finite morphism ν : E → T . 0 0 0 0 pN (2) ψ : T → O(E ) is a ring homomorphism, which takes values in the subring 0 0 ≤1 O(E ) of bounded elements. (3) Z ⊂ E (Q ) is a Zariski dense subset which accumulates at itself. 0 0 The following properties are satisfied: (1) E is equidimensional of dimension dim W = 1. For any irreducible component C ⊂ 0 0 E , κ(C) is a Zariski open subset of W . 0 0 (2) Let A denote the set of cuspidal automorphic representations π of GL (A ) such 0 0 2 Q k−2 ∞ U (N) 2 ∨ that (π ) = 0and π has the same infinitesimal character as (Sym C ) for 0,∞ some k ≥ 2 (in which case we say π has weight k), and let RA denote the set of pairs 0 0 (π ,χ ),where π ∈ A and χ = χ ⊗ χ is an accessible refinement of π such 0 0 0 0 0 0,1 0,2 0,p that χ is unramified. As in the unitary case we considered above, for (π ,χ ) ∈ RA 0,1 0 0 0 pN we have a homomorphism ψ : T → Q determined by the action of the Hecke 0 0 −1 ∞ U (N) operators on ι (π ) . There is also a character ν (π ,χ ) ∈ T (Q ) defined 0 0 0 0 in exactly the same way as in the unitary case (2.18.1). An explicit formula appears below (2.31.1). Our assumption that χ is unramified implies that this character 0,1 pN does indeed give a point of T .Now we canlet Z ⊂ Hom(T , Q ) × T (Q ) denote 0 0 0 p p the set of points of the form (ψ ,ν (π ,χ )),where (π ,χ ) ∈ RA . Then the map π 0 0 0 0 0 0 pN ψ × ν : E (Q ) → Hom(T , Q ) × T (Q ) restricts to a bijection Z → Z . 0 0 0 0 0 0 p 0 p p pN −1 (3) For any affinoid open V ⊂ T ,the map T ⊗ O(V ) → O(ν V ) is surjective. 0 0 0 0 0 0 The uniqueness of the tuple (E ,ψ ,ν , Z ) follows from [BC09, Proposition 7.2.8]. Its 0 0 0 0 existence can be proved in various ways. A construction using overconvergent modular forms is given in [Buz07]. We note that in this case, in contrast to the unitary group case, the map RA → Z is bijective – a consequence of the strong multiplicity one theorem. 0 0 We will therefore feel free to speak of the cuspidal automorphic representation π ∈ A 0 0 associated to a point lying in Z . As in the unitary group case, there is a Galois pseu- docharacter t : G → O(E ) with the property that for z ∈ Z associated to (π ,χ ), Q,Np 0 0 0 0 t = tr r . z π ,ι Let us describe explicitly the link with more classical language. We are using the normalisations of [DI95,§11].If (π ,χ ) ∈ A , then there is a cuspidal holomorphic 0 0 0 n r modular form f = q + a (f )q of level  (Np ) (for some r ≥ 1) which is an eigen- n 1 n≥2 form for all the Hecke operators T (l  Np)and U , in their classical normalisations, and l p 56 JAMES NEWTON, JACK A. THORNE we have the formulae GL (Z ) 2 l −1/2 a (f ) = eigenvalue of T on π , p a (f ) = ιχ (p). l l p 0,1 0,l Note that the central character of π is a Hecke character ψ with ψ | (z) = 0 π π R 0 0 >0 2−k 2−k z .So ψ | = ι(χ χ ) is a finite order twist of the character z → |z| .Tocon- π 0,1 0,2 0 Q vince the reader that these formulae are correct, we observe that if π is a normalised 0,l GL GL (Z ) 2 2 l 1/2 induction i μ ⊗ μ , then the eigenvalue of T on π is l (μ (l) + μ (l)) [DI95, 1 2 l 1 2 B 0,l (11.2.4)], whilst considering the central character shows that μ (l)μ (l) has (complex) 1 2 k−2 absolute value l . This is compatible with the fact that a (f ) is a sum of numbers with (k−1)/2 absolute values l . We can define a map s : E (Q ) → R, called the slope, by composing the projection p 0 to T (Q ) with the map χ → v χ .Notethatif (π ,χ ⊗ χ ) ∈ RA we have 0 p 0 0,1 0,2 0 −1/2 t 0 1 2−k (2.31.1) ν (π ,χ ) = t χ (t )χ (t ) 0 0 0 0,1 1 0,2 2 0 t 2 so the slope map sends (π ,χ ⊗ χ ) ∈ RA to 1/2 + v (χ (p)). 0 0,1 0,2 0 p 0,1 In particular, at a point z ∈ Z corresponding to a classical holomorphic modular 0 0 −1 form f , s(z ) equals the p-adic valuation of ι a (f ). Note that the corresponding pair 0 p (π ,χ ) is numerically non-critical exactly when s(z )< k − 1 and ordinary exactly when 0 0 0 s(z ) = 0. The classicality criterion of Coleman [Col96, Col97] shows that a point z ∈ k−2 × E (Q ) with κ(z) restricting to t → t on a finite index subgroup of Z and s(z)< k − 1 is necessarily in Z . pc Let Z ⊂ Z denote the subset of points corresponding to pairs (π, χ) where π 0 p is not a twist of the Steinberg representation (pc stands for potentially crystalline). We now pc pc define a ‘twin’ map τ : Z → Z .Let (π ,χ ) be the pair corresponding to a point 0 0 0 0 pc z ∈ Z .Write χ = χ ⊗ χ . Since π is not a twist of the Steinberg representation, 0 0,1 0,2 0,p GL 2 × π equals the full normalised induction i ιχ , which is irreducible. Let ψ : Q \A → 0,p 0 B Q Z be the unique finite order character which is unramified outside p and such that −1 × × × × ψ | = χ | . Then the character χ ψ | is unramified and χ = χ ψ | ⊗χ ψ | Z 0,2 0,2 Z 0,2 Z 0,1 Z p p p p pc is an accessible refinement of the twist π ⊗ ιψ . We therefore have a point τ(z) ∈ Z corresponding to the pair τ(π ,χ ) = (π ⊗ ιψ, χ ), that we call the twin of z.Notethat 0 0 0 τ = 1 and if π is unramified then τ(z) is the usual companion point appearing in the 0,p Gouvea–Mazur construction of the infinite fern [Maz97, §18]. The following lemma is an easy computation. pc Lemma 2.32. — Let z ∈ Z ,and let z = τ(z).Let s, s denote the slopes of these two points, and κ(z), κ(z ) ∈ W (Q ) their images in weight space. Then s + s = k − 1 and v (κ(z)(1 + 0 p q) − 1) = v (κ(z )(1 + q) − 1),where q = p if p is odd and q = 4 if p is even. Here is the main result of §2. SYMMETRIC POWER FUNCTORIALITY 57 Theorem 2.33. — Let (π ,χ ), (π ,χ ) ∈ RA and let n ≥ 2.Let z , z ∈ Z be the 0 0 0 0 0 0 0 0 corresponding points. Suppose that one of following two sets of conditions are satisfied: (1) The refinement χ is numerically non-critical and n-regular. (2) The refinement χ is n-regular. (3) The Zariski closures of r (G ) and r (G ) contain SL . π ,ι Q π ,ι Q 2 0 p p n−1 (4) Sym r is automorphic; π ,ι or ord (1 ) The refinement χ is ordinary. ord (2 ) π and π are not CM (so the Zariski closures of r (G ) and r (G ) contain SL ). 0 π ,ι Q π ,ι Q 2 0 0 n−1 ord (3 ) Sym r is automorphic. π ,ι n−1 If the points z ,z lie on a common irreducible component of E , then Sym r is also automor- 0 0,C π ,ι 0 p phic. Proof. — We want to apply Corollary 2.28. We first need to specify suitable data F, S, G ,U .Let F /Q be an abelian CM extension satisfying the following conditions: 2 2 • Each prime dividing Np splits in F . •[(F ) : Q] is even. • The extension F /(F ) is everywhere unramified. After extending E, we may assume that z , z ∈ E (E) and that there is an irre- 0 0 ducible component C ⊂ E containing the points z , z .Moreover, by thefirstpartof 0 0 [Con99, Theorem 3.4.2], we may assume that C is geometrically irreducible. Let W de- note the unique connected component of W containing κ(C). We can find a character χ : G → O(W) such that the determinant of the universal pseudocharacter over C −1 equals χ (χ is the product of a finite order p-unramified character and the compo- sition of with the universal character Z → O(W )). By Lemma 2.34,wecan finda finite étale morphism η : W → W and a character ψ : G  → O(W) , unramified almost c  + everywhere, such that ψψ = χ | , and such that for each place v|p of (F ) , there is a place v|v of F such that ψ | is unramified. We now let F/Q be a soluble, Galois, CM extension, containing F ,suchthat: • Each prime dividing Np splits in F. • The extension F/F is everywhere unramified. • The character ψ | is unramified away from p. Let S denote the set of places of F dividing Np. Fix as usual a set of factori- sations v = v v for v ∈ S. Fix the unitary group G as in our standard assumptions (§1). Then for each v ∈ S, thereisanisomorphism ι : G (F ) → GL (F ).Welet v 2 2 v U = U ⊂ G(A ) be an open subgroup with the property that U is hyperspecial 2 2,v 2,v v F maximal compact if v ∈ S, and U is the pre-image under ι of the subgroup U (N) of 2,v v 1 l GL (Q ) if v ∈ S has residue characteristic l (in which case F = Q ). 2 l v l 58 JAMES NEWTON, JACK A. THORNE × 2 We recall that T = Hom((F ) , G ).Let T = T × W, with κ ˜ : T → W 2 m 0 W v∈S v 0 u u u the projection map. If χ = χ ⊗ χ ∈ T (T ) denotes the universal character, then the 0 0 1 2 tuple of characters u −1 u −1 ((χ ◦ N · ψ | ◦ Art ) ⊗ (χ ◦ N · ψ | ◦ Art )) F /Q G F F /Q G F v∈S 1 v p F v 2 v p F v p v v in T (T ) determines a morphism b : T → T . Writing X for the rigid space of 2- 2 p 2 0,ps dimensional pseudocharacters of G , unramified outside Np, there is a base change morphism b : X × T → X × T covering b and sending a pair (τ, μ) to (τ | ⊗ 0,ps 2,ps 2 p G −1 ψ , b (μ)). This leads to a diagram of rigid spaces κ( ˜ μ) X × T X × T . 0,ps 2,ps 2 i 2 E × T E 0 T 2 Let C = C × W = C × T . Then the morphism C → C is finite étale. In particular, W T each irreducible component of C maps surjectively to C . Choose E /E so that the ir- reducible components of C  are geometrically irreducible (we apply [Con99,Theorem 3.4.2] again). Since C is geometrically irreducible, we still know that each irreducible component of C maps surjectively to C . Consequently, we can find points z , z of C E E 1 E lifting z , z and lying on a common geometrically irreducible component C of C .We 0 E −1 next wish to show that b ◦ i(C ) ⊂ i (E ), or equivalently that (b ◦ i) (i (E )) contains 2 2,E 2 2,E C . Since i is a closed immersion, it suffices to show that Z , the pre-image of Z in C , 2 0 satisfies b ◦ i(Z ) ⊂ i (E (Q )) (the accumulation property of Z in C is inherited from 2 2 0 p 0 the corresponding property of the subset Z ∩ C ⊂ C ). To see this, we note that for any (π, χ) ∈ Z , with lift z ∈ Z , the base change π (which exists since F/Q is soluble) is still cuspidal. Indeed, if not then r | would F π,ι G be reducible, implying that π was automorphically induced from a quadratic imaginary subfield K/Q of F/Q. This is a contradiction, since we chose F so that all primes dividing −1 Np split in F, yet K must be ramified at at least one such prime. The descent of π ⊗ ιψ to G (which exists, by [Lab11, Théorème 5.4]) gives (together with b (ν (π, χ))) a point 2 p 0 of E which equals the image of z under the map b ◦ i. We can now complete the proof. Indeed, the points b ◦ i(z ), b ◦ i(z ) lie on a common geometrically irreducible component of E  , by construction. They satisfy 2,E the conditions of Corollary 2.28 (in particular, Example 2.10 shows that our assump- tion on r (G ) in the non-ordinary case implies that all of its triangulations are non- π ,ι Q critical). We therefore conclude the existence of an automorphic representation π of n−1 G (A +) such that Sym r | r  . Our assumptions (cf. Lemma 3.5(2)) imply n F π ,ι G π ,ι 0 n SYMMETRIC POWER FUNCTORIALITY 59 n−1 that Sym r | is irreducible, and therefore that the base change of π is a cuspi- π ,ι G F n dal automorphic representation of GL (A ). Soluble descent for GL now implies that n F n n−1 Sym r is itself automorphic. π ,ι Lemma 2.34. — Let F be a CM number field. Suppose that each p-adic place of F splits in F,and let S be a set of p-adic places of F such that S S is the set of all p-adic places of F. p p Let W be a connected E-rigid space, and let χ : G → O(W) be a continuous character (continuity defined by demanding that the induced characters with values in O(U) are continuous for all affinoid admissible opens U ⊂ W,asin[Buz04, §2]), unramified almost everywhere. Then we can find a finite étale morphism η : W → W and a continuous character ψ : G → O(W) such that the following properties hold: (1) ψ is unramified almost everywhere. (2) For each v ∈ S , ψ | is unramified. p G c ∗ (3) ψψ = η (χ)| . Proof. — We first claim that we can find a finite étale morphism W → Wand a continuous character λ : G → O(W ) with the following properties: • λ is unramified almost everywhere. • χ | λλ has finite order. × × Indeed, let L : O → O(W) be defined by the formula w|p −1 L((u ) ) = χ | ◦ Art (u ). w w F v v∈S Then L is continuous, and trivial on a finite index subgroup of O (it is trivial on the norm 1 units in O ). It follows from Chevalley’s theorem [Che51, Théorème 1] that p p there is a compact open subgroup U of O such that L is trivial on (U ) := wp F p × × U × O ∩ O . F F w|p Note that if H is a product of a finite abelian group and a finite Z -module, and H ⊂ H is a finite index subgroup, then the natural map Hom(H, G ) → Hom(H , G ) m m of rigid spaces is finite étale. Maps of rigid spaces W → Hom(H, G ) biject with contin- uous characters H → O(W) . ∞,× It follows that we may extend L to a continuous character L : F \A → O(W ) , for some finite étale morphism W → W. Indeed, we apply the preceding re- mark with H the quotient of O by the closure of (U ) and H the quotient of w|p F ∞,× × p F \A by the closure of the image of U (cf. the discussion in [Buz04,§2]). We define λ by λ ◦ Art = L . The character χ | λλ hasfiniteorder becauseit F G factors through the Galois group of an abelian extension of F which is unramified at all but finitely many places and unramified at the primes above p. 60 JAMES NEWTON, JACK A. THORNE Replacing W by a connected component, we may suppose that W is connected, in which case the character χ | λλ is constant (i.e. pulled back from a morphism W → Sp E , for a finite extension E /E). Applying [BLGGT14, Lemma A.2.5], we may find a finite extension E /E and a continuous character ϕ : G → (E ) of finite order such c c  −1 that χ | λλ = ϕϕ . The proof is complete on taking W = W and ψ = ϕλ . We conclude this section with a lemma that will be used in §8. It uses the existence of the universal pseudocharacter t over E . Lemma 2.35. — Fix n ≥ 1,and let Z ⊂ E denote the set of points x satisfying one of the following conditions: (1) t is absolutely reducible; (2) t = tr ρ for an absolutely irreducible representation ρ : G → GL (Q ), and the Zariski x x x Q 2 closure of the image of ρ does not contain SL . x 2 (3) There exists a prime l|N such that t | = χ + χ for characters χ : G → Q such that x G 1 2 i Q Q l (χ /χ ) = 1 for some i = 1,..., n − 1. 1 2 Then Z is Zariski closed. Proof. — The discussion in [Che14, §4.2] shows that the locus where t is absolutely reducible is Zariski closed. If ρ : G → GL (Q ) is irreducible, then the Zariski closure x Q 2 of the image of ρ contains SL if and only if Sym ρ is irreducible. Indeed, the Zariski x 2 x closure of the image of ρ contains SL if and only if the Zariski closure G of the image x 2 x of the associated projective representation Proj ρ : G → PGL (Q ) is PGL .There are x Q 2 2 two possibilities for the group G , which is a (possibly disconnected) reductive group: the first is that it is finite, hence either dihedral or conjugate to one of A ,S ,or A .Inany of 4 4 5 these cases Sym ρ is reducible. The next is that G has a non-trivial identity component, x x which therefore contains a maximal torus of PGL . The only possibilities are therefore that either G equals the normaliser of this maximal torus (in which case Sym ρ is again x x reducible) or that G = PGL (in which case Sym ρ is irreducible). x 2 x This shows that the set Z of points satisfying conditions (1) or (2) of the Lemma is Zariski closed. Finally, if l|Nand i = 1,..., n − 1, let Z denote the set of points x 3,l,i such that t | = χ + χ for some characters χ , χ such that (χ /χ ) = 1. It remains to x G 1 2 1 2 1 2 show that Z is Zariski closed. Its complement is the set of points such that either t | 3,l,i x G is absolutely irreducible, or t | is absolutely reducible and there exists g ∈ G such x G Q Q l that the discriminant of the characteristic polynomial of g under the pseudocharacter t is non-zero. This is a union of Zariski open sets. 3. Ping pong In this section we use the rigid analytic results of §2 to prove the following theorem. We recall that we say that an automorphic representation π of GL (A ) has “weight k” 2 Q SYMMETRIC POWER FUNCTORIALITY 61 for an integer k ≥ 2if π has the same infinitesimal character as the dual of the algebraic k−2 representation Sym C . Theorem 3.1. — Fix an integer n ≥ 2.Let π be a cuspidal automorphic representation of n−1 GL (A ) which is everywhere unramified and of weight k, for some k ≥ 2. Suppose that Sym r 2 Q π ,ι is automorphic for some (equivalently, any) prime p and isomorphism ι : Q → C. Then for any every- n−1 where unramified cuspidal automorphic representation π of GL (A ) of weight l ≥ 2, Sym r is 2 Q π,ι automorphic. To prove Theorem 3.1, we will use the properties of the eigencurve E ,asdefined in §2.31. More precisely, we henceforth let p = 2, N = 1, and let E denote the eigencurve defined with respect to this particular choice of parameters. We fix an isomorphism ι : Q → C. E is supported on the connected component W ⊂ W defined by χ(−1) = 1. 0 0 2 0 u × We write χ : Z → O(W ) for the universal character. We have the following explicit result of Buzzard and Kilford on the geometry of the morphism κ : E → W and the slope map s : E (Q ) → R: Theorem 3.2. — Let w ∈ O(W ) denote the function χ (5) − 1.Then: (1) w restricts to an isomorphism between W and the open unit disc {|w| < 1}. −1 (2) Let W (b) ⊂ W denote the open subset where |8| < |w| < 1,and let E (b) = κ (W (b)). 0 0 0 Then there is a decomposition E (b) = X of E (b) as a countable disjoint union of admissible 0 i 0 i=1 open subspaces such that for each i ≥ 1, κ| : X → W (b) is an isomorphism. X i 0 −1 (3) For each i = 1, 2,...,the map s ◦κ| : W (b)(Q ) → X (Q ) → R equals the map iv ◦w. 0 i p X p p Proof. — This is almost the main theorem of [BK05], except that here we are using E denotes the full eigencurve used in the cuspidal version of the eigencurve. However, if non−ord ord [BK05], then there is a decomposition E = E E as a union of open and closed ord subspaces. This follows from the fact that the ordinary locus E in the eigencurve can also be constructed using Hida theory (see [Pil13, §6]), so is finite over W . Since E is 0 1 ord separated the open immersion E "→ E is therefore also finite, hence a closed immer- ord sion. In our particular case (p = 2, N = 1),wehave E W (the unique ordinary 1 0 non−ord family is the family of Eisenstein series) and therefore E = E , giving the statement we have here. See Lemma 7.4 of the (longer) arXiv version of [BC05] for an alternative argument. We note as well that in our normalisation, the trivial character in W corresponds to forms of weight 2, whereas in the notation of [BK05], the character x corresponds to forms of weight 2. However, this renormalisation does not change the region W (b). Before giving the proof of Theorem 3.1, we record some useful lemmas. pc Lemma 3.3. — Let z ∈ Z ∩ E (b),and suppose z ∈ X .Let z = τ(z) be the twin of z. 0 i Then z ∈ X  , where i satisfies the relation i + i = (k − 1)/v (w(z)). i p 62 JAMES NEWTON, JACK A. THORNE Proof. — By Lemma 2.32, z lies in E (b),soin X  for a unique integer i ≥ 1. 0 i Writing s, s for the slopes of these two points and k for their weights, we have k − 1 = s + s = iv (w(z)) + i v (w(z)),hence i + i = (k − 1)/v (w(z)). p p p As a sanity check, we observe that in the context of the proof of Lemma 3.3, (k − k−2 1)/v (w(z)) is always an integer. Indeed, κ(z) satisfies κ(z)(5) = 5 ζ for some k ≥ 2 p 2 and m ≥ 0. This weight lies in W (b) if and only if either k is odd, or k is even and m ≥ 1. 1−m If m ≥ 1, then v (w(z)) = 2 .If m = 0and k is odd, then v (w(z)) = 2. In either case p p we see that (k − 1)/v (w(z)) is an integer. Lemma 3.4. — Let π be an everywhere unramified cuspidal automorphic representation of GL (A ) of weight k ≥ 2. Every accessible refinement of π is numerically non-critical and n-regular 2 Q for every n ≥ 2 (recall that we have fixed p = 2 and these notions refer to the local factor at 2, π ). Proof. — Numerical non-criticality of every refinement is immediate from the fact that there are no cusp forms of level 1 that are ordinary at 2. For regularity, if we fix 1/2 1/2 a refinement χ = χ ⊗ χ then α = p ιχ (p) and β = p ιχ (p) arethe rootsofthe 1 2 1 2 2 k−1 polynomial X − a X + 2 ,with a the T -eigenvalue of the level 1 weight k normalised 2 2 2 eigenform f associated to π . We need to show that α/β is not a root of unity. Suppose α/β = ζ is a root of unity. If we fix ι : Q C, the semisimplified mod 5 Galois representation r arises up to twist from a level 1 eigenform of weight ≤ 6 (i.e. the f ,ι −1 3 5 level 1 Eisenstein series of weight 4 or 6). This shows that ι (ζ ) ≡ 2 or 2 mod m , 5 Z and therefore ζ is the product of a 5-power root of unity and ±i (since 2 has order 4 in × −1 F ). Applying a similar argument at the prime 7 with ι : Q C, we see that ι (ζ ) ≡ 5 7 7 3 5 7 2 , 2 or 2 mod m and therefore ζ is the product of a 7-power root of unity and a cube root of unity. This gives the desired contradiction. (We thank Fred Diamond for pointing out this argument to us, and thank an anonymous referee for explaining how to avoid using Hatada’s congruence which appeared in the first version of this argument.) Lemma 3.5. — Let π be a cuspidal automorphic representation of GL (A ) of weight k ≥ 2. 2 Q We temporarily let p be an arbitrary prime. Then: (1) r | is reducible if and only if π is ι-ordinary. π,ι G (2) Suppose either that r | is irreducible and π admits a 3-regular refinement, or that k > 2 and π,ι G p r is not potentially crystalline. Then the Zariski closure of r (G ) (in GL /Q ) contains SL . π,ι π,ι Q 2 2 (3) Suppose again that p = 2, and that π is everywhere unramified. Then the Zariski closure of r (G ) contains SL . π,ι Q 2 Proof. — For the first part, ι-ordinarity implies reducibility by local-global com- patibility at p,asin[Tho15, Theorem 2.4]. For the converse, if r | is reducible, π,ι G then its Jordan–Hölder factors are de Rham characters of G and therefore have the p SYMMETRIC POWER FUNCTORIALITY 63 −k form ψ ,where the ψ are Z -valued characters with finite order restriction to in- i i ertia and the k are the Hodge–Tate weights (we can assume k = 0and k = k − 1 i 1 2 in our situation). The Weil representation part of WD(r | ) is therefore equal to π,ι G 1−k −1 T −1 ψ ⊕ ψ |·| ◦ Art . Since WD(r | ) = rec (ι π ), π is a subquotient of the nor- 1 2 π,ι G p p Q Q p p p GL 1/2 3/2−k malised induction i (ιψ ◦ Art |·| ⊗ ιψ ◦ Art |·| ). It follows from [Tho15, 1 Q 2 Q B p p Lemma 2.3] that π is ι-ordinary. For the second part, we note that r | is irreducible. Indeed, if r | is re- π,ι G π,ι G Q Q p p ducible then the first part of the lemma shows that π is ι-ordinary, and [Tho15, Lemma GL 2 −1 2.3] implies that π is a subquotient of i χ ⊗ χ with v ι (χ /χ (p)) = 1 − k.If p 1 2 p 1 2 r is not potentially crystalline then WD(r | ) has N = 0 and local-global compat- π,ι π,ι G GL 2 ±1 ibility implies that if π is a subquotient of i χ ⊗ χ ,then χ /χ =| · | . This is a p 1 2 1 2 contradiction if k > 2. Thus r | is irreducible and the Zariski closure H of its image is a reductive π,ι G subgroup of GL . Let T be a maximal torus of H. Since r is Hodge–Tate regular, T is 2 π,ι regular in GL (i.e. its centralizer is a maximal torus of GL )by[Sen73, Theorem 1]. If H 2 2 does not contain SL , then it is contained in the normaliser of a maximal torus of GL and 2 2 r | is induced from a character of an index two subgroup. This forces WD(r | ) π,ι G π,ι G Q Q p p 2 2 to likewise be induced, so any refinement χ = χ ⊗ χ of π satisfies χ = χ , and this 1 2 p 1 2 Weil–Deligne representation has N = 0. This is a contradiction, since we are assuming either that there exists a 3-regular refinement or that N is non-zero. For the third part, we have already observed (see the proof of Theorem 3.2)that there are no cusp forms of level 1 that are ordinary at 2, so r | is irreducible. Suppose π,ι G that r | is induced. Then WD(r | ) = Ind ψ for some quadratic extension π,ι G π,ι G Q Q W 2 2 K/Q and character ψ : W → Q . Since this Weil–Deligne representation must be 2 K unramified, we see that K and ψ are both unramified, and therefore that ψ extends to acharacter ψ : W → Q such that WD(r | ) = ψ ⊕ (ψ ⊗ δ ). In particular, Q π,ι G K/Q 2 Q 2 the T -eigenvalue (which equals the trace of Frobenius in this representation) is 0, but, as shown in the proof of Lemma 3.4, this is impossible. Lemma 3.6. — Let i ≥ 1 be an integer, and let z ∈ Z ∩ X (Q ) be the point corresponding 0 i to a pair (π, χ). Suppose that χ is n-regular. Let z ∈ Z ∩ X (Q ) be any other point, corresponding 0 i n−1 n−1 to a pair (π ,χ ), with χ n-regular. If Sym r is automorphic, then Sym r  is also. π,ι π ,ι Proof. — We can assume n ≥ 3. We apply Theorem 2.33 (note that X i,C W (b) is irreducible): since there are no ordinary points in E , χ is numerically non- 0 C 0 critical and Lemma 3.5 implies that the Zariski closures of r (G ) and r  (G ) con- π,ι Q π ,ι Q p p tain SL (the first part of the lemma shows that we are in the locally irreducible and 3-regular case of the second part of the lemma).  64 JAMES NEWTON, JACK A. THORNE Lemma 3.7. — Let π be a cuspidal, everywhere unramified automorphic representation of GL (A ) of weight k ≥ 2,and let χ be a choice of accessible refinement. Then there exists an in- 2 Q teger m ≥ 1 such that for any integer m ≥ m , we can find a cuspidal automorphic representation π of π π GL (A ) satisfying the following conditions: 2 Q (1) π is unramified outside 2. (2) π admits two accessible refinements of distinct slopes (in particular these refinements are n-regular for every n ≥ 2). (3) There is an accessible refinement χ of π such that (π, χ) and (π ,χ ) define points z, z on the same irreducible component of E . 0,C (4) κ(z ) ∈ W (b). In particular, z ∈ X for some i ≥ 1 (notation as in the statement of Theorem 0 i 3.2). pc (5) Set z = τ(z ),and let (π ,χ ) ∈ Z be the associated pair. Then z ∈ X m . 2 −1 (6) The automorphy of any one of the three representations n−1 n−1 n−1 Sym r , Sym r , Sym r π,ι π ,ι π ,ι implies automorphy of all three. Proof. — We use Theorem 3.2. Extending E if necessary, we may assume that z ∈ E (E) and every irreducible component of E containing z is geometrically irre- 0 0 ducible. Fix one of these irreducible components and fix i such that this irreducible component contains X (such an i exists, because every irreducible component of E has i 0 Zariski open image in W , hence intersects E (b) and therefore contains a non-empty union of irreducible components of E (b)). We define m to be least integer m ≥ 1satis- 0 π π fying the inequality m +1 (2i + 2 − 3)/2 > 2i. k −2 Given m ≥ m , we choose z ∈ X (Q ) to be the point such that κ(z )(5) = 5 ,where π i m+1 k = 2i + 2 − 1. Then (k − 2)/2 > 2i = s(z ). By Coleman’s classicality criterion, pc pc z ∈ Z .If z was not in Z , its slope would be (k − 2)/2. So z ∈ Z and if (π ,χ ) 0 0 denotes the corresponding pair, then the two accessible refinements of π have distinct slopes (2i and k − 1 − 2i). Let z = τ(z ) denote the twin point, and (π ,χ ) the corresponding pair. Then z lies on X ,where i = (k − 1)/2 − i = 2 − 1, by Lemma 3.3.We’re done:the first 5 properties of (π ,χ ) follow by construction, whilst the 6th follows from Theorem 2.33, Lemma 3.4 and Lemma 3.5. We can now complete the proof of Theorem 3.1. Bouncing from z to its twin point z reminded the authors of a game of ping pong, whence the section title. Earlier versions of the argument involved longer rallies! SYMMETRIC POWER FUNCTORIALITY 65 Proof of Theorem 3.1.— Let π , π be everywhere unramified cuspidal automor- phic representations of GL (A ) of weights k , k ≥ 2, respectively. Define integers m , 2 Q 0 π 0 0 m as in Lemma 3.7 and fix an integer m ≥ max(m , m ). Combining Lemma 3.7 and π π π 0 0 n−1 Lemma 3.6 (applied with i = 2 − 1) we see that the automorphy of Sym r implies π ,ι n−1 that of Sym r . Since π was arbitrary, this completes the proof. π ,ι Part II: Raising the level Most of the remainder of this paper (§§4 – 7) is devoted to the proof of Theorem E from the introduction, namely the existence for each n ≥ 2 of a single regular algebraic, cuspidal, everywhere unramified automorphic representation π of GL (A ) such that 2 Q n−1 Sym π exists. As a guide to what follows, we now give an expanded sketch of the proof of this theorem. Fix, for the sake of argument, a regular algebraic, cuspidal, everywhere unrami- fied automorphic representation π of GL (A ). We will try to establish the existence of 2 Q n−1 n−1 Sym π by proving the automorphy of one of the Galois representations Sym r π,ι associated to a choice of prime p and isomorphism ι : Q → C, using an automor- phy lifting theorem. First, if K/Q is an imaginary quadratic extension then we can find (using e.g. [BLGGT14, Lemma A.2.5]) a (de Rham) character ω : G → Q c −1 n−1 n−1 such that ωω = (det r ) . Then the representation ρ = ω ⊗ Sym r| satisfies π,ι G c ∨ 1−n ρ ρ ⊗ , so has the potential to be associated to a RACSDC automorphic repre- sentation of GL (A ). This means we can use an automorphy lifting theorem adapted to n K n−1 such automorphic representations. (The automorphy of ρ will imply that of Sym r π,ι by quadratic descent.) We need to select π and ι so that the residual representation ρ is automorphic. For “most” ι (say, for all but finitely many primes p) the image of r will contain a conjugate π,ι n−1 of SL (F ) and Sym ρ will be irreducible, and it is not clear how to proceed. We 2 p therefore want to avoid this generic case. Here we choose π and ι so that there is an isomorphism r Ind χ for some imaginary quadratic extension K/Q and character π,ι χ : G → F . Then there is an isomorphism n n−i c i−1 ρ = ⊕ ωχ (χ ) . i=1 In particular, this residual representation is highly reducible, being a sum of n characters. Most automorphy lifting theorems in the literature require the residual representation to be irreducible; we will apply [ANT20, Theorem 1.1], an automorphy lifting theorem that does not have this requirement, but that does have some other stringent conditions. These conditions include the requirement that there exist a RACSDC automorphic rep- (A ) such that r ρ , and satisfying the following: resentation  of GL = n K ,ι •  is ι-ordinary (and so is π ). 66 JAMES NEWTON, JACK A. THORNE • There exists a prime l = p and a place v|l of K such that both π and  are l v twists of the Steinberg representation (of GL (Q ) and GL (K ), respectively). 2 l n v It is easy to arrange that the first requirement be satisfied, by choosing p to be a prime which splits in K. The second is more difficult. First it requires that π is ramified at l , whereas we have to this point asked for π to be everywhere unramified. We will thus first n−1 find a ramified π for which Sym π exists, and eventually remove the primes of ramifi- cation using the l -adic analytic continuation of functoriality results proved in the first part of the paper. The main problem is then to find a  verifying the residual automorphy of ρ such that  is a twist of the Steinberg representation. This is what will occupy us in §§4 – 6 below. The above argument is then laid out carefully in §7 in order to finally prove Theorem E. Here is how we get our hands on . By choosing an appropriate lift of the charac- × × ter χ , we can choose characters X ,..., X : K \A → C such that  = X ··· X 1 n 0 1 n is a regular algebraic and conjugate self-dual (although not cuspidal!) automorphic rep- resentation of GL (A ) whose associated residual representation is ρ . If G is a definite n K unitary group in n variables associated to the extension K/Q, quasi-split at finite places, then we can hope that  transfers to an automorphic representation of G(A ). There is 0 Q a slight wrinkle here in that such a group G does not exist if n is even, and even in the case that n is odd there is a potential obstruction to the existence of this transfer given, at least conjecturally, by Arthur’s multiplicity formula. Both of these obstacles can be avoided by replacing Q with a suitable soluble totally real extension F/Q.Inorder to avoidin- troducing additional notation in this sketch, we pretend they can be dealt with already in the case F = Q. (Actually, we will find it convenient to take  to be the box sum of two cuspidal automorphic representations of GL (A ) and GL (A ), respectively. This 2 K n−2 K means that the final form of the proof of Theorem E will be a kind of induction on n.) We thus find ourselves with an automorphic representation  of G(A ),whose 0 Q base change (in the sense of Theorem 1.2)is  . Say for the sake of argument that l splits in K, so that we can identify G(Q ) with GL (K ). If we can find another automorphic l n v representation  of G(A ), congruent modulo p to  and such that  is a twist of 1 Q 0 1,v the Steinberg representation, then we will have solved our problem. We can therefore now focus on this problem of level-raising for the definite unitary group G. There are differing approaches to this problem in the literature. First there is the purely automorphic approach, pioneered by Ribet for GL (A ) [Rib84]. Some gener- 2 Q alisations to higher rank groups of this statement do exist (see for example [Tho14a]), but nothing that is applicable in the level of generality considered here. Then there is the purely Galois theoretic approach, based on the powerful automorphy lifting theorems which are now available for Galois representations in arbitrary rank (see for example [Gee11]). We can not directly apply such results here because the only automorphy lifting theorems applicable in the residually reducible case (namely those of [ANT20]) require the existence of at least one place at which the starting automorphic representation is sufficiently non-degenerate. SYMMETRIC POWER FUNCTORIALITY 67 We solve the problem here by combining aspects of these approaches. A similar combination of techniques is used in the paper [CT17]: the idea is to first use an auto- morphic technique to replace  with a representation  such that  is so ramified 0 0,v that, in conjunction with other conditions in place,  is forced to be stable (i.e. its base change is cuspidal). (The possibility of doing this is the reason for choosing  to in fact be a box sum of two cuspidal factors, as mentioned above.) This situation is reflected in the deformation theory, where one finds that (in the big ordinary Galois deformation ring) the locus of reducible deformations is small enough that something like the tech- niques of [Gee11] can be applied to construct an automorphic lift of ρ with the required local properties. These Galois theoretic arguments are carried out in §§5, 6. What remains to be explained then is the automorphic level-raising technique de- veloped in §4. The approach to creating congruences here is based on types. We recall (using the language of [BK98]) that if s is an inertial equivalence class of G(Q ) (i.e. a supercuspidal representation of a Levi factor of G(Q ), up to unramified twist), then an s-type is a pair (U,τ), where U is an open compact subgroup of G(Q ) and τ is an irre- ducible finite-dimensional representation of U such that for each irreducible admissible representation σ of G(Q ), the supercuspidal support of σ is in class s if and only if σ | v l v v U contains τ . It can sometimes happen that two inertial equivalence classes s, s admit types −1 (U,τ) and (U ,τ ) with the property that U = U , the reduction modulo p τ of ι τ −1 is irreducible, and the reduction modulo p τ of ι τ contains τ as a Jordan–Hölder factor. This situation might be called a congruence of types. If this is the case then the theory of algebraic modular forms implies that any automorphic representation  of G(A ) such that  is of type s is congruent to another  such that  is of type s .The Q l existence of such global congruences is explained in [Vig01, §3]. It gives an efficient way to construct congruences between automorphic representations ,  such that  , are in different inertial equivalence classes, although it is not usually possible to change the Levi subgroup underlying the inertial equivalence class. Since G(Q ) GL (K ) and l n v the initial representation  is certainly not supercuspidal at l , it is not immediately clear how to use this. We therefore instead introduce an auxiliary imaginary quadratic extension E/Q in which l is inert, as well as an associated definite unitary group G , and carry out the first step of the automorphic part of the level-raising argument using algebraic modular forms on G . The importance of the group G is that there are conjugate self-dual irreducible admissible representations of GL (E ) which are not supercuspidal, but for which the 3 l associated L-packets of representations of U (Q ) contain supercuspidal elements. For 3 l carefully chosen local data, we can find use the method of types to find congruences to supercuspidal representations of U (Q ) whose base change to GL (E ) is supercuspidal. 3 l 3 l We have already constructed such congruences of types in §1.8. In terms of automorphic representations of GL (A ), this will allow us to change the Levi subgroup underlying n E 68 JAMES NEWTON, JACK A. THORNE the inertial equivalence class at l from the maximal torus of GL (E ) to the group GL × n l 3 n−3 GL . This will be enough for our intended application. 4. Raising the level – automorphic forms Let n = 2k + 1 ≥ 3 be an odd integer, and let F, p, S, G = G etc. be as in our standard assumptions (see §1). Suppose given cuspidal, conjugate self-dual automorphic representations π of GL (A ) and π of GL (A ) with the following properties: 2 2 F n−2 n−2 F (1) π = π  π is regular algebraic. 2 n−2 (2−n)/2 Consequently, π |·| and π are regular algebraic and the representations ρ = 2 n−2 r (2−n)/2 , ρ = r −1 are defined. We set ρ = r = ρ ⊕ ρ .Moreover, π and π |·| ,ι n−2 π |·| ,ι π,ι 2 n−2 2 2 n−2 π are both tempered, cf. the remark after Corollary 1.3. n−2 (2) π is ι-ordinary. (3) We are given disjoint, non-empty sets T ,T ,T of places of F with the following 1 2 3 properties: (1) For all v ∈ T = T ∪ T ∪ T , v ∈ Sand q is odd. The representation π is unram- 1 2 3 v ified outside S ∪ T. If v is a place of F lying above a place in T then, as in §§1.8, × × 1.17, we write ω( v) : k( v) → C for the unique quadratic character. th (2) For each v ∈ T , v is inert in F, q mod p is a primitive 6 root of unity, and 1 v the characteristic of k(v) is greater than n. There are characters χ ,χ ,χ ,..., v v,0 v,1 × × χ : F → C such that χ , χ are unramified and for each i = 1,..., 2k − 2, v,2k−2 v v,0 2k−2 ∼ ∼ χ | × = ω( v).Wehave π St (χ ) and π  χ . = = v,i 2, v 2 v n−2, v v,i O i=0 c nd (3) For each v ∈ T , v splits v = v v in F, q mod p is a primitive 2 root of unity, 2 v π | contains λ( v,  ) (for some order p character  as in §1.17,with 2, v GL (O ) v v 2 F n = 2), and π | contains ω( v) ◦ det. Thus π satisfies the equivalent 1 n−2, v GL (O ) v n−2 F conditions of Proposition 1.20,and π | contains λ( v,  , n). v q v c th (4) For each v ∈ T , v splits v = v v in F, q mod p is a primitive (n − 2) root of unity, 3 v π | contains λ( v,  ) (for some order p character  as in §1.17,with n−2, v GL (O ) v v n−2 F n = n − 2), and π | contains ω( v) ◦ det. Thus π satisfies the equivalent 1 2, v GL (O ) v 2 F conditions of Proposition 1.20,and π | contains λ( v,  , n). v q v Let T ={ v | v ∈ T}.Wefixfor each v ∈ T acharacter θ : C(k(v)) → C of order p 1 v (notation as in Proposition 1.12). In the rest of this section, we will prove the following theorem. + + Theorem 4.1. — With hypotheses as above, let L /F be a totally real S ∪ T-split quadratic extension, and let L = L F. Then there exists a RACSDC automorphic representation  of GL (A ) n L with the following properties: (1)  is ι-ordinary, and unramified at any place not dividing S ∪ T. (2) r = r | . ,ι π,ι G (3) For each place v ∈ T ,  | contains the representation λ( v, θ , n)| (thus satisfying the 1,L v r v| r v + v equivalent conditions of Proposition 1.19). SYMMETRIC POWER FUNCTORIALITY 69 (4) For each place v ∈ T ∪ T ,  | contains the representation λ( v,  , n) (thus satisfying 2,L 3,L v q v| v F the equivalent conditions of Proposition 1.20 with n = 2 if v ∈ T and n = n − 2 if v ∈ T ). 2 2,L 2 3,L Remark 4.2. — The places v ∈ T ∪ T play a role in ensuring that  is cuspidal 2 3 (using Lemma 4.5 below). Our set-up is adapted to the proof of Proposition 7.4, which uses an induction on the dimension to construct automorphic representations with an unramified twist of Steinberg local factor which are congruent to some very special odd- dimensional symmetric powers. We begin with two important observations. + θ Lemma 4.3. — Let v be a finite place of F which is inert in F. Then π ∈ A (GL (F )) . v n v + Proof. — The representation π is tempered because both π and π are tem- v 2, v n−2, v pered. By the main theorem of [BC11], r extends to a homomorphism r : G + → π,ι F 1−n n G (Q ) such that ν ◦ r = δ . Restricting to W and twisting by an appropri- n + p F/F v ate character, we see that the Langlands parameter of π extends to a parameter W × SL → G. Remark 4.4. — A consequence of this lemma is that for v ∈ T the character χ is 1 v non-trivial quadratic and the character χ is trivial. v,0 Lemma 4.5. — Suppose given a partition n = n +··· + n and cuspidal, conjugate self- 1 r dual automorphic representations π ,...,π of GL (A ) such that π = π  ···  π is regular n F 1 r i 1 r algebraic. Suppose moreover that the following conditions are satisfied: (1) There is an isomorphism r = r . π ,ι π,ι (2) If v ∈ T ∪ T then π | contains λ( v,  , n). 2 3 q v v v Then one of the following two statements holds: (1) We have r = 1,n = n, and so π is cuspidal. (2) After re-ordering we have r = 2,n = n − 2,n = 2.If v ∈ T then π | contains 1 2 2 GL (O ) 1, v n−2 F ω( v) ◦ det and π | contains λ( v,  ), while if v ∈ T then π | contains GL (O ) v 3 GL (O ) 2 F n−2 F 2, v 1, v v v λ( v,  ) and π | contains ω( v) ◦ det. We have isomorphisms of semisimplified residual v GL (O ) 2, v 2 F ∼ ∼ representations r −1 ρ and r (2−n)/2 ρ . = = π |·| ,ι π |·| ,ι n−2 2 Proof. — Before beginning the proof, we observe that the representations ρ , ρ 2 n−2 have the following properties: ss ss • If v ∈ T ,then ρ | is unramified and ρ | is unramified after twisting by 2 G n−2 G F F v v a ramified quadratic character. (The character  has order p.) ss • If v ∈ T ,then ρ | is unramified after twisting by a ramified quadratic char- 2 G ss acter and ρ | is unramified. n−2 G v 70 JAMES NEWTON, JACK A. THORNE We can suppose without loss of generality that r ≥ 2. Fix places v ∈ T , v ∈ T . 2 2 3 3 By Proposition 1.20, we can assume after relabelling that π | contains GL (O ) 1, v n F 3 1 v λ( v , , n ) (and in particular, n ≥ n − 2),inwhichcase (π  ···  π )| 3 v 1 1 GL (O ) 3 2, v r, v n−n F 3 3 1 v contains ω( v ) ◦ det. There is an isomorphism r  =⊕ r  (n −n)/2 ρ ⊕ ρ , π ,ι i i=1 π |·| ,ι 2 n−2 hence r ss ss ss ⊕ r  (n −n)/2 | = ρ | ⊕ ρ | . i=1 π |·| ,ι G 2 G n−2 G F F F v v v 3 3 3 r ss Since ⊕ r  (n −n)/2 | contains no unramified subrepresentation, we conclude that π |·| ,ι i=2 G i v r  (n −n)/2 contains ρ as a subrepresentation. π |·| ,ι n−2 We now look at the place v . There are two possibilities for the representation π : either π | contains λ( v , , n ), or it contains ω( v ) ◦ det. We claim GL (O ) 2 v 1 2 1, v 1, v n F 2 2 2 1 v that the first possibility does not occur. Indeed, in this case arguing as above shows that r ss ⊕ r  (n −n)/2 | contains no unramified subrepresentation, and therefore that ρ is a π |·| ,ι 2 i=2 G subrepresentation of r  (n −n)/2 . This forces n = n and r = 1, a contradiction. Therefore 1 1 π |·| ,ι we must have r = 2, n = 2, and π | contains λ( v , ). Since v , v were 2 GL (O ) 2 v 2 3 2 F 2 2, v 2 v arbitrary, this completes the proof. We now commence the proof of the theorem. Let U = U ⊂ G(A ) be an v + v F open compact subgroup with the following properties: ι (U ) v v • For each v ∈ S, π = 0. • If v ∈ S ∪ T, then U = G(O ). v F Iw (c,c),ord −1 v • If v ∈ S ,then U = ι Iw (c, c) for some c ≥ 1such that π = 0 (notation p v v v v as in [Ger19, §5.1]) and U contains no non-trivial torsion elements (note this implies that U is sufficiently small). −1 • If v ∈ T then U = ι p (notation as in §1.8). 1 v v −1 • If v ∈ T ∪ T then U = ι q (notation as in §1.17, defined with n = 2if 2 3 v v 1 v ∈ T and n = n − 2if v ∈ T ). 2 1 3 We define τ =⊗ τ(v, n) ⊗ λ( v,  , n),where τ(v, n), λ( v,  , n) are the rep- g v∈T v∈T ∪T v v 1 2 3 resentations of p , q defined in §1.8,§1.17 respectively. Thus τ is an irreducible C[U ]- v v g T module, which we view as a C[U]-module by projection to the T-component. Similarly we define λ =⊗ λ(v, θ , n) ⊗ λ( v,  , n). Fixing a sufficiently large coefficient g v∈T v v∈T ∪T v 1 2 3 −1 −1 field, we can choose O-lattices τ and λ in ι τ and ι λ , respectively. g g g g + + If L /F is an S ∪ T-split totally real quadratic extension, then we define an open compact subgroup U = U ⊂ G(A ) and representations τ , λ by the same L L,v g,L g,L recipe (where we now replace the sets S, T by their lifts S ,T to L ). i L i,L SYMMETRIC POWER FUNCTORIALITY 71 + + Proposition 4.6. — Let L /F be an S ∪ T-split totally real quadratic extension and let L = L F. Then either there exists an automorphic representation σ of G(A ) with the following properties: (1) π is the base change of σ (cf. Theorem 1.2); (2) For each place v ∈ T, σ = 0; (3) σ | contains τ . T U g or there exists an automorphic representation σ of G(A ) with the following properties: (1) Let π denote the base change of π with respect to the quadratic extension L/F. Then π is the base L L change of σ ; L,v (2) For each place v ∈ T , σ = 0; (3) σ | contains τ . T U g,L L L,T Proof. — By [Lab11, Théorème 5.1], there is an identity G M H (4.6.1)T (f ) = ι(G, H)T (f ), disc disc ∞ ∞ for any f = f ⊗ f ∈ C (G(A +)) such that f is a pseudocoefficient of discrete series. ∞ F ∞ Here the sum on the right-hand side is over representatives for equivalence classes of endoscopic data for G, represented here by the associated endoscopic group H (recall that we have fixed representative endoscopic triples in §1). The coefficients ι(G, H) are M H givenin[Lab11, Proposition 4.11], while the expression T (f ) is given in [Lab11, disc Proposition 3.4] as a formula M −1 L H (4.6.2) (| det(s − 1 | a /a H)||W (L)|) tr I ( π )(f ), L M Q H H L 0 M M π ∈ (L ) disc s L∈L /W s∈W (L) reg where (summarizing the notation of op. cit.): • M is a twisted space on a Levi of Res GL ,asin§1.5; F/F n 0 H • L is the set of standard Levi subgroups of M ; M L • W (L) is the quotient by the Weyl group W of the set of elements s in reg H H M M the twisted Weyl group W = W  θ which normalise L and such that det(s − 1 | a /a H) = 0, where a denotes the Lie algebra of the maximal Q-split L M ? subtorus of the centre of a reductive group. •  (L ) is the set of isomorphism classes of irreducible representations of the disc s twisted space L (A +) which appear as subrepresentations of the discrete spec- s F trum of L. L H • I ( π )(f ) is a certain intertwining operator, with Q a parabolic subgroup with Levi L. We fix our choice of f so that it only has non-zero traces on representations of G(F ) whose infinitesimal character is related, by twisted base change, to that of π . The argu- 0 M ment of [Shi11, Proposition 4.8] then shows that for each L ∈ L /W , there is at most 72 JAMES NEWTON, JACK A. THORNE one element s ∈ W (L) for which the corresponding summand in (4.6.2) can be non- reg zero (and a representative for s can be chosen which acts as conjugate inverse transpose on each simple factor of L). Using Proposition 1.7, linear independence of characters, the description of the discrete spectrum of general linear groups [MW89], and the Jacquet– Shalika theorem [JS81], we can combine (4.6.1)and (4.6.2) to obtain a refined identity −1 ∼ U ×U n−2 2 (4.6.3) m(σ )σ (f ) = π(f ) + (π ⊗ (π ⊗ μ ◦ det)) (f ) , i i n−2 2 where: • The sum on the left-hand side is over the finitely many automorphic represen- tations σ of G(A +) which are unramified at all places below which π is unram- i F ified, have infinitesimal character related to that of π by twisted base change, and which are related to π by (either unramified or split) base change at places v ∈ T of F , each occurring with its multiplicity m(σ ). 1 i • The twisted traces on the right-hand side are Whittaker-normalised. (These two terms arise from H = U ,L = Res GL × GL and H = U × U ,L = n F/F n−2 2 n−2 2 M , respectively. The same argument as in [Lab11, Proposition 3.7] shows that Arthur’s normalisation of the twisted trace, implicit in the term I (?) of (4.6.2), agrees with the Whittaker normalisation on the corresponding terms.) We remark that the representations π , π are tempered and that the representa- 2 n−2 tions σ (i.e. prime to T -part) are isomorphic. If v ∈ T ,thenwecan find(com- 1 1 bining Proposition 1.11 for U × U and e.g. [Hir04, Proposition 4.6]) a finite set n−2 2 {λ } of irreducible admissible representations of G(F ) and scalars d ∈ C such that v,i v,i −1 ∼ U ×U n−2 2 (π ⊗ (π ⊗ μ ◦ det)) (f ) = d λ (f ).ByProposition 1.6,Proposition n−2,v 2,v v,i v,i v v v i 1.7 and Proposition 1.11, we therefore have an identity: m(σ )σ (f ) = (v, U ,ϕ ) c τ(f ) i i,T T n U τ v 1 1 n i v|∞ v∈T τ ∈(π ) 1 v + (v, U × U ,ϕ ) d λ (f ) , n−2 2 U ×U v,i v,i v n−2 2 v|∞ v∈T Choose for each v ∈ T a representation τ ∈ (π ) such that τ | contains τ(v, n) (this 1 v v v U is possible by Corollary 1.13 and Proposition 1.14). We can assume that for each v ∈ T , λ = τ (possibly with d = 0). We conclude that there is at most one automorphic v,1 v v,1 representation σ of G(A ) with the following properties: • σ is unramified outside S ∪ T, and is related to π by split or unramified base change at all places v ∈ T ; • If v ∈ T ,then σ τ . 1 v v SYMMETRIC POWER FUNCTORIALITY 73 The representation σ occurs with multiplicity m(σ ) = (v, U ,ϕ ) c n U τ n v v|∞ v∈T + (v, U × U ,ϕ ) d . n−2 2 U ×U v,1 n−2 2 v|∞ v∈T We note that the numbers c are all non-zero, by Proposition 1.11.If m(σ ) is non-zero, then we’re done (we are in the first case in the statement of the proposition). Otherwise, c = d , which we now assume. v∈T τ v∈T v,1 1 v 1 + + + In this case, let L /F be a totally real quadratic S ∪ T-split extension, let L = L F, and let π denote the base change of π with respect to the quadratic extension L/F. If v ∈ T ,let τ = τ . Then repeating the same argument shows that there is at most 1,L v v| + one automorphic representation σ of G(A ) with the following properties: • σ is unramified outside S ∪ T , and is related to π by split or unramified base L L L change at all places v ∈ T ; 1,L • If v ∈ T ,then σ τ . 1,L v v Using the remark after Proposition 1.6, we see that the representation σ occurs with multiplicity ⎛ ⎞ ⎝ ⎠ m(σ ) = c + d = c . τ v,1 v τ v∈T v∈T v∈T 1,L 1,L 1 This is non-zero, so we’re done in this case also (and we are in the second case of the proposition). We now show how to complete the proof of Theorem 4.1, assuming first that we are in the first case of Proposition 4.6.Welet σ be the automorphic representation of Hom(F,Q ) G(A +) whose existence is asserted by Proposition 4.6.Let λ ∈ (Z ) be such ord −1 that σ contributes to S (U,ι τ ) under the isomorphism of Lemma 1.25.Let T ⊂ ord −1 End (S (U,ι τ )) be the commutative O-subalgebra generated by unramified Hecke O g j c + operators T at split places v = ww ∈ SofF ,and let m ⊂ T be the maximal ideal determined by σ . ord Then S (U, τ ˚ ⊗ k) is non-zero, by Lemma 1.24, hence (using the exactness of g O m ord ord S (U, −) as a functor on k[U]-modules, together with Proposition 1.15)S (U, λ ⊗ g O λ λ ord k) = 0, hence S (U, λ ) = 0. Applying Lemma 1.25 once again, we conclude the m g m existence of an automorphic representation  of G (A ) with the following properties: n F • r r . ,ι π,ι 74 JAMES NEWTON, JACK A. THORNE •  | contains λ . T U g •  is ι-ordinary and is unramified outside S ∪ T. + + Let  denote the base change of ,let L /F be a quadratic totally real extension as in the statement of Theorem 4.1,and let  denote base change of  with respect to the extension L/F. We claim that  satisfies the requirements of Theorem 4.1.The only points left to check are that  is cuspidal and that if v ∈ T then  satisfies L 1,L L,v condition (3) in the statement of Theorem 4.1. In fact, it is enough to show that  is cuspidal and that if v ∈ T then  | contains λ( v, θ , n)| . We first show that  is cus- 1 v r v r v v pidal. If  is not cuspidal, then Lemma 4.5 shows that  =    where  , n−2 2 n−2 2 are cuspidal, conjugate self-dual automorphic representations of GL (A ),GL (A ), n−2 F 2 F respectively. Arguing as in proof of Proposition 4.6, we obtain an identity −1 ∼ U ×U n−2 2 (4.6.4) m( ) (f ) = (f ) + ( ⊗ ( ⊗ μ ◦ det)) (f ) , i i n−2 2 where the sum on the left-hand side is over the finitely many automorphic representations of G(A +) which are unramified at all places below which  is unramified, have in- i F finitesimal character related to that of  by twisted base change, and which are related to  by (either unramified or split) base change at places v ∈ T of F . v,∞ Fix v ∈ T , and consider a test function of the form f = f ⊗ f ⊗ f ,where: 1 v ∞ • f is a coefficient for  . ∞ ∞ • f is the test function denoted φ in the statement of Proposition 1.16. v,∞ • f is the characteristic function of an open compact subgroup of G(A ). • (f ) = 0. Then  (f ) is non-negative for any i, and the left-hand side of (4.6.4)isnon-zero. We −1 ∼ U ×U n−2 2 conclude that at least one of the terms (f ) and ( ⊗ ( ⊗ μ ◦ det)) (f ) n−2 2 is non-zero. In either case Proposition 1.16 implies that the cuspidal support of  ,and therefore  , contains a supercuspidal representation # of GL (F ) such that the n−2, v 3 v T −1 semisimple residual representation attached to rec (ι #) is unramified. This contra- ss dicts Lemma 4.5, which implies that r −1 | is the sum of an unramified character |·| ,ι n−2 G and the twist of an unramified representation by a quadratic ramified character. Therefore  is cuspidal, and a similar argument now gives an identity (4.6.5) m( ) (f ) = (f ). i i With the same choice of test function we have (f ) = 0, so another application of Propo- sition 1.16 shows that  has the required property. This completes the proof of Theorem 4.1, assuming that the first case of Proposition 4.6 holds. If the second case holds, the ar- gument is very similar, except that there is no need to replace  by its base change with respect to a quadratic extension L/F. In either case, this completes the proof. SYMMETRIC POWER FUNCTORIALITY 75 5. A finiteness result for Galois deformation rings In this section we prove that certain Galois deformation rings are finite over the Iwasawa algebra (Theorem 5.2), and use this to give a criterion for a given deformation to have an irreducible specialization with useful properties (Theorem 5.7). These technical results form the basis for the arguments in §6, where we will apply our criterion to the Galois representation valued over a big ordinary Hecke algebra. The novelty of the results proved in this section is that we assume that the residual representation is reducible (in fact, to simplify the exposition we assume that this repre- sentation is a sum of characters). The main tools are the automorphy lifting theorems proved in [ANT20] and the idea of potential automorphy, for which we use [BLGGT14] as a reference. The notation and definitions we use for Galois deformation theory in the ordinary case are summarized in [ANT20, §3], and we refer to that paper in particu- lar for the notion of local and global deformation problem, and the definitions of the particular local deformation problems used below. Before getting stuck into the details, we record a useful lemma. If  is a profinite group, k is a field with the discrete topology, and ρ :  → GL (k) is a continuous rep- resentation, we say that ρ is primitive if it is not isomorphic to a representation of the form Ind σ for some finite index proper closed subgroup  ⊂  and representation σ :  → GL (k). This condition appears as a hypothesis in the automorphy lifting n/[: ] theorem proved in [ANT20]. Lemma 5.1. — Suppose that ρ = χ ⊕···⊕χ , for some continuous characters χ :  → k 1 n i such that for each i = j, χ /χ has order greater than n. Then ρ is primitive. i j Proof. — Suppose that there is an isomorphism ρ Ind σ . Then Frobenius reci- procity implies that σ contains each character χ | . These n characters are distinct: if [: ] χ | = χ | ,then (χ /χ ) = 1, which would contradict our assumption that χ /χ i j i j i j has order greater than n if i = j.Thus σ must have dimension at least n, implying that =  . It follows that ρ is primitive. Now let n ≥ 2and letF,S, p be as in our standard assumptions (see §1), and let E ⊂ Q be a coefficient field. We recall the definition of the Iwasawa algebra .If ab n ab v ∈ S , then we write  = O(I (p)) ,where I (p) denotes the inertia subgroup of p v F F v v the Galois group of the maximal abelian pro-p extension of F . We set  = ⊗  , v v∈S v the completed tensor product being over O.For each v ∈ S and i = 1,..., n there is i ab × a universal character ψ : I (p) →  . At times we will need to introduce Iwasawa v F v algebras also for extension fields F /F and for representations of degree n = n,inwhich case we will write e.g.  for the corresponding Iwasawa algebra, dropping a subscript F ,n when either F = For n = n. Let μ : G + → O be a continuous character which is de Rham and such that F ,S μ(c ) =−1 for each place v|∞ of F . Fix an integer n ≥ 2, and suppose given characters v 76 JAMES NEWTON, JACK A. THORNE × n χ ,..., χ : G → k such that for each i = 1,..., n, χ χ = μ| . We set ρ =⊕ χ ; F,S G 1 n i i F,S i=1 i then ρ extends to a homomorphism r : G → G (k) such that ν ◦ r = μ, by setting F ,S n G r(c) = (1 , 1)j ∈ G (k). We suppose that for each v ∈ S , r| is trivial. n n p G Let  be a set of finite places of F split in F and disjoint from S, and  aliftof to F. If for each v ∈ , q ≡ 1mod p and r| is trivial, then we can define the global v G deformation problem +   St S = (F/F , S ∪ , S ∪ , , r,μ, {R } ∪{R } ∪{R } ). v∈S v∈S−S v∈ v p v p v (For the convenience of the reader, we summarize the notation from [ANT20,§3].Thus the local lifting ring R represents the functor of ordinary, variable weight liftings; R v v St the functor of all liftings; and R the functor of Steinberg liftings.) If r is Schur, in the sense of [CHT08, Definition 2.1.6], then the corresponding global deformation functor is represented by an object R ∈ C .If  is empty, then we write simply S = S . S  ∅ Theorem 5.2. — Suppose that the following conditions are satisfied: (1) p > 2n. (2) For each 1 ≤ i < j ≤ n, χ /χ | has order greater than 2n. (In particular, ris Schur.) i j F(ζ ) (3) [F(ζ ) : F]= p − 1. (4)  is non-empty. Then R is a finite -algebra. Proof. — We will compare R with a deformation ring for Galois representations to G .First,fixaplace v of F prime to S ∪ , lying above a rational prime q > 2n which 2n q splits in F. After possibly enlarging k, we can find a character ψ : G → k satisfying the following conditions: −1 1−2n • ψψ = μ| . • For each v ∈ S ∪ , ψ | is unramified. • q divides the order of ψ/ψ (I ). Using the formulae in [BLGGT14, §1.1], we can write down a character 1−2n −1 + + (ψ, μ δ ) : G → G (k), F/F F 1 the tensor product representation 1−2n −1 + + r ⊗ (ψ, μ δ ) : G → G (k), F/F F n 1−2n which has multiplier , and the representations −1 1−2n r = I(r ⊗ (ψ, μ δ +)) : G + → GSp (k) 1 F/F F 2n and r = (r ) : G + → G (k). 2 1 G F 2n F SYMMETRIC POWER FUNCTORIALITY 77 These representations have the following properties: 1−2n • The multiplier character of r equals . 1−2n • The multiplier character ν ◦ r equals . G 2 2n • The representations r | and r | are both conjugate in GL (k) to ρ ⊗ ψ ⊕ 1 G 2 G 2n F F ρ ⊗ ψ . Let ρ = r | . We observe that the following conditions are satisfied: 2 2 G ker ad ρ • ζ ∈ F and F ⊂ F (ζ ). p p • ρ is primitive. • The irreducible constituents of ρ | occur with multiplicity 1. 2 F(ζ ) ker ad ρ Indeed, the condition F ⊂ F (ζ ) holds because [F(ζ ) : F]= p − 1. We have F ⊂ p p −1 F({χ /χ } ,ψ/ψ , {χ χ μ } ) = M, say, and c acts on Gal(M/F) as −1. It follows that i=j i,j i j i j F(ζ ) ∩ M has degree at most 2 over F, showing that ζ ∈ M. To see that ρ is primitive, p p it is enough (by Lemma 5.1) to show that χ /χ has order greater than 2n if i = j and that i j (χ ψ)/(χ ψ ) has order greater than 2n for any i, j . These properties hold by hypothesis in the first case and since q > 2n in the second. Finally, the constituents of ρ are, with c c c c multiplicity, χ ⊗ ψ,..., χ ⊗ ψ , χ ⊗ ψ ,..., χ ⊗ ψ . Our hypotheses include the 1 n 1 n condition that χ ⊗ ψ | = χ ⊗ ψ | if i = j.If χ ⊗ ψ | = χ ⊗ ψ | then G G G G i j i F(ζ ) F(ζ ) F(ζ ) j F(ζ ) p p p p ψ/ψ | is trivial, a contradiction. Fix an isomorphism ι : Q → C.By[BLGGT14, Theorem 3.1.2], we can find a + + Galois totally real extension L /F and a regular algebraic, self-dual, cuspidal, automor- phic representation π of GL (A +), with the following properties: 2n L • Let L = FL .Then L/F is linearly disjoint from the extension of F(ζ ) cut out ker ad ρ | L + by ρ | . In particular, [L(ζ ) : L]= p − 1, ζ ∈ L ,and L ⊂ L (ζ ). G p p p 2 F(ζ ) • Thereisanisomorphism r r | . π,ι 1 G • π is ι-ordinary. More precisely, π is of weight 0 and for each place v|p of L , π is an unramified twist of the Steinberg representation. • ρ | is primitive. 2 L • The irreducible constituents of ρ | occur with multiplicity 1. 2 L(ζ ) • For each place v of L lying above a place of , π is an unramified twist of the Steinberg representation. More precisely, [BLGGT14, Theorem 3.1.2] guarantees the existence of L satisfying the first condition and an essentially self-dual π satisfying all the remaining conditions (except possibly the last one). The last paragraph of the proof notes that the π constructed is in fact self-dual and π is an unramified twist of the Steinberg representation for each place + + v|p of L . We can moreover ensure that π is Steinberg at the places of L lying above by inserting the condition “t(P)< 0 for all places v| of L ” in the first list of conditions on [BLGGT14, p. 549]. 78 JAMES NEWTON, JACK A. THORNE + + After possibly adjoining another soluble totally real extension of F to L ,wecan assume that the following further conditions are satisfied: • ρ | is unramified at those finite places not dividing S ∪  . 2 G L L • Each place of L at which π is ramified is split over L . • For each place v ∈ S ∪  , ψ | is trivial. L L G • For each place v ∈ S , [L : Q ] > 2n(2n − 1)/2 + 1and ρ | is trivial. p,L v p G Here π denotes the base change of π . It is a RACSDC automorphic representation of GL (A ). By construction, then, π satisfies the hypotheses of [ANT20, Theorem 6.2]. 2n L L Therefore, if we define the global deformation problem + 1−2n S = (L/L , S ∪  , S ∪  , , r | , , L L L L L,2n 2 G St {R } ∪{R } ∪{R } }), v∈S v∈S −S v∈ v p,L v L p,L v L then R  is a finite  -algebra. (Here we have written  to distinguish from  = S L,2n L,2n used above.) F,n We now need to relate the rings R and R . In fact, it will be enough to construct S S a commutative diagram R /( ) R /( ) S S /( )  /( ) L,2n F,n where the top horizontal morphism is finite. We first specify the map  /( ) → L,2n /( ). It is the map that for each place w ∈ S lying above a place v of F classi- F,n p,L fies the tuple of characters v v v −1 v −1 (ψ | ,...,ψ | ,ψ | ,...,ψ | ). I I 1 L n L n I 1 I w w L L w w This endows the ring R /( ) with the structure of  -algebra. To give a map S L,2n R /( ) → R /( ), we must give a lifting of r | over R /( ) which is of type S . S S 2 G S To this end, let r denote a representative of the universal deformation (of r)to R /( ), −1 1−2n ∧ and let r = I(r ⊗ (ψ, μ δ +)) | (notation as in [BLGGT14, §1.1]). Then r F/F G G + F L is a lift of r and r | is the restriction of r| ⊗ ψ ⊕ r | ⊗ ψ to G . We need to check 2 G G G L L F F St that for each v ∈ S , r | is of type R ;and that foreach v ∈  , r | is of type R . p,L G L G L v L v v v These statements can be reduced to a universal local computation. It follows that r is of type S , and so determines a morphism R /( ) → R /( ). To complete the proof, it will be enough to show that this is a finite ring map. SYMMETRIC POWER FUNCTORIALITY 79 We can enlarge the above commutative diagram to a diagram R /( ) R /( ) S S Q /( )⊗  /( ) Q /( )⊗  /( ), t | O L,2n t O F,n 2 G where Q is the complete Noetherian local O-algebra classifying pseudocharacters of t | 2 G G lifting the restriction of t = tr r | to G ,Q is defined similarly with respect L,S ∪ 2 2 G L t L L F to the pseudocharacter t = tr r| of G ,and themap Q /( ) → Q /( ) is the G F,S∪ t | t F 2 G one classifying the natural transformation sending a pseudocharacter t lifting t to the pseudocharacter (t| ⊗ ψ) + (t | ⊗ ψ ). We deduce from [Tho15, Proposition 3.29] G G L L that the vertical arrows are finite ring maps. The map  →  is also finite, so it’s L,2n F,n enough finally to show that the map Q /( ) → Q /( ) is finite. This map can in t | t 2 G turn be written as a composite Q /( ) → Q /( ) → Q /( ), t | t | t 2 G 2 G L F where the first map classifies restriction of pseudocharacters from G to G . Since r | is F L 2 G multiplicity free, [ANT20, Proposition 2.5] (specifically, the uniqueness of the expression as a sum of pseudocharacters) implies that the second map is in fact surjective. We finally just need to show that the first map is finite, and this follows from the following general lemma. Lemma 5.3. — Let  be a topologically finitely generated profinite group, let  be a closed subgroup of finite index, and let t be a pseudocharacter of  with coefficients in k of some dimension n. Let Q be the complete Noetherian local O-algebra classifying lifts of t. Then the map Q → Q t t| t classifying restriction to  is a finite ring map. Proof. — It suffices to show that Q /(m ) is Artinian. If not, we can find a prime t Q t| ideal p of this ring of dimension 1; let A be its residue ring (which is a k-algebra), and let t be the induced pseudocharacter of  with coefficients in A. Let N =[ : ].If γ ∈ N! then γ ∈ . If we factor the characteristic polynomial of X − γ under t as (X − α ) i=1 for some elements α in the algebraic closure of Frac A, then the characteristic polyno- N! N! mial of γ under t,namely (X − α ),liesin k[X] and equals the characteristic i=1 i N! polynomial of γ under t. This shows that the elements α are in fact algebraic over k, and thus (using [Che14, Corollary 1.14]) that t can be defined over k, and must in fact equal t. This is a contradiction. n Hom(F,Q ) Corollary 5.4. — With hypotheses as in Theorem 5.2,fix λ ∈ (Z ) such that for each i = 1,..., nand τ ∈ Hom(F, Q ), we have λ =−λ . Suppose further that for each τ c,i τ,n+1−i p 80 JAMES NEWTON, JACK A. THORNE v ∈ S , [F : Q ] > n(n − 1)/2 + 1. Then there exists a homomorphism r : G → G (Z ) lifting p v p F ,S n p rsuch that r| is ordinary of weight λ,inthe senseof[Tho15, Definition 2.5]. F,S Proof. — We observe that [Tho15, Proposition 3.9, Proposition 3.14] show that for each minimal prime Q ⊂ R ,dimR /Q = dim ; consequently, there is a minimal S S prime Q of  and a finite injective algebra morphism /Q → R /Q. The corollary follows on choosing any prime of R /Q[1/p] lying above a maximal ideal of /Q [1/p] associated to the weight λ as in [Ger19, Definition 2.24]. Corollary 5.5. — With hypotheses (1) – (3) of Theorem 5.2, choose a place v ∈ S of F split in F and a lift v to F such that q ≡ 1 mod p and r| is trivial. Consider the quotient 0 v G 0 F A = R /( , {tr r (Frob ) − n} ). S S i=1,...,n ∅ ∅ v Then A is a finite -algebra. Consequently, dim R /( ) ≤ n[F : Q]+ n. Proof. — It suffices to verify that the quotient of R where the characteristic poly- 1−i nomial of Frob equals (X − q ) is a quotient of R . This in turn means check- v S 0 i=1 v {v } ur ing that the quotient A of the local unramified lifting ring R where the characteristic 0 v 1−i St polynomial of Frobenius equals (X − q ) is a quotient of R . Since A is flat over v v 0 i=1 0 0 St O, this follows from the definition of R (see [Tay08,§3]). For the statement of the next proposition, suppose given a surjection R /( ) → A in C , where A is a domain, and let r : G → G (A) denote the pushforward of (a F ,S n representative of) the universal deformation. Suppose given the following data: • A decomposition r = r ⊕ r ,where the r : G + → G (A) satisfy ν ◦ r = 1 2 i F ,S n G i i n ν ◦ r. (In other words, r| = r | ⊕ r | and if r (c) = (A , 1)j then r(c) = G G 1 G 2 G i i n F F F diag(A , A )j .) 1 2 • A subset R ⊂ S − S (consisting of places of odd residue characteristic) with the following property: for each v ∈ R we are given an integer 1 ≤ n ≤ n such that th q mod p is a primitive n root of unity and there is a decomposition r| = v G v F σ ⊕ σ ,where σ = Ind ψ with F /F the unramified extension of v,1 v,2 v,1 v,n v G v v v,n degree n and ψ an unramified character of G ,and σ is the twist of an v F v,2 v v,n unramified representation of G of dimension n − n by a ramified quadratic F v character. • An isomorphism ι : Q → C and for each v ∈ R, a character  : O → C p F v,n of order p. Thus the lifting ring R( v,  , n) is defined (notation as in §1.17). Proposition 5.6. — With the above assumptions on R /( ) → A, suppose that the following additional conditions are satisfied: SYMMETRIC POWER FUNCTORIALITY 81 (1) p > 2n. (2) For each 1 ≤ i < j ≤ n, χ /χ | has order greater than 2n. (In particular, this character is i j F(ζ ) non-trivial and ris Schur.) (3) For each v ∈ S , [F : Q ] > n(n − 1)/2 + 1. p v p (4) [F(ζ ) : F]= p − 1. (5) For each v ∈ R, both r | and r | admit a non-trivial unramified subquotient and the composite 1 G 2 G F F v v map R → R → A factors over R( v,  , n). S v Let L denote the maximal abelian pro-p extension of F unramified outside S ,and let  = S p Gal(L /F)/(c + 1).Let d denote the Z -rank of the subgroup of  generated by the elements Frob , S R p v v ∈ R.Then dim A ≤ n[F : Q]+ n − d . Proof. — Fix a place v of F split over F , prime to S, and such that q ≡ 1mod p 0 v and r| is trivial. Let I denote the ideal of A generated by the coefficients of the poly- nomial det(X − r| (Frob )) − (X − 1) . Then dim A/I ≥ dim A − n. After replacing G v F,S 0 AbyA/p,where p ⊂ A is a prime ideal minimal among those containing I, we can as- sume that r| (Frob ) is unipotent, and must show dim A ≤ n[F : Q]− d .Consider G v R F,S 0 the deformation problems (i = 1, 2): +   St S = (F/F , S ∪{v }, S ∪{ v }, , r ,μ, {R } ∪{R } ∪{R }). i 0 0 n i v∈S v∈S−S p p i v v v Let K = Frac A. We now repeat the argument of [ANT20, Lemma 3.6]: if v ∈ S ,then (since we assume [F : Q ] > n(n − 1)/2 + 1) we can appeal to [Tho15, Corollary 3.12], v p which implies the existence of an increasing filtration 1 2 n 0 ⊂ Fil ⊂ Fil ⊂· · · ⊂ Fil = K v v v • i i−1 of r| ⊗ KbyG -invariant subspaces, such that each gr Fil = Fil / Fil (i = G A F F v v v v 1,..., n) is 1-dimensional, and such that the character I (p) → K afforded by gr Fil v v i × agrees with the pushforward of the universal character ψ : I →  . Using the de- v v v composition r = r ⊕ r , we obtain induced filtrations Fil ∩(K ⊕ 0 ) of r | ⊗ K 1 2 1 G A v F and Fil ∩(0 ⊕ K ) of r ⊗ K and, applying [Tho15, Corollary 3.12] once more, we 2 A see that we can find an isomorphism  ⊗ =  =  such that, endowing A with n n n 1 2 the induced  -algebra structure, r is a lifting of r of type S for each i = 1, 2. We de- i i i i duce the existence of a surjective -algebra homomorphism R ⊗ R → A. We ob- S O S 1 2 serve that Theorem 5.2 applies to the deformation problems S and S , showing that 1 2 dim R /( ) ≤ n [F : Q]. S i × i Let ψ : G → O denote the Teichmüller lift of ψ = det r | ,and let R de- i F,S i G F,S S note the quotient of R over which the determinant of the universal deformation equals ψ .Then[Tho15, Lemma 3.36] states that there is an isomorphism R = R ⊗ O. i S O i S In particular, dim R /( ) ≤ (n − 1)[F : Q]. To complete the proof, it is enough to show that if A = A/(m ψ , m ψ ), then dim A ≤ 2[F : Q]− d = dim k ×  − d . 1 2 R R R R S S 1 2 82 JAMES NEWTON, JACK A. THORNE To this end, we observe that by construction there is a surjection k ×  → A . If # ,# : G → k ×  are the two universal characters, then the third part of 1 2 F,S Proposition 1.22 (together with our assumption that both r | and r | admit an un- 1 G 2 G F F v v n n v v ramified subquotient) implies that the relation # (Frob ) = # (Frob ) holds in A 1 v 2 v for each v ∈ R. Since  is a pro-p group, this implies that # (Frob ) = # (Frob ) in 1 v 2 v A , and hence that the map k ×  → A factors over the completed group alge- bra of the quotient of  ×  by the subgroup topologically generated by the elements (Frob , −Frob ) . This completes the proof. v v v∈R We are now in a position to prove the main theorem of this section, which guarantees the existence of generic primes in sufficiently large quotients of a certain deformation ring. For the convenience of the reader, we state our assumptions from scratch. Thus we take F, S, p as in our standard assumptions (see §1). We assume that [F(ζ ) : F]= (p − 1). We let E be a coefficient field, and suppose given an isomorphism ι : Q → C and a continuous character μ : G → O which is de Rham and such F ,S that μ(c ) =−1 for each place v|∞ of F . We fix an integer 2 ≤ n < p/2 and characters × c n χ ,..., χ : G → k such that for each i = 1,..., n, χ χ = μ| . We set ρ =⊕ χ ; F,S G 1 n i i i F,S i=1 then ρ naturally extends to a homomorphism r : G → G (k) such that ν ◦ r = μ.We F,S n G suppose for each 1 ≤ i < j ≤ n, χ /χ | has order greater than 2n. This implies that r i j F(ζ ) is Schur. We suppose that for each v ∈ S , r| is trivial and [F : Q ] > n(n − 1)/2 + 1. p G v p We suppose given a subset R = R R ⊂ S − S (consisting of places of odd 1 2 p residue characteristic) and integers 1 ≤ n ≤ n (v ∈ R) such that for each v ∈ R, q mod p v v th is a primitive n root of unity, and there is a decomposition r| = σ ⊕ σ ,where G v,1 v,2 σ = Ind ψ is induced from an unramified character of the unramified degree n v,1 v G v v,n extension of F ,and σ is the twist of an unramified representation of dimension n − n v v,2 v × × by a ramified quadratic character. We fix for each v ∈ R a character  : O → C of v,n order p. Assuming (as we may) that E is large enough, we may then (re-)define the global deformation problem S = (F/F , S, S,, r,μ, {R } ∪{R( v,  , n)} ∪{R } ). v∈S v v∈R v∈S−(S ∪R) p p v v Following [ANT20, Definition 3.7], we say that a prime p ⊂ R of dimension 1 and characteristic p is generic at p if it satisfies the following conditions: • Let A = R /p,and let r : G + → G (A) be the pushforward of (a representa- S p F ,S n tive of) the universal deformation. Then for each v ∈ S , the (pushforwards from v v ab × of the) universal characters ψ ,...,ψ : I (p) → A are distinct. 1 n F ab v v • There exists v ∈ S and σ ∈ I (p) such that the elements ψ (σ ), . . . , ψ (σ ) ∈ F 1 n A satisfy no non-trivial Z-linear relation. We say that p is generic if it is generic at p and if r | ⊗ Frac A is absolutely irreducible. p G A F,S SYMMETRIC POWER FUNCTORIALITY 83 Theorem 5.7. — With assumptions as above, let R → B be a surjection in C ,where B is a finite /( )-algebra. Let L denote the maximal abelian pro-p extension of F unramified outside S ,and let  = Gal(L /F)/(c + 1).Let d denote the Z -rank of the subgroup of  topologically p S R p p i generated by the elements Frob , v ∈ R . v i Suppose that the following conditions are satisfied: (1) Each irreducible component of Spec B has dimension strictly greater than sup({n[F : Q]+ n − d } , {n[F : Q]−[F : Q ]} ). R i=1,2 v p v∈S (2) For each direct sum decomposition r = r ⊕ r with r : G → G (k) (j = 1, 2)and n n = 0, 1 2 j F ,S n 1 2 there exists i ∈{1, 2} such that for each v ∈ R , both r | and r | admit a non-trivial i 1 G 2 G F F v v unramified subquotient. Then there exists a prime p ⊂ R of dimension 1 and characteristic p, containing the kernel of R → B, S S which is generic. Proof. — After passage to a quotient by a minimal prime, we can assume that B is a domain. The argument is now very similar to that of [ANT20, Lemma 3.9]. Indeed, by [ANT20, Lemma 3.8], we can find countable collection (I ) of ideals I ⊂ /( ) such i i≥1 i that for all i ≥ 1, dim /( , I ) ≤ sup{n[F : Q]−[F : Q ]} and if p ⊂ R is a prime i v p v∈S S of dimension 1 and characteristic p which is not generic at p,then I R ⊂ p for some i ≥ 1. i S red Let I ⊂ R be the reducibility ideal defined just before [ANT20, Lemma 3.4], and let red + I = (I , )R .Proposition 5.6 shows that dim R /I ≤ sup{n[F : Q]+ n − d } . 0 S S 0 R i=1,2 Since B is a finite /( )-algebra, we have dim B/I ≤ sup{n[F : Q]−[F : i v Q ]} .Wealsohave dim B/I ≤ sup{n[F : Q]+ n − d } . The existence of a generic p v∈S 0 R i=1,2 p i prime p containing the kernel of the map R → B thus follows from [Tho15, Lemma 1.9]. We conclude this section with a result concerning the existence of automorphic lifts of prescribed types, under the hypothesis of residual automorphy over a soluble extension. It only uses the results of [ANT20] and not the results proved earlier in this section, and is very similar in statement and proof to [BG19, Theorem 5.2.1]. We begin by re-establishing notation. We therefore let F be an imaginary CM field such that F /F is everywhere unramified. We fix a prime p and write S for the set of 0 0,p + + p-adic places of F . We fix a finite set S of finite places of F containing S . We assume 0 0,p 0 0 that each place of S splits in F , but not necessarily that each place of S − S splits in 0,p 0 0 0,p F . We choose for each v ∈ S a place v of F lying above v, and write S ={ v | v ∈ S }. 0 0 0 0 0 We fix a coefficient field E ⊂ Q . Fix an integer n ≥ 2, and suppose given a continuous representation ρ : G → GL (k) satisfying the following conditions: F ,S n 0 0 • There is an isomorphism ρ = ⊕ ρ , where each representation ρ is absolutely i i i=1 c ∨ 1−n irreducible and satisfies ρ = ρ ⊗ .Moreover, foreach 1 ≤ i < j ≤ r,we i i have ρ = ρ . i j 84 JAMES NEWTON, JACK A. THORNE Proposition 5.8. — Fix disjoint subsets T , ⊂ S consisting of prime-to-p places which 0 0 0 split in F . We assume that for each v ∈  , we have q ≡ 1 mod p and ρ| is trivial. We fix for 0 0 v G each v ∈ T a quotient R → R of the universal lifting ring of ρ| corresponding to a non-empty 0 v G v F union of irreducible components of Spec R [1/p]. We suppose that if v ∈ S and v is inert in F , then 0 0 n Hom(F ,Q ) ρ(I ) is of order prime to p. Fix a weight λ ∈ (Z ) such that for each i = 1,..., nand 0, v + τ ∈ Hom(F , Q ), λ =−λ , and suppose that for each v ∈ S , ρ| admits a lift to 0 τ c,i τ,n+1−i 0,p G p F 0, v Z which is ordinary of weight λ ,inthe senseof[Ger19, Definition 3.8]. Suppose that there exists a p v soluble CM extension F/F such that the following conditions are satisfied: (1) p > max(n, 3). For each place v|pof F, we have [F : Q ] > n(n − 1)/2 + 1 and ρ| is v p G trivial. ker ad ρ (2) F(ζ ) is not contained in F and F is not contained in F (ζ ).For each 1 ≤ i < j ≤ r, p p ρ | is absolutely irreducible and ρ | = ρ | .Moreover, ρ| is primitive and ρ(G ) G G G G F i i j F(ζ ) F(ζ ) F(ζ ) F p p p has no quotient of order p. (3) There exists a RACSDC automorphic representation π of GL (A ) and an isomorphism ι : Q → n F C such that r ρ| .Moreover, π is ι-ordinary and there exists a place v of F lying above π,ι G 0 such that π is an unramified twist of the Steinberg representation. (4) If S denotes the set of places of F lying above S , then each place of S splits in F. Then there exists a RACSDC automorphic representation π of GL (A ) satisfying the following 0 n F conditions: (1) π is unramified outside S and there is an isomorphism r ρ . 0 0 π ,ι (2) π is ι-ordinary of weight ιλ. (3) For each place v ∈ T ,r | defines a point of R . 0 π ,ι G v 0 F 0, v (4) For each place v ∈  , π is an unramified twist of the Steinberg representation. 0 0, v (5) For each place v ∈ S which is inert in F , reduction modulo p induces an isomorphism 0 0 r (I ) → r (I ). π ,ι F , v π ,ι F , v 0 0 0 0 Proof. — Let v ∈ S ,and let ρ : G → GL (Z ) be the lift of ρ| which is 0,p v F n p G 0, v F ordinary of weight λ and which exists by assumption. Thus ρ is conjugate over Q to v v an upper-triangular representation with the property that if α ,...,α : G → Q v,1 v,n F 0, v p are the characters appearing on the diagonal, then for each i = 1,..., n the character α is equal, on restriction to some open subgroup of I , to the character v,i F 0, v 1−i −1 −λ τ,n−i+1 χ : σ ∈ I → (σ) τ(Art (σ )) . λ ,i F v 0, v 0, v τ :F →Q 0, v After enlarging E, we can assume that each character α takes values in O.Weuse the v,i restricted characters α | : I (p) → O (v ∈ S , i = 1,..., n) to define a homo- v,i I (p) F 0,p F 0, v 0, v morphism  → O. −1 n Let β = α χ .Then τ =⊕ β is an inertial type and the type τ ,Hodge v,i v,i v v,i v λ ,i i=1 λ ,τ v v 2 type λ lifting ring R is defined and equidimensional of dimension 1 + n + n(n − v SYMMETRIC POWER FUNCTORIALITY 85 1)[F : Q ]/2 (see [Kis08, Theorem 3.3.4]). When τ is trivial, [Ger19, Lemma 3.10] 0, v p v λ ,τ v v shows that there is a minimal prime ideal of R such that, writing R for the corre- sponding quotient, the following properties are satisfied: • R is O-flat of dimension 1 + n + n(n − 1)[F : Q ]/2. v 0, v p λ ,τ v v • The map R → Q determined by ρ factors through R . v v v p • For every homomorphism R → Q , the corresponding Galois representation G → GL (Q ) is ordinary of weight λ , in the sense of [Ger19, Definition F n v 0, v 3.8]. • The homomorphism R ⊗  → R (completed tensor product of the tauto- O v v logical quotient map R → R and the composite  → O → R )factors over v v v the quotient R ⊗  → R (defined in e.g. [Tho15, §3.3.2]). O v v v In fact, the same proof shows that these properties hold also in the case that τ is non- trivial. Our hypotheses imply that we can extend ρ to a continuous homomorphism r : 1−n n univ G → G (k) with the property that ν ◦ r = δ .Let R denote the deformation F ,S 0 F /F 1−n n ring, defined as in [BG19, Corollary 5.1.1], of δ -polarised deformations of r, F /F where the quotients of the local lifting rings for v ∈ S are specified as follows: • If v ∈ S ,wetakethe quotient R defined above. 0,p v • If v ∈ T ,take R . 0 v St • If v ∈  , take the Steinberg lifting ring R . • If v ∈ S and v is inert in F , take the component corresponding to the functor 0 0 of lifts r of r| such that the reduction map induces an isomorphism r(I ) → + F F 0,v 0,v r(I ). 0,v univ We can invoke [BG19, Corollary 5.1.1] to conclude that R has Krull dimension at least 1. We remark that this result includes the hypothesis that r| is irreducible, but F (ζ ) 0 p + + 0 0 this is used only to know that the groups H (F , ad r) and H (F , ad r(1)) vanish, which is 0 0 true under the weaker condition that r| is Schur, which follows from our hypotheses. F (ζ ) (The vanishing of these groups implies that the deformation functor is representable and that the Euler characteristic formula gives the correct lower bound for its dimension.) We consider as well the deformation problem + 1−n n   St S =(F/F ,S, S, , r| , δ , {R } ∪{R } ∪{R } ), F G v∈S v∈S−(S ∪) v∈ F F/F v p v p v where we define S, T,  to be the sets of places of F above S ,T ,  , respectively. Then 0 0 0 univ there is a natural morphism R → R of  -algebras, which is finite (apply Lemma 5.3 S F and [Tho15, Proposition 3.29(2)]). By [ANT20, Theorem 6.2], R is a finite  -algebra. S F univ The map  → R factors through a homomorphism  → O (by construction), so F F univ R is a finite O-algebra (of Krull dimension at least 1, as we have already remarked). 86 JAMES NEWTON, JACK A. THORNE We deduce the existence of a lift r : G → G (Z ) of r arising from a homo- F ,S n p univ morphism R → Z . We can now apply [ANT20, Theorem 6.1] and soluble descent to conclude that r| is automorphic, associated to an automorphic representation π of G 0 GL (A ) with the desired properties. n F 6. Raising the level – Galois theory This section is devoted to the proof of a single theorem that will bridge the gap between Theorem 4.1 and our intended applications. Let F, S, p, G be as in our standard assumptions (see §1), and let n ≥ 3 be an odd integer such that p > 2n. Fix an isomorphism ι : Q → C. We suppose given a RACSDC automorphic representation π of GL (A ), n F and that the following conditions are satisfied: (1) π is ι-ordinary. (2) [F(ζ ) : F]= p − 1. (3) There exist characters χ ,..., χ : G → F and an isomorphism r = χ ⊕ ··· ⊕ F π,ι 1 n 1 c ∨ 1−n χ ,where foreach i = 1,..., n,wehave χ = χ and for each 1 ≤ i < j ≤ n, i i χ /χ | has order strictly greater than 2n. i j G F(ζ ) (4) For each v ∈ S , r | is trivial and [F : Q ] > n(n − 1)/2 + 1. p π,ι G v p (5) There is a set R = R R ⊂ S − S with the following properties: 1 2 p (a) The sets R and R are both non-empty and for each v ∈ R, the characteristic of 1 2 k(v) is odd. As in §1.17, we write ω( v) : k( v) →{±1} for the unique quadratic character. rd (nb) If v ∈ R ,then q mod p is a primitive 3 root of unity, and there exists a character 1 v : k → C of order p such that π | contains λ( v,  , n) (notationasin§1.17, v v q v 3 v this representation of q defined with respect to n = 3). v 1 th (c) If v ∈ R ,then q mod p is a primitive (n − 2) root of unity, and there exists a 2 v character  : k → C of order p such that π | contains λ( v,  , n) (notation v v q v n−2 v as in §1.17, this representation of q defined with respect to n = n − 2). v 1 (d) For each non-trivial direct sum decomposition r = ρ ⊕ρ , there exists i ∈{1, 2} π,ι 1 2 such that for each v ∈ R , ρ | and ρ | both admit a non-trivial unramified i G G 1 F 2 F v v subquotient. (This is the situation we will find ourselves in after applying Theorem 4.1. The sets of places R and R here will correspond to the sets T and T respectively from §4,and 1 2 1 3 it will be possible to label the characters χ so that we have the following ramification properties: • If v ∈ R ,then χ , χ , χ are unramified at v and χ ,..., χ are ramified at v 1 2 3 4 n (and the image of inertia under each of these characters has order 2). • If v ∈ R ,then χ , χ are ramified at v (and the image of inertia under each of 1 2 these characters has order 2) and χ ,..., χ are unramified at v. 3 n SYMMETRIC POWER FUNCTORIALITY 87 These properties imply condition 5(d) above, which is what we actually need for the proofs in this section.) The theorem we prove in this section is the following one: Theorem 6.1. — With assumptions as above, fix a place v of F lying above S −(S ∪ R) such St p that q ≡ 1 (mod p) and r | is trivial. Then we can find a RACSDC ι-ordinary automorphic v π,ι G St F St representation π of GL (A ) satisfying the following conditions: n F (1) There is an isomorphism r  r . π ,ι π,ι (2) For each embedding τ : F → Q ,wehave HT (r  ) = HT (r ). τ π ,ι τ π,ι (3) π is an unramified twist of the Steinberg representation. St Our proof of this theorem follows a similar template to the proof of [CT17,Theo- rem 5.1]. Briefly, we use our local conditions at the places of R, together with Theorem 5.7 to show that (after a suitable base change) we can find a generic prime in the spec- trum of the big ordinary Hecke algebra. This puts us in a position to use the “R = T ” p p theorem proved in [ANT20], which is enough to construct automorphic lifts of r (or its π,ι base change) with the desired properties. We now begin the proof. Let F /F denote the extension of F(ζ ) cut out by r | , p π,ι G F(ζ ) and let Y be a finite set of finite places of F with the following properties: a + • For each place v ∈ Y , v is split over F , prime to S, and π is unramified. • For each intermediate Galois extension F /M/Fsuch thatGal(M/F) is simple, there exists v ∈ Y which does not split in M. a a Then any Y -split finite extension L/F is linearly disjoint from F /F. After conjugation, we can find a coefficient field E such that r is valued in GL (O), and extend it to a π,ι n 1−n + + homomorphism r : G → G (O) such that ν ◦ r = δ . We write r : G → G (k) F ,S n F ,S n F/F for the reduction modulo of r. Lemma 6.2. — Let L/F be an Y -split finite CM extension. Then: (1) r| is Schur. G + L (ζ ) (2) r| is primitive. (3) Suppose moreover that L/F is soluble. Then the base change of π with respect to the extension L/F is cuspidal. (4) More generally, suppose that L/F /F is an intermediate field with L/F soluble, and let  be a 0 0 RACSDC automorphic representation of GL (A ) such that r r| . Then the base change n F ,ι G 0 F of  with respect to the extension L/F is cuspidal. Proof. — For the first part, it is enough to check that L ⊂ L (ζ ) and χ /χ | is p i j G L(ζ ) non-trivial for each 1 ≤ i < j ≤ n.Wehave [L(ζ ) : L]= p − 1, which implies L ⊂ L (ζ ), p p 88 JAMES NEWTON, JACK A. THORNE while χ /χ (G ) = χ /χ (G ), so this ratio of characters is non-trivial. The second L(ζ ) F(ζ ) i j p i j p part follows from Lemma 5.1 and the fact that χ /χ | has order greater than 2n for any i j L i = j , because L/F is linearly disjoint from the extension F /F. The third part is a special case of the fourth part, so we just prove this. Suppose for contradiction that the base change of  with respect to the extension L/F is not cuspidal. Then we can find intermediate extensions L/F /F /F such that there is a tower 2 1 0 F = M /M /.../M = F , where each extension M /M is cyclic of prime degree; 1 m m−1 0 0 i+1 i F /F is cyclic of prime degree l ; the base change  of  to F (constructed as the 2 1 F 1 iterated cyclic base change with respect to the tower M /M /.../M ) is cuspidal; but m m−1 0 the base change of  to F is not cuspidal. We will derive a contradiction. By the F 2 second part of the lemma, r| is primitive. Let σ ∈ Gal(F /F ) be a generator. Since G 2 1 the base change of  with respect to the extension F /F is not cuspidal, there exists a F 2 1 cuspidal automorphic representation $ of GL (A ) such that the base change of n/l F F 2 1 l−1 σ σ is $  $  ···  $ (see [AC89, Theorem 4.2]). We claim that $ is in fact conjugate (n/l−n)/2 self-dual. The representation $|·| is regular algebraic (by [AC89, Theorem 5.1]). l−1 σ σ Since  is conjugate self-dual, [AC89, Proposition 4.4]) shows that $$ ···$ is also conjugate self-dual. The classification of automorphic representations of GL then c,∨ σ implies that there is an isomorphism $ $ for some 0 ≤ i < l.If w is an infinite c,∨ place of F , then the purity lemma ([Clo90b, Lemma 4.9]) implies that $ $ ,hence 2 w i l−1 σ σ σ $ = $ . Since $  $  ···  $ is regular algebraic, this is possible only if i = 0 and $ is indeed conjugate self-dual. (n/l−n)/2 Therefore r is defined and there is an isomorphism $|·| ,ι r Ind r (n/l−n)/2 . ,ι $|·| ,ι F G This contradicts the second part of the lemma, which implies that r is primitive. This ,ι contradiction completes the proof. Combining Lemma 6.2 and Proposition 5.8, we see that Theorem 6.1 will follow provided we can find a soluble CM extension L/F with the following properties: • L/FisY -split. • There exists a RACSDC ι-ordinary automorphic representation π of GL (A ) n L and a place v of L lying above v such that r  r | and π is an unram- St π ,ι π,ι G ified twist of the Steinberg representation. After first replacing F by a suitable Y ∪ R-split soluble extension, we can assume in addition that S = S ∪ R ∪{v | + } and that π is unramified outside S ∪ R (use the p St F p Skinner–Wiles base change trick as in [CHT08, Lemma 4.4.1]). Lemma 6.3. — There exist infinitely many prime-to-S places v of F with the following prop- erty: v does not split in F(ζ ) and r(Frob ) is scalar. a p v a SYMMETRIC POWER FUNCTORIALITY 89 Proof. — By the Chebotarev density theorem, it is enough to find τ ∈ G such that r(τ ) is scalar and (τ) = 1. We can choose any τ ∈ G such that (τ ) = 1, and set 0 F 0 τ = τ τ . Choose a place v of F as in Lemma 6.3 which is absolutely unramified and of odd residue characteristic. We set S ={ v | + } and S ={ v }. a a F a a We will need to consider several field extensions L/F and global deformation prob- lems. We therefore introduce some new notation. We define a deformation datum to be apair D = (L, {R } ) consisting of the following data: v v∈X • AY ∪ S -split, soluble CM extension L/F. • A subset X ⊂ S − S , which may be empty. We write X for the pre-image of X in S. • For each v ∈ X, one of the following complete Noetherian local rings R ,repre- senting a local deformation problem: –For any v ∈ R such that v is split over F, the ring R = R( v,  , r| ) (no- L v v G tation as in Proposition 1.22 –wedefine  =  ). v v| –For any v ∈ S such that q ≡ 1mod p and r| is trivial, the unipotently L v G ramified local lifting ring R = R considered in [Tho15, §3.3.3]. –For any v ∈ S such that q ≡ 1mod p and r| is trivial, the Steinberg local L v G St lifting ring R = R considered in [Tho15, §3.3.4]. If D = (L, {R } ) is a deformation datum, then we can define the global deformation v v∈X problem + 1−n n S = (L/L ,S ∪ X, S ∪ X, , r| , δ ,{R } ∪{R } ). D p,L p,L L G + v∈S v v∈X L/L v p,L We write R = R ∈ C for the representing object of the corresponding deformation D S D L functor. Lemma 6.4. — If D = (L, {R } ) is a deformation datum, then each irreducible component v v∈X of R has dimension at least 1 + n[L : Q]. Proof. — This follows from [Tho15, Proposition 3.9], noting that the term 0 + H (L , ad r(1)) vanishes because r| is Schur (cf. [CHT08, Lemma 2.1.7]). L (ζ ) Givenadeformationdatum D, we define an open compact subgroup U = U ⊂ GL (O ) and a smooth O[U ]-module M = D,v n F D D v∈S −S v∈S −S L p,L L p,L ⊗ M as follows: v∈S −S D,v L p,L • If v ∈ X, then U = GL (O ) and M = O. D,v n L D,v • If v ∈ X ∩ R and R = R( v,  , r| ),then U = q and M is an O-lattice L v v G D,v v D,v −1 ∨ in ι λ( v,  , n) (notation as in §1.17). v 90 JAMES NEWTON, JACK A. THORNE 1 St • If v ∈ Xand R = R or R ,then U = Iw and M = O. v D,v v D,v v v We now define a Hecke algebra T associated to any deformation datum D = (L, {R } ). It is to be a finite  -algebra (or zero). If c ≥ 1, let U(D, c) ⊂ G(A ) v v∈X L + be the open compact subgroup defined as follows: −1 −1 U(D, c) = ι Iw (c, c) × ( ι )U v D v v v∈S v∈S −S p,L L p,L S ∪S −1 L a,L × ι K (1) × G(O ) v∈S a,L (here we are using the notation for open compact subgroups established in §1.23). Note that U(D, 1) is sufficiently small, because of our choice of v . We write ord ord T (D, c) ⊂ End (S (U(D, c), M )) O D for the O-subalgebra generated by the unramified Hecke operators T at split places v = ww ∈ S ∪ S and the diamond operators u for u ∈ Iw (1, c) (v ∈ S ). L a,L v p,L ord ord ord Following [Ger19, §2.4], we define T (D) = lim T (D, c) and H (D) = ← −c ord ord lim Hom(S (U(D, c), M ), O).Weendow T (D) with a  -algebra structure using D L ← −c thesameformula as in [Ger19, Definition 2.6.2]. We then have the following result. ord ord Proposition 6.5. —H (D) is a finite free  -module and T (D) is a finite faithful  - L L algebra, if it is non-zero. Proof. — This can be proved in the same way as [Ger19, Proposition 2.20] and [Ger19, Corollary 2.21]. The proof uses that U(D, 1) is sufficiently small. ord j We write m ⊂ T (D) for the ideal generated by m and the elements T − L w j(j−1)/2 j + q tr ∧ r(Frob ) (w a split place of L/L not lying above a place of S ∪ S ). It w L a,L is either a maximal ideal with residue field k, or the unit ideal. In either case we set ord T = T (D) , which is either a finite local  -algebra or the zero ring. (In the cases D m L we consider, it will be non-zero, but this will require proof.) For any deformation datum D, we write P for the  -subalgebra of R topo- D L D logically generated by the coefficients of the characteristic polynomials of elements of G in (a representative of) the universal deformation r .By[Tho15, Proposition 3.26], the group determinant det r | is valued in P ,and P is a complete Noetherian local S G D D -algebra. By [Tho15, Proposition 3.29], R is a finite P -algebra. L D D St Lemma 6.6. — Let D = (L, {R } ) be a deformation datum, and suppose that R = R v v∈X v for all v ∈ X. Then there is a natural surjective morphism P → T of  -algebras. D D L SYMMETRIC POWER FUNCTORIALITY 91 Proof. — The proof is the essentially the same as the proof of [Tho15,Propo- sition 4.12], but we give the details for completeness. It is enough to construct maps ord P → T (D, c) which are compatible as c ≥ 1varies. Let (D, c) denote the set of D m automorphic representations σ of G(A ) with the following properties: • r = r| . σ,ι G • σ is the trivial representation. • The subspace ∨ −1 ∞ ord ∨ −1 ∞ Hom (M ,ι σ ) ⊂ Hom (M ,ι σ ) U(D,c) U(D,c) D D where all the Hecke operators U (v ∈ S , j = 1,..., n) act with eigenvalues v,0 which are p-adic units is non-zero. Then there is an injection ord (6.6.1) T (D, c) ⊗ Q →⊕ Q m O σ ∈(D,c) p p −1 ∞ U(D,c) which sends any Hecke operator to the tuple of its eigenvalues on each (ι σ ) . We can find a coefficient field E /E with ring of integers O and for each σ ∈ (D, c),a c c homomorphism r : G + → G (O ) lifting r and such that r | = r (apply [CHT08, σ L ,S n c σ G σ,ι Lemmas 2.1.5, 2.1.7]). Let A ⊂ k ⊕ O be the subring consisting of elements (a,(a ) ) such c c σ σ σ ∈(D,c) that for each σ , a mod = a.Then A is a local ring containing the image of the map σ c c (6.6.1), and the representation r × (× r ) is valued in G (A ) and is of type S σ ∈(D,c) σ n c D (by our choice of deformation problems and level structures). Writing Q ∈ C for the S O ring classifying pseudocharacters which lift tr r| , we see that there is a commutative L,S diagram R A D c ord Q ⊗  T (D, c) S O L m L D The ring P is equal to the image of the map Q ⊗  → R . The proof is thus com- D S O L D plete on noting that the right vertical arrow is injective and the bottom horizontal arrow is surjective. We define J = ker(P → T ); this is a proper ideal if and only if T = 0. D D D D We now fix a place v of Fabove v .If L/F is a CM extension, we write v = St St St,L v | . St L Lemma 6.7. — We can find a deformation datum D = (L , ∅) with the following properties: 1 1 92 JAMES NEWTON, JACK A. THORNE (1) T = 0. (2) There exists a prime ideal p ⊂ R of dimension 1 and characteristic p which is generic, such that 1 D J ⊂ p , and such that r | is the trivial representation. D 1 p G 1 1 L ,v 1 St,L Proof. — We first claim that we can find an R ∪ Y ∪{ v }-split soluble CM extension L /F with the following properties: • For each i ∈{1, 2},let L denote the maximal abelian pro-p extension of L 0,S 0 unramified outside S ,and let  = Gal(L /L )/(c + 1).Let d denote p,L L 0,S 0 R 0 0 p i,L the Z -rank of the subgroup of  topologically generated by the elements p L Frob , v ∈ R .Then d > n + n . v i,L R 0 i,L • For each v ∈ S , [L : Q ] > n . p,L 0, v p • For each v ∈ S − (S ∪ R ), r| is trivial, and q ≡ 1mod p. L p,L L G v 0 0 0 L 0, v The third property is automatic, since S − (S ∪ R ) consists of primes above v | . L p,L L St F 0 0 0 We can construct an extension satisfying the first two properties using a similar idea to the proof of [ANT20, Theorem 7.1]. Indeed, we can find, for any odd integer d ≥ 1, a cyclic + a totally real extension M /F which is R ∪{v| + | v ∈ Y }∪ S -split and in which each d F a 2 a place v ∈ S is totally inert. If d > n and L = M · FthenL /Fwill be a R ∪ Y ∪{ v }- p 0 d 0 a split soluble CM extension which also satisfies the second point above. We need to explain how to arrange that the first point is also satisfied. By class field theory, d is equal to the i,L × c=−1 Z -rank of the subgroup of (O ⊗ Z ) topologically generated by (O ) . Since p L Z p 0 L ,R 0 i,L c=−1 O ⊗ Q decomposes as a Q [Gal(L /F )]-module with multiplicity 1, [Mai02, Z 0 L ,R 0 i,L p p × c=−1 Proposition 19] shows that this rank equals the Z-rank of (O ) , which is d|R |. L ,R 0 i,L Choosing any d > n + n therefore gives an extension with the desired properties. Let π be thebasechangeof π with respect to the extension L /F. It is cuspi- 0 0 dal by Lemma 6.2.Let D = (L , {R( v,  , r| )} ).Then D is a deformation 0 0 v G v∈R 0 L L 0, v 0 datum and the existence of π , together with Theorem 1.4,shows that T = 0. Let 0 D ur B = R /(J , m ). Then dim B ≥ n[L : Q]− n , and we may apply Theorem 5.7 D D R 0 0 0 St,L to conclude the existence of a generic prime p ⊂ R of dimension 1 and characteristic 0 D ur p which contains (J , m ). D R St,L We now make another base change. Let L /L be a CM extension with the follow- 1 0 ing properties: • L /F is soluble and Y ∪{ v }-split. 1 a • For each v ∈ R , the natural morphism R → R( v,  , r| ) factors over the L v G 1 v L 0, v ur unramified quotient R → R (cf. Proposition 1.22). v v • For each v ∈ S − S , q ≡ 1mod p and r| is trivial. L p,L v G 1 1 L 1, v Let π denote the base change of π to L .Then π is a RACSDC, ι-ordinary automor- 1 0 1 1 phic representation of GL (A ) which is unramified outside S .Thus D = (L , ∅) is a n L p,L 1 1 deformation datum and T = 0. 1 SYMMETRIC POWER FUNCTORIALITY 93 To complete the proof, it is enough to produce a commutative diagram R P T D D D 1 1 1 R P T . D D D 0 0 0 Indeed, then we can take p to be the pullback of p along the map R → R .The 1 0 D D 1 0 only arrow which has not already been constructed is the arrow T → T . This may be D D 1 0 constructed in exactly the same way as [Tho15, Proposition 4.18], using the construction of the Hecke algebra as an inverse limit as in the proof of Lemma 6.6,providedwe can prove the following statement: for any automorphic representation σ of G(A ) satisfying the following conditions: ∨ −1 ∞ ord • There exists c ≥ 1such thatHom (M ,ι σ ) = 0; U(D ,c) D 0 • σ is trivial; 0,∞ • Thereisanisomorphism r = r | ; σ ,ι π,ι G 0 L ord (in other words, such that σ contributes to S (U(D, c), M ) for some c ≥ 1), there 0 D m exists an automorphic representation σ of G(A ) satisfying the following conditions: 1 L ∨ −1 ∞ ord • There exists c ≥ 1such thatHom (M ,ι σ ) = 0; U(D ,c) D 1 • σ is trivial; 1,∞ • Thereisanisomorphism r r | . σ ,ι σ ,ι G 1 0 L To see this, we first show that the base change of any such σ to L (in the sense of 0 0 Theorem 1.2) must be cuspidal. We will show that in fact r is irreducible. Suppose σ ,ι that there is a decomposition r = ρ ⊕ ρ . By assumption, there exists v ∈ R such σ ,ι 1 2 L that both ρ | and ρ | admit an unramified subquotient. However, local-global 1 G 2 G L , v L , v 0 0 compatibility (together with Proposition 1.20)shows that r | = ρ ⊕ ρ ,where ρ is σ ,ι G 0 L , v 1 2 1 an irreducible representation of G with unramified residual representation and ρ is a 0, v 2 representation of G such that ρ is a sum of ramified characters. This is a contradiction 0, v 2 unless one of ρ and ρ is the zero representation. If μ denotes the base change of σ ,a 1 2 0 0 RACSDC ι-ordinary automorphic representation of GL (A ), then the base change of n L μ with respect to the soluble extension L /L is also cuspidal, by Lemma 6.2,and the 0 1 0 existence of σ follows from Theorem 1.4. This completes the proof. We can now complete the proof of Theorem 6.1.Werecallthatitisenough to construct a RACSDC, ι-ordinary automorphic representation π of GL (A ) such n L that r = r| and π is an unramified twist of the Steinberg representation. π ,ι G L v 1 St,L Let D and p be as in the statement of Lemma 6.7. Consider the deformation data 1 1 St 1 D = (L , {R }) and D = (L , {R }). Then there are surjections R → R 1,a 1 1,b 1 D D v v 1,b 1,a St,L St,L 1 1 94 JAMES NEWTON, JACK A. THORNE and R → R and the prime p lies in intersection of Spec R and Spec R in D D 1 D D 1,b 1 1 1,a Spec R . We see that the hypotheses of [ANT20, Theorem 4.1] are satisfied for R D D 1,b 1,b (in the notation of loc. cit., we set R ={v }, χ = 1, S(B) =∅), and conclude St,L v 1 St,L that for any minimal prime Q ⊂ R contained in p ,wehave J ⊂ Q. In particu- D 1 D 1,b 1,b lar, dim R /Q = dim  . (We remark that the essential condition for us in applying D L 1,b 1 [ANT20, Theorem 4.1] is that there is no ramification outside p; this is the reason for proving Lemma 6.7.) Lemma 6.4 and Theorem 5.2 show together that each minimal prime of R has 1,a dimension equal to dim  .Let Q ⊂ R be a minimal prime contained in p .Then L a D 1 1 1,a Q is also a minimal prime of R ,J ⊂ Q , and there exists a minimal prime Q of a D D a 0 1,b 1,b such that R /Q is a finite faithful  /Q -algebra. If Q = Q ∩ P , then there L D a L 0 1 a D 1 1,a 1 1,a are finite injective algebra maps /Q → P /Q = T /Q → R /Q . L 0 D 1 D 1 D a 1 1,a 1,b 1,a Using [Ger19, Lemma 2.25] and Theorem 1.2, we conclude the existence of an auto- morphic representation π of GL (A ) with the required properties. n L 7. Level 1 case The goal of this section is to prove Theorem E, using the level-raising results es- tablished in the last few sections. Combining this with the results of §§2 – 3, we will then be able to deduce Theorem A. Our starting point is σ , the cuspidal automorphic representation of GL (A ) of 0 2 Q weight 5 associated to the unique normalised newform 2 4 5 8 f (q) = q − 4q + 16q − 14q − 64q + ... of level  (4) and weight 5; it is the automorphic induction from the quadratic exten- sion K = Q(i) of the unique unramified Hecke character with ∞-type (4, 0).For any prime p ≡ 1 mod 4 and isomorphism ι : Q → C, σ is ι-ordinary. We observe that r = 0 σ ,ι p 0 G × −4 Ind ψ for a character ψ : G → Q (which depends on p)and det r = δ , K σ ,ι K/Q G 0 K p where δ : G →{±1} is the quadratic character with kernel G . K/Q Q K The main technical result of this section is the following theorem: Theorem 7.1. — Let n ≥ 3 be an integer. Suppose given the following data: (1) A prime p ≡ 1 (mod 48n!) and an isomorphism ι : Q → C. (2) A prime q = p. (3) A finite set X of places of K, each prime to 2pq. c 3 (4) A de Rham character ω : G → Q such that ωω = and ω| is unramified if v ∈ X . K G 0 p K Then there exists a soluble CM extension F/K with the following properties: SYMMETRIC POWER FUNCTORIALITY 95 (1) F/K is X -split. (2) There is a RACSDC, ι-ordinary automorphic representation  of GL (A ) with the following n F properties: (a) There is an isomorphism n−1 n−1 r ω | ⊗ Sym r | . ,ι G σ ,ι G F 0 F (b) For each embedding τ : F → Q , we have n−1 n−1 HT (r ) = HT (ω | ⊗ Sym r | ). τ ,ι τ G σ ,ι G F 0 F (c) There exists a place v|qof F such that  is an unramified twist of the Steinberg representation. The following lemma will be used repeatedly. Lemma 7.2. — Let n ≥ 3 be an integer, and let p ≡ 1 (mod 48n!).Let ω : G → Q be c 3 a de Rham character such that ωω = . Let F/K be a finite CM extension which is linearly disjoint from the extension of K(ζ ) cut out by n−i c n−1 i−1 r | ,and set χ = ω ⊗ ψ (ψ ) , ρ = χ ⊕· · · ⊕ χ , so that there is an isomorphism σ ,ι G 0 i 1 n K(ζ ) n−1 n−1 ρ = ω ⊗ Sym r | . σ ,ι G 0 K Then: (1) [F(ζ ) : F]= p − 1. (2) ψ/ψ | has order greater than 2n(n − 1) and for each 1 ≤ i < j ≤ n, χ /χ | has order G G i j F(ζ ) F(ζ ) p p greater than 2n. 1−n (3) For each 1 ≤ i ≤ n, χ χ = . ker ad ρ + (4) ζ ∈ F and F ⊂ F (ζ ). p p (5) ρ| is primitive. Proof. — We have [F(ζ ) : F]= p − 1 because F/K is disjoint from K(ζ )/K. To p p justify the second point, let L/K denote the extension cut out by ψ/ψ .Wemustshow that [L · F(ζ ) : F(ζ )] > 2n(n − 1).Wenotethat [L : K]≥ (p − 1)/4, because the re- p p striction of ψ/ψ to an inertia group at p has order (p − 1)/4. Moreover, L ∩ K(ζ ) has degree at most 2 (since c acts as 1 on Gal(K(ζ )/K) and as −1onGal(L/K)), so [L(ζ ) : K]≥ (p − 1) /8. Since F/K is supposed disjoint from L(ζ )/K, we have [F · L(ζ ) : K]≥ (p − 1) [F : p p K]/8. Since [F(ζ ) : F]= p − 1, we have [F(ζ ) : K]= (p − 1)[F : K]. Putting these to- p p gether we find [F · L(ζ ) : K] [F · L(ζ ) : F(ζ )]= ≥ (p − 1)/8. p p [F(ζ ) : K] p 96 JAMES NEWTON, JACK A. THORNE Since we assume p ≡ 1 (mod 48n!), we in particular have p − 1 ≥ 48n!,hence (p − 1)/8 > 2n(n − 1). i−j If 1 ≤ i < j ≤ n then χ /χ = (ψ /ψ) , so this shows the second point of the i j lemma. For the third point we compute c c n−1 n−1 1−n χ χ = (ωω | ) (ψψ | ) = . G G i i F F ker ad ρ We now come to the fourth point. To show that ζ ∈ F ,wemustfind τ ∈ G such p F that ρ(τ) is scalar but (τ) = 1. We can choose τ = τ τ for any τ ∈ G such that 0 0 F (τ ) = 1. Such a τ exists because [F(ζ ) : F]= p − 1, and ρ(τ) is scalar by the third 0 0 p + + + + part of the lemma. If F ⊂ F (ζ ) then F(ζ ) = F (ζ ) and [F(ζ ) : F]=[F (ζ ) : F ]/[F : p p p p p F ]= (p − 1)/2, contradicting the first part of the lemma. For the fifth point it is enough, by Lemma 5.1, to show that for each 1 ≤ i < j ≤ n, χ /χ has order greater than n. This follows from the second point. i j Before giving the proof of Theorem 7.1, we give a corollary which establishes the existence of the automorphic representations necessary for the proof of Theorem 7.6. Corollary 7.3. — Let n ≥ 3 be an integer. Then there exists a cuspidal automorphic representa- tion σ of GL (A ) of weight 5 with the following properties: 2 Q (1) σ is unramified away from 2 and a prime q ≡ 3 mod 4. GL (2) σ is isomorphic to a principal series representation i χ ⊗ χ ,where χ is unramified and χ 2 1 2 1 2 has conductor 4. (3) σ is an unramified twist of the Steinberg representation. n−1 (4) For any prime p and any isomorphism ι : Q → C, Sym r is automorphic. σ,ι Proof. — Choose a prime p ≡ 1 (mod 48n!) and an isomorphism ι : Q → C.It n−1 suffices to construct σ as in the statement of the corollary such that Sym r is auto- σ,ι morphic for our fixed choice of ι. avoid Let F /Q denote the extension cut out by r ⊕ , and choose a prime q sat- σ ,ι isfying q ≡ 3mod 4 (so a (f ) = 0) and q ≡−1mod p. This implies that σ satisfies the q 0 0 level-raising congruence at q. By a level-raising result for GL (A ) (e.g. [Dia89, Corol- 2 Q lary 6.9]), we can find an ι-ordinary cuspidal automorphic representation σ of GL (A ) 2 Q satisfying the following conditions: • σ has weight 5, and r r . σ,ι σ ,ι • σ is unramified at primes not dividing 2q; σ is isomorphic to a principal series GL representation i χ ⊗ χ ,where χ is unramified and χ has conductor 4; and 1 2 1 2 σ is an unramified twist of the Steinberg representation. Let ω : G → Q be a character crystalline at p and unramified at q and such that c 3 c −1 ωω = .Then (ψω)(ψω) = .Wetake X be a set of prime-to-2pq places of K at 0 SYMMETRIC POWER FUNCTORIALITY 97 which ω is unramified, and with the property that any X -split extension of K is linearly avoid disjoint from F /K. Let F/Kand π be the soluble CM extension and RACSDC automorphic repre- sentation of GL (A ) whose existence is asserted by Theorem 7.1. Thus in particular π n F is ι-ordinary, is an unramified twist of Steinberg at some place v|q of F, and there are isomorphisms n−i c n−1 n i−1 n−1 n−1 ∼ ∼ r = ω| ⊗ (⊕ ψ (ψ ) | ) = ω | ⊗ Sym r | . π,ι G G σ,ι G G i=1 F F F We now want to apply [ANT20, Theorem 6.1] (an automorphy lifting theorem) to con- n−1 n−1 clude that the representation ω ⊗ Sym r | is automorphic. This will in turn imply, σ,ι G n−1 by soluble descent, that Sym r is automorphic. The hypotheses of [ANT20,Theo- σ,ι rem 6.1] may be checked using Lemma 7.2. This concludes the proof. We first prove Theorem 7.1 in the case where n = 2k + 1 is an odd integer, using theresultsof§§4 – 6. Proposition 7.4. — Theorem 7.1 holds when n = 2k + 1 is odd. Proof. — We prove the proposition by induction on odd integers n = 2k + 1. Let p, q,X , ω be as in the statement of Theorem 7.1. Let Z denote the set of rational primes avoid below which ω is ramified, together with 2, p, q.Let F /K denote the extension of K cut out by r ⊕ . We fix a finite set X of finite places of K with the following properties: σ ,ι • X contains X . • If v ∈ Xthen v is prime to Z. In particular, ω| is unramified. avoid • For each subextension M/KofF /KwithGal(M/K) simple and non-trivial, there exists v ∈ X which does not split in M. Let q be a prime not in Z and which does not split in K, and let Y denote the set of rational primes dividing q or an element of X. We make the following observations: avoid • If F/K is a finite X-split extension, then F/K is linearly disjoint from F /K. avoid • If F /Q is a finite Y-split extension, then F /Q is linearly disjoint from F /Q 0 0 avoid and F K/K is linearly disjoint from F /K. Note in particular that Y-split extensions are linearly disjoint from K/Q.Wecan find distinct rational primes q , q , q satisfying the following conditions: 1 2 3 • For each i = 1, 2, 3, we have q ∈ Y ∪ Zand q splits in K. In particular, q is odd. i i i th • We have q > n and q mod p is a primitive 6 root of unity. The eigenvalues 1 1 G 2k ±1 of Frob on Ind ψ have ratio q mod p, while the eigenvalues of Frob on q q 1 G 1 1 th × Ind ψ have ratio which is a primitive 12k root of unity in F . G p K 98 JAMES NEWTON, JACK A. THORNE 2k • We have q ≡−1mod p and the eigenvalues of Frob on Ind ψ have ratio 2 q 2 G −1. th • The number q mod p is a primitive (n − 2) root of unity and the eigenvalues ±1 of Frob on Ind ψ have ratio q mod p. 3 G 3 To construct q , q , q we use the Chebotarev density theorem. After conjugation, we can 1 2 3 assume that r | = ψ ⊕ ψ is diagonal. Consideration of the restriction of r | to σ ,ι G σ ,ι G 0 K 0 K the inertia groups at p shows that (r ⊕ )(G ) contains the subgroup σ ,ι K 4 4 −1 × × {(diag(a ,(c/a) ), c ) | a, c ∈ F }⊂ GL (F ) × F . 2 p p p We assume p ≡ 1 (mod 48n!), hence in particular p ≡ 1 (mod 96k).Let z ∈ F be an 1+8k 16k th element of order 96k,and let x = z , y = z .Then y is a primitive 6 root of 1 1 1 1+8k 16k unity and x = y . If the prime q is chosen so that q > n and (r ⊕ )(Frob ) = 1 1 σ ,ι q 1 1 0 1 −1 4 4 8 4 (diag(x ,(y /x ) ), y ), then the eigenvalues of Frob in Ind ψ have ratio x /y = 1 1 q 1 1 1 G 1 1 G 2k 8+64k−64k 8 th z = z , a primitive 12k root of unity, while the eigenvalues of Frob in Ind ψ 1 G have ratio 16k −1 z = y = (Frob ) ≡ q (mod p). 1 q 1 4 −4 We can choose the prime q so that (r ⊕ )(Frob ) = (diag(x , x ), −1), 2 σ ,ι q 0 2 2 × 16k where x ∈ F satisfies x =−1; and we can choose q so that (r ⊕ )(Frob ) = 2 3 σ ,ι q p 2 0 3 4 4 −1 × th (diag(x ,(y /x ) ), y ),where x , y ∈ F , y is a primitive (n − 2) root of unity, and x 3 3 3 3 3 3 3 p 8 3 is chosen so that x = y . These choices of x , y are again possible because of the congru- i i 3 3 ence p ≡ 1 mod 48n!. We fix real quadratic extensions M /Q (i = 1, 2, 3) with the following properties: • M is Y ∪{p, q, q , q }-split, and q is ramified in M . 1 1 2 3 1 • M is Y ∪{p, q, q }-split, and q , q are ramified in M . 2 3 1 2 2 • M is Y ∪{p, q, q , q }-split, and q is ramified in M . 3 1 3 2 3 We write ω : G →{±1} for the quadratic character of kernel G . i Q M By a level-raising result for GL (A ) (e.g. [DT94,Theorem A],wecan finda 2 Q cuspidal, regular algebraic automorphic representation τ of GL (A ) with the following 2 Q properties: • τ is unramified outside 2, q , q . 1 2 G 2k G Q Q 2k • Thereisanisomorphism r Ind ψ and det r = det Ind ψ = τ,ι τ,ι G G K K −8k δ . K/Q • τ is an unramified twist of the Steinberg representation, and there is an iso- morphism rec τ = Ind χ ,where χ | is a character of order p.In Q q q q I q 2 W 2 2 Q Q q particular, τ is supercuspidal. 2 SYMMETRIC POWER FUNCTORIALITY 99 2k • τ is ι-ordinary and r has the same Hodge–Tate weights as Ind ψ . τ,ι For the ordinary condition in the last point, note that since p > 8k and r has Hodge– τ,ι Tate weights (0, 8k) and reducible local residual representation r | at p, τ is neces- τ,ι G sarily ι-ordinary [Edi92, Theorem 2.6]. By induction, there exists a soluble CM extension F /K with the following prop- −1 erties: • F /K is X-split. −1 • There exists a RACSDC, ι-ordinary automorphic representation π of GL (A ) with the following properties: n−2 F −1 (1) There is an isomorphism n−3 n−3 r = ω | ⊗ Sym r | . π,ι G σ ,ι G F 0 F −1 −1 (2) For each embedding τ : F → Q ,wehave −1 n−3 n−3 HT (r ) = HT (ω | ⊗ Sym r | ). τ π,ι τ G σ ,ι G E 0 E (3) There exists a place v |q of F such that π is an unramified twist of the −1 −1 v −1 Steinberg representation. We can find a soluble CM extension F /Q with the following properties: • F is Y-split. + + • The prime q is split in F , and each place of F above q is inert in F .The 1 1 0 0 0 primes q , q split in F . 2 3 0 • F /F is everywhere unramified. • For each place v|p of F , v is split over F and [F : Q ] > n(n − 1)/2 + 1. 0 0,v p • For each place v|q of F , v is split over F and q ≡ 1mod p. 0 v • There exists a crystalline character ω : G → Q , unramified outside p,such 0 F 0 p c 3 that ω ω = δ | .(Use[BLGGT14, Lemma A.2.5].) 0 K/Q G 0 F • For each place v|pq of F , the representations r | and ω | are trivial. 0 σ ,ι G 0 G 0 F F 0,v 0,v Define 2k−i−1 c,i−1 n−3 k−1 −4(k−1) ρ = ω ⊗ ω ⊗ (⊕ (Ind ψ ψ )) ⊕ ω | . 2 3 G 0 i=1 G F K 0 Then for each place v|pq of F , ρ | is trivial. 0 G 0 F 0,v We now apply Proposition 5.8 with the following choices: • F = F · F · M · M · M . 1 −1 0 1 2 3 • ρ is the residual representation defined above. + + •  is the set of places of F lying above q;T is the set of places of F lying 0 0 0 0 above q or q ;and S is the set of places of F lying above p, q, q , q ,or q . 2 3 0 1 2 3 0 100 JAMES NEWTON, JACK A. THORNE • If v|q ,then R is the fixed type deformation ring (defined as in [Sho18, Defini- 2 v 2k−2 −1 tion 3.5]) associated to the inertial type ⊕ ω(v) ◦ Art | (where as usual, F × i=0 0,v 0,v ω(v) denotes the unique quadratic character of k(v) provided that k(v) has odd characteristic). If v|q , then there is a character  : O → C of order 3 v 0,v,n−2 p such that R is the fixed type deformation ring associated to the inertial type i−1 2k−2 −1 −1 v ⊕ ι  ◦ Art | (where F /F is an unramified extension v × F 0,v,n−2 0,v i=0 0,v,n−2 0,v,n−2 of degree n − 2). • π is the twist of the base change of π with respect to the soluble CM extension n−3 n−3 F /F by the character ιω | /ω | . 1 −1 G G 0 F F 1 1 (Note that F /K is X-split, so Lemma 7.2 may be applied to r .) We conclude the exis- 1 π ,ι tence of a RACSDC, ι-ordinary automorphic representation π of GL (A ) satisfying 0 n−2 F the following conditions: • Thereisanisomorphism r ρ . π ,ι 0 0 • π is unramified outside S . 0 0 • For each place v|q of F , π is an unramified twist of the Steinberg representa- 0 0,v tion. • For each place v|q of F , there are characters χ ,χ ...,χ : F → C 1 0 v,0 v,1 v,2k−2 0,v such that π = χ  χ  ···  χ , χ is unramified,and for each i = 0,v v,0 v,1 v,2k−2 v,0 1,..., 2k − 2, χ | = ω(v). v,i F ,v • For each place v|q of F , π | contains ω(v) ◦ det. 2 0 0,v GL (O ) n−2 F 0,v • For each place v|q of F , π | contains the representation λ(v,  ) 3 0 0,v GL (O ) v n−2 F 0,v (notation as in Proposition 1.18). Let T denote the set of places of F lying above q ,and let T = T ∪ T ∪ T .Let τ i i 1 2 3 0 (n−2)/2 n−1 denote the base change of τ to F ,and let π = τ ⊗|·| ιω ω .Let π = π ⊗|· 0 2 0 1 n−2 0 −3 2 | ιω . We see that the hypotheses of Theorem 4.1 are now satisfied, and we conclude the + + existence of a T-split quadratic totally real extension L /F and a RACSDC ι-ordinary 0 0 automorphic representation  of GL (A ) satisfying the following conditions: 0 n L • The extension L /Q is soluble and Y-split. • Thereisanisomorphism 2k 2k−i c,i G G Q Q n−1 k−1 r = ω ⊗ ω ⊗ Ind ψ ⊕ ω ⊗ (⊕ (Ind ψ ψ )) ,ι 1 2 0 0 G i=1 G G K K −4k ⊕ ω . × × • For each place v|q of L , there exists a character  : O → C of order p 1 0 v 0,v,3 such that  | contains λ(v,  , n)| (notationasin§1.17). 0,v r v r v v • For each place v|q of L ,  | contains the representation λ(v,  , n) (where 3 0 0,v q v we define  =  ). v v| 0 SYMMETRIC POWER FUNCTORIALITY 101 In fact,  has the following stronger property: • For each place v|q of L ,  | contains λ(v,  , n). 1 0 0,v q v To see this, it is enough to check that no two eigenvalues α, β ∈ F of the representation n−3 2 2 (Sym Ind ψ)(Frob ) satisfy (α/β) = q (recall that if v|q is a place of L ,then q 1 0 G 1 1 L /Q is an unramified quadratic extension). Recalling the numbers x , y ∈ F , we see 0,v q 1 1 1 p ±2 8 16 i that we must check that (y /x ) = y for i = 1,..., 2k − 2. However, by construction 1 1 1 rd 8 16 th y is a primitive 3 root of unity and y /x is a primitive 6k root of unity, so we cannot 1 1 1 8 16 3i have (y /x ) = 1if1 ≤ i < 2k. 1 1 Let L = L K. Then the following conditions are satisfied: 1 0 • The extension L /K is soluble and X-split. • Let  denote the base change of  with respect to the quadratic extension 1 0 L /L .Then  is RACSDC and ι-ordinary. (It is cuspidal because L /L is 1 0 1 1 0 quadratic and n is odd, cf. [AC89, Theorem 4.2].) • For each place v of L of residue characteristic q , q , v is split over L and over 1 1 3 0 L . (The prime q splits in K.) Thus the hypotheses of Theorem 6.1 are satisfied with R (resp. R ) the set of places of 1 2 L of residue characteristic q (resp. q ), and we conclude the existence of a RACSDC ι- 1 3 ordinary automorphic representation  of GL (A ) satisfying the following conditions: n L 1 1 • Thereisanisomorphism r r | . ,ι  ,ι G 0 L • There exists a place v|q of L such that  is an unramified twist of the Stein- 1,v berg representation. Finally, let F = L M M M ,and let  be thebasechangeof  with respect to the 1 1 2 3 extension F/L . We see that the conclusion of Theorem 7.1 holds with  =  ⊗ n−1 ι(ω| /ω | ) . G 0 G F F Proof of Theorem 7.1.— If n is odd then the statement reduces to Proposition 7.4. Let m ≥ 1 be an odd integer. We will prove by induction on r ≥ 0 that the conclusion of Theorem 7.1 holds for all integers of the form n = 2 m. The case r = 0 is already known. Supposing the theorem known for a fixed r ≥ 0 r  r+1 (hence n = 2 m), we will now establish it for r + 1 (hence n = 2 m = 2n). Fix data p, q, X , ω as in the statement of Theorem 7.1. In particular p ≡ 1 (mod 48n !). Once again we enlarge X so that any X -split extension F/K is forced to be linearly disjoint from the 0 0 fixed field of ker(r ⊕ ). σ ,ι By induction, we can find a soluble CM extension F/K and a RACSDC automor- phic representation π of GL (A ) such that the following conditions are satisfied: n F n−1 n−1 • π is ι-ordinary. There is an isomorphism r ω | ⊗ Sym r | . π,ι G σ ,ι G F 0 F n−1 n−1 The representations r and ω | ⊗ Sym r have thesameHodge–Tate π,ι G σ ,ι F 0 weights (with respect to any embedding τ : F → Q ). p 102 JAMES NEWTON, JACK A. THORNE • There exists a place v|q such that π is an unramified twist of the Steinberg representation. • FisX -split. After possibly enlarging F, we can assume that the following additional conditions are satisfied: • q ≡ 1mod p and r (Frob ) is trivial. v σ ,ι v • Each place of F which is either p-adic or at which π is ramified is split over F . × × × Let , # : K \A → C be the Hecke characters of type A with r = ω and r = ψ . 0 ,ι #,ι n/2 n n/2 c n Define π = π ⊗ (|·| (# ◦ N ) ), π = π ⊗ (|·| (# ◦ N ) ).Wemakethe 1 F/K 2 F/K following observations: • π and π are cuspidal, conjugate self-dual automorphic representations of 1 2 GL (A ). n F • Let π = π  π and define r = r −n/2 ⊕ r −n/2 .Then π is regular alge- 0 1 2 π ,ι π |·| ,ι π |·| ,ι 0 0 1 2 braic and ι-ordinary. Moreover, for each finite place v of F there is an isomor- F−ss T phism WD(r | ) rec (π ), and there is an isomorphism π ,ι G 0,v 0 F v F 2n−1 2n−1 r ω | ⊗ Sym r | . π ,ι G σ ,ι G 0 F 0 F × × • There are unramified characters ξ : F → C such that π St (ξ ) and i i n i −1 n ι ξ /ξ ( ) ≡ q mod m . 1 2 v v Z r+1 −1 1−2 m • r is not isomorphic to a twist of 1 ⊕ ⊕· · · ⊕ . π ,ι We justify each of these points in turn. Since π is conjugate self-dual, the first point c −1 follows from the fact that (#)(#) =| · | (in turn a consequence of the identity c −1 (ωψ)(ωψ) = ). The second follows from the identity 2n−1 2n−1 2n−2 c c 2n−1 Sym r | = ψ ⊕ ψ ψ ⊕· · · ⊕ (ψ ) σ ,ι G 0 K n c n n−1 = (ψ ⊕ (ψ ) ) ⊗ Sym r | . σ ,ι G 0 K The third point holds by construction (q ≡ 1 (mod p) and r (Frob ) is scalar). The v π ,ι v fourth holds since otherwise r | would be a twist of the trivial representation, con- π ,ι G F(ζ ) tradicting part 2 of Lemma 7.2. We see that the hypotheses of [AT21, Theorem 5.1] are satisfied. This theorem implies that we can find a quadratic CM extension F /Fsuch thatF /K is soluble X - split, as well as a RACSDC automorphic representation π of GL (A ) satisfying the n F following conditions: • π is ι-ordinary, and there is an isomorphism n −1 n −1 r  ω | ⊗ Sym r | . π ,ι G  σ ,ι G F F SYMMETRIC POWER FUNCTORIALITY 103 • The weight of π is thesameasthatof π . • There exists a place v |v of F such that π is an unramified twist of the Steinberg representation. This existence of F and π completes the induction step, and therefore the proof of the theorem. Remark 7.5. — Weobserve that theresultsof[AT21] already suffice to prove The- orem 7.1 (and hence Theorem 7.7)when n is a power of two, without using the level raising results of Sections 4–6. We can now put everything together to deduce our main results on automorphy of symmetric powers. Theorem 7.6. — Let n ≥ 3. Then there exists a cuspidal, everywhere unramified automor- phic representation π of GL (A ) of weight k ≥ 2 such that, for any isomorphism ι : Q → C, 2 Q n−1 Sym r is automorphic. π,ι Proof. — By Corollary 7.3, we can find odd primes p = q,with q ≡ 3mod 4, and a cuspidal automorphic representation σ of GL (A ) of weight 5 satisfying the following 2 Q conditions: • σ is unramified at primes not dividing 2q; σ is isomorphic to a principal series GL representation i χ ⊗ χ ,where χ is unramified and χ has conductor 4; and 1 2 1 2 σ is an unramified twist of the Steinberg representation. n−1 • For any isomorphism ι : Q → C,Sym r is automorphic. p π,ι Now we choose an isomorphism ι : Q → C. By the second part of Lemma 3.5,the Zariski closure of r (G ) contains SL . Since σ is an unramified twist of Steinberg σ,ι Q 2 q q q it has a unique (q-adic) accessible refinement, which is numerically non-critical and n- regular. We can therefore apply Theorem 2.33 to the point of the q-adic, tame level 4 eigencurve associated to σ with its unique accessible refinement. Using the accumula- tion property of the eigencurve to find a suitable classical point in the same (geometric) irreducible component as this point, we deduce the existence of a cuspidal automorphic representation σ of GL (A ) of weight k > 2 satisfying the following conditions: 2 Q (1) σ is unramified outside 2, and σ is isomorphic to a principal series representation GL i χ ⊗ χ ,where χ is unramified and χ has conductor 4. 1 2 1 2 (2) The weight of σ satisfies k ≡ 3 mod 4 (this is possible because q ≡ 3mod 4, and we can choose any k ≡ 5mod (q − 1)q for sufficiently large α). n−1 (3) Sym r  is automorphic. σ ,ι q 104 JAMES NEWTON, JACK A. THORNE Let ι : Q → C be an isomorphism. These conditions imply that the Zariski closure of r (G ) must contain SL . Indeed, we have already observed in §3 that there are no 2- σ ,ι Q 2 ordinary cusp forms of tame level 1, so (invoking Lemma 3.5) if this Zariski closure does not contain SL then r  | must be irreducible and induced from a quadratic extension 2 σ ,ι G of Q , implying that both refinements of σ at the prime 2 have slope (k − 1)/2, an odd integer. However, Theorem 3.2 implies that there are no newforms of level 4 and odd slope (see [BK05, Corollary of Theorem B]); a contradiction. The same argument shows that the refinement χ ⊗ χ is n-regular, since the two refinements of σ have distinct 1 2 slopes. We see that (σ ,χ ⊗ χ ) satisfies the hypotheses of Theorem 2.33. Using the ac- 1 2 cumulation property of the (tame level 1, 2-adic) eigencurve, we deduce the existence of a cuspidal, everywhere unramified automorphic representation π of GL (A ) such that 2 Q n−1 Sym r is automorphic. This completes the proof. π,ι Combining Theorem 7.6 with Theorem 3.1, we deduce: Theorem 7.7. — Let n ≥ 3,and let π be a cuspidal, everywhere unramified automorphic rep- n−1 resentation of GL (A ) of weight k ≥ 2. Then for any isomorphism ι : Q → C, Sym r is 2 Q π,ι automorphic. 8. Higher levels In this section we extend our main theorem to higher levels as follows: Theorem 8.1. — Let π be a cuspidal automorphic representation of GL (A ) of weight k ≥ 2 2 Q satisfying the following two conditions: (1) For each prime l , π has non-trivial Jacquet module (equivalently, π admits an accessible refinement). l l (2) π is not a CM form. n−1 Then for any n ≥ 3 and any isomorphism ι : Q → C, Sym r is automorphic. π,ι For example, these conditions are satisfied if π is associated to a non-CM cuspidal eigenform f of level  (N) for some squarefree integer N ≥ 1; in particular, if k = 2and π is associated to a semistable elliptic curve over Q. Fix n ≥ 3 for the remainder of this section. We first prove the following special case of Theorem 8.1: Proposition 8.2. — Let π be a cuspidal automorphic representation of GL (A ) of weight 2 Q k ≥ 2 satisfying the following conditions: (1) For each prime l such that π is ramified, π has an accessible refinement which is n-regular, in the l l sense of Definition 2.23. (2) π is not a CM form. n−1 Then for any isomorphism ι : Q → C, Sym r is automorphic. π,ι p SYMMETRIC POWER FUNCTORIALITY 105 Proof. — We prove the proposition by induction on the number of primes r divid- ing the conductor N of π.The case r = 0 (equivalently, N = 1) is Theorem 7.7. Suppose therefore that r > 0 and that the theorem is known for automorphic rep- resentations of conductor divisible by strictly fewer than r primes. Let π be a cuspidal automorphic representation as in the statement of the proposition. Fix a prime p at which Q → C.Factor N = Mp ,where (M, p) = 1. π is ramified, and an isomorphism ι : Suppose first that r | is reducible. Then π is ι-ordinary and π admits an π,ι G ordinary refinement χ . After twisting by a finite order character, we can assume that (π, χ) ∈ RA (here we use the notation established for the Coleman–Mazur eigencurve in §2.31). Let C be an irreducible component of the (tame level M, p-adic) eigencurve E containing the point x corresponding to (π, χ),and let Z ⊂ E denote the Zariski 0,C 0 closed set defined in Lemma 2.35. Our hypotheses imply that x ∈ Z . We can therefore find a point x ∈ C − Z such that the image of x in W is a C 0,C p p k −2 character of the form y → y for some integer k ≥ 2. Indeed, since the image of C in W is Zariski open, we can find such a point in C . There is an affinoid neighbourhood 0,C U of this point which maps in a finite and surjective fashion onto an affinoid open in W . The image of Z ∩ U in this affinoid open is Zariski closed, and we can therefore 0,C C p p find another such point x ∈ C − Z . (In fact, the ordinary component C surjects onto a connected component of W , but we will apply the same argument for a non-ordinary 0,C component.) Choosing another point in a sufficiently small affinoid neighbourhood of x in C − Z and applying the classicality criterion, we can find a point x ∈ C −Z corresponding C C p p to an ι-ordinary cuspidal automorphic representation π of GL (A ) of weight k ≥ 2 2 Q with the following properties: (1) Let χ denote the ordinary refinement of π .Then (π ,χ ) determines a point on the same irreducible component of the (tame level M, p-adic) eigencurve E as (π, χ). 0,C (2) The level of π is prime to p. (3) For each prime l|M, each accessible refinement of π is n-regular. (4) The Zariski closure of r  (G ) (in GL /Q ) contains SL . π ,ι Q 2 2 (The latter two properties follow from the definition of the set Z in Lemma 2.35.Infact we can take x = x , since ordinary points of classical weights are classical; however, we will repeat the same argument in the next paragraph also for a non-ordinary component n−1 of the eigenvariety, in which case two steps are required.) By induction, Sym r is π ,ι automorphic. We may then apply the ordinary case of Theorem 2.33 to conclude that n−1 Sym r is automorphic. π,ι Suppose instead that r | is irreducible, and let χ be an accessible, n-regular π,ι G refinement. The existence of χ implies that the Zariski closure of r (G ) in GL /Q π,ι Q 2 contains SL , by Lemma 3.5. Again, after twisting by a finite order character, we can assume that (π, χ) ∈ RA . Repeating the same argument as in the ordinary case, we 0 106 JAMES NEWTON, JACK A. THORNE can find a cuspidal automorphic representation π of GL (A ) of weight k ≥ 2 with the 2 Q following properties: (1) π admits a non-ordinary refinement χ which is numerically non-critical and n- regular. (This again implies that the Zariski closure of r  (G ) contains SL .) π ,ι Q 2 (2) The pair (π ,χ ) determines a point on the same irreducible component of the (tame level M, p-adic) eigencurve E as (π, χ). 0,C (3) For each prime l|M, each accessible refinement of π is n-regular. (4) The level of π is prime to p. n−1 By induction, Sym r is automorphic. We can then appeal to Theorem 2.33 to con- π ,ι n−1 clude that Sym r is automorphic. π,ι In either case we are done, by induction. To reduce the general case of Theorem 8.1 to Proposition 8.2, we establish the following intermediate result. Proposition 8.3. — Let π be a cuspidal automorphic representation of GL (A ) of weight 2 Q k ≥ 2, without CM. Suppose that for each prime l , π has non-trivial Jacquet module. Then we can find a prime p, an isomorphism ι : Q → C, and another cuspidal automorphic representation π of GL (A ) of weight k with the following properties: 2 Q (1) p > max(2(n + 1), (n − 1)k). (2) The image of r contains a conjugate of SL (F ). π,ι 2 p (3) Both π and π are unramified. (4) There is an isomorphism r = r . π,ι π ,ι (5) For each prime l , π has non-trivial Jacquet module. If π is ramified, then each accessible refinement l l of π is n-regular. Proof. — We use Taylor–Wiles–Kisin patching. The idea is that if all the automor- phic representations congruent to π mod p fail to have n-regular refinements at l then the patched module will be supported on a codimension one quotient of the local defor- mation ring at l , which contradicts the numerology of the Taylor–Wiles–Kisin method. Let M denote the conductor of π.Wecan chooseaprime p satisfying (1) and (2), p > M, such that π is unramified, and satisfying the following additional condition: • For each prime l = p such that π is ramified, the universal lifting ring classifying lifts of r | of determinant equal to det r is formally smooth. π,ι G π,ι Indeed, it is sufficient that for each such prime l,the group H (Q , ad r (1)) vanishes. l π,ι Such a prime exists thanks to [Wes04, Proposition 3.2, Proposition 5.3]. Fix an additional prime q > p such that π is unramified and such that the uni- a q versal lifting ring classifying lifts of r | of determinant equal to det r is formally π,ι G π,ι smooth. This is possible by e.g. [DT94, Lemma 11]. SYMMETRIC POWER FUNCTORIALITY 107 Fix a coefficient field E/Q , large enough that there is a conjugate ρ : G → p Q GL (k) of r and such that χ = det r takes values in O. We assume moreover that 2 π,ι π,ι for each σ ∈ G , the roots of the characteristic polynomial of ρ(σ) lie in k. Let S denote the set of primes at which r is ramified (equivalently, at which ρ is ramified), together π,ι with q . We consider the global deformation problem (in the sense of [Tho16, Definition 5.6]) S = (ρ, χ, S, {O} , {D } ), v∈S v v∈S where D is the functor of lifts of ρ| of determinant χ which are Fontaine–Laffaille p G with the same Hodge–Tate weights as r , and if l ∈ S −{p} then D is the functor of π,ι l all lifts of ρ| of determinant χ . Since ρ is absolutely irreducible, the functor of de- formations of type S is represented by an object R ∈ C (cf. [Tho16, Theorem 5.9]). S O We may choose a representative ρ : G → GL (R ) of the universal deformation. S Q 2 S 1 k−2 2 We set H = H (Y , Sym O ),where U (Mq ) is the open compact subgroup U (Mq ) 1 a 1 a of GL (A ) defined in §2.31 and Y is the modular curve of level U (denoted Y(U) in 2 U [Eme06b, §4.1]). We write T ⊂ End (H) for the commutative O-subalgebra generated by the unramified Hecke operators T ,S for l ∈ S. Then there is a unique maximal ideal l l m ⊂ T with residue field k such that for each prime l ∈ S, the characteristic polynomial 2 k−1 of ρ(Frob ) equals X − T X + l S mod m. The localization H is a finite free O- l l l m module, and there is a unique strict equivalence class of liftings ρ : G → GL (T ) of m Q 2 type S such that for each prime l ∈ S, the characteristic polynomial of ρ (Frob ) equals m l 2 k−1 S the image of X − T X + l S in T [X].(See[Tho16, Proposition 6.5] for justification l l of a very similar statement in the context of Shimura curves.) In particular, there is an O-algebra morphism R → T classifying ρ , which is surjective. S m Suppose given a finite set Q of primes satisfying the following conditions: (a) Q ∩ S =∅. (b) For each q ∈ Q, q ≡ 1mod p and ρ(Frob ) has distinct eigenvalues α ,β ∈ k. q q q In this case we can define the following additional data: • The group  = (Z/qZ) (p) (i.e. the maximal p-power quotient of the q∈Q product of the units in each residue field). • The augmented global deformation problem S = (ρ, χ, S ∪ Q, {O} , {D } ), Q v∈S∪Q v v∈S∪Q where for each q ∈ Q, D is the functor all lifts of ρ| of determinant χ . q G The labelling of α , β for each q ∈ Q determines an algebra homomorphism q q O[ ]→ R in the following way: if ρ is a representative of the universal Q S S Q Q deformation, then ρ | is conjugate to a representation of the form A ⊕ B , S G q q Q Q q 108 JAMES NEWTON, JACK A. THORNE where A : G → R is a character such that A mod m is unramified q Q q R q S S Q Q and A mod m (Frob ) = α (and similarly for B ). Then A ◦ Art | × fac- q R q q q q Q S q Z tors through a homomorphism (Z/qZ) (p) → R . These homomorphisms for q ∈ Q collectively determine the algebra homomorphism O[ ]→ R . Q S k−2 1 2 • The cohomology module H = H (Y , Sym O ),where we de- Q U (Mq )∩U (Q) 1 a 2 fine ⎧ ⎫ ⎨ ⎬ ab −1 U (Q) = ∈ GL (Z) : c ≡ 0mod ( q), ad → 1 ∈  , 2 2 Q cd ⎩ ⎭ q∈Q S∪Q S∪Q and commutative O-subalgebras T ⊂ T ⊂ End (H ). By definition, O Q S∪Q T is generated by the unramified Hecke operators T ,S for l ∈ S ∪ Qand l l S∪Q S∪Q T is generated by T and the operators U for q ∈ Q. There are maximal Q Q S∪Q S∪Q ideals m ⊂ T and m ⊂ T with residue field k defined as follows: m is Q Q,1 Q the unique maximal ideal such that for each prime l ∈ S ∪ Q, the characteristic 2 k−1 polynomial of ρ(Frob ) equals X − T X + l S mod m . The ideal m is l l l Q Q,1 generated by m and the elements U − α for q ∈ Q. There is a unique strict Q q q S∪Q equivalence class of liftings ρ : G → GL (T ) of type S such that for m Q 2 m Q Q Q each l ∈ S ∪ Q, the characteristic polynomial of ρ (Frob ) equals the image of m l S∪Q S∪Q 2 k−1 X − T X + l S in T [X]. There is an O-algebra morphism R → T m m l l Q S Q classifying ρ , which is surjective. Moreover, if we view H as an R - m Q,m S Q Q,1 Q module via this map, then the two O[ ]-module structures on H ,one Q Q,m Q,1 arising from R , the other arising from the action of  via Hecke oper- S Q ators, coincide. (These statements in turn may be justified as in the proof of [Tho16, Lemma 6.8].) Finally, H is a free O[ ]-module and there is an Q,m Q Q,1 ∼ ∼ isomorphism H ⊗ O = H of R ⊗ O = R -modules. (This Q,m O[ ] m S O[ ] S Q,1 Q Q Q is again proved in a similar way to [Tho16, Lemma 6.8], using the fact that i k−2 2 H (Y , Sym (O/ ) ) is Eisenstein for i = 1, together with [KT17, U (Mq )∩U (Q) 1 a 0 Corollary 2.7], to justify the freeness.) If l ∈ S, let R ∈ C denote the universal lifting ring representing the local deformation l O problem D . By construction (if l = p) or arguing as in [CHT08, §2.4.1] (if l = p)R is l l a formally smooth O-algebra; if l = p,then R is formally smooth over O of relative dimension 3, while R has relative dimension 4. We set T = S −{p, q } and A = ⊗ R p a l∈T l (the completed tensor product being over O). The T-framed deformation rings R and T T T R are defined (see [Tho16, §5.2]) and there are canonical homomorphisms A → R S S S T T and A → R . S S By the argument of [Kis09, Proposition 3.2.5] and [Tho16, Proposition 5.10], we can find an integer q ≥ 0 with the following property: for each N ≥ 1, there exists a set Q = Q of primes satisfying conditions (a), (b) above and also: N SYMMETRIC POWER FUNCTORIALITY 109 (c) |Q |= q . N 0 (d) For each q ∈ Q , q ≡ 1mod p . T T (e) The algebra map A → R extends to a surjective algebra homomorphism S S A X ,..., X  → R ,where g = q +|T|− 1. 1 g S We choose for each N ≥ 1 a representative ρ of the universal deformation over R S S Q Q N N which lifts ρ . This choice determines an isomorphism R R ⊗ T ,where T is a S S O S Q Q N power series ring over O in 4|T|− 1 variables. We set H = H ⊗ T .Itisafree Q ,m O Q N Q ,1 N N T [ ]-module, and there is an isomorphism H ⊗ O = H of R ⊗ Q T [ ] m T [ ] N Q Q Q N N N O = R -modules. We now come to the essential point of the proof. Let l ∈ S−{p}, and fix a Frobenius lift φ ∈ G . l Q Lemma 8.4. — With our current assumptions, there is a principal ideal I ⊂ R with the l l following property: for any homomorphism f : R → Q , the resulting homomorphism ρ : G → l f Q p l GL (Q ) has the property that the eigenvalues α , β of ρ (φ ) satisfy (α /β ) = 1 for some i = 2 l l f l l l 1,..., n − 1 if and only if f (I ) = 0. Moreover, the quotient R /I has dimension strictly smaller than l l l the dimension of R . T −1 Proof. — Let (r, N) = rec (ι π ), a Weil–Deligne representation that we may as- sume is defined over E. The proof will use the fact that the Jacquet module of π is non-trivial (equivalently, that the Weil–Deligne representation (r, N) is reducible). We recall that the ring R is a formally smooth O-algebra of relative dimension 3. univ Let r : G → GL (R ) be the universal lifting. We can take I to be the ideal generated Q 2 l l l l n−1 univ by the discriminant of the characteristic polynomial of Sym r (φ ). To complete the proof of the lemma, we need to show that dim R /I < dim R . Since R is an integral l l l l domain, it is equivalent to show that I is not the zero ideal. To show this, we split into cases. If π is a twist of the Steinberg representation n−1 then the discriminant of the characteristic polynomial of Sym r(φ ) is non-zero (as the eigenvalues of r(φ ) have eigenvalues whose ratio is a non-zero power of l ), so we see that I is not the zero ideal in this case. Otherwise, N = 0and r = χ ⊕ χ is a direct sum of l 1 2 two characters of W .Let ψ : W → ET be the unramified character which sends ψ Q Q l l −1 to 1 + T; then r = χ ψ ⊕ χ ψ is a deformation of r to ET of determinant χ with 1 2 n−1 the property that the discriminant of the characteristic polynomial of Sym r (φ ) is non-zero in ET. The existence of this deformation, together with [Gee11,Proposition 2.1.5], implies that I cannot be the zero ideal in this case either. T T T T We set I = I A ⊂ A . Then dim A /I = dim A − 1. l∈S−{p} S S S S Suppose for contradiction that for each automorphic representation π contribut- ing to H for some N ≥ 1, there is a prime l ∈ Ssuch that π is ramified and there is an N l 110 JAMES NEWTON, JACK A. THORNE accessible refinement of π which is not n-regular. Then IH = 0. On the other hand, l Q a standard patching argument (cf. [Tho12, Lemma 6.10]) implies the existence of the following objects: • A ring S = T S ,..., S  and an algebra homomorphism S → R = ∞ 1 q ∞ ∞ (A /I)X ,..., X . 1 g • A finite R -module H , which is finite free as S -module. ∞ ∞ ∞ This is a contradiction. Indeed, [KT17, Lemma 2.8] shows that the dimension of H is the same, whether considered as R -or S -module. By freeness, its dimension as ∞ ∞ S -module is dim S = 4|T|+ q . On the other hand, its dimension as R -module is ∞ ∞ 0 ∞ bounded above by dim R = dim A − 1 + g = 4|T|+ q − 1. ∞ 0 We conclude that there exists an automorphic representation π contributing to H for some N ≥ 1 such that for each prime l ∈ Ssuch that π is ramified, each acces- N l sible refinement of π is n-regular. To complete the proof, we just need to explain why π l q is n-regular for each prime q ∈ Q such that π is ramified. However, our construction N q −1 shows that r | has the form C ⊕ C ,where C : I → Q has order a power of p. π ,ι I q q Q Q q q p Since p > 2n, by hypothesis, this is a fortiori n-regular. This completes the proof. We can now finish the proof of Theorem 8.1. Proof of Theorem 8.1.— Let π be a cuspidal automorphic representation of GL (A ) of weight k ≥ 2, without CM, and such that each local component π admits 2 Q l an accessible refinement. Let p, ι,and π be as in the statement of Proposition 8.3.Then n−1 Sym r  is automorphic, by Proposition 8.2. π ,ι On the other hand, our assumptions imply that the residual representation n−1 n−1 Sym r Sym r  is irreducible. We can therefore apply [BLGGT14,Theorem π,ι π ,ι n−1 4.2.1] to conclude that Sym r is automorphic. This completes the proof. π,ι Acknowledgements The influence of the papers [CT14, CT15, CT17] by Laurent Clozel and J.T. on the current work will be clear. We would like to express our warmest thanks to L.C. for many helpful discussions over the years. In particular, the idea of exploiting congruences to a theta series was first discussed during a visit to Orsay by J.T. during the preparation of these earlier papers. We also thank L.C. for helpful comments on an earlier draft of this paper. We thank an anonymous referee for their careful reading and many helpful comments. 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). SYMMETRIC POWER FUNCTORIALITY 111 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK Current address: Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK J. A. T. Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK Manuscrit reçu le 17 juillet 2020 Version révisée le 22 septembre 2021 Manuscrit accepté le 25 septembre 2021 publié en ligne le 15 octobre 2021. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Symmetric power functoriality for holomorphic modular forms

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SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS by JAMES NEWTON and JACK A. THORNE ABSTRACT Let f be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting Sym f for every n ≥ 1. We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over Q. CONTENTS Introduction ........................................................ 1 1. Definite unitary groups ................................................. 9 Part I: Analytic continuation of functorial liftings .................................... 26 2. Trianguline representations and eigenvarieties .................................... 27 3. Ping pong ........................................................ 60 Part II: Raising the level . ................................................. 65 4. Raising the level – automorphic forms ........................................ 68 5. A finiteness result for Galois deformation rings .................................... 75 6. Raising the level – Galois theory ............................................ 86 7. Level 1 case ....................................................... 94 8. Higher levels ...................................................... 104 Acknowledgements ..................................................... 110 References ......................................................... 111 Introduction Context. — Let F be a number field, and let π be a cuspidal automorphic repre- sentation of GL (A ). Langlands’s functoriality principle [Lan70, Question 5] predicts n F the existence, for any algebraic representation R : GL → GL , of a functorial lift of π n N along R; more precisely, an automorphic representation R(π) of GL (A ) which may N F be characterized by the following property: for any place v of F, the Langlands param- eter of R(π) is the image, under R, of the Langlands parameter of π . The Langlands v v parameter is defined for each place v of F using the local Langlands correspondence for GL (F ) (see [Lan89, HT01, Hen00]). n v m th The simplest interesting case is when n = 2and R = Sym is the m symmetric power of the standard representation of GL . In this case the automorphy of Sym π was proved for m = 2 by Gelbart and Jacquet [GJ78]and for m = 3, 4 by Kim and Shahidi [KS02, Kim03]. More recently, Clozel and the second author have proved the automorphy of Sym π for m ≤ 8 under the assumption that π can be realised in a space of Hilbert modular forms of regular weight [CT14, CT15, CT17]; equivalently, that the number field F is totally real and the automorphic representation π is regular algebraic, in the © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2021 https://doi.org/10.1007/s10240-021-00127-3 2 JAMES NEWTON, JACK A. THORNE sense of [Clo90b]. This includes the most classical case of automorphic representations arising from holomorphic modular forms of weight k ≥ 2. We also mention the work of th Dieulefait [Die15], which shows automorphy of the 5 symmetric power for cuspidal Hecke eigenforms of level 1 and weight k ≥ 2. On the other hand, the potential automorphy (i.e. the existence of the symmetric power lifting after making some unspecified Galois base change) of all symmetric powers for automorphic representations π associated to Hilbert modular forms was obtained by Barnet-Lamb, Gee and Geraghty [BLGG11] (the case of elliptic modular forms is due to Barnet-Lamb, Geraghty, Harris and Taylor [BLGHT11]). Results of this paper. — In this paper, we prove the automorphy of all symmetric powers for cuspidal Hecke eigenforms of level 1 and weight k ≥ 2. More precisely: Theorem A. — Let π be a regular algebraic cuspidal automorphic representation of GL (A ) 2 Q of level 1 (i.e. which is everywhere unramified). Then for each integer n ≥ 2, the symmetric power lifting n−1 Sym π exists, as a regular algebraic cuspidal automorphic representation of GL (A ). n Q In fact, we establish a more general result in which ramification is allowed: Theorem B. — Let π be a regular algebraic cuspidal automorphic representation of GL (A ) 2 Q of conductor N ≥ 1, which does not have CM. Suppose that for each prime l|N, the Jacquet module of π is non-trivial; equivalently, that π is not supercuspidal. Then for each integer n ≥ 2, the symmetric l l n−1 power lifting Sym π exists, as a regular algebraic cuspidal automorphic representation of GL (A ). n Q The class of automorphic representations described by Theorem B includes all those associated to holomorphic newforms of level  (N), for some squarefree integer N ≥ 1; in particular those associated to semistable elliptic curves over Q. We can therefore offer the following corollary in more classical language: Corollary C. — Let E be a semistable elliptic curve over Q. Then, for each integer n ≥ 2,the completed symmetric power L-function (Sym E, s) as defined in e.g. [DMW09], admits an analytic continuation to the entire complex plane. We remark that the meromorphic, as opposed to analytic, continuation of the com- pleted L-function (Sym E, s) was already known, as a consequence of the potential au- tomorphy results mentioned above. Potential automorphy results were sufficient to prove the Sato–Tate conjecture, but our automorphy results make it possible to establish effective versions of Sato–Tate (we thank Ana Caraiani and Peter Sarnak for pointing this out to us). See, for example, [Tho14b] for an unconditional result and [Mur85, BK16, RT17] for results conditional on the Riemann Hypothesis for the symmetric power L-functions. In other words, there is no quadratic Hecke character χ such that π π ⊗ χ . = SYMMETRIC POWER FUNCTORIALITY 3 Strategy. — Algebraic automorphic representations of GL (A ) are conjectured to n F admit associated Galois representations [Clo90b]. When F is totally real and π is a self-dual regular algebraic automorphic representation, these Galois representations are known to exist; their Galois deformation theory is particularly well-developed; and they admit p-adic avatars, which fit into p-adic families of overconvergent automorphic forms. We make use of all of these tools. We begin by proving the following theorem: th Theorem D. — Let n ≥ 2 be an integer and suppose that the n symmetric power lifting exists for th one regular algebraic cuspidal automorphic representation of GL (A ) of level 1. Then the n symmetric 2 Q power lifting exists for every regular algebraic cuspidal automorphic representation of GL (A ) of level 1. 2 Q We sketch the proof of Theorem D, which is based on the properties of the Coleman–Mazur eigencurve E .Werecallthatif p is a prime, the eigencurve E is a p p p-adic rigid analytic space that admits a Zariski dense set of classical points correspond- ing to pairs (f ,α) where f is a cuspidal eigenform of level 1 and some weight k ≥ 2and 2 k−1 α is a root of the Hecke polynomial X − a (f )X + p . The eigencurve admits a map κ : E → W = Hom(Z , G ) to weight space with discrete fibres; the image of (f ,α) is p p m k−2 the character x → x . We first show that for a fixed n ≥ 1, the automorphy of Sym f is a property which is “constant on irreducible components of E ”. (Here we confuse f and the automor- phic representation π that it generates in order to simplify notation.) More precisely, if (f ,α) and (f ,α ) determine points on the same irreducible component of E , then the n n automorphy of Sym f is equivalent to that of Sym f . This part of the argument, which occupies §2 of this paper, does not require a restriction to cusp forms of level 1 – see Theo- rem 2.33. It is based on an infinitesimal R = T theorem on the eigenvariety associated to a definite unitary group in n variables. Kisin (for GL )[Kis03] and Bellaïche–Chenevier (for higher rank) [BC09] have observed that such theorems are often implied by the van- ishing of adjoint Bloch–Kato Selmer groups. We are able to argue in this fashion here because we have proved the necessary vanishing results in [NT20]. To exploit this geometric property, we need to understand the irreducible com- ponents of E . This is a notorious problem. However, conjectures predict that E has a p p simple structure over a suitably thin boundary annulus of a connected component of weight space W (see e.g. [LWX17, Conjecture 1.2]). We specialise to the case p = 2, in which case Buzzard–Kilford give a beautifully simple and explicit description of the geometry of E “close to the boundary of weight space” [BK05]. More precisely, E is supported above a single connected component W ⊂ W , 2 2 which we may identify with the rigid unit disc {|w| < 1}. The main theorem of [BK05] −1 ∞ is that the pre-image κ ({|8| < |w| < 1}) decomposes as a disjoint union X of i=1 rigid annuli, each of which maps isomorphically onto {|8| < |w| < 1}.Moreover, X has the following remarkable property: if (f ,α) ∈ X is a point corresponding to a classical modular form, then the p-adic valuation v (α) (otherwise known as the slope of the pair (f ,α)) equals iv (w(κ(f , α))). p 4 JAMES NEWTON, JACK A. THORNE We can now explain the second part of the proof of Theorem D, which occupies −1 §3 of the paper. Since each irreducible component of E meets κ ({|8| < |w| < 1}),it is enough to show that each X contains a point (f ,α) such that Sym f is automorphic. This property only depends on f and not on the pair (f ,α)! Moreover, the level 1 form f −1 determines two points (f ,α), (f ,β) of κ ({|8| < |w| < 1}), which lie on components X and X satisfying i + i = (k − 1)/v (w(κ(f , α))). Starting with a well-chosen initial point i p on a given annulus X , we can jump to any other X in a finite series of swaps between i i pairs (f ,α ), (f ,β ) and moves within an annulus. We call this procedure playing ping pong, and it leads to a complete proof of Theorem D. We remark that for this second step of the proof it is essential that we work with level 1 forms, since it is only in the level 1, p = 2 case that the eigencurve E admits such a simple structure (in particular, the eigencurve is supported above a single connected com- ponent of weight space and every Galois representation appearing in E admits the same residual representation, namely the trivial 2-dimensional representation of Gal(Q/Q) over F ). We note as well that it is necessary to work with classical forms which may be ramified at the prime 2 in order for their weight characters to lie in the boundary annulus of W . We have suppressed this minor detail here. Theorem D implies that to prove Theorem A, it is enough to prove the following result: Theorem E. — For each integer n ≥ 2, there is a regular algebraic cuspidal automorphic repre- n−1 sentation π of GL (A ) of level 1 such that Sym π exists. 2 Q As in the previous works of Clozel and the second author [CT14, CT15, CT17], we achieve this by combining an automorphy lifting theorem with the construction of level-raising congruences. We aim to find f and an isomorphism ι : Q → C such that (writing r : G → GL (Q ) for the p-adic Galois representation associated to f )the f ,ι Q 2 residual representation n−1 Sym r : G → GL (F ) f ,ι Q n p is automorphic; then we hope to use an automorphy lifting theorem to verify that n−1 n−1 Sym r is automorphic, and hence that Sym f is automorphic. In contrast to the f ,ι papers just cited, where we chose r to have large image but p to be small, in order to f ,ι exploit the reducibility of the symmetric power representations of GL in small charac- teristic, here we choose r to have small image, and p to be large. f ,ι More precisely, we choose f to be congruent modulo p to a theta series, so that n−1 r = Ind ψ is induced. In this case Sym r | is a sum of characters, so its residual f ,ι f ,ι G G K automorphy can be verified using the endoscopic classification for unitary groups in n variables. The wrinkle is that the automorphy lifting theorems proved in [ANT20](gen- eralizing those of [Tho15]) require the automorphic representation π of GL (A ) (say) n K verifying residual automorphy to have a local component which is a twist of the Steinberg SYMMETRIC POWER FUNCTORIALITY 5 representation. To find such a π we need to combine the endoscopic classification with the existence of level-raising congruences. In fact, we combine two different level-raising results in order to construct the desired congruences. The first of these, original to this paper, suffices to prove Theorem E in the case that n is odd. The argument is based on a generalization of the following simple observation, which suffices to prove Theorem E in the case n = 3: let q be an odd prime power, and let U (q) denote the finite group of Lie type associated to the outer form of th GL over F .Let p be a prime such that q (mod p) is a primitive 6 root of unity. Then 3 q the unique cuspidal unipotent representation of U (q) remains irreducible on reduction modulo p, and this reduction occurs as a constituent of the reduction modulo p of a generic cuspidal representation of U (q) (see Proposition 1.15). Using the theory of depth zero types, this observation has direct consequences for the existence of congruences between automorphic representations of U . Similar arguments work for general odd n, for carefully chosen global data. We leave a discussion of the (quite intricate) details to §4. The second level-raising result, proved by Anastassiades in his thesis, allows us to n−1 2n−1 pass from the existence of Sym f to the existence of Sym f . We refer to the paper [AT21] for a more detailed discussion. It remains to extend Theorem A to the ramified case, and prove Theorem B.For this we induct on the number of primes dividing the conductor, and use an argument of ‘killing ramification’ as in the proof of Serre’s conjecture [KW09]. Thus to remove a prime l from the level we need to be able to move within a family of l -adic overconvergent modular forms to a classical form of the same tame level, but now unramified at l . This explains our assumption in Theorem B that the Jacquet module of π is non-trivial for every prime l : it implies the existence of a point associated to (a twist of) π on an l -adic eigencurve for every prime l . In a sequel to this paper [NT], we prove a new kind of automorphy lifting theorem for symmetric power Galois representations. This allows us to finally prove a version of Theorem B where the hypothesis that no local component π is supercuspidal is removed. The arguments of [NT] use only fixed weight classical automorphic forms (as opposed to overconvergent automorphic forms) but do require the results of this paper (in particular, Theorem B) as a starting point. Organization of this paper. — We begin in §1 by recalling known results on the clas- sification of automorphic representations of definite unitary groups. We make particular use of the construction of L-packets of discrete series representations of p-adic unitary groups given by Mœglin [Mœg07, Mœg14], the application of Arthur’s simple trace for- mula for definite unitary groups as explicated in [Lab11], and Kaletha’s results on the normalisation of transfer factors (in the simplest case of pure inner forms) [Kal16]. In §2 we study the interaction between the existence of symmetric power liftings of degree n with the geometry of the eigenvariety associated to a definite unitary group in n variables. The basic geometric idea is described in §2.1.In§3 we combine these results 6 JAMES NEWTON, JACK A. THORNE with the explicit description of the tame level 1, p = 2 Coleman–Mazur eigencurve to complete the proof of Theorem D. We then turn to the proof of Theorem E, which rests upon two level-raising results, only the first of which is proved here. The proof of this result is in turn split into two halves; first we give in §4 an automorphic construction of level-raising congruences using types, in the manner sketched above. Then in §6 we establish level-raising congruences of a different kind using deformation theory for residually reducible representations, as developed in [Tho15, ANT20]. These two results are applied in turn to construct our desired level-raising congruences for odd n (Proposition 7.4). A key intermediate result is a finiteness result for certain Galois deformation rings, established in §5, and which may be of independent interest. We use this to control the dimension of the locus of reducible deformations. Finally, we are in a position to prove our main theorems. In §7 we combine the preceding constructions with the main theorem of [AT21] in order to prove Theorem E and therefore Theorem A.In§8, we carry out the argument of ‘killing ramification’ in order to obtain Theorem B. The main technical challenge is to manage the hypothesis of ‘n-regularity’ which appears in our analytic continuation results (see especially Theorem 2.33). To do this we prove a result (Proposition 8.3) which takes a given automorphic representation π and constructs a congruence to an n-regular one π . This may also be of independent interest. Notation. — If F is a perfect field, we generally fix an algebraic closure F/Fand write G for the absolute Galois group of F with respect to this choice. We make the convention that a soluble extension F /F is a (finite) Galois extension with soluble Galois group Gal(F /F). When the characteristic of F is not equal to p, we write : G → Z for the p-adic th cyclotomic character. We write ζ ∈ F for a fixed choice of primitive n root of unity (when this exists). If F is a number field, then we will also fix embeddings F → F extending the map F → F for each place v of F; this choice determines a homomorphism G → G . v F F When v is a finite place, we will write O ⊂ F for the valuation ring, ∈ O for a F v v F v v fixed choice of uniformizer, Frob ∈ G for a fixed choice of (geometric) Frobenius lift, v F k(v) = O /( ) for the residue field, and q = #k(v) for the cardinality of the residue F v v field. When v is a real place, we write c ∈ G for complex conjugation. If S is a finite v F set of finite places of F then we write F /F for the maximal subextension of F unramified outside S and G = Gal(F /F). F,S S If p is a prime, then we call a coefficient field a finite extension E/Q contained inside our fixed algebraic closure Q , and write O for the valuation ring of E, ∈ O for a fixed choice of uniformizer, and k = O/( ) for the residue field. If A is a local ring, we write C for the category of complete Noetherian local A-algebras with residue field A/m . We will use this category mostly with A = EorA = O. If G is a profinite group A SYMMETRIC POWER FUNCTORIALITY 7 and ρ : G → GL (Q ) is a continuous representation, then we write ρ : G → GL (F ) for n n p the associated semisimple residual representation (which is well-defined up to conjugacy). If F is a CM number field (i.e. a totally imaginary quadratic extension of a totally + + real number field), then we write F for its maximal totally real subfield, c ∈ Gal(F/F ) for the unique non-trivial element, and δ + : Gal(F/F ) →{±1} for the unique non- F/F trivial character. If S is a finite set of finite places of F , containing the places at which F/F is ramified, we set F = F and G = Gal(F /F). S F,S S We write T ⊂ B ⊂ GL for the standard diagonal maximal torus and upper- n n n triangular Borel subgroup. Let K be a non-archimedean characteristic 0 local field, and let  be an algebraically closed field of characteristic 0. If ρ : G → GL (Q ) is a con- K n tinuous representation (which is de Rham if p equals the residue characteristic of K), then we write WD(ρ) = (r, N) for the associated Weil–Deligne representation of ρ,and F−ss WD(ρ) for its Frobenius semisimplification. We use the cohomological normalisation × ab of class field theory: it is the isomorphism Art : K → W which sends uniformizers to geometric Frobenius elements. When  = C, we have the local Langlands correspon- dence rec for GL (K): a bijection between the sets of isomorphism classes of irreducible, K n admissible C[GL (K)]-modules and Frobenius-semisimple Weil–Deligne representations over C of rank n. In general, we have the Tate normalisation rec of the local Lang- lands correspondence for GL as described in [CT14, §2.1]. When  = C,wehave T (1−n)/2 rec (π) = rec (π ⊗|·| ). If G is a reductive group over K and P ⊂ G is a parabolic subgroup and π is an G(K) admissible [P(K)]-module, then we write Ind π for the usual smooth induction. If P(K) G(K) G G = C then we write i π for the normalised induction, defined as i π = Ind π ⊗ P P P(K) 1/2 δ ,where δ : P(K) → R is the character δ (x) =| det(Ad(x)| )| (and N is the P >0 P Lie N K P P P unipotent radical of P). × × If ψ : K → C is a smooth character, then we write Sp (ψ) = (r, N) for the n n −1 −1 Weil–Deligne representation on C =⊕ C · e given by r = (ψ ◦ Art ) ⊕ (ψ|·| ◦ i=1 K −1 1−n −1 Art ) ⊕ ··· ⊕ (ψ|·| ◦ Art ) and Ne = 0, Ne = e (1 ≤ i ≤ n − 1). We write St (ψ) 1 i+1 i n K K GL −1/2 GL (K) n n for the unique irreducible quotient of i (ψ ◦ det)δ = Ind ψ ◦ det. We have B B B (K) n n rec (St (ψ)) = Sp (ψ). K n × × If F is a number field and χ : F \A → C is a Hecke character of type A (equiv- alently: algebraic), then for any isomorphism ι : Q → C thereisacontinuous character r : G → Q which is de Rham at the places v|p of F and such that for each finite χ,ι F −1 place v of F, WD(r ) ◦ Art = ι χ | . Conversely, if χ : G → Q is a continuous χ,ι F F v F v p character which is de Rham and unramified at all but finitely many places, then there × × exists a Hecke character χ : F \A → C of type A such that r = χ . In this situation 0 χ,ι we abuse notation slightly by writing χ = ιχ . If F is a CM or totally real number field and π is an automorphic representation of GL (A ), we say that π is regular algebraic if π has the same infinitesimal character as n F ∞ ∗ n an irreducible algebraic representation W of (Res GL ) . We identify X (T ) with Z F/Q n C n 8 JAMES NEWTON, JACK A. THORNE n n in the usual way, and write Z ⊂ Z for the subset of weights which are B -dominant. If ∨ n Hom(F,C) W has highest weight λ = (λ ) ∈ (Z ) , then we say that π has weight λ. τ τ ∈Hom(F,C) When F is CM, the automorphic representation π is said to be conjugate self-dual c ∨ if π π . We refer to [BLGGT14, §2.1] for the more general notion of a polarizable automorphic representation. Note that if π is conjugate self-dual, then (π, δ ) is po- F/F larized and therefore π is polarizable. If π is cuspidal, regular algebraic, and polarizable, then for any isomorphism ι : Q → C there exists a continuous, semisimple representation r : G → GL (Q ) such π,ι F n p p F−ss T −1 that for each finite place v of F, WD(r | ) rec (ι π ) (see e.g. [Car14]). (When π,ι G v v F n = 1, this is compatible with our existing notation.) We use the convention that the Hodge–Tate weight of the cyclotomic character is −1. Thus if π is of weight λ,thenfor any embedding τ : F → Q the τ -Hodge–Tate weights of r are given by π,ι HT (r ) ={λ + (n − 1), λ + (n − 2),...,λ }. τ π,ι ιτ,1 ιτ,2 ιτ,n For n ≥ 1, we define a matrix ⎛ ⎞ ⎜ ⎟ −1 ⎜ ⎟ ⎜ ⎟ = . ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ n−1 (−1) If E/F is a quadratic extension of fields of characteristic 0 then we write θ = θ : −t −1 Res GL → Res GL for the involution given by the formula θ(g) =  c(g)  . E/F n E/F n n We write U ⊂ Res GL for the fixed subgroup of θ .Then U is a quasi-split unitary n E/F n n n group. The standard pinning of GL (consisting of the maximal torus of diagonal matri- ces, Borel subgroup of upper-triangular matrices, and set {E | i = 1,..., n − 1} of root i,i+1 vectors) is invariant under the action of θ and defines an F-pinning of U ,thatwecallits standard pinning. If F is a number field or a non-archimedean local field, then we also write U for the extension of U to a group scheme over O with functor of points n n F −t −1 U (R) ={g ∈ GL (R ⊗ O ) | g =  (1 ⊗ c)(g)  }. n n O E n F n When F is a number field or a local field, we identify the dual group U = GL (C) W , n n F where W acts trivially on GL (C) and an element w ∈ W − W acts by the formula E n c F E −t −1 w · g =  g  (therefore preserving the standard pinning of GL (C)). c n n Given a partition of n (i.e. a tuple (n , n ,..., n ) of natural numbers such that 1 2 k n + n + ··· + n = n), we write L for the corresponding standard Levi sub- 1 2 k (n ,...,n ) 1 k group of GL (i.e. the block diagonal subgroup GL ×· · · × GL ⊂ GL ), and P n n n n (n ,...,n ) 1 k 1 k for the corresponding standard parabolic subgroup (i.e. block upper-triangular matrices with blocks of sizes n ,..., n ). If E is a non-archimedean characteristic 0 local field and 1 k SYMMETRIC POWER FUNCTORIALITY 9 π ,...,π are admissible representations of GL (E), ... ,GL (E), respectively, then we 1 k n n 1 k GL write π × π × ··· × π = i π ⊗ ··· ⊗ π . We write π  ···  π for the irre- 1 2 k 1 k 1 k (n ,...,n ) 1 k k k ducible admissible representation of GL (E) defined by rec ( π ) = ⊕ rec (π );it n E i E i i=1 i=1 is a subquotient of π ×· · · × π . 1 k Given a tuple (n , n ,..., n ) of natural numbers such that 2(n + ··· + n ) + n = 1 2 k 1 k−1 k n, we write M for the Levi subgroup of U given by block diagonal matrices with (n ,...,n ) n 1 k blocks of size n , n ,..., n , n , n ,..., n .Then M is a standard Levi subgroup 1 2 k−1 k k−1 1 (n ,...,n ) (with respect to the diagonal maximal torus of U ), and projection to the first k blocks gives an isomorphism M = (Res GL ×···× Res GL ) × U . We write Q (n ,...,n ) E/F n E/F n n (n ,...,n ) 1 k 1 k−1 k 1 k for the parabolic subgroup given by block upper triangular matrices (with blocks of the same sizes). If F is a non-archimedean characteristic 0 local field and π ,...,π ,π are 1 k−1 k admissible representations of GL (E), ... ,GL (E),U (F), respectively, then we write n n n k−1 k π × π × ··· × π = i π ⊗· · · ⊗ π . 1 2 k 1 k (n ,...,n ) 1. Definite unitary groups In this paper we will often use the following assumptions and notation, which we call the “standard assumptions”: • F is a CM number field such that F/F is everywhere unramified. We note + + × × × this implies that [F : Q] is even (the quadratic character of (F ) \A /O + + F F cutting out F has non-trivial restriction to F for each v|∞ but is trivial on + × (−1) ∈ (F ) ). v|∞ • p is a prime. We write S for the set of p-adic places of F . • S is a finite set of finite places of F , all of which split in F. S contains S . • For each v ∈ S, we suppose fixed a factorization v = v v in F, and write S ={ v | v ∈ S}. Let n ≥ 1 be an integer. Under the above assumptions we can fix the following data: • The unitary group G = GoverF with R-points given by the formula −t (1.0.1)G(R) ={g ∈ GL (R ⊗ F) | g = (1 ⊗ c)(g) }. n F We observe that for each finite place v of F ,G is quasi-split, while for each + + place v|∞ of F ,G(F ) is compact. We use the same formula to extend G to a reductive group scheme over O (this uses that F/F is everywhere unrami- fied). The authors apologize for using the same notation G to denote both an extension of scalars of the algebraic group G and an absolute Galois group. We hope no confusion will arise. 10 JAMES NEWTON, JACK A. THORNE • The inner twist ξ : U → G , given by the formula n,F F −1 ξ(g , g ) = (g , g  ) 1 2 1 2 n with respect to the identifications −t −1 U ={(g , g ) ∈ GL × GL | g =  g  } n,F 1 2 n n 2 n 1 n and −t G ={(g , g ) ∈ GL × GL | g = g }. F 1 2 n n 2 • Aliftof ξ to a pure inner twist (ξ, u) : U → G . We recall (see e.g. [Kal11]) n,F F 1 + that by definition, this means that u ∈ Z (F , U ) is a cocycle such that for −1σ all σ ∈ G +,wehave ξ ξ = Ad(u ). When n is odd, we define u to be the F σ 1 + cocycle inflated from Z (Gal(F/F ), U (F)), defined by the formula u = 1, n 1 u = ( , ). When n is even, we choose an element ζ ∈ F with tr (ζ ) = 0 c n n F/F 1 + and define u to be the cocycle inflated from Z (Gal(F/F ), U (F)),definedby −1 the formula u = 1, u = (ζ  ,ζ  ). (In fact, we will make essential use of this 1 c n n structure only when n is odd.) × × × • We also fix a choice of continuous character μ = μ : F \A → C such that μ| × = δ + ◦ Art + and such that if v is any place of F which is inert over F , F/F F then μ| × is unramified. + 1 + If v is a finite place of F , then the image of the cocycle u in H (F , U ) is trivial (this is true by Hilbert 90 if v splits in F, and true because det u ∈ N F if v is inert F /F v v v in F, cf. [Rog90, §1.9]). Our choice of pure inner twist (ξ, u) therefore determines a + + + U (F )-conjugacy class of isomorphisms ι : G(F ) → U (F ) (choose g ∈ U (F ) such n v n n v v v v −1 c + −1 that g g = u ;then ι is the map induced on F -points by the map Ad(g) ◦ ξ : G → c v v F + c U , which descends to F ). If v splits v = ww in F, then we have an isomorphism ι : n,F + + + G(F ) → GL (F ) (composite of inclusion G(F ) ⊂ (Res + GL )(F ) and canonical n w F/F n v v v projection (Res + GL )(F ) → GL (F )). F/F n n w + + If L /F is a finite totally real extension, then we will use the following standard notation: • We set L = L F. + + • If T is a set of places of F then we write T for the set of places of L lying above T. If w ∈ T lies above v ∈ Tand v splits v = v v in F (in particular, we suppose that we have made a choice of v|v), then we will write w for the unique place of L which lies above both w and v (in which case w splits w = w w in L). We write e.g. S for the set of places of the form w (w ∈ S ). L L We note that formation of G is compatible with base change, in the sense that the group G + is thesameasthe onegiven by formula (1.0.1) relative to the quadratic extension L SYMMETRIC POWER FUNCTORIALITY 11 L/L . The same remark applies to the pure inner twist (ξ, u). When we need to compare + + + trace formulae over F and its extension L /F (a situation that arises in §4), we will use the character μ = μ ◦ N . L F L/F 1.1. Base change and descent – first cases. — In the next few sections we summarise some results from the literature concerning automorphic representations of the group G(A +). We first give some results which do not rely on an understanding of the finer properties of L-packets for p-adic unitary groups at inert places of the extension F/F . Theorem 1.2. — Let σ be an automorphic representation of G(A +). Then there exist a parti- tion n = n + ··· + n and discrete, conjugate self-dual automorphic representations 1 k π ,...,π 1 k of GL (A ),..., GL (A ), n F n F 1 k respectively, with the following properties: (1) Let π = π  ···  π . Then for each finite place w of F below which σ is unramified, π is 1 k w unramified and is the unramified base change of σ . w| + c + −1 (2) For each place v = ww of F which splits in F, π σ ◦ ι . w v (3) For each place v|∞ of F, π has the same infinitesimal character as ⊗ W ,where W is v τ :F →C τ τ ∼ ∼ the algebraic representation of GL (F ) GL (C) such that σ W | + . = = n v n v τ G(F ) Proof. — This follows from [Lab11, Corollaire 5.3]. We call π the base change of σ .If ι : Q → C is an isomorphism, we say that σ is ι-ordinary if π is ι-ordinary at all places w|p in the sense of [Ger19, Definition 5.3]. We note that this depends only on π and the weight of π (equivalently, on σ and σ ). p p ∞ Corollary 1.3. — Let ι : Q → C be an isomorphism. Then there exists a unique continuous semisimple representation r : G → GL (Q ) with the following properties: σ,ι F n (1) For each prime-to-p place w of F below which σ is unramified, r | is unramified. σ,ι G (2) For each place v ∈ S ,r | is de Rham. p σ,ι G c + F−ss T −1 (3) For each place v = ww of F which splits in F, WD(r | ) rec (σ ◦ ι ). σ,ι G v w F w Proof. — This follows from the classification of discrete automorphic representa- tions of GL (A ) [MW89], together with the known existence of Galois representations n F attached to RACSDC (regular algebraic, conjugate self-dual, cuspidal) automorphic rep- resentations of GL (A ) (cf. [AT21, Corollary 3.4]). n F i 12 JAMES NEWTON, JACK A. THORNE We remark that if r | is multiplicity free, then the base change of σ is π = σ,ι G F(ζ ) π ···π ,where each π is a cuspidal automorphic representation of GL (A ). Indeed, 1 k i n F [MW89] shows that a non-cuspidal π would contribute a direct sum of copies of a single Galois representation twisted by powers of the cyclotomic character to r , which gives a σ,ι factor with multiplicity > 1in r | . In particular, if r | is multiplicity free then σ,ι G σ,ι G F(ζ ) F(ζ ) p p π is tempered (as each π is, by the results of [Shi11, Clo13, Car12]). Theorem 1.4. — Let π be a RACSDC automorphic representation of GL (A ). Suppose that n F π is unramified outside S. Then there exists an automorphic representation σ of G(A +) with the following properties: −1 ι (U (O )) n + (1) For each finite place v ∈ S of F, σ = 0. (2) π is the base change of σ . Proof. — This follows from [Lab11, Théorème 5.4]. 1.5. Endoscopic data and normalisation of transfer factors. — To go further we need to use some ideas from the theory of endoscopy, both for the unitary group G and for the twisted group Res GL θ . We begin by describing endoscopic data for G (cf. [Lab11, F/F n §4.2], [Rog90, §4.6]). The equivalence classes of endoscopic data for G are in bijection with pairs (p, q) of integers such that p + q = n and p ≥ q ≥ 0. Define μ = 1, μ = μ. + − We identify μ with characters of the global Weil group W using Art .Thenwecan ± F F write down an extended endoscopic triple E = (H, s,η) giving rise to each equivalence class as follows: • The group H is U × U . p q • s = diag(1,..., 1, −1,..., −1) (with p occurrences of 1 and q occurrences of −1). L L • η : H → G is given by the formulae: η : (g , g )  1 → diag(g , g )  1 ∈ GL (C) = G, 1 2 1 2 n (g , g ) ∈ GL (C) × GL (C) = H; 1 2 p q (1 , 1 )  w → diag(μ q(w)1 ,μ p(w)1 ) w(w ∈ W ) p q (−1) p (−1) q F −1 (1 , 1 )  w → diag( , )  w , p q c p q c where w ∈ W − W is any fixed element. c F F As described in [Lab11, §4.5], a choice of extended endoscopic triple E determines a E + normalisation of the local transfer factor  (v a place of F )uptonon-zeroscalar. We will fix a normalisation of local transfer factors only when n is odd, using the following observations: SYMMETRIC POWER FUNCTORIALITY 13 • The quasi-split group U , with its standard pinning, has a canonical normal- isation of transfer factors. Indeed, in this case the Whittaker normalisation of transfer factors defined in [KS99, §5] is independent of the choice of additive character and coincides with the transfer factor denoted  in [LS87]. • Our choice of pure inner twist (ξ, u) : U → G defines a normalisation of the local transfer factors for G. This normalisation of local transfer factors satisfies the adelic product formula (a very special case of [Kal18, Proposition 4.4.1]). A local transfer factor having been fixed, one can define what it means for a function H ∞ + H ∞ f ∈ C (H(F )) (resp. f ∈ C (H(A +))) to be an endoscopic transfer of a function c v c ∞ + ∞ f ∈ C (G(F )) (resp. f ∈ C (G(A +))). After the work of Waldspurger, Laumon, and c v c ∞ + ∞ Ngô, any function f ∈ C (G(F )) (resp. C (G(A +))) admits an endoscopic transfer c v c (see [Lab11, Théorème 4.3] for detailed references). We next discuss base change, or in other words, endoscopy for the twisted group Res + GL  θ . We will only require the principal extended endoscopic F/F n n L L triple (U , 1 ,η),where η : U → Res + GL is defined as follows: first, identify n n n F/F n Res + GL = (GL (C) × GL (C))  W +,where W + acts through its quotient F/F n n n F F Gal(F/F ) and an element w ∈ W + − W acts by the automorphism (g , g ) → c F F 1 2 −t −t −1 −1 L L ( g  , g  ).Then η : U → Res + GL is given by the formulae: n n n F/F n 2 1 n n t −1 η : (g)  1 → diag(g, g )  1 ∈ GL (C) × GL (C); n n (1 )  w → diag(1 , 1 )  w, (w ∈ W ); n n n F −1 (1 )  w → diag( , )  w . n c n c Following [Lab11, §4.5], we fix the trivial transfer factors in this case. By [Lab11, Lemme ∞ + U 4.1], each function φ ∈ C (Res + GL (F )  θ ) admits an endoscopic transfer φ ∈ F/F n n c v ∞ + ∞ + C (U (F )), and every function in C (U (F )) arises this way. We will follow op. cit. in n n c v c v ∞ + using the following notation: if f ∈ C (U (F )) (or more generally, if U is replaced by a n n c v ∞ + product of unitary groups) then we write f ∈ C (Res + GL (F )  θ ) for any function F/F n n c v that admits f as endoscopic transfer (with respect to the principal extended endoscopic triple defined above). If E = (H, s,η) is one of the extended endoscopic triples for G as above then, H H following [Lab11, §4.7], we set M = Res + H , and write M for the twisted space on F/F F H + M associated to the non-trivial element of Gal(F/F ). Then we may canonically identify H H M = Res + GL × GL and M = (Res + GL × GL )  (θ × θ ).Wewilluse the F/F p q F/F p q p q H ∞ same notation to describe stable base change for M . In particular, if f ∈ C (H(A +)), ∞ H then we will use f ∈ C (M (A +)) to denote a function whose endoscopic transfer (with respect to the principal extended endoscopic triple for M , defined as above) with respect to the trivial transfer factors is f (cf. [Lab11, Proposition 4.9]). Having fixed the above normalisations, we can now formulate some simple propo- sitions. 14 JAMES NEWTON, JACK A. THORNE Proposition 1.6. — Let n ≥ 1 be odd, and let v be an infinite place of F . Suppose given an extended endoscopic triple E = (H, s,η) as above and a Langlands parameter ϕ : W → H such that η ◦ ϕ is the Langlands parameter of an irreducible representation σ of G(F ).Let π be H v the (necessarily tempered, θ -invariant) irreducible admissible representation of H(F ) associated to the ∞ + Langlands parameter ϕ | ,and let f ∈ C (G(F )) be a coefficient for σ . Then there is a sign H W v v F v (v, E,ϕ ) ∈{±1} such that the identity π(f ) = (v, E,ϕ )σ (f ) = (v, E,ϕ ) holds, where H H v v H the twisted trace is Whittaker normalised (cf. [Lab11, §3.6]). Proof. — Let (ϕ ) be the L-packet of discrete series representations of H(F ) associated to ϕ . According to the main result of [Clo82], thereisasign ∈{±1} such H 1 H H that π(f ) = σ (f ). According to [Kal16, Proposition 5.10], there is 1 v,H v σ ∈(ϕ ) v v,H H asign ∈{±1} such that σ (f ) = σ (f ).Wemay take (v, E,ϕ ) = 2 2 v,H v v H σ ∈(ϕ ) v v,H H 1 2 The sign in Proposition 1.6 depends on our fixed choice of pure inner twist (be- cause it depends on the normalisation of transfer factors). We make the following basic + + but important remark, which is used in the proof of Proposition 4.6:let L /F be a finite totally real extension, and let L = L F. Then G + satisfies our standard assumptions, and comes equipped with a pure inner twist by base extension. If v is an infinite place of L , then we have the identity (v, E +,ϕ | ) = (v| +, E,ϕ ). L H W F H + ∞ + Proposition 1.7. — Let n ≥ 1 be odd, let v be a finite place of F ,and let f ∈ C (G(F )). c v Suppose given an extended endoscopic triple E = (H, s,η). (1) Suppose that v is inert in F and that f is unramified (i.e. G(O )-biinvariant). Suppose given an unramified Langlands parameter ϕ : W → H and let σ , σ be the unramified irreducible H v,H v + + representations of H(F ), G(F ) associated to the parameters ϕ , η ◦ ϕ , respectively. Let π be H H v v H + the unramified irreducible representation of M (F ) associated to ϕ | . Then there are identities H W v F H H π(f ) = σ (f ) = σ (f ), where the twisted trace is normalised so that θ fixes the unramified v,H v v v v vector of π.(If π is generic, this agrees with the Whittaker normalisation of the twisted trace.) c L (2) Suppose that v = v v splits in F. Suppose given a bounded Langlands parameter ϕ : W → H + + and let σ , σ be the representations of H(F ), G(F ) associated to the parameters ϕ , η ◦ ϕ , v,H v H H v v respectively (by the local Langlands correspondence rec for general linear groups). Let π be the F v H + H irreducible representation of M (F ) associated to ϕ | . Then there is an identity π (f ) = H W v v F v σ (f ) = σ (f ), where the twisted trace is Whittaker normalised. v,H v v Proof. — It is well-known that these identities hold up to non-zero scalar, which depends on the choice of transfer factor; the point here is that, with our choices, the scalar H H disappears. In the first part, the identity π(f ) = σ (f ) is the fundamental lemma for v,H v v stable base change [Clo90a]. The identity σ (f ) = σ (f ) is the fundamental lemma v,H v v for standard endoscopy [LN08], which holds on the nose because our transfer factors are + + identified, by the isomorphism ι : G(F ) → U (F ), with those defined in [LS87]with v n v v SYMMETRIC POWER FUNCTORIALITY 15 respect to our fixed pinning of U ; this is the ‘canonical normalisation’ of [Hal93]. If π n,F is generic then a Whittaker functional is non-zero on the unramified vector, which gives the final assertion of the first part of the proposition. H H In the second part, the equality π (f ) = σ (f ) is the fundamental lemma for v v,H v v stable base change in the split case, cf. [Rog90, Proposition 4.13.2] (where the result is stated for U(3) but the proof is valid in general). The equality σ (f ) = σ (f ) holds v,H v v because σ can be expressed as the normalised induction of a character twist of σ v v,H + + + (after choosing an appropriate embedding H → G and a parabolic subgroup of G F F F v v v containing H + ) and because the correspondence f → f can in this case be taken to be F v the corresponding character twist of the constant term along H (cf. [Rog90, Lemma 4.13.1] and [Shi11, §§3.3–3.4], noting that our normalisation of transfer factors at the place v in this case agrees on the nose with the analogue of the factor written as  in loc. cit., as follows from the definition in [LS87]). 1.8. L-packets and types for p-adic unitary groups. — Let v be a place of F inert in F. In this section we follow Mœglin [Mœg07, Mœg14] in defining L-packets of tempered representations for the group G(F ) (equivalently, given our choice of pure inner twist, U (F )). We write A(GL (F )) for the set of isomorphism classes of irreducible admissible n v representations of GL (F ) over C,and A (GL (F )) for its subset of tempered rep- n v t n v + + θ resentations. We define A(U (F )) and A (U (F )) similarly. We write A (GL (F )) n t n n v v v and A (GL (F )) for the respective subsets of θ -invariant representations (so e.g. n v θ θ A (GL (F )) is the set of irreducible representations of GL (F ) such that π := π ◦ θ = n v n v π ). Using the local Langlands correspondence rec for GL (F ) (and the Jacobson– F n v Morozov theorem), we can identify A(GL (F )) with the set of GL (C)-conjugacy classes n v n of Langlands parameters, i.e. the set of GL (C)-conjugacy classes of continuous homo- morphisms ϕ : W × SL (C) → GL (C) satisfying the following conditions: F 2 n • ϕ| is semisimple; • ϕ| is algebraic. SL (C) Then A (GL (F )) is identified with the set of parameters ϕ such that the ϕ(W ) is t n v F relatively compact, and A (GL (F )) is identified with the set of conjugate self-dual pa- n v θ θ rameters. We write A (GL (F )) ⊂ A (GL (F )) for the subset of parameters ϕ which n v + n v t t + + extend to a homomorphism ϕ : W × SL (C) → U .Suchanextension,ifitex- 2 n F F v v ists, is unique up to GL (C)-conjugacy (see e.g. [GGP12, Theorem 8.1]). The existence of such an extension ϕ can be equivalently phrased as follows: fix a decomposition l j w ∨ i c n ϕ =⊕ ρ ⊕ σ ⊕ (τ ⊕ τ ) ,where: i∈I j∈J k∈K k i j k • The integers l , m , n are all non-zero. i j k • Each representation ρ , σ , τ is irreducible and no two are isomorphic. i j k w ,∨ w ,∨ c c ∼ ∼ • For each i we have ρ = ρ and for each j we have σ = σ .For each k we i j i j w ,∨ have τ  τ . k 16 JAMES NEWTON, JACK A. THORNE n−1 • For each i, ρ is conjugate self-dual of sign (−1) and for each j , σ is conjugate i j self-dual of sign (−1) , in the sense of [GGP12,p.10]. + + Then an extension ϕ exists if and only if each integer m is even. If the extension ϕ F F v v is discrete, in the sense that Cent(GL (C), im ϕ ) is finite, then l = 1for each i ∈ I n i and the sets J, K are empty. If the parameter ϕ corresponding to a representation π ∈ A (GL (F )) is discrete, then we say that π is θ -discrete. n v + Let S denote the set of equivalence classes of pairs ((n ,..., n ), (π ,...,π )), n 1 k 1 k where (n ,..., n ) is a partition of n and π ,...,π are supercuspidal representations of 1 k 1 k GL (F ), ...,GL (F ), respectively. Two such pairs are said to be equivalent if they n v n v 1 k are isomorphic after permutation of the indices {1,..., k}. Thus we may think of an element of S as a formal sum of supercuspidal representations. We recall (see e.g. [BZ77]) that to any π ∈ A(GL (F )) we may associate the supercuspidal support sc(π) ∈ S , n v n defined by the condition that π occurs as an irreducible subquotient of the representation π × π × ··· × π (notation for induction as defined at the beginning of this paper). 1 2 k Mœglin associates to any element τ ∈ A (U (F )) its extended cuspidal support t n esc(τ ) ∈ S . We do not recall the definition here but note that its definition can be re- duced to the case where τ is supercuspidal, in the following sense: suppose that τ is a subquotient of a representation π × ··· × π × τ = i π ⊗ ··· ⊗ π ⊗ τ , 1 k−1 0 1 k−1 0 (n ,...,n ) 1 k where τ is a supercuspidal representation of U (F ).Then esc(τ ) = sc(π ) +··· + 0 n 1 k v θ θ sc(π ) + esc(τ ) + sc(π ) + ··· + sc(π ). k−1 0 k−1 1 + θ Proposition 1.9. — If τ ∈ A (U (F )) then there is a unique element π ∈ A (GL (F )) t n τ n v + v t such that esc(τ ) = sc(π ). Proof. — [Mœg07, Lemme 5.4] states that there is a unique element π = π ∈ A (GL (F )) such that esc(τ ) = sc(π). We need to explain why in fact π ∈ τ n v A (GL (F )) .[Mœg07, Théorème 5.7] states that this is true when τ is square- n v + integrable. In general, we can find a Levi subgroup M ⊂ U and an irreducible (n ,...,n ) n 1 k square-integrable representation π ⊗ ··· ⊗ π ⊗ τ of M (F ) such that τ is 1 k−1 0 (n ,...,n ) 1 k v a subquotient of i π ⊗ ··· ⊗ π ⊗ τ (see [Wal03, Proposition III.4.1]). Then 1 k−1 0 (n ,...,n ) 1 k θ θ θ π = (π × π ) × (π × π ) × ··· × (π × π ) × π , so the result follows from the τ 1 2 k−1 τ 1 2 k−1 square-integrable case. According to the proposition, there is a well-defined map + θ BC : A (U (F )) → A (GL (F )) t n n v + v t defined by BC(τ ) = π (which might be called stable base change). τ SYMMETRIC POWER FUNCTORIALITY 17 Proposition 1.10. — The map BC is surjective, and it has finite fibres. Proof. — The image of BC contains the θ -discrete representations, and the fibres of BC above such representations are finite, by [Mœg07, Théorème 5.7]. The general case can again be reduced to this one. θ −1 If π ∈ A (GL (F )) , then we define (π) = BC (π). By definition, the sets n v + (π) partition A (U (F )) and therefore deserve to be called L-packets. The following t n proposition is further justification for this. Proposition 1.11. — Let π ∈ A (GL (F )) , and fix an extension π to the twisted group n v + × ∞ GL (F )  θ . Then there are constants c ∈ C such that for any f ∈ C (GL (F )  θ): n v τ n v π(f ) = c τ(f ). τ ∈(π) Proof. — When π is θ -discrete, this is the content of [Mœg07, Proposition 5.5]. In general, (π) admits the following explicit description: decompose π = π × π × π , 1 2 where π ∈ A (GL (F )) and π ∈ A (GL (F )) is θ -discrete. Then (π) is the set 1 t n v 2 n−2n v + 1 t 1 of Jordan–Hölder factors of the induced representations π × τ as τ varies over the set 1 2 2 of elements of (π ). Using the compatibility of transfer with normalised constant terms along a parabolic (see [Mor10, Lemma 6.3.4]) we thus have an identity θ ∼ U π(f ) = (π × π × π ) (f ) = c (π × τ )(f ) 1 2 τ 1 2 1 2 τ ∈(π ) 2 2 for some constants c ∈ C . To prove the proposition, it is enough to show that if τ ,τ ∈ τ 2 2 2 (π ) are non-isomorphic then the induced representations π × τ , π × τ have no 2 1 2 1 Jordan–Hölder factors in common. This follows from [Wal03, Proposition III.4.1]. We now introduce some particular representations of U . These are built out of depth zero supercuspidal representations of U . Accordingly we first introduce some cus- pidal representations of the finite group of Lie type U (k(v)): • We write τ(v) for the unique cuspidal unipotent representation of U (k(v)) (see [Lus77,§9]). • Let k /k(v) be a degree 3 extension, and define C = ker(N : Res G → Res G ). k k( v)/k k k( v)/k(v) m k /k(v) m 3 3 3 3 Then there is a unique U (k(v))-conjugacy class of embeddings C → U (as 3 3,k(v) can be proved using e.g. [DL76, Corollary 1.14]). th Let p be a prime such that q is a primitive 6 root of unity modulo p, and let θ : C(k(v)) → C be a character of order p. Then we write λ(v, θ) 18 JAMES NEWTON, JACK A. THORNE 3,k(v) for the (negative of the) Deligne–Lusztig induction −R θ.Then λ(v, θ) is a cuspidal irreducible representation of U (k(v)) (note that C is not contained in any proper k(v)-rational parabolic of U ). 3,k(v) We define C = Res G . Then the homomorphism C(k( v)) → C(k(v)), z → z/z , k k( v)/k( v) m × c is surjective, and we define a character θ : C(k( v)) → C of order p by θ(z) = θ(z/z ). There is a unique GL (k( v))-conjugacy class of embeddings C → GL , and we write 3 3,k( v) GL 3,k( v) λ( v,θ) for the Deligne–Lusztig induction R θ.Then λ( v,θ) is a cuspidal irreducible representation of GL (k( v)). We now assume that the residue characteristic of k(v) is odd. U (F ) Proposition 1.12. — (1) Let τ = c-Ind τ(v) (compact induction). Then τ is a su- v v U (O ) percuspidal irreducible admissible representation of U (F ) and BC(τ ) = St (χ)  1,where 3 v 2 χ : F → C is the unique non-trivial quadratic unramified character. U (F ) (2) Let λ (θ) = c-Ind λ(v, θ). Then λ (θ) is a supercuspidal irreducible admissible repre- v v U (O ) sentation of U (F ). × × (3) Extend λ( v,θ) to a representation of F GL (O ) by making F act trivially, and let λ (θ) = 3 F v v v v GL (F ) 3 v c-Ind λ( v,θ).Then λ (θ) is a supercuspidal irreducible admissible representation of × v F GL (O ) 3 F v v GL (F ),and BC(λ (θ)) = λ (θ). 3 v v v Proof. — If μ is a cuspidal irreducible representation of U (k(v)),then 0 3 U (F ) v + c-Ind μ is a supercuspidal, irreducible admissible representation of U (F ) (see 0 3 U (O ) v 3 + [MP96, Proposition 6.6] – we will return to this theme shortly). The essential point there- fore is to calculate the extended cuspidal support in each case, which can be done using theresultsof[LS20] (which require the assumption that k(v) has odd characteristic). Indeed §8in op. cit. explains how to compute the reducibility points Red(π) (defined in [Mœg07, §4]) of a depth 0 supercuspidal representation, at least up to unramified twist. We compute that Red(τ ) ={(1, 3/2), (χ, 1)} or {(1, 1), (χ, 3/2)} which corresponds to BC(τ ) = St (1)  χ or BC(τ ) = St (χ)  1. Since BC(τ ) ∈ A (GL (F )) ,the v 2 v 2 v 3 v + second alternative holds. For λ (θ), we deduce that Red(λ (θ)) ={(ρ, 1)},where ρ is v v a conjugate self-dual unramified twist of λ (θ). We again conclude by sign considera- tions. Corollary 1.13. — Let n = 2k + 1 be an odd integer, and consider a representation 2k−2 θ π = St (χ)  1  ( χ ) ∈ A (GL (F )) , 2 i n v + i=1 t where χ : F → C is the unique non-trivial quadratic unramified character and for each i = 1,..., 2k − 2, χ : F → C is a character such that χ | has order 2. We can assume, after i i O w ,∨ relabelling, that χ = χ (i = 1,..., k − 1), and then (π) contains each irreducible subquo- 2k−1−i tient of the induced representation χ × χ ×· · · × χ × τ . 1 2 k−1 v SYMMETRIC POWER FUNCTORIALITY 19 w ,∨ Proof. — First we explain why we can relabel the characters so that χ = χ . 2k−1−i Considering the explicit description of A (GL (F )) , we need to explain why conjugate n v + self-dual characters must appear with even multiplicity amongst the χ . Suppose χ is i 1 conjugate self-dual. We know that χ | is the non-trivial quadratic character, so there are two possibilities for χ determined by χ ( ) =−1 or 1 (this value is also the sign of 1 1 v χ ). If the sign is −1, the multiplicity of χ is one of the even exponents m . If the sign is 1 1 j +1, dimension reasons force its multiplicity to be even. The rest of the Corollary follows from the definition of (π) in terms of extended supercuspidal supports. Note that we do not claim that (π) contains only the subquotients of this induced representation – this is not true even when k = 1. To exploit Corollary 1.13 we need to introduce some results from the theory of types. We state only the results we need, continuing to assume that n = 2k + 1 is odd. Let p denote the standard parahoric subgroup of U (O ) associated to the partition v n (1, 1,..., 1, 3); in other words, the pre-image under the reduction modulo map U (O ) → U (k(v)) of Q (k(v)). Projection to the Levi factor gives a surjective n F n (1,1,...,1,3) × k−1 homomorphism p → M (k(v)) = (k( v) ) × U (k(v)). v (1,1,...,1,3) 3 × k−1 Given a cuspidal representation σ(v) of M (k(v)) (k( v) ) × U (k(v)), (1,1,...,1,3) 3 the pair (p , σ (v)) defines a depth zero unrefined minimal K-type in the sense of × k−1 [MP96]. In this case we write E(σ ) for the set of irreducible representations of (F ) × + k−1 + + + U (O ) ⊂ M (F ) whose restriction to M (O ) = (O ) × U (O ) 3 (1,1,...,1,3) (1,1,...,1,3) 3 F v F F F v v v v is isomorphic to (the inflation of) σ(v). We have the following result. Proposition 1.14. — Let σ(v) be a cuspidal irreducible representation of M (k(v)). (1,1,...,1,3) Then: M (F ) (1,1,...,1,3) v (1) For any σ ∈ E(σ (v)), the compact induction c-Ind σ (v) is irreducible and su- k−1 (F ) ×U (O + ) percuspidal. (2) Let π be an irreducible admissible representation of U (F ). Then π | contains σ(v) if and only n p v v if π is a subquotient of an induced representation M (F ) U (1,1,...,1,3) n v i c-Ind σ (v) Q k−1 (1,1,...,1,3) (F ) ×U (O ) 3 + for some σ ∈ E(σ (v)). Proof. — See [MP96, Proposition 6.6] and [MP96, Theorem 6.11]. We now describe explicitly the two types that we need. Recall that we are assuming that the characteristic of k(v) is odd. Let ω( v) : k( v) →{±1} denote the unique non- trivial quadratic character of k( v) . • The representation τ(v, n) of p inflated from the representation ω( v) ⊗ ··· ⊗ ω( v) ⊗ τ(v) of M (k(v)). (1,1,...,1,3) 20 JAMES NEWTON, JACK A. THORNE • The representation λ(v, θ, n) of p inflated from the representation ω( v) ⊗ ··· ⊗ ω( v) ⊗ λ(v, θ) of M (k(v)) (where θ as above is a character of C(k(v)) of order p,and (1,1,...,1,3) th we assume q mod p is a primitive 6 root of unity). These types are introduced because they are related by a congruence modulo p, because th of our assumption that q mod p is a primitive 6 root of unity: Proposition 1.15. — Fix an isomorphism Q → C and use this to view τ(v, n) and λ(v, θ, n) as representations with coefficients in Q .Then: (1) τ(v, n) is irreducible. (2) λ(v, θ, n) contains τ(v, n) as a Jordan–Hölder factor with multiplicity 1. (As usual, overline denotes semi-simplified residual representation over F .) Proof. — The modular irreducibility of cuspidal unipotent representations is a gen- eral phenomenon (see [DM18]). The proposition is a statement about representations of U (k(v)), which can be proved by explicit computation with Brauer characters; see [Gec90, Theorem 4.2] (although note that there is a typo in the proof: the right-hand side of the first displayed equation should have χ in place of χ 2 ). 1 q −q The following proposition will be a useful tool for exploiting the type (p ,λ(v,θ,n)). ∞ + We introduce an associated test function φ(v,θ, n) ∈ C (U (F )): it is the function sup- c v ported on p and inflated from the character of λ(v, θ, n) .If π is an admissible rep- resentation of U (F ),then π(φ(v, θ, n)) is (up to a positive real scalar depending on normalisation of measures) the dimension of the space Hom (λ(v, θ, n), π | ). p p v v Proposition 1.16. — Assume that the characteristic of k(v) is greater than n. Let φ = φ(v,θ, n),and let E = (H, s,η) be one of our fixed endoscopic triples for U , with H = U × U . n p q θ θ Suppose given representations π , π in A (GL (F )) , A (GL (F )) , respectively, such that p q p v + q v + t t ∼ H (π ⊗ π ) (φ ) = 0. Then sc(π ) + sc(π ) = λ (θ) + χ + ··· + χ ,where χ ,...,χ : p q p q v 1 2k−2 1 2k−2 F → C are characters such that for each i = 1,..., 2k − 2, χ | = ω( v). v O Proof. — By Proposition 1.11, there is an identity ∼ H H (π ⊗ π ) (φ ) = c c (τ ⊗ τ )(φ ) p q τ τ p q p q τ ∈(π ) p p τ ∈(π ) q q × H for some constants c , c ∈ C .Now,[KV12, Theorem 2.2.6] shows that φ can be τ τ p q k(v) −1 ⊗(k−1) taken to be a weighted sum of inflations to H(O ) of characters R (θ ⊗ω( v) ) F C v i SYMMETRIC POWER FUNCTORIALITY 21 k−1 associated to conjugacy classes of embeddings C : C × Res G → H .(Ourap- i k( v)/k(v) k(v) peal to this reference is the reason for the additional assumption on the characteristic of ∼ H k(v) in the statement of the theorem.) If (π ⊗ π ) (φ ) = 0, then there exists a sum- p q mand on the right-hand side such that τ ⊗ τ contains the inflation to H(O ) of the p q F k(v) ⊗(k−1) (irreducible) representation with character −R (θ ⊗ ω( v) ). Taking into account the compatibility between parabolic induction and Deligne–Lusztig induction, the tran- sitivity of Deligne–Lusztig induction [Lus76], and Proposition 1.14, we see that for one of the representations τ , τ (the one for the factor of even rank), the extended cuspidal p q support is a sum of characters of F , each of which is the twist of an unramified character by a ramified quadratic character; and for the other of the representations τ , τ ,the ex- p q tended cuspidal support is a sum of such characters, together with λ (θ). This completes the proof. 1.17. Types for the general linear group. — In this section we record some analogues of the results of the previous section for general linear groups. Let 2 ≤ n ≤ n be an integer. Let v be a finite place of F. We assume that the characteristic of k( v) is odd. We have already introduced the notation ω( v) for the unique non-trivial quadratic character of k( v) . We introduce a further representation of the finite group GL (k( v)) of Lie type: • Let k /k( v) be an extension of degree n , and suppose that q mod p is a prim- n 1 v th × × itive n root of unity modulo p.Let  : k → C be a character of order p. 1 n Then  is distinct from its conjugates by Gal(k /k ), and we write λ( v, ) = n v GL n −1 1 (−1) R  for the Deligne–Lusztig induction. Then λ( v, ) is an Res G k /k( v) m irreducible representation of GL (k( v)). The notation λ( v, ) thus generalises that introduced in the previous section (where n = 3and  = θ ). Proposition 1.18. — Let π be an irreducible admissible representation of GL (F ),and let n v F /F denote an unramified extension of degree n . Then the following are equivalent: v,n v 1 (1) The restriction of π to GL (O ) contains λ( v, ). n F 1 v (2) There exists a continuous character χ : F → C such that χ | × =  and rec π v,n O v v,n −1 Ind (χ ◦ Art ). In particular, π is supercuspidal. W F F v,n v,n 1 Proof. — This follows from the results of [Hen92] (see especially §3.4 of that paper) and [MP96]. Let n = n − n . We write q ⊂ GL (O ) for the standard parahoric subgroup as- 2 1 v n F sociated to the partition (n , n ), i.e. the pre-image under the reduction modulo map 1 2 v GL (O ) → GL (k( v)) of P (k( v)). We write λ( v, , n) for the irreducible repre- n F n (n ,n ) v 1 2 sentation of q inflated from the representation λ( v, ) ⊗ (ω( v) ◦ det) of L (k( v)). v (n ,n ) 1 2 22 JAMES NEWTON, JACK A. THORNE We write r ⊂ q for the standard parahoric subgroup associated to the partition v v (n , 1, 1,..., 1). Then we have the following analogue of Proposition 1.14: Proposition 1.19. — Let π be an irreducible admissible representation of GL (F ). Then the n v following are equivalent: • π | contains λ( v, , n)| . r r v v • sc(π) = π +χ +···+χ ,where π satisfies the equivalent conditions of Proposition 1.18 1 1 n 1 and χ ,...,χ : F → C are characters such that for each i = 1,..., n , χ | = 1 n 2 i 2 v O ω( v). Proof. — This once again follows from the results of [MP96]. The pair (q , λ( v, , n)) is not in general a type (because λ( v, , n) is not a cus- pidal representation of L (k( v)) unless n = 1). Nevertheless, we have the following (n ,n ) 2 1 2 proposition: Proposition 1.20. — Let π be an irreducible admissible representation of GL (F ). Then the n v following are equivalent: (1) The restriction of π to q contains λ( v, , n). (2) There exist irreducible admissible representations π of GL (F ) (i = 1, 2) such that π = π i n v 1 π , the restriction of π to GL (O ) contains λ( v, ), and the restriction of π to GL (O ) 2 1 n F 2 n F 1 v 2 v contains ω( v) ◦ det. We note that in the situation of the proposition, π is the twist of an unramified representation by a quadratic ramified character. Proof. — Let P = P ,L = L ,and let N denote the unipotent radical of P. (n ,n ) (n ,n ) P 1 2 1 2 Abbreviate λ = λ( v, , n) and λ = λ| .If π is an irreducible admissible repre- N L(O ) P F sentation of GL (F ) then we define π = Hom (λ, π | ). We first show that for any n v q q v v λ P admissible representation π of GL (F ), the natural projection π → π (restriction of n v projection to unnormalised Jacquet module) is an isomorphism. Indeed, it is surjective by [Vig98, II.10.1, 1)]. To show that it is injective, let μ = λ| and let R = P , r (n ,1,1,...,1) v 1 N the unipotent radical of R. Then the pair (r , μ) is a depth zero unrefined minimal R v K-type in the sense of [MP96]. We now have a commutative diagram λ P π π μ R π π , R SYMMETRIC POWER FUNCTORIALITY 23 where the left vertical arrow is the natural inclusion and the right vertical arrow is the natural projection to co-invariants. The lower horizontal arrow is an isomorphism, by [Mor99, Lemma 3.6]. We conclude that the top horizontal arrow is injective, and there- fore an isomorphism. Suppose now that π is an irreducible admissible representation of GL (F ) and n v λ μ that π = 0. Then π = 0, so by Proposition 1.19, π is an irreducible subquotient of an induced representation π = π × χ × ··· × χ , where the inducing data is as in the 1 1 n statement of that proposition. Computation of the Jacquet module (using the geometric lemma [BZ77, Lemma 2.12]) shows that (π ) has dimension 1; therefore π must be isomorphic to the unique irreducible subquotient of π which contains λ. This is π × π , 1 2 where π is the unique irreducible subquotient of χ ×··· × χ such that π | 2 1 n 2 GL (O ) 2 n F 2 v contains ω( v) ◦ det (note that π × π is irreducible, by [Zel80, Proposition 8.5]). 1 2 Suppose instead that π = π  π = π × π ,with π , π as in the statement of the 1 2 1 2 1 2 P λ proposition. Then the geometric lemma shows that π = 0, hence π = 0. We now introduce the local lifting ring associated to the inertial type which is the analogue, on the Galois side, of the pair (q , λ( v, , n)) introduced above. We recall that th k /k( v) is an extension of degree n , q mod p is a primitive n root of unity modulo p, n 1 v 1 1 × × and  : k → C is a character of order p.Let ι : Q → C be an isomorphism, so that n p −1 × ι  : k → Q is a character with trivial reduction modulo p. Fix a coefficient field 1 p E and suppose given a representation ρ : G → GL (k) of the form ρ = σ ⊕ σ , F n v,1 v,2 v v v where: • Let F /F be the unramified extension of degree n and residue field k .Then v,n v 1 n 1 1 there is an unramified character ψ : G → k and an isomorphism σ = F v,1 v v,n Ind ψ . G v v,n −1 • σ | ⊗ω( v) ◦ Art is trivial. (In other words, σ is the twist of an unramified v,2 I v,2 F F v v representation by a ramified quadratic character.) We recall that C denotes the category of complete Noetherian local O-algebras with residue field O/ = k. Lemma 1.21. — Let R ∈ C and let ρ : G → GL (R) be a continuous lift of ρ (i.e. a O v F n v v continuous homomorphism such that ρ mod m = ρ ). Then there are continuous lifts σ : G → v R v,i F v v GL (R) of σ (i = 1, 2) with the property that σ ⊕ σ is 1 + M (m )-conjugate to ρ . n v,i v,1 v,2 n R v Moreover, each σ is itself unique up to 1 + M (m )-conjugacy. v,i n R Proof. — The splitting exists and is unique because the groups i i H (F , Hom(σ , σ )) and H (F , Hom(σ , σ )) vanish for i = 0, 1. Compare v v,1 v,2 v v,2 v,1 [Sho18, Lemma 2.3].  24 JAMES NEWTON, JACK A. THORNE Let R ∈ C denote the universal lifting ring, i.e. the representing object of the functor of all continuous lifts of ρ . We write R( v, , ρ ) for the quotient of v v R associated by [Sho18, Definition 3.5] to the inertial type τ : I → GL (Q ), v F n v v p i−1 1 −1 q −1 −1 ⊕n τ =⊕ (ι  ◦ Art ) ⊕ (ω( v) ◦ Art ) . We record the following properties of i=1 F F v v R( v, , ρ ). Proposition 1.22. — (1) The ring R( v, , ρ ) is reduced, p-torsion-free, and is supported on a union of irreducible components of R . In particular, Spec R( v, , ρ ) is O-flat and equidi- mensional of dimension 1 + n . (2) Let x : R → Q be a homomorphism, and let ρ : G → GL (Q ) be the pushforward of x F n v p v p the universal lifting, with its associated direct sum decomposition ρ = σ ⊕ σ . Then x factors x x,1 x,2 through R( v, , ρ ) if and only if there is an isomorphism σ = Ind ψ for a character v x,1 x v,n −1 −1 ψ : G → Q such that ψ | ◦ Art = ι  and σ | ⊗ ω( v) ◦ Art is trivial. x F x I F x,2 I v,n F v,n F F 1 v,n 1 v v (3) Let σ : G → GL (R( v, , ρ )) be the representation associated to the universal lifting by v,1 F n v v 1 Lemma 1.21. There exists α ∈ R( v, , ρ )/( ) such that for any Frobenius lift φ ∈ G , v v F v v 1 n det(X − σ (φ )) ≡ (X − α ) mod . v,1 v (4) Let L /F be a finite extension such that τ | is trivial, and let R denote the universal lifting ring v v v I L L of ρ| . Then the natural morphism R → R( v, , ρ ) (classifying restriction of the universal L v v v ur lifting to G ) factors over the quotient R → R that classifies unramified liftings of ρ | . L G v L L v L v v Proof. — The first two properties follow from [Sho18, Proposition 3.6]. For the n n 1 1 third, let φ be a Frobenius lift. We note that det(X−σ (φ )) = X +(−1) det σ (φ ). v v,1 v v,1 v Indeed, this can be checked at Q -points, at which σ is irreducible, induced from v,1 −1 −1 acharacter of G which extends ι  ◦ Art . Reducing modulo and apply- v,n F v,n ing Hensel’s lemma, we find that there is an element α ∈ R( v, , ρ )/( ) such that i−1 n 1   n 1 det(X − σ (φ )) ≡ (X − q α ) mod .If α = (α ) then det(X − σ (φ )) = v,1 v v v,1 i=1 v v v v (X − α ) . For the fourth part of the lemma, we need to show that the universal lifting is unramified on restriction to G . Since R( v, , ρ ) is reduced, it suffices to check this at each geometric generic point. At such a point σ is irreducible, induced from a charac- v,1 ter of G , while σ is a quadratic ramified twist of an unramified representation. The F v,2 v,n result follows. 1.23. Algebraic modular forms. — Finally, we define notation for algebraic modular forms on the group G. Retaining our standard assumptions, fix a coefficient field E ⊂ Q containing the image of each embedding F → Q , with ring of integers O,and let I n I denote the set of embeddings τ : F → E inducing a place of S .Given λ = (λ ) ∈ (Z ) , p τ τ we write V for the E[ GL (F )]-module denoted W in [Ger19, Definition 2.3]; λ n v λ v∈S it is the restriction to GL (F ) of a tensor product of highest weight representations of n v GL (E). We write V ⊂ V for the O[ GL (O )]-submodule denoted M in loc. n λ λ n F λ v∈S v cit.;itisan O-lattice. SYMMETRIC POWER FUNCTORIALITY 25 In this paper we will only consider algebraic modular forms with respect to open compact subgroups U ⊂ G(A ) which decompose as a product U = U ,and such −1 that for each v ∈ S ,U ⊂ ι GL (O ). Given such a subgroup, together with a finite p v n F v v set  of finite places of F and a smooth O[U ]-module M, finite as O-module, we + ∞ define S (U, M) to be the set of functions f : G(F )\G(A ) → V ⊗ Msuch that for λ λ O each u ∈ Uand g ∈ G(A ), u · f (gu) = f (g).(Here U actson V ⊗ M via projection to λ O U × U .) If λ = 0, we drop it from the notation and simply write S(U, M). We recall the definition of some useful open compact subgroups and Hecke oper- ators (see [Ger19, §2.3] for more details): + c • For any place v of F which splits v = ww in F, the maximal compact subgroup −1 GL (O ).If v ∈  ∪ S ,U = ι GL (O ),and 1 ≤ j ≤ n, then the unramified n F p v n F w w Hecke operator T given by the double coset operator Id 0 w j j −1 v T = ι GL (O ) GL (O ) × U n F n F w w w w 0Id n−j acts on S (U, M). + c • For any place v of F which splits v = ww in F, the Iwahori subgroup Iw ⊂ GL (O ) of matrices which are upper-triangular modulo . n F w • For any place v ∈ S and c ≥ b ≥ 0with c ≥ 1, the subgroup Iw (b, c) ⊂ p v GL (O ) of matrices which are upper-triangular and unipotent upper- n F v v −1 triangular modulo .If U = ι Iw (b, c) for each v ∈ S and 1 ≤ j ≤ n, v v p v v then the re-normalised Hecke operator U of [Ger19, Definition 2.8] acts v,λ on S (U, M). (This Hecke operator depends on our choice of uniformizer . λ v However, the ordinary part of S (U, M), defined below using these operators, is independent of choices.) + c • For any place v of F which splits v = ww in F, the principal congruence subgroup K (1) = ker(GL (O ) → GL (k( v))). v n F n −1 When U = ι Iw (b, c) for each v ∈ S , there is a canonical direct sum decomposition v v p n−ord ord ord S (U, M) = S (U, M) ⊕ S (U, M) with the property that S (U, M) is the largest λ λ λ submodule of S (U, M) where each operator U (v ∈ S , j = 1,..., n)actsinvertible λ p v,λ ([Ger19, Definition 2.13]). We recall some basic results about the spaces S (U, M). We say that U is suffi- ∞ + −1 ciently small if for g ∈ G(A ),the group G(F ) ∩ gUg is trivial. We have the following simple lemma (cf. [Ger19, p. 1351]): Lemma 1.24. — Suppose that U is sufficiently small and that M is O-flat. Then for any c c c ≥ 1,the naturalmap S (U, M) ⊗ O/ → S (U, M/( )) is an isomorphism. λ O λ After fixing an isomorphism ι : Q → C, we can describe the spaces S (U, M) in classical terms ([Ger19, Lemma 2.5]): 26 JAMES NEWTON, JACK A. THORNE Lemma 1.25. — Let ι : Q → C be an isomorphism. Then there is an isomorphism ∨ ∞ S (U, M) ⊗ C = ⊕ m(σ ) Hom ((M ⊗ C) ,σ ) λ O,ι σ U O,ι respecting the action of Hecke operators at finite places away from  ∪ S , where the sum runs over automorphic representations σ of G(A +) such that for each embedding τ : F → C inducing a place v + + of F , σ is the restriction to G(F ) of the dual of the irreducible algebraic representation of GL (C) v n of highest weight λ −1 . ι τ Part I: Analytic continuation of functorial liftings The first part of this paper (§§2 – 3) is devoted to the proof of Theorem D from th the introduction, which shows that the automorphy of the n symmetric power for one th cuspidal Hecke eigenform of level 1 implies the automorphy of the n symmetric power for all cuspidal Hecke eigenforms of level 1. As described in the introduction, the proof has two main ingredients. The first, which is the main result of §2, is that automorphy of symmetric powers can be prop- agated along irreducible components of the Coleman–Mazur eigencurve. The second ingredient, which is explained in §3, uses the main result of [BK05] and has already been sketched in the introduction. Here we make some further introductory remarks on §2.Bymakingasuitable(in particular, soluble) base change to a CM field, we translate ourselves to the setting of definite unitary groups. We start from a classical point z of an eigenvariety for a rank th 2 unitary group, E , such that the n symmetric power of the associated Galois repre- sentation is known to be automorphic. We use Emerton’s construction of eigenvarieties (involving his locally analytic Jacquet functor), and our point of view on eigenvarieties and Galois representations is particularly influenced by those of [BC09]and [BHS17]. Like the authors of [BHS17], we rely in an essential way on the results of [KPX14], which make it possible to spread out pointwise triangulations to global triangulations. We consider the diagram: E X × T 2 ps,2 2 Sym n+1 E X × T n+1 ps,n+1 n+1 Here, E is an eigenvariety for a rank n + 1 unitary group, X is a certain rigid space n ps,d of d -dimensional p-adic Galois pseudocharacters and T is a rigid space parameterising characters of a p-adic torus. Our eigenvarieties come equipped with maps to these char- acter varieties as part of their construction; combining this with the existence of a family SYMMETRIC POWER FUNCTORIALITY 27 of Galois pseudocharacters over the eigenvariety interpolating the global Langlands cor- respondence at classical points gives the closed immersions i appearing in the diagram. th The map Sym corresponds to taking the n symmetric power of the 2-dimensional pseu- docharacter. Our task is to show that if C is an irreducible component of E containing z ,then 2 0 Sym (i (C)) is contained in the image of i . A classicality result (Lemma 2.30) will then 2 n be used to show that for another classical point z of C , its symmetric power Sym (i (z )) 1 2 1 is actually the image of a classical point of E . n+1 To show that Sym (i (C)) is indeed contained in the image of i , we combine a 2 n simple lemma in rigid geometry (Lemma 2.2) with information coming from the local ge- ometry of a certain natural locally closed neighbourhood of Sym (i (z )) in X ×T 2 0 ps,n+1 n+1 which contains open subspaces of both E and Sym (i (C)). This subspace is essentially n+1 2 the trianguline variety, but since we work with spaces of pseudocharacters instead of rep- resentations we restrict to open neighbourhoods in which our pseudocharacters are abso- lutely irreducible and hence naturally lift to representations. Our results on the vanishing of adjoint Selmer groups [NT20] are used to compare E and the trianguline variety. n+1 We proceed in a similar way to the proof of [BC09, Corollary 7.6.11], which shows that vanishing of an adjoint Selmer group implies that i induces an isomorphism between n+1 completed local rings of the eigenvariety and the trianguline variety. 2. Trianguline representations and eigenvarieties Throughout this section, we let p be a prime and let E ⊂ Q be a coefficient field. We write C for the completion of Q .If X is a quasi-separated E-rigid space we let X (Q ) = X (E ), where the union is over finite extensions of E. We can naturally p E ⊂Q view X (Q ) as a subset of the set of closed points of the rigid space X (where base ex- tension of a quasi-separated rigid space is as defined in [BGR84, §9.3.6], see also [Con99, §3.1]). 2.1. An ‘analytic continuation’ lemma. — Suppose given a diagram of E-rigid spaces Y GX , where α is a closed immersion. We identify X with a subspace of G.Let x ∈ Y be a point such that β(x) ∈ X . −1 Lemma 2.2. — Suppose that β (X ) contains an affinoid open neighbourhood of x. Then for each irreducible component C of Y containing x, we have β(C) ⊂ X . 28 JAMES NEWTON, JACK A. THORNE −1 Proof. — We observe that β (X ) ∩C is a Zariski closed subset of C which contains −1 a non-empty affinoid open subset. This forces β (X ) ∩ C = C (apply [Con99, Lemma 2.2.3]), hence β(C) ⊂ X . 2.3. A Galois deformation space. — Let F, S, p be as in our standard assumptions (§1). We assume that E contains the image of every embedding τ : F → Q . 2.3.1. Trianguline deformations – infinitesimal geometry. — This section has been greatly influenced by works of Bellaïche and Chenevier [BC09, Che11]. We use the formalism of families of (ϕ,  )-modules, as in [KPX14]. Thus if v ∈ S and X is an F p E-rigid space, one can define the Robba ring R ;if V is a family of representations of X,F G over X, then the functor D of [KPX14, Theorem 2.2.17] associates to V the family v rig D (V) of (ϕ,  )-modules over X which is, locally on X, finite free over R . We refer F X,F rig v v to [HS16, §2] for the definitions of these objects, as well as more detailed references. If X = Sp A, where A is an E-affinoid algebra, we write R = R .If δ : F → A is a X,F A,F v v v continuous character, we have a rank one (ϕ,  )-module R (δ ) defined by [KPX14, F A,F v v v Construction 6.2.4]. We will also have cause to mention the (ϕ, )-cohomology groups H (−) which are defined in [KPX14, §2.3]. ϕ,γ Let v ∈ S ,and let ρ : G → GL (E) be a continuous representation. If δ = p v F n v n × (δ ,...,δ ) : (F ) → E is a continuous character, we call a triangulation of ρ of v,1 v,n v parameter δ an increasing filtration of D (ρ ) by direct summand (ϕ,  )-stable R - v v F E,F rig v v submodules such that the successive graded pieces are isomorphic to R (δ ), ... , E,F v,1 R (δ ). We say that ρ is trianguline of parameter δ if it admits a triangulation of E,F v,n v v parameter δ .If δ satisfies δ ( ) ∈ O for each i, then we say that δ is an ordinary v v v,i v v −1 parameter. Equivalently, δ is ordinary if δ ◦ Art extends to a continuous character of v v,i G for each i. For an ordinary parameter δ , ρ is trianguline of parameter δ if and only F v v v −1 if ρ has a filtration with successive graded pieces isomorphic to δ ◦ Art ,...,δ ◦ v v,1 v,n −1 Art . We say that the character δ is regular if for all 1 ≤ i < j ≤ n,wehave δ /δ = x v v,i v,j Hom (F ,E) Q v a a v v,τ for any a = (a ) ∈ Z , where by definition x (y) = τ(y) . Note that the v v,τ τ ≥0 τ a a − a v v v,τ characters x satisfy |x (p)| = p , so there is an affinoid cover of the rigid space Hom(F , G ) with each open containing only finitely many x . n reg We define T = Hom((F ) , G ), a smooth rigid space over E, and write T ⊂ v m v v T for the Zariski open subspace of regular characters (Zariski open by the finiteness × n observation in the preceding paragraph). We define W = Hom((O ) , G ) and write v m r : T → W for the natural restriction map. v v v Lemma 2.4. — Let ρ : G → GL (E) be a continuous representation. Then for any δ ∈ v F n v reg T (E), ρ admits at most one triangulation of parameter δ . If such a triangulation exists, then ρ is v v v strictly trianguline of parameter δ in the sense of [KPX14, Definition 6.3.1]. v SYMMETRIC POWER FUNCTORIALITY 29 Proof. — Suppose ρ admits a triangulation of parameter δ ,so D (ρ ) is v v v rig equipped with an increasing filtration Fil . Following [KPX14, Definition 6.3.1], we need 0 −1 to show that for each 0 ≤ i ≤ n the cohomology group H D (ρ )/ Fil (δ ) v i v,i+1 ϕ,γ rig is one-dimensional. It follows from [KPX14, Proposition 6.2.8] that 0 −1 H gr D (ρ ) (δ ) vanishes when i < j and is one-dimensional when i = j . v v,i ϕ,γ j rig The vanishing holds precisely because δ is regular. A dévissage completes the proof. Definition 2.5. — If δ : F → E is a continuous character (hence locally Q -analytic) we let the tuple (wt (δ)) be such that the derivative of δ is the map τ τ ∈Hom (F ,E) F → E x → −wt (δ)τ (x). τ ∈Hom (F ,E) Q v We can extend this discussion to Artinian local rings. Let C denote the category of Artinian local E-algebras with residue field E. If A ∈ C ,then R = R ⊗ A. If A,F E,F E E v v ρ : G → GL (A) is a continuous representation, then D (ρ ) is a free R -module. v F n v A,F v rig v If δ ∈ T (A), we call a triangulation of ρ of parameter δ an increasing filtration of v v v v D (ρ ) by direct summand (ϕ,  )-stable R -submodules such that the successive v F A,F rig v v graded pieces are isomorphic to R (δ ), ... , R (δ ). A,F v,1 A,F v,n v v If ρ : G → GL (E) is a continuous representation and F is a triangulation of v F n v parameter δ ∈ T (E), then we write D : C → Sets for the functor which associates v v ρ ,F ,δ v v v E to any A the set of equivalence classes of triples (ρ , F ,δ ),where: v v v • ρ : G → GL (A) is a lifting of ρ , continuous with respect to the p-adic topol- F n v v v ogy on A. • δ ∈ T (A) is a lifting of δ . v v • F is a triangulation of ρ of parameter δ which lifts F (note that there is a v v v † † † canonical isomorphism D (ρ ) ⊗ E = D (ρ ),as D commutes with base A v rig v rig rig change). Triples (ρ , F ,δ ), (ρ , F ,δ ) are said to be equivalent if there exists g ∈ 1 + M (m ) n A v v v v v v which conjugates ρ to ρ and takes F to F . v v v v We write D for the functor of equivalence classes of lifts ρ : G → GL (A). ρ F n v v Thus forgetting the triangulation determines a natural transformation D → D . ρ ,F ,δ ρ v v v v reg Proposition 2.6. — Suppose that δ ∈ T (E). Then the natural transformation D → v ρ ,F ,δ v v v v D is relatively representable, and injective on A-points for every A ∈ C .If ρ is absolutely irreducible, ρ v then both functors are pro-representable, in which case there is a surjective morphism R → R ρ ρ ,F ,δ v v v v of (pro-)representing objects. Proof. — If F = Q , this is contained in [BC09, Proposition 2.3.6] and [BC09, v p Proposition 2.3.9]. The general case is given by [Nak13, Lemma 2.35, Proposition 2.37, 30 JAMES NEWTON, JACK A. THORNE Corollary 2.38] (noting that Nakamura works with Berger’s category of B-pairs, which is equivalent to the category of (ϕ, )-modules over the Robba ring). A consequence of Proposition 2.6 is that when δ is regular, D (E[ ]) can be v ρ ,F ,δ v v v identified with a subspace of D (E[ ]) = H (F , ad ρ ). Since F is moreover uniquely ρ v v v determined by δ (Lemma 2.4), this subspace depends only on δ , when it is defined. We v v write H (F , ad ρ ) for this subspace. We observe that there is a natural transforma- v v tri,δ tion D → Spf O , which sends a triple (ρ , F ,δ ) to the character r (δ ). ρ ,F ,δ W ,r (δ ) v v v v v v v v v v v Evaluating on E[ ]-points, we obtain an E-linear map H (F , ad ρ ) → T W , v v r (δ ) v tri,δ v v where T W denotes the Zariski tangent space of W at the point r (δ ). This map r (δ ) v v v v v v appears in the statement of the following lemma: Lemma 2.7. — Let ρ : G → GL (E) be a continuous representation. Suppose that: v F n (1) ρ is de Rham. reg (2) ρ is trianguline of parameter δ ∈ T (E). v v (3) For each τ ∈ Hom (F , E), we have Q v wt (δ )<wt (δ )< ··· <wt (δ ). τ v,1 τ v,2 τ v,n We note that the labelled weights wt coincide with the labelled Hodge–Tate weights of ρ (cf. [KPX14, τ v Lemma 6.2.12]). Then the natural map 1 1 ker H (F , ad ρ) → T W → H (F , ad ρ) v r (δ ) v v v v tri,δ has image contained in 1 1 1 H (F , ad ρ) = ker(H (F , ad ρ) → H (F , ad ρ ⊗ B )). v v v Q d R g p Proof. — We must show that if (ρ , F ,δ ) ∈ D (E[ ]) is an element in the ρ ,F ,δ v v v v v v kernel of the map to T W ,then ρ is de Rham. When F = Q , this follows from r (δ ) v v p v v v [BC09, Proposition 2.3.4]; in general it follows from modifying their argument as in [HS16, Proposition 2.6] (the coefficients in this latter result are assumed to be a field, whilst we need coefficients E[ ], but the same proof works with any Artin local E-algebra as coefficient ring). We say that a triangulation of a representation ρ : G → GL (Q ) of parameter v F n v p δ is non-critical if for each τ ∈ Hom (F , E), the labelled weights are an increasing v Q v sequence of integers: wt (δ )<wt (δ )< ··· <wt (δ ). τ v,1 τ v,2 τ v,n SYMMETRIC POWER FUNCTORIALITY 31 In other words, if δ satisfies condition (3) of Lemma 2.7. We now give a criterion for a de Rham representation to have a triangulation satisfying this condition. This generalizes [HS16, Lemma 2.9], which treats the crystalline case. Lemma 2.8. — Let v ∈ S ,and let ρ : G → GL (Q ) be a de Rham representation p v F n satisfying the following conditions: (1) There exists an increasing filtration of the associated Weil–Deligne representation WD(ρ ) (by sub- Weil–Deligne representations) with associated gradeds given by characters χ ,...,χ : W → v,1 v,n F Q . (2) For each embedding τ : F → E,the τ -Hodge–Tate weights of ρ are distinct. v v (3) For each embedding τ : F → E,let k < ··· < k denote the strictly increasing sequence of v τ,1 τ,n τ -Hodge–Tate weights of ρ . Then we have for all τ ∈ Hom (F , E): v Q v v (χ (p)) < k + k p v,1 τ,2 τ ,1 τ =τ and for all i = 2,..., n − 1: i−1 v ((χ ...χ )(p)) < k + k  + k  . p v,1 v,i τ,i+1 τ ,i τ ,j τ =τ τ j=1 Then ρ is trianguline of parameter δ , where for each i = 1,..., n, δ : F → Q is defined by the v v v,i v p −k τ,i formula δ (x) = (χ ◦ Art (x)) τ(x) . In particular, the pair (ρ ,δ ) satisfies condition v,i v,i F v v (3) of Lemma 2.7. Proof. — The filtration of WD(ρ ) determines an increasing filtration 0 = M ⊂ v 0 M ⊂ M ⊂ ··· ⊂ M = D (ρ ) of D (ρ ) by sub-(ϕ, N, G )-modules (via the equiv- 1 2 n pst v pst v F alence of categories of [BS07, Proposition 4.1]). The main result of [Ber08] states that there is an equivalence of tensor categories between the category of filtered (ϕ, N, G )- modules and a certain category of (ϕ,  )-modules (restricting to the usual equivalence between weakly admissible filtered (ϕ, N, G )-modules and (ϕ,  )-modules associated F F v v to de Rham representations). We thus obtain a triangulation of the associated (ϕ,  )- module of ρ . A filtered (ϕ, N, G )-module M of rank 1 is determined up to isomorphism by its associated character χ : W → Q and the (unique) integers a such that F τ v p gr (M ⊗ Q ) = 0. The corresponding rank-1 (ϕ,  )-module is the one F F v F ⊗ Q ,τ ⊗id v v Q p p p −a associated to the character δ : F → Q given by the formula δ = x (χ ◦ Art ) (cf. v p v [KPX14, Example 6.2.6(3)]). What we therefore need to verify is that if a ∈ Z are the τ,i τ,i integers for which gr (M /M ⊗ Q ) = 0, then a = k . i i−1 F ,τ τ,i τ,i v,0 p 32 JAMES NEWTON, JACK A. THORNE This follows from hypothesis (3) of the lemma, together with the fact that D (ρ ) pst v is a weakly admissible filtered (ϕ, N, G )-module, as we now explain. We show by induc- tion on i that the jumps in the induced Hodge–de Rham filtration of M areasclaimed. For M , if these jumps are k then we have for each τ 1 τ,j ⎛ ⎞ 1 1 ⎝ ⎠ k + k  > v (χ ( )) τ,2 τ ,1 p v,1 v [F : Q ] [F : Q ] v p v,0 p τ =τ = t (M ) ≥ t (M ) = k  . N 1 H 1 τ ,j [F : Q ] v p Since the sequences k are strictly increasing, this is possible only if j = 1for each τ .In τ,i τ general, if the jumps of M areasexpectedand M /M has jumps k then we have i−1 i i−1 τ,j for each τ ⎛ ⎞ i−1 ⎝ ⎠ k + k + k > t (M ) τ,i+1 τ ,i τ ,j N i [F : Q ] v p j=1 τ =τ τ ⎛ ⎞ i−1 ⎝ ⎠ ≥ t (Fil ) = k + k . H i τ ,j  τ ,j [F : Q ] v p j=1 τ τ Once again this is possible only if j = i for each τ . Definition 2.9. — We say that a character δ ∈ T (Q ) is numerically non-critical if it satisfies v v the following conditions: (1) For each τ ∈ Hom (F , E), the labelled weights wt (δ ), ..., wt (δ ) are an increasing Q v τ v,1 τ v,n sequence of integers. (2) For each τ ∈ Hom (F , E),and foreach i = 1,..., n − 1, we have Q v v ((δ ...δ )(p)) < wt (δ ) − wt (δ ). p v,1 v,i τ v,i+1 τ v,i Following [BC09, Remark 2.4.6], we may reformulate Lemma 2.8 as follows: let ρ : G → GL (Q ) be a Hodge–Tate regular de Rham representation, and suppose v F n that WD(ρ ) is equipped with an increasing filtration such that the associated gradeds are given by characters χ ,...,χ : W → Q .Let k < ··· < k be the strictly increas- v,1 v,n F τ,1 τ,n v p ing sequences of τ -Hodge–Tate weights, and let δ ∈ T (Q ) be the character defined by −k τ,i the formula δ (x) = (χ ◦ Art (x)) τ(x) .Thenif δ is numerically non-critical, v,i v,i F v the representation ρ admits a non-critical triangulation with parameter δ . v v The most important case for us is that of 2-dimensional de Rham representations of G , and their symmetric powers. In this case the possible triangulations admit a par- p SYMMETRIC POWER FUNCTORIALITY 33 ticularly explicit description (cf. [Col08]; this description can be easily justified, including the case p = 2, using the results of [Ber08]): Example 2.10. — Let ρ : G → GL (Q ) be a de Rham representation with Hodge–Tate Q 2 ss weights k < k such that WD(ρ) = χ ⊕ χ with χ : W → Q . Assume for simplicity that 1 2 1 2 i Q χ = χ . Then we have the following possibilities: 1 2 −1 (1) If ρ is not potentially crystalline, then we can choose χ , χ so that χ = χ (|·| ◦ Art ).In 1 2 1 2 p this case ρ has a unique triangulation. It is non-critical, of parameter −k −k 1 2 δ = (x χ ◦ Art , x χ ◦ Art ). 1 Q 2 Q p p (2) If ρ is potentially crystalline and irreducible, or reducible and indecomposable, ρ has two triangula- tions. Both of these are non-critical, and their respective parameters are −k −k 1 2 δ = (x χ ◦ Art , x χ ◦ Art ) 1 Q 2 Q p p and −k −k 1 2 δ = (x χ ◦ Art , x χ ◦ Art ). 2 Q 1 Q p p (3) If ρ is decomposable, we can assume ρ = ψ ⊕ ψ where ψ has Hodge–Tate weight k and 1 2 i i WD(ψ ) = χ . In this case ρ admits two triangulations. The non-critical triangulation has pa- i i rameter −k −k 1 2 δ = (x χ ◦ Art , x χ ◦ Art ) 1 Q 2 Q p p and the critical triangulation has parameter −k −k 2 1 δ = (x χ ◦ Art , x χ ◦ Art ) 2 Q 1 Q p p (see for example [Ber17, Example 3.7] for the crystalline case). reg reg We now consider the global situation. We define T = T , T = T , v∈S v∈S v p p and W = W . We write r = r : T → W for the product of restriction maps. v v v∈S v∈S p p Let G = (GL × GL )  {±1} denote the group scheme defined in [CHT08, §2.1], ν : n n 1 G G → GL its character, and suppose given a continuous homomorphism ρ : G + → n 1 F ,S 1−n n G (E) such that ν ◦ ρ = δ . We write ad ρ for the E[G + ]-module given by n G + F ,S F/F adjoint action of G on the Lie algebra of GL . We write D : C → Sets for the functor n n ρ which associates to each A ∈ C the set of ker(GL (A) → GL (E))-conjugacy classes of n n 1−n n lifts ρ : G + → G (A) of ρ such that ν ◦ ρ = δ . F ,S n G + F/F If  ⊂ G is a subgroup, then ρ() ⊂ G (E) = GL (E) × GL (E), and we follow F,S n 1 [CHT08] in writing ρ| :  → GL (E) for the projection to the first factor. If v ∈ S, then there is a natural functor D → D , given by restriction ρ → ρ | . ρ ρ| G G F F v v 34 JAMES NEWTON, JACK A. THORNE reg Let δ = (δ ) ∈ T (E) be such that for each v ∈ S , ρ = ρ| is trianguline of v v∈S p v G p F parameter δ .Wedefine afunctor D = D × D ρ,F,δ ρ D ρ ,F ,δ ρ v v v v∈S v v∈S 1 + 1 + and H (F /F , ad ρ) = D (E[ ]) ⊂ H (F /F , ad ρ). S ρ,F,δ S tri,δ Proposition 2.11. — Keeping assumptions as above, suppose further that the following conditions are satisfied: (1) For each v ∈ S , ρ is de Rham. p v (2) For each v ∈ S, WD(ρ| ) is generic, in the sense of [All16, Definition 1.1.2]. (3) For each v ∈ S and for each τ ∈ Hom (F , E),wehave p Q v wt (δ )<wt (δ )< ··· <wt (δ ). τ v,1 τ v,2 τ v,n In other words, the triangulation of ρ with parameter δ is non-critical. v v 1 + (4) H (F , ad ρ) = 0. 1 + Then dim H (F /F , ad ρ) ≤ dim W . E S E tri,δ Proof. — Our assumption that WD(ρ| ) is generic means that for each v ∈ S , G p 1 1 1 1 H (F , ad ρ) = H (F , ad ρ) and for each v ∈ S − S ,H (F , ad ρ) = H (F , ad ρ) (see v v p v v f g f 1 + [All16, Remark 1.2.9]). Lemma 2.7 then implies that the map H (F /F , ad ρ) → tri,δ 1 + T W is injective (its kernel being contained in H (F , ad ρ) = 0). r(δ) 2.11.1. Trianguline representations – global geometry. — We fix a continuous pseu- docharacter τ : G → k of dimension n ≥ 1 which is conjugate self-dual, in the sense F,S ∨ 1−n that τ ◦ c = τ ⊗ . (We define pseudocharacters following Chenevier [Che14], where they are called determinants. For a summary of this theory, including what it means for a pseudocharacter of G to be conjugate self-dual, see [NT20,§2].) Let R denote F,S ps the universal pseudodeformation ring representing the functor of lifts of τ to conjugate self-dual pseudocharacters over objects of C (cf. [NT20, §2.19]). If v ∈ S ,let R de- O p ps,v note the pseudodeformation ring of τ | . We write X for the rigid generic fibre of G ps R ,and X for the rigid generic fibre of R . Then there is a natural morphism ps ps,v ps,v X → X = X of rigid spaces over E. We recall that to any representation ps ps,p ps,v v∈S ρ : G → GL (E) such that tr ρ = τ , and which is conjugate self-dual in the sense that F,S n c ∨ 1−n ρ ρ ⊗ , is associated a closed point tr ρ ∈ X (E). Conversely, if t ∈ X (E),then ps ps there exists a semi-simple conjugate self-dual representation ρ : G → GL (Q ) such F,S n that tr ρ = t, and this representation is unique up to isomorphism. If ρ : G → GL (E) is an absolutely irreducible representation such that ρ F,S n ∨ 1−n ρ ⊗ , then there is a homomorphism ρ : G + → G (E) such that ρ | = ρ and 1 F ,S n 1 G F,S 1−n a ν ◦ ρ = δ . The integer a ∈{0, 1} is uniquely determined by ρ,and anytwo G 1 F/F SYMMETRIC POWER FUNCTORIALITY 35 such extensions are conjugate by an element of GL (Q ).(See[CHT08, Lemma 2.1.4].) The following lemma extends this to objects of C : Lemma 2.12. — Let ρ : G → GL (E) be an absolutely irreducible representation such that F,S n c ∨ 1−n ρ ρ ⊗ , and fix an extension ρ as in the previous paragraph. Let A ∈ C . Then the following 1 E sets are in canonical bijection: (1) The set of ker(GL (A) → GL (E))-conjugacy classes of liftings ρ : G → GL (A) of ρ n n F,S n ∨ 1−n such that tr ρ ◦ c = tr(ρ ) ⊗ . (2) The set of ker(GL (A) → GL (E))-conjugacy classes of liftings ρ : G + → G (A) of ρ n n F ,S n 1 1−n such that ν ◦ ρ = δ . 1 F/F Proof. — There is an obvious map sending ρ to tr ρ | . We need to check that 1 1 F,S this is bijective (at the level of conjugacy classes). To check injectivity, let ρ ,ρ : G + → F ,S 1 1 G (A) be two such homomorphisms and suppose that ρ | , ρ | are ker(GL (A) → n G G n 1 F,S 1 F,S GL (E))-conjugate. We must show that ρ , ρ are themselves conjugate. We may suppose 1 1 that in fact ρ | = ρ | . In this case Schur’s lemma (cf. [CHT08, Lemma 2.1.8]), G G 1 F,S 1 F,S −1 applied to ρ (c) ρ (c),shows that ρ , ρ are equal. 1 1 1 1 Now suppose given ρ : G → GL (A) lifting ρ , and such that tr ρ ◦ c = tr(ρ ) ⊗ F,S n 1−n .Let J ∈ GL (E) be defined by ρ (c) = (J, −ν ◦ ρ (c))j (cf. [CHT08, Lemma n 1 G 1 c −t −1 1−n 2.1.1]), so that ρ(σ ) = Jρ(σ) J ⊗ for all σ ∈ G .Then[Che14, Example 3.4] F,S c   −t  −1 implies the existence of a matrix J ∈ GL (A) lifting J such that ρ (σ ) = J (ρ (σ )) (J ) for all σ ∈ G .By[CHT08, Lemma 2.1.1], this implies the existence of a homo- F,S morphism ρ : G + → G (A) lifting ρ and such that ρ | = ρ . This completes the F ,S n 1 G 1 1 F,S proof. v−irr There is a Zariski open subspace X ⊂ X consisting of those points at which ps,v ps,v p−irr v−irr the universal pseudocharacter is absolutely irreducible. We write X = X ps,p ps,v v∈S p−irr p−irr and X for the pre-image of X . Thus again there is a canonical morphism ps ps,p p−irr p−irr X → X . According to [Che14, §4.2], there exists an Azumaya algebra A over ps ps,p v−irr u × u X and a homomorphism ρ : G → A such that tr ρ is the universal pseudochar- ps,v v v v acter. Lemma 2.13. — Let ρ : G → GL (E) be an absolutely irreducible representation, cor- v F n v−irr responding to a point z = tr ρ ∈ X (E). Then there exists an affinoid open neighbourhood ps,v v−irr z ∈ U ⊂ X (E) and an isomorphism A| M (O ). U n U ps,v Proof. — Let U be an open affinoid neighbourhood of z.The stalk O is a U,z Henselian local ring ([FvdP04, Proposition 7.1.8]). Thus the stalk A is an Azumaya algebra over a Henselian local ring which is split over the closed point; it is therefore split, i.e. there exists an isomorphism A M (O ). After shrinking U , this extends to an z n U,z isomorphism A(U) M (O(U)), as desired. n 36 JAMES NEWTON, JACK A. THORNE p−irr Lemma 2.14. — (1) Let z ∈ X (E) be the closed point corresponding to the isomorphism ps,p class of a tuple (ρ ) of absolutely irreducible representations ρ : G → GL (E). Then there v v∈S v F n p v is a canonical isomorphism p−irr T X ⊕ H (F , ad ρ ). z ps,p v∈S v v p−irr (2) Let z ∈ X (E) be the closed point determined by a representation ρ : G + → G (E) such that ps F ,S n for each v ∈ S , ρ| is absolutely irreducible. Then there is a canonical isomorphism p G p−irr T X = H (G , ad ρ), z F ,S ps which has the property that the diagram p−irr T X H (G , ad ρ) z ps F ,S p−irr T X ⊕ H (F , ad ρ ) z ps,p v∈S v v commutes. Proof. — The first part follows from [Che14, §4.1], which states that the completed v−irr local ring of X at the E-point corresponding to an absolutely irreducible represen- ps,v tation ρ : G → GL (E) pro-represents the functor D . The second follows from this v F n ρ v v and Lemma 2.12. Proposition 2.15. — Let v ∈ S ,and let ρ : G → GL (E) be an absolutely irreducible p v F n reg v−irr reg representation which is trianguline of parameter δ ∈ T (E).Let z ∈ X × T be the closed point v ps,v v corresponding to the pair (ρ ,δ ).Then: v v v−irr reg (1) There exists an affinoid open neighbourhood U ⊂ X × T of z over which there exists a ps,v v universal representation ρ : G → GL (O(U )).Let V ⊂ U denote the set of points (ρ ,δ ) F n v v v v v v such that ρ is trianguline of parameter δ ,and let Z ⊂ U denote the Zariski closure of V . Then v v v v v V is the set of points of a Zariski open subspace of Z . v v (2) The Zariski tangent space of Z at z is contained in the subspace H (F , ad ρ ) of the Zariski v v v tri,δ v−irr reg tangent space of X × T at z. ps,v v v−irr reg Proof. — By Lemma 2.13, there is an affinoid neighbourhood U ⊂ X × T ps,v v of z over which there exists a universal representation ρ : G → GL (O(U )).Wecan F n v v v assume without loss of generality that U is connected. By [KPX14, Corollary 6.3.10], there is a reduced rigid space Z over E and a proper birational morphism f : Z → Z having the following properties: SYMMETRIC POWER FUNCTORIALITY 37 • For every point z ∈ Z , the absolutely irreducible representation ρ  is trian- f (z ) guline. ∗ u • There is an increasing filtration 0 = F ⊂ F ⊂··· ⊂ F = D (f (ρ ))) by 0 1 n rig coherent (ϕ,  )-stable R -submodules. F Z ,F v v −1 • There exists a Zariski closed subset Z ⊂ Z such that Z ∩ f (V ) =∅ and b b for each z ∈ Z − Z , the pullback of F to D (ρ  ) is a triangulation of • f (z ) rig parameter δ . f (z ) • Over Z − Z , F is in fact a filtration by local direct summand R - • Z ,F b v submodules. • The map f factors through a proper birational morphism f : Z → Z ,where Z is the normalisation of Z .Moreover, f factors as the composition of a se- v v quence of proper birational morphisms between normal rigid spaces Z = Z → Z → ··· Z = Z m m−1 1 v where each morphism Z → Z is glued, locally on the target, from analyti- i i−1 fications of birational projective schemes over Spec(A),with Sp(A) ⊂ Z an i−1 affinoid open. Note that the final point is a consequence of the construction in the proof of [KPX14, −1 Theorem 6.3.9]. The third point actually implies that Z = Z f (V ),hence Z = v v f (Z ) V . Since f is proper this shows that V is Zariski open in Z . v v v Let z ˜ ,..., z ˜ ∈ Z be the closed points of the normalisation with image in Z 1 m v v equal to z.For each 1 ≤ j ≤ m,let z be a closed point of the preimage of z ˜ in Z . We denote by z the image of z in any of the Z ,for 1 ≤ i ≤ m. We claim that the map j i O → O on completed local rings is injective; indeed, it follows from the final point Z ,z ˜ Z ,z v j j in the itemized list above that we need to show injectivity for each map O → O . Z ,z Z ,z i j i+1 j Each of these maps coincides with the map on complete local rings A → O associated x X,x with a (projective, birational) morphism of schemes X → Spec(A), where A is the ring of functions on an open affinoid neighbourhood of z ∈ Z , x ∈ Spec(A) is the maximal j i ideal given by z and x ∈ X is a closed point mapping to x. The complete local ring A j x is a domain (by normality and excellence of A )and themap A → O is injective (by x x X,x dominance of X → Spec(A)), so [GD71, Corollaire 3.9.8] gives the desired injectivity. The map O → O is the normalisation of O , so it is also injective. Putting Z ,z Z ,z v Z ,z ˜ v i=1 v i everything together, we have shown that the map O → O is injective. Z ,z Z ,z v i=1 v i After possibly extending E, we can assume that all of the points z in Z have residue field E. The existence of a global triangulation over Z − Z implies that for each i = 1,..., m, there is a classifying map R → O ,where F is the unique ρ ,F ,δ Z ,z v v v v Here we mean the relative analytification defined by Köpf [Köp74], see also [Con06, Example 2.2.11]. 38 JAMES NEWTON, JACK A. THORNE triangulation of ρ of parameter δ . This implies the existence of a commutative diagram v v O O U ,z Z ,z v v R O  , ρ ,F ,δ Z ,z v v v i=1 where the left vertical arrow is surjective (Proposition 2.6) and the top horizontal arrow is surjective. We have already noted that the right vertical arrow is injective. These facts together imply that the top horizontal arrow factors through a surjective map R → ρ ,F ,δ v v v O . This implies the desired result at the level of Zariski tangent spaces. Z ,z Remark 2.16. — Prompted by a referee, we note that the definition of ‘Zariski dense’ given in [KPX14, Definition 6.3.2] is somewhat non-standard. In this paper (and in other references we cite such as [BHS17]), a subset Z of a rigid space X is called Zariski dense if the smallest closed analytic subset of X which contains Z is X. In [KPX14, Definition 6.3.2] the stronger condition is imposed that Z is Zariski dense (in the usual sense) in each member of some admissible affinoid cover of X. When we apply [KPX14, Corollary 6.3.10] in the above proof, we have a Zariski dense subset of an affinoid, so there is no discrepancy between the definitions in this case. Eugen Hellmann has explained to us that the crucial result [KPX14, Theorem 6.3.9] does in fact hold with the weaker, standard definition of Zariski dense. Since it may be of interest, we sketch the argument. We start with X, δ,Mas in [KPX14, Theorem 6.3.9] and suppose we have a Zariski dense (in the usual sense) subset X ⊂ X satisfying the assumptions of loc. cit. We alg may assume that X is normal and connected, and will show that X can be enlarged to alg a subset which is Zariski dense in the stronger sense of [KPX14]. i ∨ i ∨ There are coherent sheaves H (M (δ)),H (M (δ)/t ) on X, which are lo- ϕ,γ ϕ,γ K K cally free over a non-empty (hence dense) Zariski open subset U ⊂ X. At points z in 0 ∨ 0 ∨ the Zariski dense subset X ∩ U, the fibre H (M (δ)) ⊗ k(z) = H (M (δ )) alg O z ϕ,γ X ϕ,γ z K K has dimension one and the map M → R (π )(δ ) dual to a non-zero element of z k(z) K z this fibre is surjective. The latter condition is equivalent to non-vanishing of the map 0 ∨ 0 ∨ H (M (δ)) ⊗ k(z) → H (M (δ)/t ) ⊗ k(z) for every p-adic embedding σ . O σ O X X ϕ,γ ϕ,γ K K These conditions hold over a Zariski open subset U ⊂ U. Since U contains X ∩ U, alg it is also Zariski dense in X. Moreover, U contains a Zariski dense subset of every affi- noid open V ⊂ X. Indeed, the intersection V ∩ U with the Zariski open and dense subset U contains an affinoid open subset of V. Repeating this step, the intersection V ∩ U also contains an affinoid open subset of V. We have shown that we obtain the desired enlargement of X by adjoining U . alg SYMMETRIC POWER FUNCTORIALITY 39 2.17. The unitary group eigenvariety. — Now let F, S, p, G = G be as in our standard assumptions (§1). We continue to assume that E contains the image of every embedding τ : F → Q . In particular, the reductive group Res + G splits over E. F /Q n Let U = U ⊂ G (A ) be an open compact subgroup such that for every n n,v n + v F + + finite place v ∈ SofF ,U is hyperspecial maximal compact subgroup of G (F ).We n,v n ∞,S S 1 n n −1 S S define T = O[T ,..., T ,(T ) ]⊂ O[U \G (A )/U ] to be the algebra generated n w w w n F n c + by the unramified Hecke operators at split places v = ww of F not lying in S. These operators were defined in §1.23. We write T ⊂ B = T N ⊂ GL for the usual maximal torus and upper triangular n n n n n Borel subgroup, and define E-rigid spaces W = Hom( T (O ), G ) n n F m v∈S and T = Hom( T (F ), G ). n n v m v∈S Restriction of characters determines a morphism r : T → W of rigid spaces. Note that n n the spaces T , W may be canonically identified with the spaces T , W of the previous n n section. We fix a choice of isomorphism ι : Q → C.If π is an automorphic represen- tation of G (A ) with π = 0, there is a corresponding semisimple Galois represen- n F tation r : G → GL (Q ) (cf. Corollary 1.3), which satisfies local-global compati- π,ι F,S n −1 ∞ U S bility at each place of F. The space ι (π ) is naturally an isotypic T -module, which therefore determines a homomorphism ψ : T → Q .Wecallan accessible re- n p finement of π a choice χ = (χ ) for each v ∈ S of a (necessarily smooth) character v v∈S p χ : T (F ) → Q which appears as a subquotient of the normalised Jacquet module v n v −1/2 −1 −1 ι r (π ) = ι (π δ ); equivalently, for which there is an embedding of π into N v v,N (F ) v n n v B GL the normalised induction i ιχ .Notethat χ ∈ T (Q ). v n B p Lemma 2.18. — Let π be an automorphic representation of G (A ),and let χ = (χ ) n v v∈S F p T −1 be an accessible refinement of π.Thenfor each v ∈ S , there is an increasing filtration of rec (ι π ) p v by sub-Weil–Deligne representations with graded pieces (1−n)/2 (1−n)/2 χ |·| ◦ Art ,...,χ |·| ◦ Art . v,1 F v,n F v v Proof. — Since π admits an accessible refinement, it is a subquotient of a principal k (n −1)/2 series representation. Suppose that rec (π ) =⊕ Sp (ψ |·| ) for some charac- F v i v i=1 n ters ψ : F → C . By the Langlands classification, π is isomorphic to a subquotient of i v v 40 JAMES NEWTON, JACK A. THORNE the normalised induction GL = i St (ψ ) ⊗ ··· ⊗ St (ψ ), n 1 n k P 1 k where P ⊂ GL is the standard parabolic subgroup corresponding to the partition n = n + n + ··· + n . It will therefore suffice to show the stronger statement that if α = 1 2 k α ⊗ ··· ⊗ α is a subquotient of the normalised Jacquet module of , then there is an 1 n k (n −1)/2 increasing filtration of ⊕ Sp (ψ |·| ) by sub-Weil–Deligne representations with i=1 n (n −1)/2 graded pieces given by α ◦ Art ,...,α ◦ Art . We recall that each Sp (ψ |·| ) 1 F n F i v v comes with a standard basis e , e ,..., e . We concatenate and relabel these bases so that 1 2 n k (n −1)/2 e , e ,..., e is a basis for ⊕ Sp (ψ |·| ) with e j−1 ,... e j the standard basis 1 2 n i i=1 n i 1+ n n i i i=1 i=1 (n −1)/2 for Sp (ψ |·| ). We first treat the case k = 1, n = n. After twisting we can assume that ψ = 1. Then (n−1)/2 (1−n)/2 the normalised Jacquet module of St equals |·| ⊗· · · ⊗|·| , while there is a (n−1)/2 unique invariant flag of Sp (|·| ) given by Fil = span(e ,..., e ) (i = 1,..., n) which i 1 i has the desired graded pieces. Now we return to the general case. Using [Zel80, Theorem 1.2], we see that the irreducible subquotients of the normalised Jacquet module of  are precisely the charac- −1 −1 ters β ⊗ ··· ⊗ β ,where w ∈ S is any permutation which is increasing on each w (1) w (n) of the sets {1,..., n }, {n + 1,..., n + n }, ... , {n +· · · + n + 1,..., n + ··· + n }, 1 1 1 2 1 k−1 1 k (n −1)/2 (1−n )/2 i i and (β ,...,β ) is the concentration of the tuples (ψ |·| ,...,ψ |·| ) for 1 n i i i = 1,..., k. We see that the increasing filtration of k (n −1)/2 ⊕ Sp (ψ |·| ) i=1 n given by Fil = span(e −1 ,..., e −1 ) is a filtration by sub-Weil–Deligne representations j w (1) w (j) which has the desired property. This completes the proof. If χ is an accessible refinement of π,thenwewrite ν(π, χ) ∈ T (Q ) for the char- acter −1/2 −1 (2.18.1) ν(π, χ) = κ(π) · (χ ι δ ) , v v∈S B p where κ(π) ∈ T (E) is the (B -dominant) Q -algebraic character which is the highest n n p −1 weight of ι π .If κ(π) = (κ ≥ κ ≥ ··· ≥ κ ) then the labelled Hodge– ∞ v τ,1 τ,2 τ,n τ :F →Q Tate weights of r | , in increasing order, are (−κ < −κ + 1 < ··· < −κ + n − π,ι G τ,1 τ,2 τ,n 1) . τ :F →Q We write j : T → T forthe mapdefinedbythe formula n n n −1 1−n j (ν) = ν · (1, ◦ Art ,..., ◦ Art ). n v v F F v v The reason for introducing this map is that if π is an automorphic representation of (1−n)/2 G (A +) and χ is an accessible refinement, then the parameter δ associated to χ|·| n F SYMMETRIC POWER FUNCTORIALITY 41 by the formula of Lemma 2.8 satisfies j (ν(π, χ)) = δ. We call the accessible refinement χ numerically non-critical or ordinary if δ is. Note that this property depends on the pair (π, χ) and not just on χ . 2.18.1. Emerton’s eigenvariety construction. — We now describe the construction, fol- lowing Emerton [Eme06b], of the (tame level U ) eigenvariety for G . We use Emerton’s n n construction because we do not want to restrict to considering π with Iwahori-fixed vec- tors at places in S (as is done, for example, in [BC09]) and it seems to us that Emerton’s representation-theoretic viewpoint is the most transparent way to handle this level of generality. We recall the set-up of §1.23, so for each dominant weight λ we have a module n S S (U , O/ ) of algebraic modular forms, which has a natural action of T . When λ is λ n trivial we omit it from the notation. We define p p s S(U , O) := lim lim S(U U , O/ ) n n ← − − → s U and p p S(U , E) := S(U , O) ⊗ E, n n p p so S(U , E) is an E-Banach space (with unit ball S(U , O)), equipped with an admissible n n continuous representation of G (F ). For dominant weights λ, we can consider the space ∨ p V −alg + S of locally V -algebraic vectors S(U , E) .Wehavea (G (F ) × T )-equivariant iso- λ n p n morphism p ∨ p V −alg lim S (U U , O) ⊗ V S(U , E) λ p O n λ n − → (see [Eme06b, Corollary 2.2.25]). We can also consider the space of locally Q -analytic p an vectors S(U , E) , and apply Emerton’s locally analytic Jacquet functor J to this locally n n analytic representation of G (F ). We thereby obtain an essentially admissible locally p an analytic representation J S(U , E) of T (F ), and by duality a coherent sheaf B n v n n v∈S M on T , equipped with an action of T .Wedenoteby A ⊂ End(M ) the coherent n n n n O -algebra subsheaf generated by T . Now we can define the eigenvariety, an E-rigid space, as a relative rigid analytic spectrum E := Sp A − → T n n n equipped with the canonical finite morphism ν . 42 JAMES NEWTON, JACK A. THORNE −1 We define another finite morphism ν : E → T by twisting ν by δ (see Remark n n 2.21). By construction, we also have a ring homomorphism ψ : T → O(E ),soweobtain a map on points: ∗ S ψ × ν : E (Q ) → Hom(T , Q ) × T (Q ). n n p n p p For E /E finite (with E ⊂ Q ), a point (ψ ,ν ) ∈ Hom(T , E ) × T (E ) is in the image of 0 0 n p n ψ × ν if and only if the eigenspace p  an J S(U , E ) [ψ ,ν δ ] B 0 0 B n n n is non-zero, or in other words if there is a non-zero T(F )-equivariant map v∈S p  an ν δ → J S(U , E ) [ψ ]. 0 B B 0 n n n We define the subset Z ⊂ E (Q ) of classical points to be those for which there is more- n n over a non-zero map to the Jacquet module of the locally algebraic vectors: p  alg ν δ → J S(U , E ) [ψ ]. 0 B B 0 n n n S × Lemma 2.19. — For any characters ψ : T → E and χ : T(F ) → E , we have n v∈S p an Hom χ, J S(U , E) [ψ ] T(F ) B v n n v∈S p an = Hom χ, J S(U , E) [ψ ] . T(F ) B v n n v∈S Proof. — This can be seen using Emerton’s canonical lift [Eme06a,Proposition p an 3.4.9], which identifies both sides of the equality with the same eigenspace in S(U , E) . Alternatively, we can use the left exactness of the Jacquet functor. In the latter argument S p an we need to use the fact that T acts on S(U , E) via a Noetherian ring and we then n n deduce that passing to an eigenspace for a (finitely generated) ideal in this ring commutes with the Jacquet functor. We now relate the classical points Z to refined automorphic representations. Let ∞ U A denote the set of automorphic representations π of G (A ) such that (π ) = 0, n n F let RA denote the set of pairs (π, χ) where π ∈ A and χ is an accessible refine- n n ment of π,and let Z ⊂ Hom(T , Q ) × T (Q ) denote the set of points of the form n n n p p (ψ ,ν(π,χ)),where (π, χ) ∈ RA . We note in particular the existence of the surjective π n map γ : RA → Z . n n n Lemma 2.20. — The map ψ × ν restricts to a bijection Z → Z . n n SYMMETRIC POWER FUNCTORIALITY 43 Proof. — If z ∈ Z ∩ E (E ) is a classical point defined over E , it follows from the n n above discussion on locally algebraic vectors that z arises from a non-zero map p  ∗ ∨ ν(z)δ → J lim S (U U , E )[ψ (z)]⊗  V B B λ p E n n n λ − → for some dominant weight λ. It follows from [Eme06a, Prop. 4.3.6] that such maps cor- respond bijectively with non-zero maps ∨ −1 p  ∗ ν(z)δ (λ ) → J lim S (U U , E )[ψ (z)] , B B λ p n n n − → ∨ ∨ where λ is the highest weight of V . By Lemma 1.25,wehave (π, χ) ∈ RA where π has highest weight ιλ , ψ = n ∞ π −1/2 ∗ ∨ ψ (z) and ν(z) = λ χδ = ν(π, χ) . This shows that we do indeed have an induced map Z → Z , and it is easy to see that this is a bijection. n n Remark 2.21. — An accessible refinement χ is numerically non-critical if and only 1/2 if for every v ∈ S the character ν(π, χ) δ = κ(π) χ δ has non-critical slope, in the p v B v v −1/2 1/2 sense of [Eme06a, Defn. 4.4.3]. The renormalisation (replacing χ δ with χ δ )ap- v v B B n n 1/2 pears in Emerton’s eigenvariety construction because χ δ is a smooth character ap- −1 pearing in the (non-normalised) Jacquet module ι π , whilst Bellaïche–Chenevier v,N (F ) n v normalise things to be compatible with the Hecke action on Iwahori-fixed vectors (see [BC09, Prop. 6.4.3]). Our next task is to recall some well known properties of the eigenvariety E (cf. [Bre15, §7]), variants of which are established by numerous authors in slightly differ- ent contexts (e.g. [Che04, Buz07, Eme06b, Loe11]). We follow the exposition of [BHS17] which establishes these properties for the patched eigenvariety. In order to at least sketch the proofs of these properties in our context, we first introduce a ‘spectral variety’ which will turn out to be a Fredholm hypersurface over W . n−1 We fix the element z = (z ) ∈ T (F ) with z = diag( ,..., , 1), v v∈S n v v v p v∈S v and let Y be the closed subgroup of T (F ) generated by T (O ) and z. n v n F v∈S v∈S p p The rigid space Y = Hom(Y, G ) is then identified with W × G .Asin[BHS17, §3.3], m n m p an it follows from [Eme06a, Proposition 3.2.27] that J S(U , E) has dual equal to the n n space of global sections of a coherent sheaf N on W × G .Wedefine Y to be the n n m z −1 schematic support (cf. above Définition 3.6 in [BHS17]) of N (δ ). This rigid space comes equipped with a closed immersion Y "→ W × G . The twist in the definition of z n m Y is there to ensure that this closed immersion is compatible with the map ν.Indeed,the map from E given by composing ν with the restriction map to W × G factors through n n m a finite map f : E → Y , giving us a commutative diagram: n z 44 JAMES NEWTON, JACK A. THORNE E T n n Y W × G z n m Now we state our proposition summarising the key properties of the eigenvari- ety E . Proposition 2.22. — The tuple (E ,ψ,ν,Z ) has the following properties: n n (1) E is a reduced E-rigid space, equipped with a finite morphism ν : E → T . We write κ for the n n n induced map κ : E → W . n n (2) Z ⊂ E (Q ) is a Zariski dense subset which accumulates at every point of Z (in other words, each n n n point of Z admits a basis of affinoid neighbourhoods V such that V ∩ Z is Zariski dense in V), n n ∗ S and the map ψ × ν : E (Q ) → Hom(T , Q ) × T (Q ) restricts to a bijection Z → Z . n n n n p n p p S −1 (3) For any affinoid open V ⊂ T , the map T ⊗ O(V) → O(ν V) is surjective. (4) E is equidimensional of dimension equal to dim W . For any irreducible component C ⊂ E , κ(C) n n n is a Zariski open subset of W . (5) Let z ∈ E (Q ) be a point, and suppose that δ = j (ν(z)) factors as δ = δ δ ,where δ is a n n alg sm alg strictly dominant algebraic character and δ is smooth, and that δ is numerically non-critical. Then sm z ∈ Z . ≤1 (6) ψ takes values in the subring O(E ) of bounded elements. Proof. — First we note that a tuple satisfying the first three properties is unique – we will not actually use this fact, but it can be proved in the same way as [BC09,Proposition 7.2.8] (our context is slightly different, as we equip our eigenvarieties with a map to T instead of W × G ). We also note that it is not essential for our purposes to show that E n m n is reduced (this is the most delicate of the listed properties); we could instead replace E with its underlying reduced subspace. Now we summarise how to verify these properties. Property (5) follows from Emer- ton’s ‘classicality criterion’ for his Jacquet functor [Eme06a, Theorem 4.4.5] (cf. Remark 2.21). Property (3) holds by construction. Property (4) can be established as in [BHS17, §3.3] using the spectral variety Y . More precisely, (the proof of) Lemma 3.10 in this reference shows that the closed analytic subset of W × G underlying Y is a Fredholm hypersurface, and Y has an admisible n m z z cover by affinoids (U ) on which the map to W is finite and surjective with image an i∈I n open affinoid W ⊂ W . Moreover, each U is disconnected from its complement in the i n inverse image of W and (U , N ) is a finite projective O (W )-module. i n W i i n Having established the existence of a good affinoid cover of the spectral variety, −1 we set U = f (U ). Since f is a finite map, (U ) is an admissible affinoid cover of i i i∈I E . It can then be shown, as in [BHS17, Proposition 3.11], that each affinoid O (U ) is n E i n SYMMETRIC POWER FUNCTORIALITY 45 isomorphic to a O (W )-algebra of endomorphisms of the finite projective O (W )- W i W i n n module (U , M ).Wecan nowprove Property (4)asin[BHS17, Corollaire 3.12]: this i n shows that E is equidimensional of dimension equal to W , without embedded compo- n n nents, and each irreducible component maps surjectively to an irreducible component of Y . Since irreducible components of Fredholm hypersurfaces are again Fredholm hyper- surfaces, the image of such an irreducible component is Zariski open in W (cf. [BHS17, Corollaire 3.13]). Now to establish property (2), using property (5), it suffices to show that points z ∈ E (Q ) with numerically non-critical δ = j (ν(z)) accumulate at any point z with κ(z ) n n 0 0 locally algebraic (cf. [BHS17, Théorème 3.19]). Using the good affinoid cover described in the previous paragraph, we have an affinoid neighbourhood U of z which is a finite cover of an affinoid W ⊂ W . In fact, such U form a neighbourhood basis at z (cf. [Taï16, n 0 Theorem 2.1.1, Lemma 2.1.2]). The valuations v (δ (p)) are bounded as z varies in U p v,i (with δ = j (ν(z))). It follows from the description in Definition 2.9 that there is a subset −1 of W accumulating at κ(z ) such that κ () ∩ U consists entirely of points with n 0 −1 numerically non-critical δ. The subset κ () ∩ U is Zariski dense in U. Finally, to establish property (1) it remains to prove that E is reduced. Since we showed that E is without embedded components, it suffices to prove that every irre- ducible component of E contains a reduced point. Using (4) and the Zariski density of algebraic characters in W , it suffices to show that E is reduced at every point z with n n 0 κ(z ) algebraic. We use a good affinoid neighbourhood U = Sp(B) of z as in the pre- 0 0 vious paragraph, with W = κ(U) = Sp(A). The finite A-algebra B is identified with a sub-A-algebra of End (M),where M = (U, M ) is a finite projective A-module. As A n in the proof of [Che05, Proposition 3.9], it now suffices to show that for w in a Zariski dense subset of W, the Hecke algebra T and T (F ) act semisimply on the fibre n v v∈S M ⊗ k(w) — we use the fact that an endomorphism of a projective A-module which vanishes in the fibres at a Zariski dense subset of points in W necessarily vanishes. The proof of [BHS17, Corollaire 3.20] shows that we can achieve this by choosing w so that their pre-images in U have très classique associated characters δ [BHS17, Définition 3.17] (this is a condition on characters with algebraic image in W which can be guaranteed by a ‘numerical’ condition as in the proof of [BHS17, Théorème 3.19], in particular it gives a Zariski-dense and self-accumulating subset of E ). We can replace the reference + S to [CEG 16] in the proof with the well-known assertion that the Hecke algebra T acts semisimply on lim S (U U , E) for dominant λ. Finally, property (6) follows from the λ p − → S p p fact that the T -action stabilizes the unit ball S(U , O) ⊂ S(U , E). n n n The properties established in Proposition 2.22 imply the existence of a conjugate self-dual Galois pseudocharacter T : G → O(E ) with the property that for any point n F,S n z ∈ Z corresponding to a pair (π, χ),T = tr r . This is proved as in [BC09,Proposi- n n,z π,ι ≤1 tion 7.5.4] and [Che04, Proposition 7.1.1]. The key points are that O(E ) is compact ≤1 [BC09, Lemma 7.2.11] and the map O(E ) → C given by the evaluation maps n p z∈Z n 46 JAMES NEWTON, JACK A. THORNE at each z ∈ Z is a continuous injection (by Zariski density of Z and reducedness of E ). n n n Then [Che14, Example 2.32] is used to glue together the pseudocharacters tr r to form π,ι the continuous pseudocharacter T . The pseudocharacter T determines an admissible cover E = E (τ ) as a dis- n n τ n n joint union of finitely many open subspaces indexed by G -orbits of pseudocharacters τ : G → F , over which the residual pseudocharacter satisfies the condition T = τ n F,S p n,z n (cf. [Che14, Theorem 3.17]). Fix a pseudocharacter τ : G → F . Extending E if necessary, we may assume n F,S p that τ takes values in k. We recall some of the E-rigid spaces of Galois representations defined in §2.11.1, now decorated with n subscripts. Thus X is the space of conjugate ps,n p−irr self-dual deformations of τ , X is its Zariski open subspace of pseudocharacters which n ps,n irr are irreducible at the p-adic places. We also have the subspace X of pseudocharacters ps,n which are (globally) irreducible. The existence of T determines a morphism λ : E(τ ) → n n X , and the morphism i = λ × (j ◦ ν) : E (τ ) → X × T is a closed immersion, by ps,n n n n ps,n n point (3) in the list of defining properties of E . Now assume that n ≥ 3, and let τ : G → F be a conjugate self-dual pseu- 2 F,S p n−1 docharacter of dimension 2. Let τ = Sym τ ;then τ is a conjugate self-dual pseu- n 2 n docharacter of G of dimension n. Taking symmetric powers of pseudocharacters de- F,S termines a morphism σ : X → X . On the other hand, we can define a map n,g ps,2 ps,n σ : T → T by the formula n,p 2 n n−1 n−2 n−1 ((δ ,δ )) → ((δ ,δ δ ,...,δ )) . v,1 v,2 v∈S v,2 v∈S p v,1 v,1 v,2 p We write σ = σ × σ : X × T → X × T for the product of these two morphisms. n n,g n,p ps,2 2 ps,n n We have constructed a diagram σ ◦i i n 2 n E (τ ) X × T E (τ ). 2 2 ps,n n n n Compare Lemma 2.2. Definition 2.23. — Let π be an automorphic representation of G (A +),let χ = (χ ) be 2 F v v∈S an accessible refinement of π,and let n ≥ 2. We say that χ is n-regular if for each v ∈ S the character χ = χ ⊗ χ satisfies (χ /χ ) = 1 for 1 ≤ i ≤ n − 1. v v,1 v,2 v,1 v,2 Theorem 2.24. — Let (π ,χ ) ∈ RA satisfy tr r = τ ,and let z = γ (π ,χ ) ∈ 2 2 2 π ,ι 2 2 2 2 2 E (τ )(Q ). Suppose that: 2 2 (1) The refinement χ is numerically non-critical and n-regular. (2) There exists (π ,χ ) ∈ RA such that (σ ◦ i )(z ) = i (z ),where z = γ (π ,χ ). n n n n 2 2 n n n n n n (3) For each v ∈ S , the Zariski closure of r (G ) (in GL /Q ) contains SL . p π ,ι F 2 2 2 v Then each irreducible component C of E (τ ) containing z satisfies (σ ◦ i )(C) ⊂ i (E (τ ) ). 2 2 C 2 n 2 n n n C p p Proof. — Extending E (the field over which E is defined) if necessary, we may as- sume that z ∈ E (τ )(E) and r takes values in GL (E).By[Con99, Theorem 3.4.2] 2 2 2 π ,ι 2 2 SYMMETRIC POWER FUNCTORIALITY 47 (which says that an irreducible component of E (τ ) is contained in the base change of 2 2 C an irreducible component of E (τ )), it suffices to show that each irreducible component 2 2 C of E (τ ) containing z satisfies (σ ◦ i )(C) ⊂ i (E (τ )). By Lemma 2.2,itisenoughto 2 2 2 n 2 n n n −1 show that (σ ◦ i ) (i (E (τ ))) contains an affinoid open neighbourhood of z .Toprove n 2 n n n 2 this, we will use a number of the results established so far. Since χ is a numerically non-critical refinement, the parameter δ of the associ- 2 2 ated triangulation is non-critical, in the sense that for each τ ∈ Hom(F, E), the sequence of τ -weights of δ is strictly increasing (Lemma 2.8). Passing to symmetric powers, we see that r is trianguline of parameter δ = σ (δ ),and that δ is non-critical (although it is π ,ι n n,p 2 n not necessarily numerically non-critical). reg The n-regularity of χ implies that δ ∈ T (E). We are going to apply Propo- 2 n 1 + sition 2.11 to conclude that dim H (F /F , ad r ) ≤ dim W = dim E (τ ).Note E S π ,ι n n n tri,δ that if v ∈ SthenWD(r | ) is generic, because it is pure: the base change of π π ,ι G n n F to GL (A ) exists (for example, by [Lab11, Corollaire 5.3]) and is cuspidal (because n F r is irreducible), so we can appeal to the main theorem of [Car12] (which estab- π ,ι lishes the general case; under various additional hypotheses, purity was established in [HT01, TY07, Shi11, Clo13]). If v ∈ S , let us write f : X × T → X × T for p v ps,n n ps,n,v v the natural restriction map. By Proposition 2.15 and Lemma 2.14,wecan findfor each v ∈ S an affinoid open neighbourhood U ⊂ X × T of the point f (z ) such that the p v ps,n,v v v n following properties hold: v−irr reg u • In fact, U ⊂ X × T and there exists a universal representation ρ : G → v F ps,n,v v v v GL (O(U )) over U . n v v • Let Z ⊂ U denote the Zariski closure of the set V ⊂ U of points correspond- v v v v ing to pairs (ρ ,δ ) such that ρ is trianguline of parameter δ .Thenthe Zariski v v v v tangent space of Z at f (z ) is contained in H (F , ad r | ). v v n v π ,ι G tri,δ n F We can then find an affinoid open neighbourhood U ⊂ X × T of the point z such ps,n n n that the following properties hold: −1 u • U ⊂∩ f (U ) and there exists a universal representation ρ : G → v∈S v F,S p v GL (O(U)) over U . −1 • Let Z = U ∩ (∩ f (Z )).Then Z is a closed analytic subset of U and the v∈S v 1 + Zariski tangent space of Z at the point z is contained in H (F /F , ad r ). n S π ,ι tri,δ 1 + By the main theorem of [NT20], we have H (F , ad r ) = 0, so Proposition π ,ι 2.11 implies that the Zariski tangent space of Z at point z has dimension at most dim W . Let U = E (τ ) ∩ U.Then U is an affinoid open neighbourhood of z in E (τ ).We n n n n n note that if z = γ (π ,χ ) ∈ U ,where χ is a numerically non-critical refinement, then n n n n z ∈ Z (by Lemma 2.8, and the definition of Z ). Such points accumulate at z , implying that every irreducible component of U containing the point z is contained in Z.In particular, Z contains an affinoid open neighbourhood of z in U ,sowehave dim Z ≥ n 48 JAMES NEWTON, JACK A. THORNE dim E (τ ) = dim W . It follows that dim Z = dim W ,that O is a regular local ring, n n n n Z,z and that Z is smooth at the point z . Consequently, U and Z are locally isomorphic at z , E (τ ) is smooth at the point z ,and E (τ ) has a unique irreducible component n n n n n n passing through z . Applying Lemma 2.2, we can also deduce that the unique irreducible component Z of Z containing z is contained in U . −1 Now let U = (σ ◦ i ) (U),and let g = (σ ◦ i )| : U → U.Then U is an n 2 n 2 U −1 admissible open of E (τ ),and g (Z) ⊂ U is a non-empty closed analytic subset. Let 2 2 (π ,χ ) ∈ RA be a pair such that χ satisfies the analogue of property (1) in the theorem. 2 2 2 Arguing again as in the second paragraph of the proof, we see that if the point (σ ◦ i )(γ (π ,χ )) lies in U , then it in fact lies in Z . Since such points accumulate at z , 2 2 2 2 2 −1 we see that g (Z ) contains each irreducible component of U which passes through z (and hence contains an affinoid open neighbourhood of z ). Since Z ⊂ U we deduce −1 that (σ ◦ i ) (E (τ )) contains an affinoid open neighbourhood of z . This completes n 2 n n 2 the proof. Remark 2.25. — Note that assumption (3) on the image of the local Galois rep- resentation ensures that all symmetric powers remain locally irreducible. We need this to apply the results of §2.11.1. The authors expect that, with some effort, this material could be adjusted to allow locally reducible (but globally irreducible) families of Galois representations. We also prove a version of this result in the ordinary case. We first note a well- known consequence of Hida theory: Lemma 2.26. — The Zariski closure of the classical points with ordinary refinements ord E (τ ) ⊂ E (τ ) is a union of connected components of E (τ ) which are finite over W ,and every n n n n n n n ord ord classicalpointof E (τ ) has an ordinary refinement. All points of E (τ ) with dominant locally n n n n algebraic image in W are classical. ord Proof. — We can identify E (τ ) with the generic fibre of the formal spectrum of n n S,ord ∞ Hida–Hecke algebra (a localization of the ring denoted by T (U (p ), O) in [Ger19, §2]), since this is naturally a Zariski closed subspace of X × T in which the classical ps,n n points with ordinary refinements are Zariski dense. We deduce from Hida theory that ord E (τ ) is finite over W and equidimensional of dimension dim W .Moreover, themap n n n n ord ◦ ν : E (τ ) → T factors through the open subspace T ⊂ T classifying unitary charac- n n n n ters of T (F ). n v v∈S On the other hand, we claim that every point of E (τ ) × T is contained in n n ν,T n n ord ord E (τ ) . Assuming this, these (reduced) subspaces of E (τ ) are equal and E (τ ) is n n n n n n an open and closed subspace of E (τ ). The final part of the lemma follows from the n n classicality theorem in Hida theory [Ger19, Lemma 2.25]. ◦ ord It remains to show the claimed inclusion of E (τ ) × T in E (τ ) . Suppose n n ν,T n n n n z is an E-point of E (τ ) × T (extending scalars deals with the general case). The n n ν,T n n SYMMETRIC POWER FUNCTORIALITY 49 p an ∗ character ν(z)δ then appears in the eigenspace J S(U , E) [ψ (z)]. This charac- B B n n n p an ∗ ter therefore also appears in J S(U , E) [ψ (z)] , by Lemma 2.19. Applying [Sor17, n n p ∗ Corollary 6.4] to S(U , O)[ψ (z)] (note that Sorensen’s Jacquet modules are twisted by −1 δ compared to ours), we deduce that the unitary character ν(z) appears in the ordinary p ∗ ord part Ord S(U , O)[ψ (z)]. This shows that z is a point of E (τ ) . B n n Theorem 2.27. — Let (π ,χ ) ∈ RA satisfy tr r = τ ,and let z = γ (π ,χ ) ∈ 2 2 2 π ,ι 2 2 2 2 2 E (τ )(Q ). Suppose that: 2 2 (1) The refinement χ is ordinary. (2) There exists (π ,χ ) ∈ RA such that (σ ◦ i )(z ) = i (z ),where z = γ (π ,χ ). n n n n 2 2 n n n n n n (3) The Zariski closure of r (G ) contains SL . π ,ι F 2 Then each irreducible component C of E (τ ) containing z satisfies (σ ◦ i )(C) ⊂ i (E (τ ) ). 2 2 C 2 n 2 n n n C p p Proof. — Extending E if necessary, we may assume that z ∈ E (τ )(E) and r 2 2 2 π ,ι HT−reg takes values in GL (E).Wedenoteby T ⊂ T the Zariski open subset where for each 2 n v ∈ S and τ ∈ Hom (F , E) the labelled weights wt (δ ) are distinct for i = 1,..., n. p Q v τ n,v,i By Lemma 2.12 and (a global variant of) Lemma 2.13, there is an open affinoid neighbourhood irr reg z = (tr r ,δ ) ∈ U ⊂ X × T n π ,ι n n ps,n n and a universal representation ρ : G → GL (O(U)) such that the induced representa- F,S n tion (ρ )ˆ : G → GL (O(U)ˆ ) with coefficients in the completed local ring at z extends z F n z n n n u u 1−n n to a homomorphism (ρ )ˆ : G + → G (O(U)ˆ ) with ν ◦ (ρ )ˆ = δ . z F ,S n z G z + n n n n F/F Since χ is ordinary, the parameter δ = σ (δ ) is ordinary. We denote by 2 n n,p 2 FL (O ) − → U the rigid space (equipped with a proper map to U ) classifying S -tuples p p (F ) of full flags in O . We consider the closed subspace v v∈S p U ord n Z ⊂ FL (O ) whose points z correspond to flags F which are G -stable (under the ρ -action) for each v F i −1 v ∈ S and the action of G on gr (F ) is given by δ ◦ Art where δ is the parameter p F v z,v,i z HT−reg ord of α(z). Since our parameters lie in T , Z → U is a closed immersion (it is a proper monomorphism). u ord Using the existence of (ρ )ˆ , we can view the tangent space T Z as a sub- z z n n space of H (G , ad r ). By a similar argument to Proposition 2.11, it follows from e.g. F ,S π ,ι [Ger19, Lemma 3.9] (which gives the analogue of Lemma 2.7 in the ordinary case) and ord the main theorem of [NT20] that the map T Z → T W is injective. On the other z r(δ ) n n n ord ord hand, E (τ ) ∩ U , a subspace of Z containing z , is equidimensional of dimension n n n ord ord dim W . We deduce that Z is smooth at z ,and that E (τ ) is locally isomorphic to Z n n n n at z . We complete the proof in the same way as Theorem 2.24. n 50 JAMES NEWTON, JACK A. THORNE We restate Theorem 2.24 and Theorem 2.27 in a way that does not make explicit reference to E . Corollary 2.28. — Let (π ,χ ), (π ,χ ) ∈ RA ,and let z , z ∈ E (Q ) be the corre- 2 2 2 2 2 2 2 2 p sponding points of the eigenvariety. Suppose that one of the following two sets of conditions are satisfied: (1) The refinement χ is numerically non-critical and n-regular. (2) For each v ∈ S , every triangulation of r | is non-critical. The refinement χ is n-regular. p π ,ι G F 2 (3) For each v ∈ S , the Zariski closures of the images of r | and r | contain SL . p π ,ι G π ,ι G 2 2 F F v 2 v (4) There exists an automorphic representation π of G (A +) such that n n F n−1 Sym r r . π ,ι π ,ι 2 n (5) The points z ,z lie on a common irreducible component of E ; 2 2,C 2 p or ord (1 ) The refinement χ is ordinary. ord (2 ) The Zariski closure of the image of r | contains SL . π ,ι G 2 2 F ord (3 ) There exists an automorphic representation π of G (A ) such that n n F n−1 Sym r = r . π ,ι π ,ι 2 n ord (4 ) The points z ,z lie on a common irreducible component of E (this implies that the refinement 2 2,C 2 p χ is also ordinary, by Lemma 2.26). Then there exists an automorphic representation π of G (A ) such that n F n−1 Sym r r  . π ,ι π ,ι ∞ ∞ U Proof. — Choose U ⊂ G (A ) so that (π ) = 0and take τ = tr r .Then n n 2 π ,ι F n 2 n−1 n−2 n−1 (σ ◦ i )(z ) ∈ i (E (τ )(Q )). We claim that setting χ = χ ⊗χ χ ⊗· · · ⊗χ n 2 2 n n n n,v 2,v,2 p 2,v,1 2,v,1 2,v,2 for v ∈ S defines an accessible refinement χ of π .Fix v ∈ S . To temporarily simplify p n n p notation, we write χ = χ ⊗ χ for χ . The representation π is isomorphic to either 1 2 2,v 2,v GL −1/2 2 St (ιχ |·| ) or to an irreducible parabolic induction i ιχ . In the first case, 2 1 T −1 n−1 T −1 n−1 (1−n)/2 ∼ ∼ rec (ι π ) Sym rec (ι π ) Sp (χ |·| ) + = + = n,v 2,v n 1 F F v v and χ is the unique accessible refinement of π . In the second case, n,v n,v n−1 T −1 n−1 T −1 n−1−i i (1−n)/2 −1 ∼ ∼ rec +(ι π ) = Sym rec +(ι π ) = χ χ |·| ◦ Art . n,v 2,v + 1 2 F F F v v i=0 By [Con99, Theorem 3.4.2], this assumption is equivalent to requiring that there is a finite extension of coefficient fields E /Esuch that z , z lie on a common geometrically irreducible component of E  . 2 2,E 2 SYMMETRIC POWER FUNCTORIALITY 51 Note that π is generic (the base change of π to GL (A ) is cuspidal, since r is irre- n,v n n F π ,ι ducible). We can now use the characterisation of generic representations in the Bernstein– Zelevinsky classification [Zel80, Theorem 9.7] and the compatibility with local Lang- lands [Rod82, §4.4]. It follows that no pair of characters in the above direct sum decom- GL position have ratio equal to the norm character, so the parabolic induction i ιχ is n,v irreducible and isomorphic to π . In particular, χ is an accessible refinement of π . n,v n,v n,v Taking the above discussion into account, it is straightforward to see that (σ ◦ i )(z ) is associated to the pair (π ,χ ) ∈ RA . Thus the hypotheses of Theorem 2.24 or 2 2 n n n 2.27 are satisfied, and for any (π ,χ ) as in the statement of the corollary there exists a 2 2 n−1 point z ∈ E (τ )(Q ) such that T = tr Sym r . It remains to show that z ∈ Z ,or n n n,z π ,ι n n p n in other words that z is associated to a classical automorphic representation. In the ordinary case, this follows from Lemma 2.26. In the remaining case, Lemma n−1 2.29 shows that for each v ∈ S , every triangulation of Sym r | is non-critical. It p π ,ι G 2 v follows from Lemma 2.30 that z ∈ Z . Lemma 2.29. — Let v ∈ S ,and let ρ : G → GL (Q ) be a continuous, regular de Rham p v F 2 v p representation such that WD(ρ ) has two distinct characters χ , χ as Jordan–Hölder factors, which v 1 2 satisfy (χ /χ ) = 1 for each i = 1,..., n − 1. Suppose moreover that every triangulation of ρ is 1 2 v n−1 non-critical. Then every triangulation of Sym ρ is non-critical. Proof. — We begin by describing the data of the triangulation of ρ in a bit more detail. Let K = F and let L/K be a Galois extension over which ρ becomes semi-stable. v v Let L be the maximal unramified extension of L/Q . After enlarging E, we can assume 0 p that every embedding of L in Q lands in E, and that ρ is defined over E. The filtered (ϕ, N, Gal(L/K))-module D associated to ρ consists of the following data: (1) A free L ⊗ E-module D of rank 2, equipped with a σ ⊗ 1-semilinear endomor- 0 Q phism ϕ. (2) An L ⊗ E-linear endomorphism N of D satisfying the relation Nϕ = pϕN. 0 Q (3) An L -semilinear, E-linear action of the group Gal(L/K) on D that commutes with the action of both ϕ and N. (4) A decreasing, Gal(L/K)-stable, filtration Fil D of D = D ⊗ L. • L L L For each embedding τ : L → E, we write l ⊂ D = D ⊗ E for the image of the τ τ L L⊗E,τ rank 1 step of the filtration Fil .Wecan defineanactionofthe group W on D by the • K −α(g) formula g · v = (g mod W ) ◦ ϕ ,where α(g) is the power of the absolute arithmetic Frobenius induced by g on the residue field of K. This action preserves the factors of the product decomposition D = D ,where t ranges over embeddings t : L → Eand D = D ⊗ E. Moreover, the isomorphism 0 t L ⊗E,t class of the Weil–Deligne representation D is independent of t. The data of a triangu- lation of ρ is equivalent to the data of a choice of character appearing in some (hence v 52 JAMES NEWTON, JACK A. THORNE n−1 every) D . If N is non-zero on D , then there is a unique N-stable line in Sym D . t t t n−1 Hence there is a unique triangulation of Sym ρ , induced by the unique (non-critical) triangulation of ρ , and it is also non-critical. From now on we assume that N = 0, and we proceed as indicated in [Che11, Example 3.26]. We can choose a basis e , e for D as L ⊗ E-module such that the projection of 1 2 0 Q the vectors e , e to each D is a basis of eigenvectors for the group W . 1 2 t K Having made this choice of basis, each line l is spanned by a linear combination of e , e . Our assumption that every triangulation of ρ is non-critical is equivalent to the 1 2 v requirement that l maybespannedbyavector e + a e ,where a ∈ E for all τ . Indeed, τ 1 τ 2 τ if l is spanned by e for some i, then the triangulation corresponding to the submodule τ i of D spanned by e will fail the condition required for non-criticality with respect to the embedding τ . Having made these normalisations, the condition that every triangulation of n−1 Sym ρ be non-critical is equivalent to the following statement: let I ⊂{0,..., n − 1} i |I| be a subset, and let a x ∈ E[x] be a polynomial, which is equal to (1 + a x) Q(x) i τ i∈I for some polynomial Q(x) ∈ E[x] of degree at most n − 1 −|I|,then Q(x) = 0. Polyno- mials of the latter form correspond to elements of the |I|th step of the Hodge filtration n−1 on Sym D and the statement implies that this Hodge filtration is in general position compared to the filtration induced by every triangulation. Replacing the variable x with −a x, we can assume that a =−1. As in [Che11, Example 3.26], the vanishing of the τ τ |I| successive derivatives at 1 of a x gives a non-degenerate linear system of |I| equa- i∈I tions satisfied by the a , and therefore the a are all zero. Non-degeneracy is checked i i by noticing that the determinant of the linear system is the Vandermonde determinant (i − j). i<j∈I p−irr Lemma 2.30. — Let z ∈ E (τ )(Q ) be a point with i (z) = (tr r ,δ) ∈ X (Q ) × n n n z ps,n p p reg T (Q ). Suppose that δ = j (ν(z)) = δ δ with δ algebraic and δ smooth. Suppose moreover n alg sm alg sm that, for each v ∈ S , every triangulation of r | is non-critical. Then z ∈ Z (in particular, δ is p z G n alg strictly dominant). Proof. — After extending E, we may assume that z ∈ E (τ )(E) and r takes values n n z in GL (E). We note that, since δ is locally algebraic, it follows from property (5)ofthe p−irr −1 reg eigenvariety that the subset of numerically non-critical classical points in i (X ×T ) ps,n n n accumulates at z. It follows from [KPX14, Corollary 6.3.10], applied as in Proposition p−irr −1 reg 2.15, that there is a connected affinoid neighbourhood U of z in i (X × T ),over ps,n n n which there exist representations ρ : G → GL (O(U)) for each v ∈ S with trace equal F n p v v to the restriction to G of the universal pseudocharacter and a non-empty Zariski open and dense subspace V ⊂ U such that for every z ∈ V with i (z ) = (tr r ,δ ), r is trian- n SYMMETRIC POWER FUNCTORIALITY 53 guline of parameter δ . Now we can apply [BHS17, Lemme 2.11] to deduce that δ is alg strictly dominant and r is trianguline of parameter δ. We now argue as in [BHS17, Prop. 3.28] (which is itself similar to the argument of [Che11, Prop. 4.2]). The idea of the argument is to show that failure of classicality would entail the existence of a ‘companion point’ to z, with the same associated Galois representation and a locally algebraic weight which is not strictly dominant. This would contradict [BHS17, Lemma 2.11]. Let η = ν(z)δ = η η ,with η dominant algebraic (since δ is strictly domi- B alg sm alg alg nant) and η smooth. By the construction of E and Lemma 2.19 we have a non-zero sm n space of morphisms p an ∗ 0 = Hom η, J S(U , E) [ψ (z)] . T (F ) B n v n n v∈S Now we use some of the work of Orlik–Strauch [OS15], with notation as in [Bre15, §2]. We denote by g the Q -Lie algebra of G (F ) = GL (F ) and de- n p n n v v∈S v v∈S p p note by b ⊂ g the lower triangular Borel. We define a locally analytic representation n n + n of G (F ) (see [Bre15, Thm. 2.2] for the definition of the functor F ): v∈S p B G G −1 −1 −1 n n F (ηδ ) := F U(g ) ⊗ η ,η δ . n,E sm B U(b ) alg B n n,E n B B n n −1 Note that U(g ) ⊗ η has a unique simple submodule (isomorphic to the n,E U(b ) alg n,E −1 ∨ unique simple quotient of U(g ) ⊗ η ), the algebraic representation V(η ) with n,E alg U(b ) alg n,E −1 −1 lowest (with respect to B )weight η . It follows from [Bre15, Thm. 2.2] that F (ηδ ) alg B G −1 has a locally algebraic quotient isomorphic to V(η ) ⊗ Ind η δ . alg E sm By [Bre15, Thm. 4.3], there is a non-zero space of morphisms n −1 p an ∗ 0 = Hom F (ηδ ), S(U , E) [ψ (z)] . G (F ) n v B n v∈S B n p n G −1 The Jordan–Hölder factors of F (ηδ ) can be described using [Bre15,Thm.2.2] and standard results on the Jordan–Hölder factors of Verma modules (see [Bre15, Cor. 4.6]). Suppose λ ∈ T (E) is an algebraic character. Denote by M the unique simple n λ −1 submodule of the dual Verma module U(g ) ⊗ λ . Then the Jordan–Hölder n,E U(b ) n,E G −1 factors of F (ηδ ) areall of theform JH(w, π) = F M ,π w·η alg with P a parabolic subgroup of G (F ) containing B , π a Jordan–Hölder factor n n n v∈S v −1 of the parabolic induction of η δ from B to the Levi of P ,and w an element of the sm n n We caution the reader that the version of this paper currently available on the arXiv contains a less general result than the published version, to which we appeal here. In particular, it restricts to Galois representations which are known in advance to be crystalline. 54 JAMES NEWTON, JACK A. THORNE Weyl group of (Res G ) × E, acting by the ‘dot action’ on η .Here P is maximal F /Q n Q alg n for M , in the sense of [Bre15,§2]. w·η alg p an ∗ We claim that there cannot be a non-zero morphism JH(w, π)→ S(U ,E) [ψ (z)] for w = 1. Suppose, for a contradiction, that there is such a map. It follows from [Bre16, ∗ ∗ Cor. 3.4] that we have ψ (z) = ψ (z ) (and hence an isomorphism of Galois represen- tations r = r ) for a point z ∈ E (τ )(E) with j (ν(z )) locally algebraic but not strictly z z n n n dominant (its algebraic part matches the algebraic part of j (w · η )). The argument in n alg the first paragraph of this proof, using [BHS17, Lemma 2.11], then gives a contradiction. We deduce from this that any map n −1 p an ∗ F (ηδ ) → S(U , E) [ψ (z)] B n B n −1 factors through the locally algebraic quotient V(η ) ⊗ Ind η δ . Applying [Bre15, alg E sm Thm. 4.3] again, we deduce that we have equalities p an ∗ Hom η, J S(U , E) [ψ (z)] T (F ) B n v n n v∈S −1 p an ∗ = Hom F (ηδ ), S(U , E) [ψ (z)] G (F ) B n n v n v∈S B p n −1 p alg ∗ = Hom F (ηδ ), S(U , E) [ψ (z)] G (F ) B n n v n v∈S B p n p alg ∗ = Hom η, J S(U , E) [ψ (z)] . T (F ) B n v n v∈S n In particular, our point z arises from a non-zero map p alg ∗ η → J S(U , E) [ψ (z)] . n n Applying [Eme06a, Prop. 4.3.6] and computing locally algebraic vectors as in §2.18.1 we see that such a map corresponds to a non-zero map of smooth representations −1/2 p ∗ η → J lim S (U U , E)[ψ (z)] and hence a pair (π ,ι ◦ η δ ) ∈ RA with sm B η p n sm n n n B alg n − →U corresponding classical point equal to z. We therefore have z ∈ Z . 2.31. Application to the eigencurve. — Thus far in this section we have found it conve- nient to phrase our arguments in terms of automorphic forms on unitary groups. Since our intended application will rely on particular properties of the Coleman–Mazur eigen- curve for GL , we now show how to deduce what we need for the eigencurve from what we have done so far. We first introduce the version of the eigencurve that we use. Fix an integer N ≥ 1, × × × prime to p.Let T = Hom(Q /Z × Q , G ); it is the E-rigid space parameterising char- 0 m p p p × 2 × acters χ = χ ⊗ χ of (Q ) such that χ is unramified. Let W = Hom(Z , G ), 0 0,1 0,2 0,1 0 m p p and write r : T → W for the morphism given by r (χ ⊗ χ ) = χ /χ | = 0 0 0 0 0,1 0,2 0,1 0,2 p SYMMETRIC POWER FUNCTORIALITY 55 pN −1 χ | ×.Wedenotethe map r ◦ ν : E → W by κ.Let T = O[{T , S } ] de- 0 0 0 0 l l lpN 0,2 Z 0 note the polynomial ring in unramified Hecke operators at primes not dividing Np. l 0 l 0 Here T and S are the double coset operators for the matrices and .Let l l 01 0 l U (N) = U (N) ⊂ GL (Z) = GL (Z ) be defined by 1 1 l 2 2 l l l ab U (N) = ∈ GL (Z ) : c, d − 1 ∈ NZ . 1 l 2 l l cd The eigencurve is a tuple (E ,ψ ,ν , Z ),where: 0 0 0 0 (1) E is a reduced E-rigid space, equipped with a finite morphism ν : E → T . 0 0 0 0 pN (2) ψ : T → O(E ) is a ring homomorphism, which takes values in the subring 0 0 ≤1 O(E ) of bounded elements. (3) Z ⊂ E (Q ) is a Zariski dense subset which accumulates at itself. 0 0 The following properties are satisfied: (1) E is equidimensional of dimension dim W = 1. For any irreducible component C ⊂ 0 0 E , κ(C) is a Zariski open subset of W . 0 0 (2) Let A denote the set of cuspidal automorphic representations π of GL (A ) such 0 0 2 Q k−2 ∞ U (N) 2 ∨ that (π ) = 0and π has the same infinitesimal character as (Sym C ) for 0,∞ some k ≥ 2 (in which case we say π has weight k), and let RA denote the set of pairs 0 0 (π ,χ ),where π ∈ A and χ = χ ⊗ χ is an accessible refinement of π such 0 0 0 0 0 0,1 0,2 0,p that χ is unramified. As in the unitary case we considered above, for (π ,χ ) ∈ RA 0,1 0 0 0 pN we have a homomorphism ψ : T → Q determined by the action of the Hecke 0 0 −1 ∞ U (N) operators on ι (π ) . There is also a character ν (π ,χ ) ∈ T (Q ) defined 0 0 0 0 in exactly the same way as in the unitary case (2.18.1). An explicit formula appears below (2.31.1). Our assumption that χ is unramified implies that this character 0,1 pN does indeed give a point of T .Now we canlet Z ⊂ Hom(T , Q ) × T (Q ) denote 0 0 0 p p the set of points of the form (ψ ,ν (π ,χ )),where (π ,χ ) ∈ RA . Then the map π 0 0 0 0 0 0 pN ψ × ν : E (Q ) → Hom(T , Q ) × T (Q ) restricts to a bijection Z → Z . 0 0 0 0 0 0 p 0 p p pN −1 (3) For any affinoid open V ⊂ T ,the map T ⊗ O(V ) → O(ν V ) is surjective. 0 0 0 0 0 0 The uniqueness of the tuple (E ,ψ ,ν , Z ) follows from [BC09, Proposition 7.2.8]. Its 0 0 0 0 existence can be proved in various ways. A construction using overconvergent modular forms is given in [Buz07]. We note that in this case, in contrast to the unitary group case, the map RA → Z is bijective – a consequence of the strong multiplicity one theorem. 0 0 We will therefore feel free to speak of the cuspidal automorphic representation π ∈ A 0 0 associated to a point lying in Z . As in the unitary group case, there is a Galois pseu- docharacter t : G → O(E ) with the property that for z ∈ Z associated to (π ,χ ), Q,Np 0 0 0 0 t = tr r . z π ,ι Let us describe explicitly the link with more classical language. We are using the normalisations of [DI95,§11].If (π ,χ ) ∈ A , then there is a cuspidal holomorphic 0 0 0 n r modular form f = q + a (f )q of level  (Np ) (for some r ≥ 1) which is an eigen- n 1 n≥2 form for all the Hecke operators T (l  Np)and U , in their classical normalisations, and l p 56 JAMES NEWTON, JACK A. THORNE we have the formulae GL (Z ) 2 l −1/2 a (f ) = eigenvalue of T on π , p a (f ) = ιχ (p). l l p 0,1 0,l Note that the central character of π is a Hecke character ψ with ψ | (z) = 0 π π R 0 0 >0 2−k 2−k z .So ψ | = ι(χ χ ) is a finite order twist of the character z → |z| .Tocon- π 0,1 0,2 0 Q vince the reader that these formulae are correct, we observe that if π is a normalised 0,l GL GL (Z ) 2 2 l 1/2 induction i μ ⊗ μ , then the eigenvalue of T on π is l (μ (l) + μ (l)) [DI95, 1 2 l 1 2 B 0,l (11.2.4)], whilst considering the central character shows that μ (l)μ (l) has (complex) 1 2 k−2 absolute value l . This is compatible with the fact that a (f ) is a sum of numbers with (k−1)/2 absolute values l . We can define a map s : E (Q ) → R, called the slope, by composing the projection p 0 to T (Q ) with the map χ → v χ .Notethatif (π ,χ ⊗ χ ) ∈ RA we have 0 p 0 0,1 0,2 0 −1/2 t 0 1 2−k (2.31.1) ν (π ,χ ) = t χ (t )χ (t ) 0 0 0 0,1 1 0,2 2 0 t 2 so the slope map sends (π ,χ ⊗ χ ) ∈ RA to 1/2 + v (χ (p)). 0 0,1 0,2 0 p 0,1 In particular, at a point z ∈ Z corresponding to a classical holomorphic modular 0 0 −1 form f , s(z ) equals the p-adic valuation of ι a (f ). Note that the corresponding pair 0 p (π ,χ ) is numerically non-critical exactly when s(z )< k − 1 and ordinary exactly when 0 0 0 s(z ) = 0. The classicality criterion of Coleman [Col96, Col97] shows that a point z ∈ k−2 × E (Q ) with κ(z) restricting to t → t on a finite index subgroup of Z and s(z)< k − 1 is necessarily in Z . pc Let Z ⊂ Z denote the subset of points corresponding to pairs (π, χ) where π 0 p is not a twist of the Steinberg representation (pc stands for potentially crystalline). We now pc pc define a ‘twin’ map τ : Z → Z .Let (π ,χ ) be the pair corresponding to a point 0 0 0 0 pc z ∈ Z .Write χ = χ ⊗ χ . Since π is not a twist of the Steinberg representation, 0 0,1 0,2 0,p GL 2 × π equals the full normalised induction i ιχ , which is irreducible. Let ψ : Q \A → 0,p 0 B Q Z be the unique finite order character which is unramified outside p and such that −1 × × × × ψ | = χ | . Then the character χ ψ | is unramified and χ = χ ψ | ⊗χ ψ | Z 0,2 0,2 Z 0,2 Z 0,1 Z p p p p pc is an accessible refinement of the twist π ⊗ ιψ . We therefore have a point τ(z) ∈ Z corresponding to the pair τ(π ,χ ) = (π ⊗ ιψ, χ ), that we call the twin of z.Notethat 0 0 0 τ = 1 and if π is unramified then τ(z) is the usual companion point appearing in the 0,p Gouvea–Mazur construction of the infinite fern [Maz97, §18]. The following lemma is an easy computation. pc Lemma 2.32. — Let z ∈ Z ,and let z = τ(z).Let s, s denote the slopes of these two points, and κ(z), κ(z ) ∈ W (Q ) their images in weight space. Then s + s = k − 1 and v (κ(z)(1 + 0 p q) − 1) = v (κ(z )(1 + q) − 1),where q = p if p is odd and q = 4 if p is even. Here is the main result of §2. SYMMETRIC POWER FUNCTORIALITY 57 Theorem 2.33. — Let (π ,χ ), (π ,χ ) ∈ RA and let n ≥ 2.Let z , z ∈ Z be the 0 0 0 0 0 0 0 0 corresponding points. Suppose that one of following two sets of conditions are satisfied: (1) The refinement χ is numerically non-critical and n-regular. (2) The refinement χ is n-regular. (3) The Zariski closures of r (G ) and r (G ) contain SL . π ,ι Q π ,ι Q 2 0 p p n−1 (4) Sym r is automorphic; π ,ι or ord (1 ) The refinement χ is ordinary. ord (2 ) π and π are not CM (so the Zariski closures of r (G ) and r (G ) contain SL ). 0 π ,ι Q π ,ι Q 2 0 0 n−1 ord (3 ) Sym r is automorphic. π ,ι n−1 If the points z ,z lie on a common irreducible component of E , then Sym r is also automor- 0 0,C π ,ι 0 p phic. Proof. — We want to apply Corollary 2.28. We first need to specify suitable data F, S, G ,U .Let F /Q be an abelian CM extension satisfying the following conditions: 2 2 • Each prime dividing Np splits in F . •[(F ) : Q] is even. • The extension F /(F ) is everywhere unramified. After extending E, we may assume that z , z ∈ E (E) and that there is an irre- 0 0 ducible component C ⊂ E containing the points z , z .Moreover, by thefirstpartof 0 0 [Con99, Theorem 3.4.2], we may assume that C is geometrically irreducible. Let W de- note the unique connected component of W containing κ(C). We can find a character χ : G → O(W) such that the determinant of the universal pseudocharacter over C −1 equals χ (χ is the product of a finite order p-unramified character and the compo- sition of with the universal character Z → O(W )). By Lemma 2.34,wecan finda finite étale morphism η : W → W and a character ψ : G  → O(W) , unramified almost c  + everywhere, such that ψψ = χ | , and such that for each place v|p of (F ) , there is a place v|v of F such that ψ | is unramified. We now let F/Q be a soluble, Galois, CM extension, containing F ,suchthat: • Each prime dividing Np splits in F. • The extension F/F is everywhere unramified. • The character ψ | is unramified away from p. Let S denote the set of places of F dividing Np. Fix as usual a set of factori- sations v = v v for v ∈ S. Fix the unitary group G as in our standard assumptions (§1). Then for each v ∈ S, thereisanisomorphism ι : G (F ) → GL (F ).Welet v 2 2 v U = U ⊂ G(A ) be an open subgroup with the property that U is hyperspecial 2 2,v 2,v v F maximal compact if v ∈ S, and U is the pre-image under ι of the subgroup U (N) of 2,v v 1 l GL (Q ) if v ∈ S has residue characteristic l (in which case F = Q ). 2 l v l 58 JAMES NEWTON, JACK A. THORNE × 2 We recall that T = Hom((F ) , G ).Let T = T × W, with κ ˜ : T → W 2 m 0 W v∈S v 0 u u u the projection map. If χ = χ ⊗ χ ∈ T (T ) denotes the universal character, then the 0 0 1 2 tuple of characters u −1 u −1 ((χ ◦ N · ψ | ◦ Art ) ⊗ (χ ◦ N · ψ | ◦ Art )) F /Q G F F /Q G F v∈S 1 v p F v 2 v p F v p v v in T (T ) determines a morphism b : T → T . Writing X for the rigid space of 2- 2 p 2 0,ps dimensional pseudocharacters of G , unramified outside Np, there is a base change morphism b : X × T → X × T covering b and sending a pair (τ, μ) to (τ | ⊗ 0,ps 2,ps 2 p G −1 ψ , b (μ)). This leads to a diagram of rigid spaces κ( ˜ μ) X × T X × T . 0,ps 2,ps 2 i 2 E × T E 0 T 2 Let C = C × W = C × T . Then the morphism C → C is finite étale. In particular, W T each irreducible component of C maps surjectively to C . Choose E /E so that the ir- reducible components of C  are geometrically irreducible (we apply [Con99,Theorem 3.4.2] again). Since C is geometrically irreducible, we still know that each irreducible component of C maps surjectively to C . Consequently, we can find points z , z of C E E 1 E lifting z , z and lying on a common geometrically irreducible component C of C .We 0 E −1 next wish to show that b ◦ i(C ) ⊂ i (E ), or equivalently that (b ◦ i) (i (E )) contains 2 2,E 2 2,E C . Since i is a closed immersion, it suffices to show that Z , the pre-image of Z in C , 2 0 satisfies b ◦ i(Z ) ⊂ i (E (Q )) (the accumulation property of Z in C is inherited from 2 2 0 p 0 the corresponding property of the subset Z ∩ C ⊂ C ). To see this, we note that for any (π, χ) ∈ Z , with lift z ∈ Z , the base change π (which exists since F/Q is soluble) is still cuspidal. Indeed, if not then r | would F π,ι G be reducible, implying that π was automorphically induced from a quadratic imaginary subfield K/Q of F/Q. This is a contradiction, since we chose F so that all primes dividing −1 Np split in F, yet K must be ramified at at least one such prime. The descent of π ⊗ ιψ to G (which exists, by [Lab11, Théorème 5.4]) gives (together with b (ν (π, χ))) a point 2 p 0 of E which equals the image of z under the map b ◦ i. We can now complete the proof. Indeed, the points b ◦ i(z ), b ◦ i(z ) lie on a common geometrically irreducible component of E  , by construction. They satisfy 2,E the conditions of Corollary 2.28 (in particular, Example 2.10 shows that our assump- tion on r (G ) in the non-ordinary case implies that all of its triangulations are non- π ,ι Q critical). We therefore conclude the existence of an automorphic representation π of n−1 G (A +) such that Sym r | r  . Our assumptions (cf. Lemma 3.5(2)) imply n F π ,ι G π ,ι 0 n SYMMETRIC POWER FUNCTORIALITY 59 n−1 that Sym r | is irreducible, and therefore that the base change of π is a cuspi- π ,ι G F n dal automorphic representation of GL (A ). Soluble descent for GL now implies that n F n n−1 Sym r is itself automorphic. π ,ι Lemma 2.34. — Let F be a CM number field. Suppose that each p-adic place of F splits in F,and let S be a set of p-adic places of F such that S S is the set of all p-adic places of F. p p Let W be a connected E-rigid space, and let χ : G → O(W) be a continuous character (continuity defined by demanding that the induced characters with values in O(U) are continuous for all affinoid admissible opens U ⊂ W,asin[Buz04, §2]), unramified almost everywhere. Then we can find a finite étale morphism η : W → W and a continuous character ψ : G → O(W) such that the following properties hold: (1) ψ is unramified almost everywhere. (2) For each v ∈ S , ψ | is unramified. p G c ∗ (3) ψψ = η (χ)| . Proof. — We first claim that we can find a finite étale morphism W → Wand a continuous character λ : G → O(W ) with the following properties: • λ is unramified almost everywhere. • χ | λλ has finite order. × × Indeed, let L : O → O(W) be defined by the formula w|p −1 L((u ) ) = χ | ◦ Art (u ). w w F v v∈S Then L is continuous, and trivial on a finite index subgroup of O (it is trivial on the norm 1 units in O ). It follows from Chevalley’s theorem [Che51, Théorème 1] that p p there is a compact open subgroup U of O such that L is trivial on (U ) := wp F p × × U × O ∩ O . F F w|p Note that if H is a product of a finite abelian group and a finite Z -module, and H ⊂ H is a finite index subgroup, then the natural map Hom(H, G ) → Hom(H , G ) m m of rigid spaces is finite étale. Maps of rigid spaces W → Hom(H, G ) biject with contin- uous characters H → O(W) . ∞,× It follows that we may extend L to a continuous character L : F \A → O(W ) , for some finite étale morphism W → W. Indeed, we apply the preceding re- mark with H the quotient of O by the closure of (U ) and H the quotient of w|p F ∞,× × p F \A by the closure of the image of U (cf. the discussion in [Buz04,§2]). We define λ by λ ◦ Art = L . The character χ | λλ hasfiniteorder becauseit F G factors through the Galois group of an abelian extension of F which is unramified at all but finitely many places and unramified at the primes above p. 60 JAMES NEWTON, JACK A. THORNE Replacing W by a connected component, we may suppose that W is connected, in which case the character χ | λλ is constant (i.e. pulled back from a morphism W → Sp E , for a finite extension E /E). Applying [BLGGT14, Lemma A.2.5], we may find a finite extension E /E and a continuous character ϕ : G → (E ) of finite order such c c  −1 that χ | λλ = ϕϕ . The proof is complete on taking W = W and ψ = ϕλ . We conclude this section with a lemma that will be used in §8. It uses the existence of the universal pseudocharacter t over E . Lemma 2.35. — Fix n ≥ 1,and let Z ⊂ E denote the set of points x satisfying one of the following conditions: (1) t is absolutely reducible; (2) t = tr ρ for an absolutely irreducible representation ρ : G → GL (Q ), and the Zariski x x x Q 2 closure of the image of ρ does not contain SL . x 2 (3) There exists a prime l|N such that t | = χ + χ for characters χ : G → Q such that x G 1 2 i Q Q l (χ /χ ) = 1 for some i = 1,..., n − 1. 1 2 Then Z is Zariski closed. Proof. — The discussion in [Che14, §4.2] shows that the locus where t is absolutely reducible is Zariski closed. If ρ : G → GL (Q ) is irreducible, then the Zariski closure x Q 2 of the image of ρ contains SL if and only if Sym ρ is irreducible. Indeed, the Zariski x 2 x closure of the image of ρ contains SL if and only if the Zariski closure G of the image x 2 x of the associated projective representation Proj ρ : G → PGL (Q ) is PGL .There are x Q 2 2 two possibilities for the group G , which is a (possibly disconnected) reductive group: the first is that it is finite, hence either dihedral or conjugate to one of A ,S ,or A .Inany of 4 4 5 these cases Sym ρ is reducible. The next is that G has a non-trivial identity component, x x which therefore contains a maximal torus of PGL . The only possibilities are therefore that either G equals the normaliser of this maximal torus (in which case Sym ρ is again x x reducible) or that G = PGL (in which case Sym ρ is irreducible). x 2 x This shows that the set Z of points satisfying conditions (1) or (2) of the Lemma is Zariski closed. Finally, if l|Nand i = 1,..., n − 1, let Z denote the set of points x 3,l,i such that t | = χ + χ for some characters χ , χ such that (χ /χ ) = 1. It remains to x G 1 2 1 2 1 2 show that Z is Zariski closed. Its complement is the set of points such that either t | 3,l,i x G is absolutely irreducible, or t | is absolutely reducible and there exists g ∈ G such x G Q Q l that the discriminant of the characteristic polynomial of g under the pseudocharacter t is non-zero. This is a union of Zariski open sets. 3. Ping pong In this section we use the rigid analytic results of §2 to prove the following theorem. We recall that we say that an automorphic representation π of GL (A ) has “weight k” 2 Q SYMMETRIC POWER FUNCTORIALITY 61 for an integer k ≥ 2if π has the same infinitesimal character as the dual of the algebraic k−2 representation Sym C . Theorem 3.1. — Fix an integer n ≥ 2.Let π be a cuspidal automorphic representation of n−1 GL (A ) which is everywhere unramified and of weight k, for some k ≥ 2. Suppose that Sym r 2 Q π ,ι is automorphic for some (equivalently, any) prime p and isomorphism ι : Q → C. Then for any every- n−1 where unramified cuspidal automorphic representation π of GL (A ) of weight l ≥ 2, Sym r is 2 Q π,ι automorphic. To prove Theorem 3.1, we will use the properties of the eigencurve E ,asdefined in §2.31. More precisely, we henceforth let p = 2, N = 1, and let E denote the eigencurve defined with respect to this particular choice of parameters. We fix an isomorphism ι : Q → C. E is supported on the connected component W ⊂ W defined by χ(−1) = 1. 0 0 2 0 u × We write χ : Z → O(W ) for the universal character. We have the following explicit result of Buzzard and Kilford on the geometry of the morphism κ : E → W and the slope map s : E (Q ) → R: Theorem 3.2. — Let w ∈ O(W ) denote the function χ (5) − 1.Then: (1) w restricts to an isomorphism between W and the open unit disc {|w| < 1}. −1 (2) Let W (b) ⊂ W denote the open subset where |8| < |w| < 1,and let E (b) = κ (W (b)). 0 0 0 Then there is a decomposition E (b) = X of E (b) as a countable disjoint union of admissible 0 i 0 i=1 open subspaces such that for each i ≥ 1, κ| : X → W (b) is an isomorphism. X i 0 −1 (3) For each i = 1, 2,...,the map s ◦κ| : W (b)(Q ) → X (Q ) → R equals the map iv ◦w. 0 i p X p p Proof. — This is almost the main theorem of [BK05], except that here we are using E denotes the full eigencurve used in the cuspidal version of the eigencurve. However, if non−ord ord [BK05], then there is a decomposition E = E E as a union of open and closed ord subspaces. This follows from the fact that the ordinary locus E in the eigencurve can also be constructed using Hida theory (see [Pil13, §6]), so is finite over W . Since E is 0 1 ord separated the open immersion E "→ E is therefore also finite, hence a closed immer- ord sion. In our particular case (p = 2, N = 1),wehave E W (the unique ordinary 1 0 non−ord family is the family of Eisenstein series) and therefore E = E , giving the statement we have here. See Lemma 7.4 of the (longer) arXiv version of [BC05] for an alternative argument. We note as well that in our normalisation, the trivial character in W corresponds to forms of weight 2, whereas in the notation of [BK05], the character x corresponds to forms of weight 2. However, this renormalisation does not change the region W (b). Before giving the proof of Theorem 3.1, we record some useful lemmas. pc Lemma 3.3. — Let z ∈ Z ∩ E (b),and suppose z ∈ X .Let z = τ(z) be the twin of z. 0 i Then z ∈ X  , where i satisfies the relation i + i = (k − 1)/v (w(z)). i p 62 JAMES NEWTON, JACK A. THORNE Proof. — By Lemma 2.32, z lies in E (b),soin X  for a unique integer i ≥ 1. 0 i Writing s, s for the slopes of these two points and k for their weights, we have k − 1 = s + s = iv (w(z)) + i v (w(z)),hence i + i = (k − 1)/v (w(z)). p p p As a sanity check, we observe that in the context of the proof of Lemma 3.3, (k − k−2 1)/v (w(z)) is always an integer. Indeed, κ(z) satisfies κ(z)(5) = 5 ζ for some k ≥ 2 p 2 and m ≥ 0. This weight lies in W (b) if and only if either k is odd, or k is even and m ≥ 1. 1−m If m ≥ 1, then v (w(z)) = 2 .If m = 0and k is odd, then v (w(z)) = 2. In either case p p we see that (k − 1)/v (w(z)) is an integer. Lemma 3.4. — Let π be an everywhere unramified cuspidal automorphic representation of GL (A ) of weight k ≥ 2. Every accessible refinement of π is numerically non-critical and n-regular 2 Q for every n ≥ 2 (recall that we have fixed p = 2 and these notions refer to the local factor at 2, π ). Proof. — Numerical non-criticality of every refinement is immediate from the fact that there are no cusp forms of level 1 that are ordinary at 2. For regularity, if we fix 1/2 1/2 a refinement χ = χ ⊗ χ then α = p ιχ (p) and β = p ιχ (p) arethe rootsofthe 1 2 1 2 2 k−1 polynomial X − a X + 2 ,with a the T -eigenvalue of the level 1 weight k normalised 2 2 2 eigenform f associated to π . We need to show that α/β is not a root of unity. Suppose α/β = ζ is a root of unity. If we fix ι : Q C, the semisimplified mod 5 Galois representation r arises up to twist from a level 1 eigenform of weight ≤ 6 (i.e. the f ,ι −1 3 5 level 1 Eisenstein series of weight 4 or 6). This shows that ι (ζ ) ≡ 2 or 2 mod m , 5 Z and therefore ζ is the product of a 5-power root of unity and ±i (since 2 has order 4 in × −1 F ). Applying a similar argument at the prime 7 with ι : Q C, we see that ι (ζ ) ≡ 5 7 7 3 5 7 2 , 2 or 2 mod m and therefore ζ is the product of a 7-power root of unity and a cube root of unity. This gives the desired contradiction. (We thank Fred Diamond for pointing out this argument to us, and thank an anonymous referee for explaining how to avoid using Hatada’s congruence which appeared in the first version of this argument.) Lemma 3.5. — Let π be a cuspidal automorphic representation of GL (A ) of weight k ≥ 2. 2 Q We temporarily let p be an arbitrary prime. Then: (1) r | is reducible if and only if π is ι-ordinary. π,ι G (2) Suppose either that r | is irreducible and π admits a 3-regular refinement, or that k > 2 and π,ι G p r is not potentially crystalline. Then the Zariski closure of r (G ) (in GL /Q ) contains SL . π,ι π,ι Q 2 2 (3) Suppose again that p = 2, and that π is everywhere unramified. Then the Zariski closure of r (G ) contains SL . π,ι Q 2 Proof. — For the first part, ι-ordinarity implies reducibility by local-global com- patibility at p,asin[Tho15, Theorem 2.4]. For the converse, if r | is reducible, π,ι G then its Jordan–Hölder factors are de Rham characters of G and therefore have the p SYMMETRIC POWER FUNCTORIALITY 63 −k form ψ ,where the ψ are Z -valued characters with finite order restriction to in- i i ertia and the k are the Hodge–Tate weights (we can assume k = 0and k = k − 1 i 1 2 in our situation). The Weil representation part of WD(r | ) is therefore equal to π,ι G 1−k −1 T −1 ψ ⊕ ψ |·| ◦ Art . Since WD(r | ) = rec (ι π ), π is a subquotient of the nor- 1 2 π,ι G p p Q Q p p p GL 1/2 3/2−k malised induction i (ιψ ◦ Art |·| ⊗ ιψ ◦ Art |·| ). It follows from [Tho15, 1 Q 2 Q B p p Lemma 2.3] that π is ι-ordinary. For the second part, we note that r | is irreducible. Indeed, if r | is re- π,ι G π,ι G Q Q p p ducible then the first part of the lemma shows that π is ι-ordinary, and [Tho15, Lemma GL 2 −1 2.3] implies that π is a subquotient of i χ ⊗ χ with v ι (χ /χ (p)) = 1 − k.If p 1 2 p 1 2 r is not potentially crystalline then WD(r | ) has N = 0 and local-global compat- π,ι π,ι G GL 2 ±1 ibility implies that if π is a subquotient of i χ ⊗ χ ,then χ /χ =| · | . This is a p 1 2 1 2 contradiction if k > 2. Thus r | is irreducible and the Zariski closure H of its image is a reductive π,ι G subgroup of GL . Let T be a maximal torus of H. Since r is Hodge–Tate regular, T is 2 π,ι regular in GL (i.e. its centralizer is a maximal torus of GL )by[Sen73, Theorem 1]. If H 2 2 does not contain SL , then it is contained in the normaliser of a maximal torus of GL and 2 2 r | is induced from a character of an index two subgroup. This forces WD(r | ) π,ι G π,ι G Q Q p p 2 2 to likewise be induced, so any refinement χ = χ ⊗ χ of π satisfies χ = χ , and this 1 2 p 1 2 Weil–Deligne representation has N = 0. This is a contradiction, since we are assuming either that there exists a 3-regular refinement or that N is non-zero. For the third part, we have already observed (see the proof of Theorem 3.2)that there are no cusp forms of level 1 that are ordinary at 2, so r | is irreducible. Suppose π,ι G that r | is induced. Then WD(r | ) = Ind ψ for some quadratic extension π,ι G π,ι G Q Q W 2 2 K/Q and character ψ : W → Q . Since this Weil–Deligne representation must be 2 K unramified, we see that K and ψ are both unramified, and therefore that ψ extends to acharacter ψ : W → Q such that WD(r | ) = ψ ⊕ (ψ ⊗ δ ). In particular, Q π,ι G K/Q 2 Q 2 the T -eigenvalue (which equals the trace of Frobenius in this representation) is 0, but, as shown in the proof of Lemma 3.4, this is impossible. Lemma 3.6. — Let i ≥ 1 be an integer, and let z ∈ Z ∩ X (Q ) be the point corresponding 0 i to a pair (π, χ). Suppose that χ is n-regular. Let z ∈ Z ∩ X (Q ) be any other point, corresponding 0 i n−1 n−1 to a pair (π ,χ ), with χ n-regular. If Sym r is automorphic, then Sym r  is also. π,ι π ,ι Proof. — We can assume n ≥ 3. We apply Theorem 2.33 (note that X i,C W (b) is irreducible): since there are no ordinary points in E , χ is numerically non- 0 C 0 critical and Lemma 3.5 implies that the Zariski closures of r (G ) and r  (G ) con- π,ι Q π ,ι Q p p tain SL (the first part of the lemma shows that we are in the locally irreducible and 3-regular case of the second part of the lemma).  64 JAMES NEWTON, JACK A. THORNE Lemma 3.7. — Let π be a cuspidal, everywhere unramified automorphic representation of GL (A ) of weight k ≥ 2,and let χ be a choice of accessible refinement. Then there exists an in- 2 Q teger m ≥ 1 such that for any integer m ≥ m , we can find a cuspidal automorphic representation π of π π GL (A ) satisfying the following conditions: 2 Q (1) π is unramified outside 2. (2) π admits two accessible refinements of distinct slopes (in particular these refinements are n-regular for every n ≥ 2). (3) There is an accessible refinement χ of π such that (π, χ) and (π ,χ ) define points z, z on the same irreducible component of E . 0,C (4) κ(z ) ∈ W (b). In particular, z ∈ X for some i ≥ 1 (notation as in the statement of Theorem 0 i 3.2). pc (5) Set z = τ(z ),and let (π ,χ ) ∈ Z be the associated pair. Then z ∈ X m . 2 −1 (6) The automorphy of any one of the three representations n−1 n−1 n−1 Sym r , Sym r , Sym r π,ι π ,ι π ,ι implies automorphy of all three. Proof. — We use Theorem 3.2. Extending E if necessary, we may assume that z ∈ E (E) and every irreducible component of E containing z is geometrically irre- 0 0 ducible. Fix one of these irreducible components and fix i such that this irreducible component contains X (such an i exists, because every irreducible component of E has i 0 Zariski open image in W , hence intersects E (b) and therefore contains a non-empty union of irreducible components of E (b)). We define m to be least integer m ≥ 1satis- 0 π π fying the inequality m +1 (2i + 2 − 3)/2 > 2i. k −2 Given m ≥ m , we choose z ∈ X (Q ) to be the point such that κ(z )(5) = 5 ,where π i m+1 k = 2i + 2 − 1. Then (k − 2)/2 > 2i = s(z ). By Coleman’s classicality criterion, pc pc z ∈ Z .If z was not in Z , its slope would be (k − 2)/2. So z ∈ Z and if (π ,χ ) 0 0 denotes the corresponding pair, then the two accessible refinements of π have distinct slopes (2i and k − 1 − 2i). Let z = τ(z ) denote the twin point, and (π ,χ ) the corresponding pair. Then z lies on X ,where i = (k − 1)/2 − i = 2 − 1, by Lemma 3.3.We’re done:the first 5 properties of (π ,χ ) follow by construction, whilst the 6th follows from Theorem 2.33, Lemma 3.4 and Lemma 3.5. We can now complete the proof of Theorem 3.1. Bouncing from z to its twin point z reminded the authors of a game of ping pong, whence the section title. Earlier versions of the argument involved longer rallies! SYMMETRIC POWER FUNCTORIALITY 65 Proof of Theorem 3.1.— Let π , π be everywhere unramified cuspidal automor- phic representations of GL (A ) of weights k , k ≥ 2, respectively. Define integers m , 2 Q 0 π 0 0 m as in Lemma 3.7 and fix an integer m ≥ max(m , m ). Combining Lemma 3.7 and π π π 0 0 n−1 Lemma 3.6 (applied with i = 2 − 1) we see that the automorphy of Sym r implies π ,ι n−1 that of Sym r . Since π was arbitrary, this completes the proof. π ,ι Part II: Raising the level Most of the remainder of this paper (§§4 – 7) is devoted to the proof of Theorem E from the introduction, namely the existence for each n ≥ 2 of a single regular algebraic, cuspidal, everywhere unramified automorphic representation π of GL (A ) such that 2 Q n−1 Sym π exists. As a guide to what follows, we now give an expanded sketch of the proof of this theorem. Fix, for the sake of argument, a regular algebraic, cuspidal, everywhere unrami- fied automorphic representation π of GL (A ). We will try to establish the existence of 2 Q n−1 n−1 Sym π by proving the automorphy of one of the Galois representations Sym r π,ι associated to a choice of prime p and isomorphism ι : Q → C, using an automor- phy lifting theorem. First, if K/Q is an imaginary quadratic extension then we can find (using e.g. [BLGGT14, Lemma A.2.5]) a (de Rham) character ω : G → Q c −1 n−1 n−1 such that ωω = (det r ) . Then the representation ρ = ω ⊗ Sym r| satisfies π,ι G c ∨ 1−n ρ ρ ⊗ , so has the potential to be associated to a RACSDC automorphic repre- sentation of GL (A ). This means we can use an automorphy lifting theorem adapted to n K n−1 such automorphic representations. (The automorphy of ρ will imply that of Sym r π,ι by quadratic descent.) We need to select π and ι so that the residual representation ρ is automorphic. For “most” ι (say, for all but finitely many primes p) the image of r will contain a conjugate π,ι n−1 of SL (F ) and Sym ρ will be irreducible, and it is not clear how to proceed. We 2 p therefore want to avoid this generic case. Here we choose π and ι so that there is an isomorphism r Ind χ for some imaginary quadratic extension K/Q and character π,ι χ : G → F . Then there is an isomorphism n n−i c i−1 ρ = ⊕ ωχ (χ ) . i=1 In particular, this residual representation is highly reducible, being a sum of n characters. Most automorphy lifting theorems in the literature require the residual representation to be irreducible; we will apply [ANT20, Theorem 1.1], an automorphy lifting theorem that does not have this requirement, but that does have some other stringent conditions. These conditions include the requirement that there exist a RACSDC automorphic rep- (A ) such that r ρ , and satisfying the following: resentation  of GL = n K ,ι •  is ι-ordinary (and so is π ). 66 JAMES NEWTON, JACK A. THORNE • There exists a prime l = p and a place v|l of K such that both π and  are l v twists of the Steinberg representation (of GL (Q ) and GL (K ), respectively). 2 l n v It is easy to arrange that the first requirement be satisfied, by choosing p to be a prime which splits in K. The second is more difficult. First it requires that π is ramified at l , whereas we have to this point asked for π to be everywhere unramified. We will thus first n−1 find a ramified π for which Sym π exists, and eventually remove the primes of ramifi- cation using the l -adic analytic continuation of functoriality results proved in the first part of the paper. The main problem is then to find a  verifying the residual automorphy of ρ such that  is a twist of the Steinberg representation. This is what will occupy us in §§4 – 6 below. The above argument is then laid out carefully in §7 in order to finally prove Theorem E. Here is how we get our hands on . By choosing an appropriate lift of the charac- × × ter χ , we can choose characters X ,..., X : K \A → C such that  = X ··· X 1 n 0 1 n is a regular algebraic and conjugate self-dual (although not cuspidal!) automorphic rep- resentation of GL (A ) whose associated residual representation is ρ . If G is a definite n K unitary group in n variables associated to the extension K/Q, quasi-split at finite places, then we can hope that  transfers to an automorphic representation of G(A ). There is 0 Q a slight wrinkle here in that such a group G does not exist if n is even, and even in the case that n is odd there is a potential obstruction to the existence of this transfer given, at least conjecturally, by Arthur’s multiplicity formula. Both of these obstacles can be avoided by replacing Q with a suitable soluble totally real extension F/Q.Inorder to avoidin- troducing additional notation in this sketch, we pretend they can be dealt with already in the case F = Q. (Actually, we will find it convenient to take  to be the box sum of two cuspidal automorphic representations of GL (A ) and GL (A ), respectively. This 2 K n−2 K means that the final form of the proof of Theorem E will be a kind of induction on n.) We thus find ourselves with an automorphic representation  of G(A ),whose 0 Q base change (in the sense of Theorem 1.2)is  . Say for the sake of argument that l splits in K, so that we can identify G(Q ) with GL (K ). If we can find another automorphic l n v representation  of G(A ), congruent modulo p to  and such that  is a twist of 1 Q 0 1,v the Steinberg representation, then we will have solved our problem. We can therefore now focus on this problem of level-raising for the definite unitary group G. There are differing approaches to this problem in the literature. First there is the purely automorphic approach, pioneered by Ribet for GL (A ) [Rib84]. Some gener- 2 Q alisations to higher rank groups of this statement do exist (see for example [Tho14a]), but nothing that is applicable in the level of generality considered here. Then there is the purely Galois theoretic approach, based on the powerful automorphy lifting theorems which are now available for Galois representations in arbitrary rank (see for example [Gee11]). We can not directly apply such results here because the only automorphy lifting theorems applicable in the residually reducible case (namely those of [ANT20]) require the existence of at least one place at which the starting automorphic representation is sufficiently non-degenerate. SYMMETRIC POWER FUNCTORIALITY 67 We solve the problem here by combining aspects of these approaches. A similar combination of techniques is used in the paper [CT17]: the idea is to first use an auto- morphic technique to replace  with a representation  such that  is so ramified 0 0,v that, in conjunction with other conditions in place,  is forced to be stable (i.e. its base change is cuspidal). (The possibility of doing this is the reason for choosing  to in fact be a box sum of two cuspidal factors, as mentioned above.) This situation is reflected in the deformation theory, where one finds that (in the big ordinary Galois deformation ring) the locus of reducible deformations is small enough that something like the tech- niques of [Gee11] can be applied to construct an automorphic lift of ρ with the required local properties. These Galois theoretic arguments are carried out in §§5, 6. What remains to be explained then is the automorphic level-raising technique de- veloped in §4. The approach to creating congruences here is based on types. We recall (using the language of [BK98]) that if s is an inertial equivalence class of G(Q ) (i.e. a supercuspidal representation of a Levi factor of G(Q ), up to unramified twist), then an s-type is a pair (U,τ), where U is an open compact subgroup of G(Q ) and τ is an irre- ducible finite-dimensional representation of U such that for each irreducible admissible representation σ of G(Q ), the supercuspidal support of σ is in class s if and only if σ | v l v v U contains τ . It can sometimes happen that two inertial equivalence classes s, s admit types −1 (U,τ) and (U ,τ ) with the property that U = U , the reduction modulo p τ of ι τ −1 is irreducible, and the reduction modulo p τ of ι τ contains τ as a Jordan–Hölder factor. This situation might be called a congruence of types. If this is the case then the theory of algebraic modular forms implies that any automorphic representation  of G(A ) such that  is of type s is congruent to another  such that  is of type s .The Q l existence of such global congruences is explained in [Vig01, §3]. It gives an efficient way to construct congruences between automorphic representations ,  such that  , are in different inertial equivalence classes, although it is not usually possible to change the Levi subgroup underlying the inertial equivalence class. Since G(Q ) GL (K ) and l n v the initial representation  is certainly not supercuspidal at l , it is not immediately clear how to use this. We therefore instead introduce an auxiliary imaginary quadratic extension E/Q in which l is inert, as well as an associated definite unitary group G , and carry out the first step of the automorphic part of the level-raising argument using algebraic modular forms on G . The importance of the group G is that there are conjugate self-dual irreducible admissible representations of GL (E ) which are not supercuspidal, but for which the 3 l associated L-packets of representations of U (Q ) contain supercuspidal elements. For 3 l carefully chosen local data, we can find use the method of types to find congruences to supercuspidal representations of U (Q ) whose base change to GL (E ) is supercuspidal. 3 l 3 l We have already constructed such congruences of types in §1.8. In terms of automorphic representations of GL (A ), this will allow us to change the Levi subgroup underlying n E 68 JAMES NEWTON, JACK A. THORNE the inertial equivalence class at l from the maximal torus of GL (E ) to the group GL × n l 3 n−3 GL . This will be enough for our intended application. 4. Raising the level – automorphic forms Let n = 2k + 1 ≥ 3 be an odd integer, and let F, p, S, G = G etc. be as in our standard assumptions (see §1). Suppose given cuspidal, conjugate self-dual automorphic representations π of GL (A ) and π of GL (A ) with the following properties: 2 2 F n−2 n−2 F (1) π = π  π is regular algebraic. 2 n−2 (2−n)/2 Consequently, π |·| and π are regular algebraic and the representations ρ = 2 n−2 r (2−n)/2 , ρ = r −1 are defined. We set ρ = r = ρ ⊕ ρ .Moreover, π and π |·| ,ι n−2 π |·| ,ι π,ι 2 n−2 2 2 n−2 π are both tempered, cf. the remark after Corollary 1.3. n−2 (2) π is ι-ordinary. (3) We are given disjoint, non-empty sets T ,T ,T of places of F with the following 1 2 3 properties: (1) For all v ∈ T = T ∪ T ∪ T , v ∈ Sand q is odd. The representation π is unram- 1 2 3 v ified outside S ∪ T. If v is a place of F lying above a place in T then, as in §§1.8, × × 1.17, we write ω( v) : k( v) → C for the unique quadratic character. th (2) For each v ∈ T , v is inert in F, q mod p is a primitive 6 root of unity, and 1 v the characteristic of k(v) is greater than n. There are characters χ ,χ ,χ ,..., v v,0 v,1 × × χ : F → C such that χ , χ are unramified and for each i = 1,..., 2k − 2, v,2k−2 v v,0 2k−2 ∼ ∼ χ | × = ω( v).Wehave π St (χ ) and π  χ . = = v,i 2, v 2 v n−2, v v,i O i=0 c nd (3) For each v ∈ T , v splits v = v v in F, q mod p is a primitive 2 root of unity, 2 v π | contains λ( v,  ) (for some order p character  as in §1.17,with 2, v GL (O ) v v 2 F n = 2), and π | contains ω( v) ◦ det. Thus π satisfies the equivalent 1 n−2, v GL (O ) v n−2 F conditions of Proposition 1.20,and π | contains λ( v,  , n). v q v c th (4) For each v ∈ T , v splits v = v v in F, q mod p is a primitive (n − 2) root of unity, 3 v π | contains λ( v,  ) (for some order p character  as in §1.17,with n−2, v GL (O ) v v n−2 F n = n − 2), and π | contains ω( v) ◦ det. Thus π satisfies the equivalent 1 2, v GL (O ) v 2 F conditions of Proposition 1.20,and π | contains λ( v,  , n). v q v Let T ={ v | v ∈ T}.Wefixfor each v ∈ T acharacter θ : C(k(v)) → C of order p 1 v (notation as in Proposition 1.12). In the rest of this section, we will prove the following theorem. + + Theorem 4.1. — With hypotheses as above, let L /F be a totally real S ∪ T-split quadratic extension, and let L = L F. Then there exists a RACSDC automorphic representation  of GL (A ) n L with the following properties: (1)  is ι-ordinary, and unramified at any place not dividing S ∪ T. (2) r = r | . ,ι π,ι G (3) For each place v ∈ T ,  | contains the representation λ( v, θ , n)| (thus satisfying the 1,L v r v| r v + v equivalent conditions of Proposition 1.19). SYMMETRIC POWER FUNCTORIALITY 69 (4) For each place v ∈ T ∪ T ,  | contains the representation λ( v,  , n) (thus satisfying 2,L 3,L v q v| v F the equivalent conditions of Proposition 1.20 with n = 2 if v ∈ T and n = n − 2 if v ∈ T ). 2 2,L 2 3,L Remark 4.2. — The places v ∈ T ∪ T play a role in ensuring that  is cuspidal 2 3 (using Lemma 4.5 below). Our set-up is adapted to the proof of Proposition 7.4, which uses an induction on the dimension to construct automorphic representations with an unramified twist of Steinberg local factor which are congruent to some very special odd- dimensional symmetric powers. We begin with two important observations. + θ Lemma 4.3. — Let v be a finite place of F which is inert in F. Then π ∈ A (GL (F )) . v n v + Proof. — The representation π is tempered because both π and π are tem- v 2, v n−2, v pered. By the main theorem of [BC11], r extends to a homomorphism r : G + → π,ι F 1−n n G (Q ) such that ν ◦ r = δ . Restricting to W and twisting by an appropri- n + p F/F v ate character, we see that the Langlands parameter of π extends to a parameter W × SL → G. Remark 4.4. — A consequence of this lemma is that for v ∈ T the character χ is 1 v non-trivial quadratic and the character χ is trivial. v,0 Lemma 4.5. — Suppose given a partition n = n +··· + n and cuspidal, conjugate self- 1 r dual automorphic representations π ,...,π of GL (A ) such that π = π  ···  π is regular n F 1 r i 1 r algebraic. Suppose moreover that the following conditions are satisfied: (1) There is an isomorphism r = r . π ,ι π,ι (2) If v ∈ T ∪ T then π | contains λ( v,  , n). 2 3 q v v v Then one of the following two statements holds: (1) We have r = 1,n = n, and so π is cuspidal. (2) After re-ordering we have r = 2,n = n − 2,n = 2.If v ∈ T then π | contains 1 2 2 GL (O ) 1, v n−2 F ω( v) ◦ det and π | contains λ( v,  ), while if v ∈ T then π | contains GL (O ) v 3 GL (O ) 2 F n−2 F 2, v 1, v v v λ( v,  ) and π | contains ω( v) ◦ det. We have isomorphisms of semisimplified residual v GL (O ) 2, v 2 F ∼ ∼ representations r −1 ρ and r (2−n)/2 ρ . = = π |·| ,ι π |·| ,ι n−2 2 Proof. — Before beginning the proof, we observe that the representations ρ , ρ 2 n−2 have the following properties: ss ss • If v ∈ T ,then ρ | is unramified and ρ | is unramified after twisting by 2 G n−2 G F F v v a ramified quadratic character. (The character  has order p.) ss • If v ∈ T ,then ρ | is unramified after twisting by a ramified quadratic char- 2 G ss acter and ρ | is unramified. n−2 G v 70 JAMES NEWTON, JACK A. THORNE We can suppose without loss of generality that r ≥ 2. Fix places v ∈ T , v ∈ T . 2 2 3 3 By Proposition 1.20, we can assume after relabelling that π | contains GL (O ) 1, v n F 3 1 v λ( v , , n ) (and in particular, n ≥ n − 2),inwhichcase (π  ···  π )| 3 v 1 1 GL (O ) 3 2, v r, v n−n F 3 3 1 v contains ω( v ) ◦ det. There is an isomorphism r  =⊕ r  (n −n)/2 ρ ⊕ ρ , π ,ι i i=1 π |·| ,ι 2 n−2 hence r ss ss ss ⊕ r  (n −n)/2 | = ρ | ⊕ ρ | . i=1 π |·| ,ι G 2 G n−2 G F F F v v v 3 3 3 r ss Since ⊕ r  (n −n)/2 | contains no unramified subrepresentation, we conclude that π |·| ,ι i=2 G i v r  (n −n)/2 contains ρ as a subrepresentation. π |·| ,ι n−2 We now look at the place v . There are two possibilities for the representation π : either π | contains λ( v , , n ), or it contains ω( v ) ◦ det. We claim GL (O ) 2 v 1 2 1, v 1, v n F 2 2 2 1 v that the first possibility does not occur. Indeed, in this case arguing as above shows that r ss ⊕ r  (n −n)/2 | contains no unramified subrepresentation, and therefore that ρ is a π |·| ,ι 2 i=2 G subrepresentation of r  (n −n)/2 . This forces n = n and r = 1, a contradiction. Therefore 1 1 π |·| ,ι we must have r = 2, n = 2, and π | contains λ( v , ). Since v , v were 2 GL (O ) 2 v 2 3 2 F 2 2, v 2 v arbitrary, this completes the proof. We now commence the proof of the theorem. Let U = U ⊂ G(A ) be an v + v F open compact subgroup with the following properties: ι (U ) v v • For each v ∈ S, π = 0. • If v ∈ S ∪ T, then U = G(O ). v F Iw (c,c),ord −1 v • If v ∈ S ,then U = ι Iw (c, c) for some c ≥ 1such that π = 0 (notation p v v v v as in [Ger19, §5.1]) and U contains no non-trivial torsion elements (note this implies that U is sufficiently small). −1 • If v ∈ T then U = ι p (notation as in §1.8). 1 v v −1 • If v ∈ T ∪ T then U = ι q (notation as in §1.17, defined with n = 2if 2 3 v v 1 v ∈ T and n = n − 2if v ∈ T ). 2 1 3 We define τ =⊗ τ(v, n) ⊗ λ( v,  , n),where τ(v, n), λ( v,  , n) are the rep- g v∈T v∈T ∪T v v 1 2 3 resentations of p , q defined in §1.8,§1.17 respectively. Thus τ is an irreducible C[U ]- v v g T module, which we view as a C[U]-module by projection to the T-component. Similarly we define λ =⊗ λ(v, θ , n) ⊗ λ( v,  , n). Fixing a sufficiently large coefficient g v∈T v v∈T ∪T v 1 2 3 −1 −1 field, we can choose O-lattices τ and λ in ι τ and ι λ , respectively. g g g g + + If L /F is an S ∪ T-split totally real quadratic extension, then we define an open compact subgroup U = U ⊂ G(A ) and representations τ , λ by the same L L,v g,L g,L recipe (where we now replace the sets S, T by their lifts S ,T to L ). i L i,L SYMMETRIC POWER FUNCTORIALITY 71 + + Proposition 4.6. — Let L /F be an S ∪ T-split totally real quadratic extension and let L = L F. Then either there exists an automorphic representation σ of G(A ) with the following properties: (1) π is the base change of σ (cf. Theorem 1.2); (2) For each place v ∈ T, σ = 0; (3) σ | contains τ . T U g or there exists an automorphic representation σ of G(A ) with the following properties: (1) Let π denote the base change of π with respect to the quadratic extension L/F. Then π is the base L L change of σ ; L,v (2) For each place v ∈ T , σ = 0; (3) σ | contains τ . T U g,L L L,T Proof. — By [Lab11, Théorème 5.1], there is an identity G M H (4.6.1)T (f ) = ι(G, H)T (f ), disc disc ∞ ∞ for any f = f ⊗ f ∈ C (G(A +)) such that f is a pseudocoefficient of discrete series. ∞ F ∞ Here the sum on the right-hand side is over representatives for equivalence classes of endoscopic data for G, represented here by the associated endoscopic group H (recall that we have fixed representative endoscopic triples in §1). The coefficients ι(G, H) are M H givenin[Lab11, Proposition 4.11], while the expression T (f ) is given in [Lab11, disc Proposition 3.4] as a formula M −1 L H (4.6.2) (| det(s − 1 | a /a H)||W (L)|) tr I ( π )(f ), L M Q H H L 0 M M π ∈ (L ) disc s L∈L /W s∈W (L) reg where (summarizing the notation of op. cit.): • M is a twisted space on a Levi of Res GL ,asin§1.5; F/F n 0 H • L is the set of standard Levi subgroups of M ; M L • W (L) is the quotient by the Weyl group W of the set of elements s in reg H H M M the twisted Weyl group W = W  θ which normalise L and such that det(s − 1 | a /a H) = 0, where a denotes the Lie algebra of the maximal Q-split L M ? subtorus of the centre of a reductive group. •  (L ) is the set of isomorphism classes of irreducible representations of the disc s twisted space L (A +) which appear as subrepresentations of the discrete spec- s F trum of L. L H • I ( π )(f ) is a certain intertwining operator, with Q a parabolic subgroup with Levi L. We fix our choice of f so that it only has non-zero traces on representations of G(F ) whose infinitesimal character is related, by twisted base change, to that of π . The argu- 0 M ment of [Shi11, Proposition 4.8] then shows that for each L ∈ L /W , there is at most 72 JAMES NEWTON, JACK A. THORNE one element s ∈ W (L) for which the corresponding summand in (4.6.2) can be non- reg zero (and a representative for s can be chosen which acts as conjugate inverse transpose on each simple factor of L). Using Proposition 1.7, linear independence of characters, the description of the discrete spectrum of general linear groups [MW89], and the Jacquet– Shalika theorem [JS81], we can combine (4.6.1)and (4.6.2) to obtain a refined identity −1 ∼ U ×U n−2 2 (4.6.3) m(σ )σ (f ) = π(f ) + (π ⊗ (π ⊗ μ ◦ det)) (f ) , i i n−2 2 where: • The sum on the left-hand side is over the finitely many automorphic represen- tations σ of G(A +) which are unramified at all places below which π is unram- i F ified, have infinitesimal character related to that of π by twisted base change, and which are related to π by (either unramified or split) base change at places v ∈ T of F , each occurring with its multiplicity m(σ ). 1 i • The twisted traces on the right-hand side are Whittaker-normalised. (These two terms arise from H = U ,L = Res GL × GL and H = U × U ,L = n F/F n−2 2 n−2 2 M , respectively. The same argument as in [Lab11, Proposition 3.7] shows that Arthur’s normalisation of the twisted trace, implicit in the term I (?) of (4.6.2), agrees with the Whittaker normalisation on the corresponding terms.) We remark that the representations π , π are tempered and that the representa- 2 n−2 tions σ (i.e. prime to T -part) are isomorphic. If v ∈ T ,thenwecan find(com- 1 1 bining Proposition 1.11 for U × U and e.g. [Hir04, Proposition 4.6]) a finite set n−2 2 {λ } of irreducible admissible representations of G(F ) and scalars d ∈ C such that v,i v,i −1 ∼ U ×U n−2 2 (π ⊗ (π ⊗ μ ◦ det)) (f ) = d λ (f ).ByProposition 1.6,Proposition n−2,v 2,v v,i v,i v v v i 1.7 and Proposition 1.11, we therefore have an identity: m(σ )σ (f ) = (v, U ,ϕ ) c τ(f ) i i,T T n U τ v 1 1 n i v|∞ v∈T τ ∈(π ) 1 v + (v, U × U ,ϕ ) d λ (f ) , n−2 2 U ×U v,i v,i v n−2 2 v|∞ v∈T Choose for each v ∈ T a representation τ ∈ (π ) such that τ | contains τ(v, n) (this 1 v v v U is possible by Corollary 1.13 and Proposition 1.14). We can assume that for each v ∈ T , λ = τ (possibly with d = 0). We conclude that there is at most one automorphic v,1 v v,1 representation σ of G(A ) with the following properties: • σ is unramified outside S ∪ T, and is related to π by split or unramified base change at all places v ∈ T ; • If v ∈ T ,then σ τ . 1 v v SYMMETRIC POWER FUNCTORIALITY 73 The representation σ occurs with multiplicity m(σ ) = (v, U ,ϕ ) c n U τ n v v|∞ v∈T + (v, U × U ,ϕ ) d . n−2 2 U ×U v,1 n−2 2 v|∞ v∈T We note that the numbers c are all non-zero, by Proposition 1.11.If m(σ ) is non-zero, then we’re done (we are in the first case in the statement of the proposition). Otherwise, c = d , which we now assume. v∈T τ v∈T v,1 1 v 1 + + + In this case, let L /F be a totally real quadratic S ∪ T-split extension, let L = L F, and let π denote the base change of π with respect to the quadratic extension L/F. If v ∈ T ,let τ = τ . Then repeating the same argument shows that there is at most 1,L v v| + one automorphic representation σ of G(A ) with the following properties: • σ is unramified outside S ∪ T , and is related to π by split or unramified base L L L change at all places v ∈ T ; 1,L • If v ∈ T ,then σ τ . 1,L v v Using the remark after Proposition 1.6, we see that the representation σ occurs with multiplicity ⎛ ⎞ ⎝ ⎠ m(σ ) = c + d = c . τ v,1 v τ v∈T v∈T v∈T 1,L 1,L 1 This is non-zero, so we’re done in this case also (and we are in the second case of the proposition). We now show how to complete the proof of Theorem 4.1, assuming first that we are in the first case of Proposition 4.6.Welet σ be the automorphic representation of Hom(F,Q ) G(A +) whose existence is asserted by Proposition 4.6.Let λ ∈ (Z ) be such ord −1 that σ contributes to S (U,ι τ ) under the isomorphism of Lemma 1.25.Let T ⊂ ord −1 End (S (U,ι τ )) be the commutative O-subalgebra generated by unramified Hecke O g j c + operators T at split places v = ww ∈ SofF ,and let m ⊂ T be the maximal ideal determined by σ . ord Then S (U, τ ˚ ⊗ k) is non-zero, by Lemma 1.24, hence (using the exactness of g O m ord ord S (U, −) as a functor on k[U]-modules, together with Proposition 1.15)S (U, λ ⊗ g O λ λ ord k) = 0, hence S (U, λ ) = 0. Applying Lemma 1.25 once again, we conclude the m g m existence of an automorphic representation  of G (A ) with the following properties: n F • r r . ,ι π,ι 74 JAMES NEWTON, JACK A. THORNE •  | contains λ . T U g •  is ι-ordinary and is unramified outside S ∪ T. + + Let  denote the base change of ,let L /F be a quadratic totally real extension as in the statement of Theorem 4.1,and let  denote base change of  with respect to the extension L/F. We claim that  satisfies the requirements of Theorem 4.1.The only points left to check are that  is cuspidal and that if v ∈ T then  satisfies L 1,L L,v condition (3) in the statement of Theorem 4.1. In fact, it is enough to show that  is cuspidal and that if v ∈ T then  | contains λ( v, θ , n)| . We first show that  is cus- 1 v r v r v v pidal. If  is not cuspidal, then Lemma 4.5 shows that  =    where  , n−2 2 n−2 2 are cuspidal, conjugate self-dual automorphic representations of GL (A ),GL (A ), n−2 F 2 F respectively. Arguing as in proof of Proposition 4.6, we obtain an identity −1 ∼ U ×U n−2 2 (4.6.4) m( ) (f ) = (f ) + ( ⊗ ( ⊗ μ ◦ det)) (f ) , i i n−2 2 where the sum on the left-hand side is over the finitely many automorphic representations of G(A +) which are unramified at all places below which  is unramified, have in- i F finitesimal character related to that of  by twisted base change, and which are related to  by (either unramified or split) base change at places v ∈ T of F . v,∞ Fix v ∈ T , and consider a test function of the form f = f ⊗ f ⊗ f ,where: 1 v ∞ • f is a coefficient for  . ∞ ∞ • f is the test function denoted φ in the statement of Proposition 1.16. v,∞ • f is the characteristic function of an open compact subgroup of G(A ). • (f ) = 0. Then  (f ) is non-negative for any i, and the left-hand side of (4.6.4)isnon-zero. We −1 ∼ U ×U n−2 2 conclude that at least one of the terms (f ) and ( ⊗ ( ⊗ μ ◦ det)) (f ) n−2 2 is non-zero. In either case Proposition 1.16 implies that the cuspidal support of  ,and therefore  , contains a supercuspidal representation # of GL (F ) such that the n−2, v 3 v T −1 semisimple residual representation attached to rec (ι #) is unramified. This contra- ss dicts Lemma 4.5, which implies that r −1 | is the sum of an unramified character |·| ,ι n−2 G and the twist of an unramified representation by a quadratic ramified character. Therefore  is cuspidal, and a similar argument now gives an identity (4.6.5) m( ) (f ) = (f ). i i With the same choice of test function we have (f ) = 0, so another application of Propo- sition 1.16 shows that  has the required property. This completes the proof of Theorem 4.1, assuming that the first case of Proposition 4.6 holds. If the second case holds, the ar- gument is very similar, except that there is no need to replace  by its base change with respect to a quadratic extension L/F. In either case, this completes the proof. SYMMETRIC POWER FUNCTORIALITY 75 5. A finiteness result for Galois deformation rings In this section we prove that certain Galois deformation rings are finite over the Iwasawa algebra (Theorem 5.2), and use this to give a criterion for a given deformation to have an irreducible specialization with useful properties (Theorem 5.7). These technical results form the basis for the arguments in §6, where we will apply our criterion to the Galois representation valued over a big ordinary Hecke algebra. The novelty of the results proved in this section is that we assume that the residual representation is reducible (in fact, to simplify the exposition we assume that this repre- sentation is a sum of characters). The main tools are the automorphy lifting theorems proved in [ANT20] and the idea of potential automorphy, for which we use [BLGGT14] as a reference. The notation and definitions we use for Galois deformation theory in the ordinary case are summarized in [ANT20, §3], and we refer to that paper in particu- lar for the notion of local and global deformation problem, and the definitions of the particular local deformation problems used below. Before getting stuck into the details, we record a useful lemma. If  is a profinite group, k is a field with the discrete topology, and ρ :  → GL (k) is a continuous rep- resentation, we say that ρ is primitive if it is not isomorphic to a representation of the form Ind σ for some finite index proper closed subgroup  ⊂  and representation σ :  → GL (k). This condition appears as a hypothesis in the automorphy lifting n/[: ] theorem proved in [ANT20]. Lemma 5.1. — Suppose that ρ = χ ⊕···⊕χ , for some continuous characters χ :  → k 1 n i such that for each i = j, χ /χ has order greater than n. Then ρ is primitive. i j Proof. — Suppose that there is an isomorphism ρ Ind σ . Then Frobenius reci- procity implies that σ contains each character χ | . These n characters are distinct: if [: ] χ | = χ | ,then (χ /χ ) = 1, which would contradict our assumption that χ /χ i j i j i j has order greater than n if i = j.Thus σ must have dimension at least n, implying that =  . It follows that ρ is primitive. Now let n ≥ 2and letF,S, p be as in our standard assumptions (see §1), and let E ⊂ Q be a coefficient field. We recall the definition of the Iwasawa algebra .If ab n ab v ∈ S , then we write  = O(I (p)) ,where I (p) denotes the inertia subgroup of p v F F v v the Galois group of the maximal abelian pro-p extension of F . We set  = ⊗  , v v∈S v the completed tensor product being over O.For each v ∈ S and i = 1,..., n there is i ab × a universal character ψ : I (p) →  . At times we will need to introduce Iwasawa v F v algebras also for extension fields F /F and for representations of degree n = n,inwhich case we will write e.g.  for the corresponding Iwasawa algebra, dropping a subscript F ,n when either F = For n = n. Let μ : G + → O be a continuous character which is de Rham and such that F ,S μ(c ) =−1 for each place v|∞ of F . Fix an integer n ≥ 2, and suppose given characters v 76 JAMES NEWTON, JACK A. THORNE × n χ ,..., χ : G → k such that for each i = 1,..., n, χ χ = μ| . We set ρ =⊕ χ ; F,S G 1 n i i F,S i=1 i then ρ extends to a homomorphism r : G → G (k) such that ν ◦ r = μ, by setting F ,S n G r(c) = (1 , 1)j ∈ G (k). We suppose that for each v ∈ S , r| is trivial. n n p G Let  be a set of finite places of F split in F and disjoint from S, and  aliftof to F. If for each v ∈ , q ≡ 1mod p and r| is trivial, then we can define the global v G deformation problem +   St S = (F/F , S ∪ , S ∪ , , r,μ, {R } ∪{R } ∪{R } ). v∈S v∈S−S v∈ v p v p v (For the convenience of the reader, we summarize the notation from [ANT20,§3].Thus the local lifting ring R represents the functor of ordinary, variable weight liftings; R v v St the functor of all liftings; and R the functor of Steinberg liftings.) If r is Schur, in the sense of [CHT08, Definition 2.1.6], then the corresponding global deformation functor is represented by an object R ∈ C .If  is empty, then we write simply S = S . S  ∅ Theorem 5.2. — Suppose that the following conditions are satisfied: (1) p > 2n. (2) For each 1 ≤ i < j ≤ n, χ /χ | has order greater than 2n. (In particular, ris Schur.) i j F(ζ ) (3) [F(ζ ) : F]= p − 1. (4)  is non-empty. Then R is a finite -algebra. Proof. — We will compare R with a deformation ring for Galois representations to G .First,fixaplace v of F prime to S ∪ , lying above a rational prime q > 2n which 2n q splits in F. After possibly enlarging k, we can find a character ψ : G → k satisfying the following conditions: −1 1−2n • ψψ = μ| . • For each v ∈ S ∪ , ψ | is unramified. • q divides the order of ψ/ψ (I ). Using the formulae in [BLGGT14, §1.1], we can write down a character 1−2n −1 + + (ψ, μ δ ) : G → G (k), F/F F 1 the tensor product representation 1−2n −1 + + r ⊗ (ψ, μ δ ) : G → G (k), F/F F n 1−2n which has multiplier , and the representations −1 1−2n r = I(r ⊗ (ψ, μ δ +)) : G + → GSp (k) 1 F/F F 2n and r = (r ) : G + → G (k). 2 1 G F 2n F SYMMETRIC POWER FUNCTORIALITY 77 These representations have the following properties: 1−2n • The multiplier character of r equals . 1−2n • The multiplier character ν ◦ r equals . G 2 2n • The representations r | and r | are both conjugate in GL (k) to ρ ⊗ ψ ⊕ 1 G 2 G 2n F F ρ ⊗ ψ . Let ρ = r | . We observe that the following conditions are satisfied: 2 2 G ker ad ρ • ζ ∈ F and F ⊂ F (ζ ). p p • ρ is primitive. • The irreducible constituents of ρ | occur with multiplicity 1. 2 F(ζ ) ker ad ρ Indeed, the condition F ⊂ F (ζ ) holds because [F(ζ ) : F]= p − 1. We have F ⊂ p p −1 F({χ /χ } ,ψ/ψ , {χ χ μ } ) = M, say, and c acts on Gal(M/F) as −1. It follows that i=j i,j i j i j F(ζ ) ∩ M has degree at most 2 over F, showing that ζ ∈ M. To see that ρ is primitive, p p it is enough (by Lemma 5.1) to show that χ /χ has order greater than 2n if i = j and that i j (χ ψ)/(χ ψ ) has order greater than 2n for any i, j . These properties hold by hypothesis in the first case and since q > 2n in the second. Finally, the constituents of ρ are, with c c c c multiplicity, χ ⊗ ψ,..., χ ⊗ ψ , χ ⊗ ψ ,..., χ ⊗ ψ . Our hypotheses include the 1 n 1 n condition that χ ⊗ ψ | = χ ⊗ ψ | if i = j.If χ ⊗ ψ | = χ ⊗ ψ | then G G G G i j i F(ζ ) F(ζ ) F(ζ ) j F(ζ ) p p p p ψ/ψ | is trivial, a contradiction. Fix an isomorphism ι : Q → C.By[BLGGT14, Theorem 3.1.2], we can find a + + Galois totally real extension L /F and a regular algebraic, self-dual, cuspidal, automor- phic representation π of GL (A +), with the following properties: 2n L • Let L = FL .Then L/F is linearly disjoint from the extension of F(ζ ) cut out ker ad ρ | L + by ρ | . In particular, [L(ζ ) : L]= p − 1, ζ ∈ L ,and L ⊂ L (ζ ). G p p p 2 F(ζ ) • Thereisanisomorphism r r | . π,ι 1 G • π is ι-ordinary. More precisely, π is of weight 0 and for each place v|p of L , π is an unramified twist of the Steinberg representation. • ρ | is primitive. 2 L • The irreducible constituents of ρ | occur with multiplicity 1. 2 L(ζ ) • For each place v of L lying above a place of , π is an unramified twist of the Steinberg representation. More precisely, [BLGGT14, Theorem 3.1.2] guarantees the existence of L satisfying the first condition and an essentially self-dual π satisfying all the remaining conditions (except possibly the last one). The last paragraph of the proof notes that the π constructed is in fact self-dual and π is an unramified twist of the Steinberg representation for each place + + v|p of L . We can moreover ensure that π is Steinberg at the places of L lying above by inserting the condition “t(P)< 0 for all places v| of L ” in the first list of conditions on [BLGGT14, p. 549]. 78 JAMES NEWTON, JACK A. THORNE + + After possibly adjoining another soluble totally real extension of F to L ,wecan assume that the following further conditions are satisfied: • ρ | is unramified at those finite places not dividing S ∪  . 2 G L L • Each place of L at which π is ramified is split over L . • For each place v ∈ S ∪  , ψ | is trivial. L L G • For each place v ∈ S , [L : Q ] > 2n(2n − 1)/2 + 1and ρ | is trivial. p,L v p G Here π denotes the base change of π . It is a RACSDC automorphic representation of GL (A ). By construction, then, π satisfies the hypotheses of [ANT20, Theorem 6.2]. 2n L L Therefore, if we define the global deformation problem + 1−2n S = (L/L , S ∪  , S ∪  , , r | , , L L L L L,2n 2 G St {R } ∪{R } ∪{R } }), v∈S v∈S −S v∈ v p,L v L p,L v L then R  is a finite  -algebra. (Here we have written  to distinguish from  = S L,2n L,2n used above.) F,n We now need to relate the rings R and R . In fact, it will be enough to construct S S a commutative diagram R /( ) R /( ) S S /( )  /( ) L,2n F,n where the top horizontal morphism is finite. We first specify the map  /( ) → L,2n /( ). It is the map that for each place w ∈ S lying above a place v of F classi- F,n p,L fies the tuple of characters v v v −1 v −1 (ψ | ,...,ψ | ,ψ | ,...,ψ | ). I I 1 L n L n I 1 I w w L L w w This endows the ring R /( ) with the structure of  -algebra. To give a map S L,2n R /( ) → R /( ), we must give a lifting of r | over R /( ) which is of type S . S S 2 G S To this end, let r denote a representative of the universal deformation (of r)to R /( ), −1 1−2n ∧ and let r = I(r ⊗ (ψ, μ δ +)) | (notation as in [BLGGT14, §1.1]). Then r F/F G G + F L is a lift of r and r | is the restriction of r| ⊗ ψ ⊕ r | ⊗ ψ to G . We need to check 2 G G G L L F F St that for each v ∈ S , r | is of type R ;and that foreach v ∈  , r | is of type R . p,L G L G L v L v v v These statements can be reduced to a universal local computation. It follows that r is of type S , and so determines a morphism R /( ) → R /( ). To complete the proof, it will be enough to show that this is a finite ring map. SYMMETRIC POWER FUNCTORIALITY 79 We can enlarge the above commutative diagram to a diagram R /( ) R /( ) S S Q /( )⊗  /( ) Q /( )⊗  /( ), t | O L,2n t O F,n 2 G where Q is the complete Noetherian local O-algebra classifying pseudocharacters of t | 2 G G lifting the restriction of t = tr r | to G ,Q is defined similarly with respect L,S ∪ 2 2 G L t L L F to the pseudocharacter t = tr r| of G ,and themap Q /( ) → Q /( ) is the G F,S∪ t | t F 2 G one classifying the natural transformation sending a pseudocharacter t lifting t to the pseudocharacter (t| ⊗ ψ) + (t | ⊗ ψ ). We deduce from [Tho15, Proposition 3.29] G G L L that the vertical arrows are finite ring maps. The map  →  is also finite, so it’s L,2n F,n enough finally to show that the map Q /( ) → Q /( ) is finite. This map can in t | t 2 G turn be written as a composite Q /( ) → Q /( ) → Q /( ), t | t | t 2 G 2 G L F where the first map classifies restriction of pseudocharacters from G to G . Since r | is F L 2 G multiplicity free, [ANT20, Proposition 2.5] (specifically, the uniqueness of the expression as a sum of pseudocharacters) implies that the second map is in fact surjective. We finally just need to show that the first map is finite, and this follows from the following general lemma. Lemma 5.3. — Let  be a topologically finitely generated profinite group, let  be a closed subgroup of finite index, and let t be a pseudocharacter of  with coefficients in k of some dimension n. Let Q be the complete Noetherian local O-algebra classifying lifts of t. Then the map Q → Q t t| t classifying restriction to  is a finite ring map. Proof. — It suffices to show that Q /(m ) is Artinian. If not, we can find a prime t Q t| ideal p of this ring of dimension 1; let A be its residue ring (which is a k-algebra), and let t be the induced pseudocharacter of  with coefficients in A. Let N =[ : ].If γ ∈ N! then γ ∈ . If we factor the characteristic polynomial of X − γ under t as (X − α ) i=1 for some elements α in the algebraic closure of Frac A, then the characteristic polyno- N! N! mial of γ under t,namely (X − α ),liesin k[X] and equals the characteristic i=1 i N! polynomial of γ under t. This shows that the elements α are in fact algebraic over k, and thus (using [Che14, Corollary 1.14]) that t can be defined over k, and must in fact equal t. This is a contradiction. n Hom(F,Q ) Corollary 5.4. — With hypotheses as in Theorem 5.2,fix λ ∈ (Z ) such that for each i = 1,..., nand τ ∈ Hom(F, Q ), we have λ =−λ . Suppose further that for each τ c,i τ,n+1−i p 80 JAMES NEWTON, JACK A. THORNE v ∈ S , [F : Q ] > n(n − 1)/2 + 1. Then there exists a homomorphism r : G → G (Z ) lifting p v p F ,S n p rsuch that r| is ordinary of weight λ,inthe senseof[Tho15, Definition 2.5]. F,S Proof. — We observe that [Tho15, Proposition 3.9, Proposition 3.14] show that for each minimal prime Q ⊂ R ,dimR /Q = dim ; consequently, there is a minimal S S prime Q of  and a finite injective algebra morphism /Q → R /Q. The corollary follows on choosing any prime of R /Q[1/p] lying above a maximal ideal of /Q [1/p] associated to the weight λ as in [Ger19, Definition 2.24]. Corollary 5.5. — With hypotheses (1) – (3) of Theorem 5.2, choose a place v ∈ S of F split in F and a lift v to F such that q ≡ 1 mod p and r| is trivial. Consider the quotient 0 v G 0 F A = R /( , {tr r (Frob ) − n} ). S S i=1,...,n ∅ ∅ v Then A is a finite -algebra. Consequently, dim R /( ) ≤ n[F : Q]+ n. Proof. — It suffices to verify that the quotient of R where the characteristic poly- 1−i nomial of Frob equals (X − q ) is a quotient of R . This in turn means check- v S 0 i=1 v {v } ur ing that the quotient A of the local unramified lifting ring R where the characteristic 0 v 1−i St polynomial of Frobenius equals (X − q ) is a quotient of R . Since A is flat over v v 0 i=1 0 0 St O, this follows from the definition of R (see [Tay08,§3]). For the statement of the next proposition, suppose given a surjection R /( ) → A in C , where A is a domain, and let r : G → G (A) denote the pushforward of (a F ,S n representative of) the universal deformation. Suppose given the following data: • A decomposition r = r ⊕ r ,where the r : G + → G (A) satisfy ν ◦ r = 1 2 i F ,S n G i i n ν ◦ r. (In other words, r| = r | ⊕ r | and if r (c) = (A , 1)j then r(c) = G G 1 G 2 G i i n F F F diag(A , A )j .) 1 2 • A subset R ⊂ S − S (consisting of places of odd residue characteristic) with the following property: for each v ∈ R we are given an integer 1 ≤ n ≤ n such that th q mod p is a primitive n root of unity and there is a decomposition r| = v G v F σ ⊕ σ ,where σ = Ind ψ with F /F the unramified extension of v,1 v,2 v,1 v,n v G v v v,n degree n and ψ an unramified character of G ,and σ is the twist of an v F v,2 v v,n unramified representation of G of dimension n − n by a ramified quadratic F v character. • An isomorphism ι : Q → C and for each v ∈ R, a character  : O → C p F v,n of order p. Thus the lifting ring R( v,  , n) is defined (notation as in §1.17). Proposition 5.6. — With the above assumptions on R /( ) → A, suppose that the following additional conditions are satisfied: SYMMETRIC POWER FUNCTORIALITY 81 (1) p > 2n. (2) For each 1 ≤ i < j ≤ n, χ /χ | has order greater than 2n. (In particular, this character is i j F(ζ ) non-trivial and ris Schur.) (3) For each v ∈ S , [F : Q ] > n(n − 1)/2 + 1. p v p (4) [F(ζ ) : F]= p − 1. (5) For each v ∈ R, both r | and r | admit a non-trivial unramified subquotient and the composite 1 G 2 G F F v v map R → R → A factors over R( v,  , n). S v Let L denote the maximal abelian pro-p extension of F unramified outside S ,and let  = S p Gal(L /F)/(c + 1).Let d denote the Z -rank of the subgroup of  generated by the elements Frob , S R p v v ∈ R.Then dim A ≤ n[F : Q]+ n − d . Proof. — Fix a place v of F split over F , prime to S, and such that q ≡ 1mod p 0 v and r| is trivial. Let I denote the ideal of A generated by the coefficients of the poly- nomial det(X − r| (Frob )) − (X − 1) . Then dim A/I ≥ dim A − n. After replacing G v F,S 0 AbyA/p,where p ⊂ A is a prime ideal minimal among those containing I, we can as- sume that r| (Frob ) is unipotent, and must show dim A ≤ n[F : Q]− d .Consider G v R F,S 0 the deformation problems (i = 1, 2): +   St S = (F/F , S ∪{v }, S ∪{ v }, , r ,μ, {R } ∪{R } ∪{R }). i 0 0 n i v∈S v∈S−S p p i v v v Let K = Frac A. We now repeat the argument of [ANT20, Lemma 3.6]: if v ∈ S ,then (since we assume [F : Q ] > n(n − 1)/2 + 1) we can appeal to [Tho15, Corollary 3.12], v p which implies the existence of an increasing filtration 1 2 n 0 ⊂ Fil ⊂ Fil ⊂· · · ⊂ Fil = K v v v • i i−1 of r| ⊗ KbyG -invariant subspaces, such that each gr Fil = Fil / Fil (i = G A F F v v v v 1,..., n) is 1-dimensional, and such that the character I (p) → K afforded by gr Fil v v i × agrees with the pushforward of the universal character ψ : I →  . Using the de- v v v composition r = r ⊕ r , we obtain induced filtrations Fil ∩(K ⊕ 0 ) of r | ⊗ K 1 2 1 G A v F and Fil ∩(0 ⊕ K ) of r ⊗ K and, applying [Tho15, Corollary 3.12] once more, we 2 A see that we can find an isomorphism  ⊗ =  =  such that, endowing A with n n n 1 2 the induced  -algebra structure, r is a lifting of r of type S for each i = 1, 2. We de- i i i i duce the existence of a surjective -algebra homomorphism R ⊗ R → A. We ob- S O S 1 2 serve that Theorem 5.2 applies to the deformation problems S and S , showing that 1 2 dim R /( ) ≤ n [F : Q]. S i × i Let ψ : G → O denote the Teichmüller lift of ψ = det r | ,and let R de- i F,S i G F,S S note the quotient of R over which the determinant of the universal deformation equals ψ .Then[Tho15, Lemma 3.36] states that there is an isomorphism R = R ⊗ O. i S O i S In particular, dim R /( ) ≤ (n − 1)[F : Q]. To complete the proof, it is enough to show that if A = A/(m ψ , m ψ ), then dim A ≤ 2[F : Q]− d = dim k ×  − d . 1 2 R R R R S S 1 2 82 JAMES NEWTON, JACK A. THORNE To this end, we observe that by construction there is a surjection k ×  → A . If # ,# : G → k ×  are the two universal characters, then the third part of 1 2 F,S Proposition 1.22 (together with our assumption that both r | and r | admit an un- 1 G 2 G F F v v n n v v ramified subquotient) implies that the relation # (Frob ) = # (Frob ) holds in A 1 v 2 v for each v ∈ R. Since  is a pro-p group, this implies that # (Frob ) = # (Frob ) in 1 v 2 v A , and hence that the map k ×  → A factors over the completed group alge- bra of the quotient of  ×  by the subgroup topologically generated by the elements (Frob , −Frob ) . This completes the proof. v v v∈R We are now in a position to prove the main theorem of this section, which guarantees the existence of generic primes in sufficiently large quotients of a certain deformation ring. For the convenience of the reader, we state our assumptions from scratch. Thus we take F, S, p as in our standard assumptions (see §1). We assume that [F(ζ ) : F]= (p − 1). We let E be a coefficient field, and suppose given an isomorphism ι : Q → C and a continuous character μ : G → O which is de Rham and such F ,S that μ(c ) =−1 for each place v|∞ of F . We fix an integer 2 ≤ n < p/2 and characters × c n χ ,..., χ : G → k such that for each i = 1,..., n, χ χ = μ| . We set ρ =⊕ χ ; F,S G 1 n i i i F,S i=1 then ρ naturally extends to a homomorphism r : G → G (k) such that ν ◦ r = μ.We F,S n G suppose for each 1 ≤ i < j ≤ n, χ /χ | has order greater than 2n. This implies that r i j F(ζ ) is Schur. We suppose that for each v ∈ S , r| is trivial and [F : Q ] > n(n − 1)/2 + 1. p G v p We suppose given a subset R = R R ⊂ S − S (consisting of places of odd 1 2 p residue characteristic) and integers 1 ≤ n ≤ n (v ∈ R) such that for each v ∈ R, q mod p v v th is a primitive n root of unity, and there is a decomposition r| = σ ⊕ σ ,where G v,1 v,2 σ = Ind ψ is induced from an unramified character of the unramified degree n v,1 v G v v,n extension of F ,and σ is the twist of an unramified representation of dimension n − n v v,2 v × × by a ramified quadratic character. We fix for each v ∈ R a character  : O → C of v,n order p. Assuming (as we may) that E is large enough, we may then (re-)define the global deformation problem S = (F/F , S, S,, r,μ, {R } ∪{R( v,  , n)} ∪{R } ). v∈S v v∈R v∈S−(S ∪R) p p v v Following [ANT20, Definition 3.7], we say that a prime p ⊂ R of dimension 1 and characteristic p is generic at p if it satisfies the following conditions: • Let A = R /p,and let r : G + → G (A) be the pushforward of (a representa- S p F ,S n tive of) the universal deformation. Then for each v ∈ S , the (pushforwards from v v ab × of the) universal characters ψ ,...,ψ : I (p) → A are distinct. 1 n F ab v v • There exists v ∈ S and σ ∈ I (p) such that the elements ψ (σ ), . . . , ψ (σ ) ∈ F 1 n A satisfy no non-trivial Z-linear relation. We say that p is generic if it is generic at p and if r | ⊗ Frac A is absolutely irreducible. p G A F,S SYMMETRIC POWER FUNCTORIALITY 83 Theorem 5.7. — With assumptions as above, let R → B be a surjection in C ,where B is a finite /( )-algebra. Let L denote the maximal abelian pro-p extension of F unramified outside S ,and let  = Gal(L /F)/(c + 1).Let d denote the Z -rank of the subgroup of  topologically p S R p p i generated by the elements Frob , v ∈ R . v i Suppose that the following conditions are satisfied: (1) Each irreducible component of Spec B has dimension strictly greater than sup({n[F : Q]+ n − d } , {n[F : Q]−[F : Q ]} ). R i=1,2 v p v∈S (2) For each direct sum decomposition r = r ⊕ r with r : G → G (k) (j = 1, 2)and n n = 0, 1 2 j F ,S n 1 2 there exists i ∈{1, 2} such that for each v ∈ R , both r | and r | admit a non-trivial i 1 G 2 G F F v v unramified subquotient. Then there exists a prime p ⊂ R of dimension 1 and characteristic p, containing the kernel of R → B, S S which is generic. Proof. — After passage to a quotient by a minimal prime, we can assume that B is a domain. The argument is now very similar to that of [ANT20, Lemma 3.9]. Indeed, by [ANT20, Lemma 3.8], we can find countable collection (I ) of ideals I ⊂ /( ) such i i≥1 i that for all i ≥ 1, dim /( , I ) ≤ sup{n[F : Q]−[F : Q ]} and if p ⊂ R is a prime i v p v∈S S of dimension 1 and characteristic p which is not generic at p,then I R ⊂ p for some i ≥ 1. i S red Let I ⊂ R be the reducibility ideal defined just before [ANT20, Lemma 3.4], and let red + I = (I , )R .Proposition 5.6 shows that dim R /I ≤ sup{n[F : Q]+ n − d } . 0 S S 0 R i=1,2 Since B is a finite /( )-algebra, we have dim B/I ≤ sup{n[F : Q]−[F : i v Q ]} .Wealsohave dim B/I ≤ sup{n[F : Q]+ n − d } . The existence of a generic p v∈S 0 R i=1,2 p i prime p containing the kernel of the map R → B thus follows from [Tho15, Lemma 1.9]. We conclude this section with a result concerning the existence of automorphic lifts of prescribed types, under the hypothesis of residual automorphy over a soluble extension. It only uses the results of [ANT20] and not the results proved earlier in this section, and is very similar in statement and proof to [BG19, Theorem 5.2.1]. We begin by re-establishing notation. We therefore let F be an imaginary CM field such that F /F is everywhere unramified. We fix a prime p and write S for the set of 0 0,p + + p-adic places of F . We fix a finite set S of finite places of F containing S . We assume 0 0,p 0 0 that each place of S splits in F , but not necessarily that each place of S − S splits in 0,p 0 0 0,p F . We choose for each v ∈ S a place v of F lying above v, and write S ={ v | v ∈ S }. 0 0 0 0 0 We fix a coefficient field E ⊂ Q . Fix an integer n ≥ 2, and suppose given a continuous representation ρ : G → GL (k) satisfying the following conditions: F ,S n 0 0 • There is an isomorphism ρ = ⊕ ρ , where each representation ρ is absolutely i i i=1 c ∨ 1−n irreducible and satisfies ρ = ρ ⊗ .Moreover, foreach 1 ≤ i < j ≤ r,we i i have ρ = ρ . i j 84 JAMES NEWTON, JACK A. THORNE Proposition 5.8. — Fix disjoint subsets T , ⊂ S consisting of prime-to-p places which 0 0 0 split in F . We assume that for each v ∈  , we have q ≡ 1 mod p and ρ| is trivial. We fix for 0 0 v G each v ∈ T a quotient R → R of the universal lifting ring of ρ| corresponding to a non-empty 0 v G v F union of irreducible components of Spec R [1/p]. We suppose that if v ∈ S and v is inert in F , then 0 0 n Hom(F ,Q ) ρ(I ) is of order prime to p. Fix a weight λ ∈ (Z ) such that for each i = 1,..., nand 0, v + τ ∈ Hom(F , Q ), λ =−λ , and suppose that for each v ∈ S , ρ| admits a lift to 0 τ c,i τ,n+1−i 0,p G p F 0, v Z which is ordinary of weight λ ,inthe senseof[Ger19, Definition 3.8]. Suppose that there exists a p v soluble CM extension F/F such that the following conditions are satisfied: (1) p > max(n, 3). For each place v|pof F, we have [F : Q ] > n(n − 1)/2 + 1 and ρ| is v p G trivial. ker ad ρ (2) F(ζ ) is not contained in F and F is not contained in F (ζ ).For each 1 ≤ i < j ≤ r, p p ρ | is absolutely irreducible and ρ | = ρ | .Moreover, ρ| is primitive and ρ(G ) G G G G F i i j F(ζ ) F(ζ ) F(ζ ) F p p p has no quotient of order p. (3) There exists a RACSDC automorphic representation π of GL (A ) and an isomorphism ι : Q → n F C such that r ρ| .Moreover, π is ι-ordinary and there exists a place v of F lying above π,ι G 0 such that π is an unramified twist of the Steinberg representation. (4) If S denotes the set of places of F lying above S , then each place of S splits in F. Then there exists a RACSDC automorphic representation π of GL (A ) satisfying the following 0 n F conditions: (1) π is unramified outside S and there is an isomorphism r ρ . 0 0 π ,ι (2) π is ι-ordinary of weight ιλ. (3) For each place v ∈ T ,r | defines a point of R . 0 π ,ι G v 0 F 0, v (4) For each place v ∈  , π is an unramified twist of the Steinberg representation. 0 0, v (5) For each place v ∈ S which is inert in F , reduction modulo p induces an isomorphism 0 0 r (I ) → r (I ). π ,ι F , v π ,ι F , v 0 0 0 0 Proof. — Let v ∈ S ,and let ρ : G → GL (Z ) be the lift of ρ| which is 0,p v F n p G 0, v F ordinary of weight λ and which exists by assumption. Thus ρ is conjugate over Q to v v an upper-triangular representation with the property that if α ,...,α : G → Q v,1 v,n F 0, v p are the characters appearing on the diagonal, then for each i = 1,..., n the character α is equal, on restriction to some open subgroup of I , to the character v,i F 0, v 1−i −1 −λ τ,n−i+1 χ : σ ∈ I → (σ) τ(Art (σ )) . λ ,i F v 0, v 0, v τ :F →Q 0, v After enlarging E, we can assume that each character α takes values in O.Weuse the v,i restricted characters α | : I (p) → O (v ∈ S , i = 1,..., n) to define a homo- v,i I (p) F 0,p F 0, v 0, v morphism  → O. −1 n Let β = α χ .Then τ =⊕ β is an inertial type and the type τ ,Hodge v,i v,i v v,i v λ ,i i=1 λ ,τ v v 2 type λ lifting ring R is defined and equidimensional of dimension 1 + n + n(n − v SYMMETRIC POWER FUNCTORIALITY 85 1)[F : Q ]/2 (see [Kis08, Theorem 3.3.4]). When τ is trivial, [Ger19, Lemma 3.10] 0, v p v λ ,τ v v shows that there is a minimal prime ideal of R such that, writing R for the corre- sponding quotient, the following properties are satisfied: • R is O-flat of dimension 1 + n + n(n − 1)[F : Q ]/2. v 0, v p λ ,τ v v • The map R → Q determined by ρ factors through R . v v v p • For every homomorphism R → Q , the corresponding Galois representation G → GL (Q ) is ordinary of weight λ , in the sense of [Ger19, Definition F n v 0, v 3.8]. • The homomorphism R ⊗  → R (completed tensor product of the tauto- O v v logical quotient map R → R and the composite  → O → R )factors over v v v the quotient R ⊗  → R (defined in e.g. [Tho15, §3.3.2]). O v v v In fact, the same proof shows that these properties hold also in the case that τ is non- trivial. Our hypotheses imply that we can extend ρ to a continuous homomorphism r : 1−n n univ G → G (k) with the property that ν ◦ r = δ .Let R denote the deformation F ,S 0 F /F 1−n n ring, defined as in [BG19, Corollary 5.1.1], of δ -polarised deformations of r, F /F where the quotients of the local lifting rings for v ∈ S are specified as follows: • If v ∈ S ,wetakethe quotient R defined above. 0,p v • If v ∈ T ,take R . 0 v St • If v ∈  , take the Steinberg lifting ring R . • If v ∈ S and v is inert in F , take the component corresponding to the functor 0 0 of lifts r of r| such that the reduction map induces an isomorphism r(I ) → + F F 0,v 0,v r(I ). 0,v univ We can invoke [BG19, Corollary 5.1.1] to conclude that R has Krull dimension at least 1. We remark that this result includes the hypothesis that r| is irreducible, but F (ζ ) 0 p + + 0 0 this is used only to know that the groups H (F , ad r) and H (F , ad r(1)) vanish, which is 0 0 true under the weaker condition that r| is Schur, which follows from our hypotheses. F (ζ ) (The vanishing of these groups implies that the deformation functor is representable and that the Euler characteristic formula gives the correct lower bound for its dimension.) We consider as well the deformation problem + 1−n n   St S =(F/F ,S, S, , r| , δ , {R } ∪{R } ∪{R } ), F G v∈S v∈S−(S ∪) v∈ F F/F v p v p v where we define S, T,  to be the sets of places of F above S ,T ,  , respectively. Then 0 0 0 univ there is a natural morphism R → R of  -algebras, which is finite (apply Lemma 5.3 S F and [Tho15, Proposition 3.29(2)]). By [ANT20, Theorem 6.2], R is a finite  -algebra. S F univ The map  → R factors through a homomorphism  → O (by construction), so F F univ R is a finite O-algebra (of Krull dimension at least 1, as we have already remarked). 86 JAMES NEWTON, JACK A. THORNE We deduce the existence of a lift r : G → G (Z ) of r arising from a homo- F ,S n p univ morphism R → Z . We can now apply [ANT20, Theorem 6.1] and soluble descent to conclude that r| is automorphic, associated to an automorphic representation π of G 0 GL (A ) with the desired properties. n F 6. Raising the level – Galois theory This section is devoted to the proof of a single theorem that will bridge the gap between Theorem 4.1 and our intended applications. Let F, S, p, G be as in our standard assumptions (see §1), and let n ≥ 3 be an odd integer such that p > 2n. Fix an isomorphism ι : Q → C. We suppose given a RACSDC automorphic representation π of GL (A ), n F and that the following conditions are satisfied: (1) π is ι-ordinary. (2) [F(ζ ) : F]= p − 1. (3) There exist characters χ ,..., χ : G → F and an isomorphism r = χ ⊕ ··· ⊕ F π,ι 1 n 1 c ∨ 1−n χ ,where foreach i = 1,..., n,wehave χ = χ and for each 1 ≤ i < j ≤ n, i i χ /χ | has order strictly greater than 2n. i j G F(ζ ) (4) For each v ∈ S , r | is trivial and [F : Q ] > n(n − 1)/2 + 1. p π,ι G v p (5) There is a set R = R R ⊂ S − S with the following properties: 1 2 p (a) The sets R and R are both non-empty and for each v ∈ R, the characteristic of 1 2 k(v) is odd. As in §1.17, we write ω( v) : k( v) →{±1} for the unique quadratic character. rd (nb) If v ∈ R ,then q mod p is a primitive 3 root of unity, and there exists a character 1 v : k → C of order p such that π | contains λ( v,  , n) (notationasin§1.17, v v q v 3 v this representation of q defined with respect to n = 3). v 1 th (c) If v ∈ R ,then q mod p is a primitive (n − 2) root of unity, and there exists a 2 v character  : k → C of order p such that π | contains λ( v,  , n) (notation v v q v n−2 v as in §1.17, this representation of q defined with respect to n = n − 2). v 1 (d) For each non-trivial direct sum decomposition r = ρ ⊕ρ , there exists i ∈{1, 2} π,ι 1 2 such that for each v ∈ R , ρ | and ρ | both admit a non-trivial unramified i G G 1 F 2 F v v subquotient. (This is the situation we will find ourselves in after applying Theorem 4.1. The sets of places R and R here will correspond to the sets T and T respectively from §4,and 1 2 1 3 it will be possible to label the characters χ so that we have the following ramification properties: • If v ∈ R ,then χ , χ , χ are unramified at v and χ ,..., χ are ramified at v 1 2 3 4 n (and the image of inertia under each of these characters has order 2). • If v ∈ R ,then χ , χ are ramified at v (and the image of inertia under each of 1 2 these characters has order 2) and χ ,..., χ are unramified at v. 3 n SYMMETRIC POWER FUNCTORIALITY 87 These properties imply condition 5(d) above, which is what we actually need for the proofs in this section.) The theorem we prove in this section is the following one: Theorem 6.1. — With assumptions as above, fix a place v of F lying above S −(S ∪ R) such St p that q ≡ 1 (mod p) and r | is trivial. Then we can find a RACSDC ι-ordinary automorphic v π,ι G St F St representation π of GL (A ) satisfying the following conditions: n F (1) There is an isomorphism r  r . π ,ι π,ι (2) For each embedding τ : F → Q ,wehave HT (r  ) = HT (r ). τ π ,ι τ π,ι (3) π is an unramified twist of the Steinberg representation. St Our proof of this theorem follows a similar template to the proof of [CT17,Theo- rem 5.1]. Briefly, we use our local conditions at the places of R, together with Theorem 5.7 to show that (after a suitable base change) we can find a generic prime in the spec- trum of the big ordinary Hecke algebra. This puts us in a position to use the “R = T ” p p theorem proved in [ANT20], which is enough to construct automorphic lifts of r (or its π,ι base change) with the desired properties. We now begin the proof. Let F /F denote the extension of F(ζ ) cut out by r | , p π,ι G F(ζ ) and let Y be a finite set of finite places of F with the following properties: a + • For each place v ∈ Y , v is split over F , prime to S, and π is unramified. • For each intermediate Galois extension F /M/Fsuch thatGal(M/F) is simple, there exists v ∈ Y which does not split in M. a a Then any Y -split finite extension L/F is linearly disjoint from F /F. After conjugation, we can find a coefficient field E such that r is valued in GL (O), and extend it to a π,ι n 1−n + + homomorphism r : G → G (O) such that ν ◦ r = δ . We write r : G → G (k) F ,S n F ,S n F/F for the reduction modulo of r. Lemma 6.2. — Let L/F be an Y -split finite CM extension. Then: (1) r| is Schur. G + L (ζ ) (2) r| is primitive. (3) Suppose moreover that L/F is soluble. Then the base change of π with respect to the extension L/F is cuspidal. (4) More generally, suppose that L/F /F is an intermediate field with L/F soluble, and let  be a 0 0 RACSDC automorphic representation of GL (A ) such that r r| . Then the base change n F ,ι G 0 F of  with respect to the extension L/F is cuspidal. Proof. — For the first part, it is enough to check that L ⊂ L (ζ ) and χ /χ | is p i j G L(ζ ) non-trivial for each 1 ≤ i < j ≤ n.Wehave [L(ζ ) : L]= p − 1, which implies L ⊂ L (ζ ), p p 88 JAMES NEWTON, JACK A. THORNE while χ /χ (G ) = χ /χ (G ), so this ratio of characters is non-trivial. The second L(ζ ) F(ζ ) i j p i j p part follows from Lemma 5.1 and the fact that χ /χ | has order greater than 2n for any i j L i = j , because L/F is linearly disjoint from the extension F /F. The third part is a special case of the