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Symmetric power functoriality for holomorphic modular forms, II

Symmetric power functoriality for holomorphic modular forms, II SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II by JAMES NEWTON and JACK A. THORNE ABSTRACT Let f be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of the symmetric power lifting Sym f for every n ≥ 1. CONTENTS 1. Introduction ....................................................... 117 2. An automorphy lifting theorem for symmetric power representations ....................... 122 3. Killing ramification . .................................................. 139 Acknowledgements ..................................................... 148 Appendix A: The case of weight one forms ........................................ 149 References ......................................................... 150 1. Introduction Let F be a number field, and let π be a cuspidal automorphic representation of GL (A ). Langlands’s functoriality principle predicts the existence, for any n ≥ 1, of an 2 F automorphic representation Sym π of GL (A ), characterized by the requirement that n+1 F for any place v of F, the Langlands parameter of (Sym π) is the image of the Langlands parameter of π under the nth symmetric power Sym : GL → GL of the standard v 2 n+1 representation of GL . For a more detailed discussion of the context surrounding this problem, including known results by other authors, we refer the reader to the introduction of [NT21], of which this paper is a continuation. In that paper we studied the problem of symmetric power functoriality in the case that F = Q and π is regular algebraic (in which case π corresponds to a twist of a cuspidal Hecke eigenform f of weight k ≥ 2, cf. [Gel75,§3]). We established the existence of the symmetric power liftings Sym π under the assump- tion that there is no prime p such that π is supercuspidal. (This includes the case that f has level SL (Z).) In this paper we remove this assumption, proving the following theorem: Theorem A. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ). 2 Q Suppose that π is non-CM. Then for each integer n ≥ 1, Sym π exists, as a regular algebraic, cuspidal automorphic representation of GL (A ). n+1 Q In the ‘missing’ cases of π which are holomorphic limit of discrete series at ∞ or n n CM, the existence of Sym π for all n is well known, although of course Sym π is usually © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2021 https://doi.org/10.1007/s10240-021-00126-4 118 JAMES NEWTON, JACK A. THORNE not cuspidal. The most difficult case of icosahedral weight one eigenforms ([Kim04,The- orem 6.4]) requires Kim and Shahidi’s results on tensor product and symmetric power functoriality. We provide some details in Appendix A. Using the modularity of elliptic curves over Q [BCDT01], we deduce the following corollary: Corollary B. — Let E be an elliptic curve over Q without complex multiplication. 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. Our strategy to prove Theorem A is inspired by the proof of Serre’s conjecture [KW09a]. There one takes as given Serre’s conjecture in the level 1 case (i.e. for every prime number p, the residual modularity of odd irreducible representations ρ : Gal(Q/Q) → GL (F ) 2 p unramified outside p), proved in [Kha06], and hopes to reduce the general case to this one by induction on the number of primes away from p at which ρ is ramified. Here we associate to any regular algebraic, cuspidal automorphic representation π of GL (A ) the set sc(π ) of primes p such that π is supercuspidal. Fixing n ≥ 1, we 2 Q p prove the existence of Sym π by induction on the cardinality of the set sc(π ),the case |sc(π )|= 0 being exactly the main result of [NT21]. Our induction argument uses congruences between automorphic representations. If p is a prime and ι : Q → C is an isomorphism, then there is an associated Galois representation r : Gal(Q/Q) → GL (Q ) π,ι 2 and its mod p reduction r : Gal(Q/Q) → GL (F ). π,ι 2 p If π is another regular algebraic, cuspidal automorphic representation of GL (A ),and 2 Q thereisanisomorphism r = r π,ι π ,ι (in other words, a congruence modulo p between π and π ), then passage to symmetric powers gives an isomorphism n n Sym r = Sym r . π,ι π ,ι n n If Sym π is known to exist, and the image of the representation Sym r  is sufficiently π ,ι non-degenerate (for example, irreducible), then automorphy lifting theorems (such as SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 119 those proved in [BLGGT14]) can be used to deduce the automorphy of Sym r ,hence π,ι the existence of Sym π . If p is a prime such that π is supercuspidal, it may be possible to choose π so that sc(π ) = sc(π ) −{p}, opening the way to an induction argument. This idea of ‘killing ramification’ plays a significant role in [KW09a]. The difficulty in applying this approach here is that if p ≤ n then the representation Sym r is never irreducible, so the automorphy lifting theorems proved in [BLGGT14] π,ι do not apply. (The automorphy lifting theorems for residually reducible representations proved in [ANT20] apply only for ordinary representations, a possibility which is ruled out if π is supercuspidal.) This approach might perhaps yield the existence of Sym π when p > n for every p ∈ sc(π ), but to get a result like Theorem A a new idea is required. Here we prove a new kind of automorphy lifting theorem, Theorem 2.1, specially tailored to the problem of symmetric powers (although we hope that these ideas will also be useful for other cases of Langlands functoriality). We consider the morphism P → R, where R is the universal deformation ring of the (supposed irreducible) representation r and P is the universal pseudodeformation ring of the pseudocharacter associated to π,ι the symmetric power Sym r ; the morphism P → R is the universal one classifying the π,ι pseudocharacter of Sym of the universal deformation of r . A version of the Taylor– π,ι Wiles–Kisin patching argument upgrades this to a commutative diagram P R ∞ ∞ P R, where P , R are ‘patched deformation rings’ and the vertical arrows are surjections. ∞ ∞ Since r is assumed to be irreducible, the arguments of [Kis09b, Kis09a] show that R π,ι ∞ is a domain which acts faithfully on a space of patched (rank-2) modular forms. The essential additional ingredient is the main result of [NT20], which shows that Spec P is regular (of dimension 0) at the point corresponding to the pseudocharacter of the representation Sym r ; this in turn implies that Spec P is regular at the image of π ,ι ∞ this point in Spec P , and allows us to deduce that the image of Spec R → Spec P is ∞ ∞ ∞ contained in the support of a space of patched (rank-(n + 1)) modular forms, leading to a proof of Theorem 2.1. Our a priori knowledge about the ring P obviates the need to kill the dual Selmer group of Sym r . π,ι To actually prove Theorem A, we combine Theorem 2.1 with a modified version of the ‘killing ramification’ technique of [KW09a], based on a variation of the notion of ‘good dihedral’ representation introduced in that paper. This is not quite routine since we need our ‘good dihedral’ automorphic representations π to have the property that, if q is the good dihedral prime, then there is an isomorphism ι : Q → C such that r has q π,ι q q 120 JAMES NEWTON, JACK A. THORNE large image. We achieve this by introducing Steinberg type ramification at another auxil- iary prime r, which is acceptable since the presence of r does not affect the set sc(π ).We call an automorphic representation π that comes equipped with the requisite auxiliary primes ‘seasoned’ (see Definition 3.6). Notation. If F is a perfect field, we generally fix an algebraic closure F/F and write G for the absolute Galois group of F with respect to this choice. When the characteristic of F is not equal to p, we write  : G → Z for the p-adic cyclotomic character. We th 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 v v each place v of F; this choice determines a homomorphism G → G . When v is a F F finite place, we will write W ⊂ G for the Weil group, O ⊂ F for the valuation ring, F F F v v v v ∈ O for a fixed choice of uniformizer, Frob ∈ G for a fixed choice of (geometric) v F v F v v Frobenius lift, k(v) = O /( ) for the residue field, and q = #k(v) for the cardinality F v v × × of the residue field. If R is a ring and α ∈ R , then we write ur : W → R for the α F unramified character which sends Frob to α. When v is a real place, we write c ∈ G v v F for complex conjugation. If S is a finite set of finite places of F then we write F /Ffor 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. We write C for the category of complete Noetherian local O-algebras with residue field k.If G is a profinite group and ρ : G → GL (Q ) is a continuous representation, then we write ρ : G → GL (F ) for the associated semisimple residual representation (which is well- n p defined up to conjugacy). If F is a number field, v is a finite place of F, and ρ, ρ : G → GL (O ) are continuous representations, which are potentially crystalline if v|p,then n Q we use the notation ρ ∼ ρ established in [BLGGT14, §1] (which indicates that these two representations define points on a common component of a suitable deformation ring). 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 corre- spondence rec for GL (K): a bijection between the sets of isomorphism classes of ir- K n reducible, admissible C[GL (K)]-modules and Frobenius-semisimple Weil–Deligne rep- resentations over C of rank n. In general, we have the Tate normalisation of the local Langlands correspondence for GL as described in [CT14, §2.1]. When = C,wehave T (1−n)/2 rec (π ) = rec (π ⊗|·| ). K SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 121 × × 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 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 n F ∞ as an irreducible algebraic representation W of (Res GL ) . F/Q n C If π is cuspidal, regular algebraic, and polarizable, in the sense of [BLGGT14], then for any isomorphism ι : Q → C there exists a continuous, semisimple represen- F−ss tation r : G → GL (Q ) such that for each finite place v of F, WD(r | ) = π,ι F n π,ι G p Fv T −1 rec (ι π ) (see e.g. [Car14]). (When n = 1, this is compatible with our existing nota- tion.) We use the convention that the Hodge–Tate weight of the cyclotomic character is −1. We use special terminology in the case n = 2: if k ≥ 2 is an integer, we say that π k−2 2 ∨ has weight k if we can take W = (⊗ Sym C ) . (If F is totally real, then the τ ∈Hom(F,C) cuspidal automorphic representations of weight k are those which are associated to cusp- idal Hilbert modular forms of parallel weight k.) In this case the Hodge–Tate weights of k−1 r with respect to any embedding τ : F → Q are {0, k − 1} and the character  det r π,ι π,ι has finite order. If F is a number field, G is a reductive group over F, v is a finite place of F, and U is an open compact subgroup of G(F ), then we write H(G(F ), U ) for the convo- v v v v lution algebra of compactly supported U -biinvariant functions f : G(F ) → Z (convo- v v lution defined with respect to the Haar measure on G(F ) which gives U volume 1). v v Then H(G(F ), U ) is a free Z-module, with basis given by the characteristic functions v v [U g U ] of double cosets for g ∈ U \G(F )/U . v v v v v v v If 1 ≤ i ≤ n,let α = diag( ,..., , 1,..., 1) ∈ GL (F ) (where there are i ,i v v n v occurrences of  on the diagonal). We define (i) T =[GL (O )α GL (O )]∈ H(GL (F ), GL (O )). n F  ,i n F n v n F v v v v v We write Iw ⊂ GL (O ) for the standard Iwahori subgroup (elements which are upper- v n F × n triangular modulo  )and Iw ⊂ Iw for the kernel of the natural map Iw → (k(v) ) v v,1 v v given by reduction modulo  , then projection to the diagonal. If U ⊂ Iw is a subgroup v v v containing Iw ,and 1 ≤ i ≤ n,thenwedefine v,1 (i) U =[U α U ]∈ H(GL (F ), U ). v  ,i v n v v v 122 JAMES NEWTON, JACK A. THORNE 2. An automorphy lifting theorem for symmetric power representations Let p be a prime and let F be a totally real field. Fix an isomorphism ι : Q → C. Let n ≥ 1. Let π be a regular algebraic, cuspidal automorphic representation of GL (A ) 2 F satisfying the following conditions: • π has weight 2 and is non-CM. • For each place v|p of F, r | is not ordinary, in the sense of [Tho16, §5.1]. π,ι G Note that, together with the assumption that π has weight 2, this implies that r | is potentially Barsotti–Tate. π,ι G • Let Proj r : G → PGL (F ) denote the projective representation associated π,ι F 2 p to r . Then there exists a ≥ 1such that p > max(5, 2n − 1) and there is a π,ι sandwich a a PSL (F ) ⊂ Proj r (G ) ⊂ PGL (F ), 2 p π,ι F 2 p up to conjugacy in PGL (F ). 2 p We impose the final condition to ensure that we can choose Taylor–Wiles primes such n−1 that the image of the corresponding Frobenius element under Sym r is regular π,ι semisimple. The aim of this section is prove the following theorem: Theorem 2.1. — Suppose that there exists another regular algebraic, cuspidal automorphic rep- resentation π of GL (A ) such that the following conditions are satisfied: 2 F (1) π hasweight2andisnon-CM. (2) For each place v|pof F,r  | is not ordinary. π ,ι G (3) There is an isomorphism r  r . π ,ι π,ι (4) For each place v  pof F, π is a character twist of the Steinberg representation if and only if π is. n−1 (5) Sym r  is automorphic. π ,ι n−1 Then Sym r is automorphic. π,ι We begin with a preliminary reduction. Let E/Q be a coefficient field. After pos- sibly enlarging E, we can find conjugates r, r of r , r respectively which take values in π,ι π ,ι GL (O). We can also assume that the eigenvalues of each element in the image of r lie in k. After passage to a soluble totally real extension, we can assume that the following additional conditions are satisfied: (6) [F : Q] is even. −1 (7) det r = det r  =  . π,ι π ,ι (8) For each place v|p of F, π and π are unramified. v SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 123 (9) For each finite place v  p of F, π and π are Iwahori-spherical. The number of places such that π is ramified is even. (10) Let S denote the set of p-adic places of F and the set of places v such that π is ramified. Let S = S ∪ .Thenfor each v ∈ S, r| is trivial. v p G For each v ∈ , q ≡ 1mod p and π ,π are isomorphic to the Steinberg v v representation (not just up to twist — note that condition (7) already implies that any such twist is by a quadratic character). (11) There exists an everywhere unramified CM quadratic extension K/F, with each place v ∈ S split in K. Let  = π .Then  is RACSDC (i.e. regular algebraic, conjugate self-dual, and cus- n−1 pidal). We will show that the representation Sym r is automorphic; this will imply ,ι Theorem 2.1, by soluble descent. We let  = π .Then  is also RACSDC, there is an n−1 isomorphism r = r , and Sym r is automorphic. We write  for the RACSDC ,ι  ,ι  ,ι n−1 automorphic representation of GL (A ) such that r = Sym r . n K  ,ι  ,ι Recall that C denotes the category of complete Noetherian local O-algebras with residue field k.If v is a place of F, we write R ∈ C for the object representing the functor Lift : C → Sets which associates to A ∈ C the set of homomorphisms v O O −1 r : G → GL (A) lifting r| (i.e. such that r mod m = r)suchthat det r =  | .We F 2 G A G v F F v v introduce certain quotients of R : • If v ∈ S , the smallest reduced O-torsion-free quotient R of R such that if p v F : R → Q is a homomorphism such that the pushforward of the universal v p lifting to Q is crystalline of Hodge–Tate weights {0, 1} (with respect to any em- bedding F → Q ) and is not ordinary, then f factors through R .By[Kis09a, v v Corollary 2.3.13], R is a domain of dimension 4 +[F : Q ]. v v p • If v ∈ , the smallest reduced O-torsion-free quotient R of R such that if f : R → Q is a homomorphism such that the pushforward of the universal v p −1 lifting to Q is an extension of  by the trivial character, then f factors through R .By[Kis09b, Corollary 2.6.7], R is a domain of dimension 4. v v −1, • If v ∈ S (the set of infinite places of F), the quotient R of R denoted R ∞ v in [Kis09a, Proposition 2.5.6]. Then R is a domain of dimension 3. If Q is a finite set of finite places of F, disjoint from S, then we write Def : C → Sets Q O for the functor which associates to A ∈ C the set of 1 + M (m )-conjugacy classes of lifts O 2 A r : G → GL (A) of r satisfying the following conditions: F 2 −1 •  r is unramified outside S ∪ Q, and det r =  . • For each v ∈ S ∪ S , the homomorphism R → A determined by r| factors ∞ G v v through the quotient R → R introduced above. Our assumption on the image of r implies that the functor Def is represented by an ob- ject R ∈ C . If Q is empty then we write R = R . We also introduce some variants. Let Q O ∅ 124 JAMES NEWTON, JACK A. THORNE R = ⊗ R .Then R is an O-flat domain of Krull dimension 1 + 3[F : Q]+ 3|S| loc v∈S∪S v loc (cf. [Kis09b, Lemma 3.4.12]). We write Def : C → Sets for the functor of 1 + M (m )- O 2 A conjugacy of tuples ( r, {A } ),where  r is as above and A ∈ 1 + M (m ),and v v∈S∪S v 2 A −1 γ ∈ 1 + M (m ) acts by γ · ( r, {A } ) = (γ rγ , {γ A } ). This functor is rep- 2 A v v∈S∪S v v∈S∪S ∞ ∞ −1 resented by an object denoted R ∈ C . The tuple of representations (A  r| A ) O G v v∈S∪S F ∞ Q v v is independent of the choice of representative for a given conjugacy class, and the univer- sal property of R determines a homomorphism R → R . loc v Q The objects in this paragraph will only be used in the case p = 2. We write Def : C → Sets for the functor of 1 + M (m )-conjugacy classes of lifts r : G → GL (A) of r O 2 A F 2 satisfying the following conditions: −1 •  r is unramified outside S ∪ Q, and if v ∈ Sthendet r| =  | . G G F F v v • For each v ∈ S ∪ S , the homomorphism R → A determined by r| factors ∞ G v Fv through the quotient R → R introduced above. We write Def for the functor of 1 + M (m )-conjugacy classes of tuples ( r, {A } ), 2 A v v∈S∪S Q ∞ where r is as above and A ∈ 1 + M (m ). Then the functors Def and Def are repre- v 2 A Q Q , , sented by objects R ,R ∈ C and there is again a natural morphism R → R . O loc Q Q Q n−1  n−1 Let t = det Sym (r| ) and t = det Sym (r | ) denote the group determi- G G K K nants over O (in the sense of [Che14]) associated to these two symmetric power represen- tations, and let t denote the group determinant over k which is their common reduction [0,n−1] modulo  . We introduce the object P ∈ C which is the quotient R of R intro- O S t,S duced in [NT20, §2.19]. Informally, P represents the functor of conjugate self-dual group 1−n determinants of G lifting t which have similitude character  and are semistable K,S with Hodge–Tate weights in the interval [0, n − 1]. Lemma 2.2. — Let A ∈ C be Artinian, let v ∈ S ,and let  r : G → GL (A) be a O p F 2 −1 lift of r of determinant  | such that the associated homomorphism R → A factors through F v n n−1 R → R .View A as an A[G ]-module via the representation Sym  r : G → GL (A) (we v F F n v v v n−1 2 n equip Sym A with its standard ordered basis, cf. [BLGG11, Definition 3.3.1]). Then A is iso- morphic, as Z [G ]-module, to a subquotient of a lattice in a crystalline (in particular, semistable) p F Q [G ]-module with all Hodge–Tate weights in the interval [0, n − 1]. p F Proof. — It follows from the construction in [Kis09b, Kis09a] that the dual of A is isomorphic, as Z [G ]-module, to the generic fibre of a finite flat group scheme p F over O .By[BBM82, Théorème 3.1.1], there exists a lattice L in a crystalline Q [G ]- F p F v v representation with Hodge–Tate weights in the interval [0, 1] such that A is isomorphic, n−1 2 as Z [G ]-module, to a quotient of L. It follows that Sym A is isomorphic, as Z [G ]- p F p F v v n−1 n−1 module, to a quotient of Sym L. The proof is complete on noting that Sym Lis Z Z p p n−1 a lattice in (Sym L)[1/p], a crystalline Q [G ]-representation with all Hodge–Tate p F Z v weights in the interval [0, n − 1].  SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 125 Let A ∈ C ,and let r : G → GL (A) be a lift of r which determines a map R → A. O F 2 n−1 Lemma 2.2 shows that the pseudocharacter associated to Sym ( r| ) satisfies condition (2.16.1) of [NT20]. In particular there are morphisms P → O associated to the pseu- docharacters t, t . Taking the pseudocharacter of the symmetric power of the universal deformation over R determines a morphism P → Rin C . We will study this morphism using the Taylor–Wiles method. In this paper we call a Taylor–Wiles datum of level N ≥ 1 a tuple (Q, Q,(α ,β ) ),where: v v v∈Q • Q is a finite set of places of F, split in K, such that for each v ∈ Q, q ≡ 1mod p . • For each v ∈ Q, we have fixed a factorisation v =  v v in K such that Q ={ v | v ∈ Q}. • For each v ∈ Q, α ,β ∈ k are eigenvalues of r(Frob ). We require that v v v n−1 n−2 n−1 α ,α β ,...,β are distinct elements of k. v v v We note that this last condition is stronger than the one typically appearing in applications to automorphy of 2-dimensional Galois representations and is specially adapted to our purposes here. univ Let r : G → GL (R ) denote a representative of the universal deformation. F 2 Q univ If v ∈ Qthen r | is conjugate (in GL (R )) to a unique representation of the G 2 Q Q v form A ⊕ B ,where A mod m = ur and B mod m = ur . The characters v v v R α v R β Q v Q v × univ × A , B : G → R are independent of the choice of r . We write  = k(v) (p) v v F Q v Q Q v∈Q (i.e. product of maximal p-power quotients of k(v) ). The collection of characters A ◦ Art | (v ∈ Q) determine a homomorphism O[ ]→ R with the property that v F O Q Q the natural map R → R factors through a canonical isomorphism R ⊗ O R. Q Q O[ ] [0,n−1] We write P ∈ C for the quotient of the quotient R of R introduced in Q O S∪Q t,S∪Q [NT20, §2.19] corresponding to pseudodeformations t of t with the following additional properties: • For each v ∈ Q, t| factors through the maximal Hausdorff abelian quo- ab 1 tient G → G . By [ANT20, Proposition 2.5] (cf. [NT20, Lemma 4.28], in [ANT20] we generalise [BC09, Proposition 1.5.1] to arbitrary characteristic), n−i i−1 there are unique characters χ ,...,χ lifting ur such that t| is the v,1 v,n G α β K v v  v pseudocharacter associated to χ ⊕ ··· ⊕ χ . v,1 v,n n+1−2i • Let m =n/2 and let i ∈{1,..., n}.If n = 2m is even, then χ | = χ | . v,i I I K v,m K v  v (n+1−2i)/2 If n = 2m + 1 is odd, then χ | = χ | . v,i I I K v,m K v  v The characters χ (v ∈ Q) give P the structure of O[ ]-algebra, and again there v,m Q Q is a universal morphism P → R . We remark that this need not be a morphism of Q Q O[ ]-algebras when n is odd, although it is when n is even. It is possible that our condition on the eigenvalues of Frob necessarily entails that t is abelian at  v.However,we haven’t verified this and it doesn’t cost us anything to build this in to the definition of P . Q 126 JAMES NEWTON, JACK A. THORNE To carry out the patching argument, we need to introduce spaces of automorphic forms. We first discuss automorphic forms on a definite quaternion algebra over F, follow- ing the set-up of [Kis09a, §3.1] and [KW09b, §7]. Let D be a definite quaternion algebra over F, ramified precisely at the infinite places and at the places of . Fix a choice of max- imal order O and for each finite place v ∈ , an identification O ⊗ O = M (O ). D D O F 2 F F v v ∞ × If U = U ⊂ (D ⊗ A ) is an open subgroup, then we write H (U) for the set of v F D v F ∞ × × ∞ × ∞ × functions f : (D ⊗ A ) → O such that for all γ ∈ D , z ∈ (A ) , g ∈ (D ⊗ A ) , O O F F F F F and u ∈ U, we have f (γ gzu) = f (g). We define × × U = (O ⊗ O ) × (D ⊗ F ) . 0 D O F F v F v v∈ v∈ If (Q, Q, {α ,β } ) is a Taylor–Wiles datum of level N ≥ 1, then we define U (Q; N) = v v v∈Q 1 U (Q; N) ⊂ U (Q) = U (Q) by U (Q; N) = U (Q) = U if v ∈ Q, and 1 v 0 0 v 1 v 0 v 0,v v v U (Q) = Iw and 0 v v ab −1 × N U (Q; N) = ∈ Iw : ad → 1 ∈ k(v) (p)/(p ) 1 v v cd if v ∈ Q. Thus U (Q; N) ⊂ U (Q) is a normal subgroup with quotient 1 0 U (Q)/U (Q; N) =  /(p ). 0 1 Q We introduce Hecke operators. If v ∈ ∪ Q is a finite place of F, then the un- (1) (2) ramified Hecke operators T , T act on H (U (Q)) and H (U (Q; N)).If v ∈ Q D 0 D 1 v v (1) univ then the operator U acts on H (U (Q)) and H (U (Q; N)). We write T for D 0 D 1 D, ∪Q univ,Q (1) (2) the polynomial ring over O in the indeterminates T , T (v ∈ ∪ Q) and T = v v D, ∪Q univ (1) T [{U } ]. v∈Q D, ∪Q univ There is a unique maximal ideal m ⊂ T of residue field k such that for all D, 2 (1) finite places v ∈ S of F, the characteristic polynomial of r(Frob ) equals X − T X + (2) (1) (2) q T mod m and for each v ∈ S ,T ∈ m and T − 1 ∈ m .If (Q, Q,(α ,β ) ) v D p D D v v v∈Q v v v univ,Q is a Taylor–Wiles datum, then we write m for the maximal ideal of T generated D,Q D, ∪Q univ (1) by m ∩ T and the elements U − α (v ∈ Q). D v D, ∪Q × ∞ × × If χ : F \(A ) / det U (Q; N) → O is a quadratic character and f ∈ H (U (Q; N)), then we define f ⊗ χ ∈ H (U (Q; N)) by the formula (f ⊗ χ)(g) = D 1 D 1 χ(det(g))f (g). We observe that if p = 2and f ∈ H (U (Q; N)) then f ⊗ χ ∈ D 1 m D,Q H (U (Q; N)) . D 1 m D,Q Let m ≥ 0denote the p-adic valuation of the least common multiple of the × ∞ × −1 × exponents of the Sylow p-subgroups of the finite groups F \(U(A ) ∩ t D t) for ∞ × t ∈ (D ⊗ A ) ; this number is finite, see [KW09b, §7.2]. F SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 127 Proposition 2.3. — Let N ≥ 1 and let (Q, Q,(α ,β ) ) be a Taylor–Wiles datum of v v v∈Q level N + m . (1) The maximal ideals m and m are in the support of H (U ) and H (U (Q)), D D,Q D 0 D 0 respectively. univ N N (2) H (U (Q; N)) is a T ⊗ O[ /(p )]-module free as O[ /(p )]- D 1 m O Q Q D,Q D, ∪Q module, and there is an isomorphism H (U (Q; N)) ⊗ O = H (U ) D 1 m D 0 m O[ /(p )] D,Q Q D univ of T -modules. D, ∪Q (3) There exists a structure on H (U (Q; N)) of R -module such that for any repre- D 1 m Q D,Q univ sentative r of the universal deformation of r and for each finite place v ∈ S ∪ Q of F, univ (1) univ (2) tr r (Frob ) acts as T and det r (Frob ) acts as q T . Moreover, the O[ ]- v v v Q Q v Q v module structure induced by the map O[ ]→ R agrees with the one in the second part Q Q of the lemma. Proof. — The first part is a consequence of the Jacquet–Langlands correspondence (and the existence of π ). The second part is [KW09b, Corollary 7.5]. The third part is proved in thesameway as [KW09b, Lemma 9.1] (cf. also [Kis09a, Lemma 3.2.7]); note (1) that since T ∈ m for each v ∈ S , only automorphic representations which are non- D p ordinary at each v ∈ S can contribute to H (U (Q; N)) . p D 1 m We next discuss automorphic forms on a definite unitary group of rank n.We therefore fix a unitary group G over F, split by K/F, as in [NT20, §4.1], together with an extension of G to a reductive group scheme over O . We recall that G comes equipped with isomorphisms ι : G → Res GL for each place v ofFwhich splits v = w O O /O n F K F v w v ww in K. Moreover, for each place v  ∞ of F, G is quasi-split, while for each place v|∞ of F, G(F ) is compact. If V = V ⊂ G(A ) is an open compact subgroup, then we write H (V) v + G v F for the set of functions f : G(F)\G(A )/V → O.Wedefine V = V by choosing 0 0,v −1 V = G(O ) if v ∈ and V = ι Iw if v ∈ .If (Q, Q, {α ,β } ) is a Taylor– 0,v F 0,v  v v v v∈Q v  v Wiles datum of level N ≥ 1, then we define V (Q; N) = V (Q; N) ⊂ V (Q) = 1 1 v 0 −1 V (Q) by V (Q; N) = V (Q) = V if v ∈ Q, and V (Q) = ι Iw and 0 v 1 v 0 v 0,v 0 v  v −1 n+1−2i × N V (Q; N) = ι {(a ) ∈ Iw | a → 1 ∈ k( v) (p)/(p )} 1 v ij  v v ii i=1 if v ∈ Qand n is even, and (n+1−2i)/2 −1 × N V (Q; N) = ι {(a ) ∈ Iw | a → 1 ∈ k( v) (p)/(p )} 1 v ij  v v ii i=1 if v ∈ Qand n is odd. Thus V (Q; N) ⊂ V (Q) is a normal subgroup with quotient 1 0 V (Q)/V (Q; N)  /(p ). 0 1 Q 128 JAMES NEWTON, JACK A. THORNE We introduce Hecke operators for the group G. If v ∈ S ∪ Q is a place of F which c −1 (i) splits v = ww in K, then the unramified Hecke operators ι T (i = 1,... n)act on the w w −1 (i) spaces H (V (Q)) and H (V (Q; N)).If v ∈ Q then the operators ι U (i = 1,..., n) G 0 G 1 univ act on the spaces H (V (Q)) and H (V (Q; N)). We write T for the polynomial G 0 G 1 G,S∪Q (i) ring over O in the indeterminates T (where w is a place of K split over F, not lying univ,Q i=1,...,n univ (i) above S ∪ Qand 1 ≤ i ≤ n), and T = T [{U } ].Thus H (V (Q)) and G 0 G,S∪Q G,S∪Q  v∈Q univ,Q H (V (Q; N)) are T -modules. G 1 G,S∪Q univ There is a unique maximal ideal m ⊂ T of residue field k such that for all G,S finite places w of K, split over F and not lying above S, the characteristic polynomial n−1 i i(i−1)/2 (i) n−i of Sym r(Frob ) equals (−1) q T X mod m .If (Q, Q,(α ,β ) ) is a w G v v v∈Q i=0 w w univ,Q Taylor–Wiles datum, then we write m for the maximal ideal of T generated by G,Q G,S∪Q univ m ∩ T and the elements G,S∪Q i(1−i)/2 (i) n−j j−1 U − q α β v v j=1 for v ∈ Q, i = 1,..., n. The unitary group G comes with a determinant map det : G → U ,where U = 1 1 ker(N : Res G → G ).If K/F K/F m m ∞ × θ : U (F)\U (A )/ det V (Q; N) → O 1 1 1 is a character and f ∈ H (V (Q; N)) then we define f ⊗ θ ∈ H (V (Q; N)) by the G 1 G 1 formula (f ⊗ θ)(g) = θ(det(g))f (g).If f ∈ H (V (Q; N)) and θ is trivial, then G 1 m G,Q f ⊗ θ ∈ H (V (Q; N)) . We will use this construction only in conjunction with the G 1 m G,Q following lemma. Lemma 2.4. — Suppose that n is even and that p = 2. Suppose that × ∞ × × χ : F \(A ) / det U (Q; N) → O is a quadratic character. Then there exists a unique character ∞ × θ : U (F)\U (A )/ det V (Q; N) → O χ 1 1 1 ∞ × c c such that for all z ∈ (A ) we have θ (z/z ) = χ(zz ). Proof. — There is a short exact sequence of F-groups 0 → G → Res G → U → 0, m K/F m 1 where the last map is z → z/z . By Hilbert 90 we have a short exact sequence × ∞ × × ∞ × ∞ 0 → F \(A ) → K \(A ) → U (F)\U (A ) → 0. 1 1 F K F SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 129 ∞ × This shows that there is a unique character θ : U (F)\U (A ) → O such that χ 1 1 c c ∞ × θ (z/z ) = χ(zz ) for all z ∈ (A ) . We need to check that θ is trivial on det V (Q; N). χ χ 1 This can be checked locally at each finite place of F. At places v ∈ Q it follows from the fact that χ is unramified at v.If v ∈ Q then we see, using that n is even and identify- × × ing U (F ) with F ,that det V (Q; N) is contained in the subgroup of O with square 1 v 1 v v F image in k( v) . Since χ is quadratic, θ annihilates this subgroup, and we’re done. Let m ≥ 0denote the p-adic valuation of the least common multiple of the expo- −1 ∞ nents of the Sylow p-subgroups of the finite groups G(F) ∩ tV t (t ∈ G(A )). Proposition 2.5. — Let N ≥ 1 and let (Q, Q,(α ,β ) )) be a Taylor–Wiles datum of v v v∈Q level N + m .Then: (1) The maximal ideals m , m are in the support of H (V ) and H (V (Q)),respec- G G,Q G 0 G 0 tively. univ N N (2) H (V (Q; N)) is a T ⊗ O[ /(p )]-module free as O[ /(p )]- G 1 m O Q Q G,Q G,S∪Q module, and there is an isomorphism H (V (Q; N)) ⊗ O = H (V ) G 1 m O[ /(p )] G 0 m G,Q Q G univ of T -modules. G,S∪Q univ (3) There exists a structure on H (V (Q; N)) of P -module such that if  : G → G 1 m Q K G,Q i P (i = 1,..., n) are the coefficients of the universal characteristic polynomial, defined as in [Che14, §1.10], then for any finite place v ∈ S ∪ Q of F which splits v = ww in univ i(i−1)/2 (i) K,  (Frob ) acts on H (V (Q; N)) as q T . Moreover, the O[ ]- w G 1 m Q i G,Q w w module structure on H (V (Q; N)) induced by the map O[ ]→ P agrees with G 1 m Q Q G,Q the one in the second part of the lemma. Proof. — The first part may be deduced, as in [NT20, §4.3], from [Lab11, Théorème 5.4] (and the existence of  ). The other parts are proved in a very sim- ilar way to the second and third parts of Proposition 2.3, as we now explain. We × n begin by constructing a more familiar set of objects. Let  = k(v) (p) .Let Q v∈Q V (Q; N) = V (Q; N) ⊂ V (Q) be the subgroup defined by V (Q; N) = V (Q) v 0 v 0 v 1 v 1 1 if v ∈ Qand −1 × N V (Q; N) = ι {(a ) ∈ Iw |∀i = 1,..., n, a → 1 ∈ k( v) (p)/(p )} v ij  v ii 1  v if v ∈ Q. Thus V (Q; N) ⊂ V (Q) is a normal subgroup with quotient V (Q)/ 0 0 V (Q; N)  /(p ). We write P ∈ C for the quotient of the ring R introduced O S∪Q 1 Q Q in [NT20, §2.19] corresponding to pseudodeformations whose restriction to G factors ab through G for each v ∈ Q. As in the case of P ,[ANT20, Proposition 2.5] again shows (i)  × that for each v ∈ Q there are unique characters A : G → (P ) (i = 1,..., n)such v Q (i) (1) (n) that A mod m = ur n−i i−1 and det(A ⊕ ··· ⊕ A ) is the restriction to G of the P K v α β v v  v Q v v universal pseudocharacter over G . We now claim that the following statements hold: K 130 JAMES NEWTON, JACK A. THORNE univ  N  N • H (V (Q; N)) is a T ⊗ O[ /(p )]-module free as O[ /(p )]- G m O 1 G,Q G,S∪Q Q Q module, and there is an isomorphism H (V (Q; N)) ⊗ N O G m O[ /(p )] 1 G,Q univ H (V ) of T -modules. G 0 m G G,S∪Q • There is a unique structure on H (V (Q; N)) of P -module such that for G m 1 G,Q Q c univ any place v ∈ S ∪ QofFwhich splits v = ww in K,  (Frob ) acts on i(i−1)/2 (i) H (V (Q; N)) as q T . G m G,Q 1 w w • The two induced O[ ]-module structures on H (V (Q; N)) (one by the G m G,Q Q 1 isomorphism V (Q)/V (Q; N) =  /(p ), the other by the map  → P 1 Q Q Q (i) associated to the tuple of characters A ◦ Art | )are thesame. v  v O To prove the first point, we need to explain why H (V (Q; N)) is an O[ /(p )]- G m 1 G,Q Q module with the claimed coinvariants. The action of  /(p ) is induced by the ac- tion of V (Q) via the isomorphism V (Q)/V (Q; N) =  /(p ) (which therefore 0 0 1 Q univ commutes with the action of T ). The freeness follows because V (Q)/V (Q; N) G,S∪Q 1 acts freely on the quotient G(F)\G(A )/V (Q; N), because V (Q; N) contains all F 1 1 the p-torsion elements of V (Q); compare the proof of [KW09b, Lemma 7.4] and [BHKT19, Lemma 8.18]. The freeness of this action implies that there is an isomor- univ phism H (V (Q; N)) ⊗ O = H (V (Q)) of T -modules. To com- G m O[ /(p )] G 0 m 1 G,Q G,Q G,S∪Q plete the proof of the first point, we need to explain why there is an isomorphism H (V (Q)) = H (V ) . This follows from [NT20, Proposition 3.1]. The second G 0 m G 0 m G,Q G part is proved in the same way as [NT20, Lemma 4.7]. The third part is proved using [NT20, Lemma 4.7] and [ANT20, Proposition 2.5] (in particular, the uniqueness of the decomposition of residually multiplicity-free pseudocharacters). We now need to explain why the above claims imply the properties in the statement of the proposition. There are canonical quotient morphisms P → P and →  . The proof is complete on noting that trace induces an isomorphism H (V (Q; N)) ⊗ N O[ /(p )] = H (V (Q; N)) and that the map G m O[ /(p )] Q G 1 m 1 G,Q G,Q P → P factors through an isomorphism P ⊗ O[ ] P . Q O[ ] Q Q Q Q We are now ready to prove Theorem 2.1. We will need to treat the cases p > 2and p = 2separately. Proof of Theorem 2.1,case p > 2.— Define H = H (V ) and H = H (U ) . G G 0 m D D 0 m G D The proof of the theorem will be based on the following proposition: Proposition 2.6. — We can find an integer q ≥ 0 with the following property: let W = OY ,..., Y , Z ,..., Z . Then we can find the following data: 1 q 1 4|S∪S |−1 (1) Complete Noetherian local W -algebras P , R equipped with isomorphisms P ⊗ ∞ ∞ ∞ ∞ W ∼ ∼ O P and R ⊗ O R in C . = = ∞ W O (2) A surjection R X ,..., X  → R in C , where g = q +|S ∪ S |− 1. loc 1 g ∞ O ∞ SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 131 (3) A P -module H and an R -module H such that both H , H are finite ∞ G,∞ ∞ D,∞ G,∞ D,∞ free W -modules, complete with isomorphisms H ⊗ O H (as P-module) and ∞ G,∞ W G H ⊗ O H (as R-module). D,∞ W D (4) A morphism P → R of O-algebras making the diagram ∞ ∞ P R ∞ ∞ P R commute. Before giving the proof of Proposition 2.6, we show how it implies the theorem. Let p ⊂ P denote the kernel of the morphism P → O associated to t.Itisenoughto show that p is in the support of H as P-module. Indeed, this would imply (using [Lab11, n−1 Corollaire 5.3] and the irreducibility of Sym r| ) the existence of a RACSDC au- n−1 tomorphic representation  of GL (A ) such that r Sym (r| ). By descent n n K  ,ι G n K (in the form of e.g. [BLGHT11, Lemma 1.5]), this would imply the sought-after auto- n−1 morphy of Sym r. Equivalently, we must show that if p is the pre-image of p un- der the morphism P → P, then p is in the support of H as P -module. (Since ∞ ∞ G,∞ ∞ Supp H = Supp H ∩ Spec P, intersection taken inside Spec P .) G G,∞ ∞ P P The P -module H is a Cohen–Macaulay module (i.e. it is finite and the di- ∞ G,∞ mension of its support is equal to its depth), since H is a finite free W -module. Ap- G,∞ ∞ plying [Sta13, Tag 0BUS], it follows that each irreducible component of Supp H G,∞ has dimension q + 4|S ∪ S |. Similarly, we see that each irreducible component of Supp H has dimension q + 4|S ∪ S |. Since R is a quotient of R X ,..., X , D,∞ ∞ ∞ loc 1 g a domain of Krull dimension q + 4|S ∪ S |, we see that R X ,..., X  → R is an ∞ loc 1 g ∞ isomorphism, that R is a domain, and that H is a faithful R -module. ∞ D,∞ ∞ Let p ⊂ P denote the kernel of the morphism P → O associated to t ,and let p denote the pre-image of p under the morphism P → P. Then p ∈ Supp H ,by ∞ G,∞ ∞ P hypothesis, and therefore dim(P  ) ≥ q + 4|S ∪ S |− 1. We claim that the Zariski ∞,(p ) ∞ tangent space to the local ring P  has dimension at most q + 4|S ∪ S |− 1. Indeed, ∞,(p ) ∞ it suffices to note that the quotient P  /(Y ,..., Y , Z ,..., Z ) = P  equals ∞,(p ) 1 q 1 4|S∪S |−1 (p ) its residue field E, by the vanishing of the adjoint Bloch–Kato Selmer group of r  (i.e. ,ι by [NT20, Theorem A, Proposition 2.21, Example 2.34]). We deduce that P is ∞,(p ) a regular local ring of dimension q + 4|S ∪ S |− 1, so there is a unique irreducible component Z of Spec P containing the point p , which has dimension q + 4|S ∪ S | ∞ ∞ and is contained in Supp H . G,∞ Since Spec R is irreducible and the image of the morphism Spec R → Spec P ∞ ∞ ∞ contains p , we find that the morphism Spec R → Spec P factors through Z. In par- ∞ ∞ ticular, p lies in Z, hence in Supp H . This completes the proof of the theorem. ∞ G,∞ ∞ 132 JAMES NEWTON, JACK A. THORNE The proof of Proposition 2.6 is based on a patching argument. We first prove a lemma which shows that there are enough Taylor–Wiles data. The argument is very similar (and essentially identical in the case n = 2) to the proof of [DDT97,Theorem 2.49]. We spell out the details here just to show that the condition that the numbers n−i i−1 α β (i = 1,..., n) are distinct does not cause any difficulty. v v 1 0 Lemma 2.7. — Let q = dim H (F /F, ad r(1)),and let g = q +|S ∪ S |− 1. Then k S ∞ for any N ≥ 1, we can find a Taylor–Wiles datum (Q, Q,(α ,β ) ) of level N ≥ 1 such that v v v∈Q there is a surjection R X ,..., X  → R of R -algebras. loc 1 g loc Proof. — Let (Q, Q,(α ,β ) ) be a Taylor–Wiles datum. A standard computa- v v v∈Q tion (compare e.g. [Kis09b, Proposition 3.2.5], [Tho16, Proposition 5.10]), shows that there is a surjection R X ,..., X  → R where g = λ +|Q|+|S ∪ S |− 1and λ loc 1 g Q ∞ Q is the dimension of the group ⎛ ⎞ 1 0 1 0 ⎝ ⎠ ker H (F /F, ad r (1)) → H (F , ad r(1)) . S v v∈Q We therefore need to show that for any N ≥ 1, we can find a Taylor–Wiles datum of level Nsuch that |Q|= q and λ = 0. By induction, and the Chebotarev density theorem, it is enough to show the following claim: 1 0 • Let [φ]∈ H (F /F, ad r(1)) be non-zero. Then for any N ≥ 1, there exists σ ∈ G such that φ(σ ) ∈ (σ − 1) ad r(1) and the eigenvalues α, β of r(σ ) satisfy F(ζ ) (α/β) = 1for i = 1,..., n − 1. Let L/F denote the extension cut out by Proj r. Recall our assumption that  = Gal(L/F) is conjugate in PGL (F ) either to PSL (F a ) or PGL (F a ) for some p > max(5, 2n − 1). 2 p 2 p 2 p In either case [, ] PSL (F a ) is a non-abelian simple group that acts absolutely irre- 2 p 0 0 ducibly on ad .Moreover, [CPS75, Table (4.5)] shows that H (PSL (F a ), ad ) = 0. The 2 p inflation-restriction exact sequence implies that H (L(ζ N )/F, ad r(1)) = 0. Another ap- plication of inflation-restriction shows that Res [φ]= 0; we may identify this restric- L(ζ )/F 0 0 tion with a non-zero G -equivariant homomorphism f : G → ad r(1). Since ad r F L(ζ ) is absolutely irreducible as a k[PSL (F )]-module, f (G ) spans ad r(1) as k-vector 2 p L(ζ ) space. To prove the claim above, choose any σ ∈ G such that the eigenvalues α, β ∈ k F(ζ ) of r(σ ) satisfy (α/β) = 1for i = 1,..., n − 1. (This is possible since if t generates F a,then 2 a t has multiplicative order ≥ (p − 1)/2 > n − 1and Proj r(G ) contains an element F(ζ ) −1 2 which is conjugate to diag(t, t ) = diag(t , 1).) If φ(σ ) ∈ (σ − 1) ad r(1) then we’re done. Suppose instead that φ(σ ) ∈ (σ − 0 0 1) ad r(1). Since f (G ) spans ad r(1),wecan find τ ∈ G such that f (τ ) ∈ (σ − L(ζ ) L(ζ ) N N p p SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 133 1) ad r(1).Then τσ ∈ G ,Proj r(τ σ ) = Proj r(σ ), and the cocycle relation shows F(ζ ) that φ(τ σ ) = φ(τ ) + φ(σ ) ∈ (τ σ − 1) ad r(1). In either case we have established the claim; this completes the proof. 1 0 Now we give the proof of Proposition 2.6.Let q = dim H (F /F, ad r(1)) and k S g = q +|S ∪ S |− 1. For each N ≥ 1, fix a Taylor–Wiles datum (Q , Q ,(α ,β ) ) ∞ N N v v v∈Q of level N + max(m , m ) such that there exists a surjection R X ,..., X  → R D G loc 1 g of R -algebras. Fix v ∈ Sand define T = O{Z } /(Z ).Weview T loc 0 v,i,j v∈S∪S ,1≤i,j≤2 v ,1,1 ∞ 0 as an augmented O-algebra via the augmentation which sends each Z to 0. Choose v,i,j for each N ≥ 1 a surjection OY ,..., Y  → O[ /(p )]. Then we get surjections 1 q Q N N W → T ⊗ O[ /(p )]→ O[ /(p )]. ∞ O Q Q N N Define R = R ,H = T ⊗ H (U (Q; N)) ,P = T ⊗ P ,H = N D,N O D 1 m N O Q G,N Q D,Q N univ T ⊗ H(V (Q; N)) . We fix a representative r : G → GL (R) for the universal O 1 m F 2 G,Q univ deformation over R, and representatives r : G → GL (R ) for the universal defor- F 2 Q univ mations over R lifting r for each N ≥ 1. These choices determine isomorphisms ∼ ∼ R = T ⊗ Rand R = T ⊗ R , which classify the universal S ∪ S -framed liftings O N O Q ∞ univ univ (r , {1 + (Z )} ) (resp. (r , {1 + (Z )} )). Thus each ring R , P has v,i,j v∈S∪S v,i,j v∈S∪S N N ∞ Q ∞ ∼ ∼ aW -algebra structure, and there are isomorphisms R ⊗ O R, P ⊗ O P. = = ∞ N W N W ∞ ∞ Moreover, Proposition 2.3 and Proposition 2.5 show that the modules H , H are G,N D,N finite free as T ⊗ O[ /(p )]-modules. Finally, completed tensor product with T pro- O Q motes the morphism P → R to a morphism P → R . Q Q N N N N To patch these objects together we now carry out a diagonalisation argument along very similar lines to the proof of [Kis09b, Proposition 3.3.1]. By [NT20, Lemma 2.16], we can find an integer g ≥ 0and for eachN ≥ 1 a surjection OX ,..., X  → P of 0 1 g N O-algebras. Let a ⊂ W denote the kernel of the augmentation W → O,and forany ∞ ∞ N ≥ 1let a denote the kernel of the map W → O[ /(p )]. Choose a sequence N ∞ Q (b ) of open ideals of W satisfying the following conditions: N N≥1 ∞ • For each N ≥ 1, b ⊂ b . N+1 N • For each N ≥ 1, a ⊂ b . N N •∩ b = 0. N≥1 N Let s = max(dim H [1/p], dim H [1/p]).Let r = length (W /b ) . Then the se- E D E G N ∞ N quence (r ) is non-decreasing and N N≥1 length H /(b ) = length H /(b ) ≤ r , D,N N D,N N N R W N ∞ so H /(b ) has a natural structure of R /m -module. Similarly, H /(b ) has a D,N N N G,N N natural structure of P /m -module. Thus for every pair of integers N ≥ M ≥ 1we have, N 134 JAMES NEWTON, JACK A. THORNE by passage to quotient from the data constructed above, a diagram (of rings and modules) H /(b ) H /(b ) D,N M D M r r M M R X ,..., X ⊗ W R /m R/m loc 1 g O ∞ N R R r r M M OX ,..., X ⊗ W P /m P/m 1 g O ∞ N 0 P P H /(b ) H /(b ) G,N M G M where the horizontal arrows are all surjective. Keeping M fixed, the cardinalities of the rings and modules appearing in this diagram (excepting those in the first column) are uniformly bounded as N varies. By the pigeonhole principle, we can therefore find an increasing sequence (N ) of integers N ≥ Msuch that for each M ≥ 1 there is a M M≥1 M commutative diagram of O-algebras r r M M R X ,..., X ⊗ W R /m R/m loc 1 g O ∞ N M+1 R R M+1 r r M M R /m R/m M R R r r M M OX ,..., X ⊗ W P /m P/m 1 g O ∞ N 0 M+1 P P M+1 r r M M P /m P/m , M P P where the morphisms from the back square of the cube to the front are isomorphisms, and there are commutative diagrams of modules H /(b ) H /(b ) D,N M D M M+1 H /(b ) H /(b ) D,N M D M H /(b ) H /(b ) G,N M G M M+1 H /(b ) H /(b ), G,N M G M M SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 135 compatible with the module structures arising from the previous commutative cube, and where the arrows from back to front are again isomorphisms. We define r r M M R = lim R /m and P = lim P /m . ∞ N ∞ N M R M P N N M M ← − ← − M M and similarly H = lim H /(b ) and H = lim H /(b ). D,∞ D,N M G,∞ G,N M M M ← − ← − M M By passage to inverse limit, there is a diagram (of rings and modules) H H D,∞ D R X ,..., X ⊗ W R R loc 1 g O ∞ ∞ OX ,..., X ⊗ W P P 1 g O ∞ ∞ H H . G,∞ G To complete the proof of the proposition, it remains to show that these objects have the following properties: • The morphism R X ,..., X  → R is surjective. loc 1 g ∞ • The modules H ,H are finite free over W and the morphism H → D,∞ G,∞ ∞ D,∞ H (resp. H → H ) factors over an isomorphism H /(a) = H (resp. D G,∞ G D,∞ D H /(a) H ). G,∞ G • The morphism R → R (resp. P → P) factors over an isomorphism R /(a) ∞ ∞ ∞ R (resp. P /(a) = P). The first point holds because R X ,..., X  is a complete local ring and each map loc 1 g R X ,..., X  → R /m is surjective. The second follows from e.g. Nakayama’s loc 1 g N M R lemma and the freeness of the modules H /b ,H /b . For the third point, we D,N M G,N M M M r r M M recall that R /(a) = R, and consequently R /(m , a) = R/m . We therefore need N N M M R R to show that the natural map r r M M lim R /m /(a) → lim R /(m , a) N N M R M R N N M M ← − ← − M M is an isomorphism, or equivalently that the ideal aR of R is closed in the m -adic ∞ ∞ R topology. This is true since R is a Noetherian ring. The same proof applies to the ring P . ∞ 136 JAMES NEWTON, JACK A. THORNE Now we treat the case p = 2. Proof of Theorem 2.1,case p = 2.— Define H = H(V ) and H = H (U ) . G 0 m D D 0 m G D The proof of the theorem in this case will be based on the following proposition, incor- porating ideas from the 2-adic patching argument given in [KW09b, Kis09a]. If A ∈ C , we follow [KW09b, §2.1] in writing Sp for the functor Sp : C → Sets represented A A by A. We write G : C → Groups for the functor which sends A ∈ C to the group m O O × × ker(A → (A/m ) ). Proposition 2.8. — We can find an integer q ≥|S ∪ S |− 2 with the following property: let W = OY ,..., Y , Z ,..., Z  and let γ = 2 −|S ∪ S |+ q. Then we can find the ∞ 1 q 1 4|S∪S |−1 ∞ following data: (1) Complete Noetherian local W -algebras P , R equipped with isomorphisms P ⊗ ∞ ∞ ∞ ∞ W ∼ ∼ O P and R ⊗ O R in C . = = ∞ W O (2) A complete Noetherian local O-algebra R and surjections R X ,..., X  → R loc 1 g ∞ ∞ and R → R in C , where g = 2q + 1. ∞ O (3) A P -module H and an R -module H such that both H , H are finite ∞ G,∞ ∞ D,∞ G,∞ D,∞ free W -modules, together with isomorphisms H ⊗ O = H (as P-module) and ∞ G,∞ W G H ⊗ O H (as R-module). D,∞ W D (4) A morphism P → R of O-algebras making the diagram ∞ ∞ P R ∞ ∞ P R commute. γ γ γ (5) A free action of G on Sp and a G -equivariant morphism δ : Sp → G ,where m R m R m ∞ ∞ G acts on itself by the square of the identity. These objects have the following additional properties: −1 γ (6) We have δ (1) = Sp . The induced action of G [2](O) on R lifts to H . ∞ D,∞ R m (7) There exists an action of G [2] on Sp such that the morphism Sp → Sp is m P R P ∞ ∞ ∞ γ γ G [2]-equivariant, and the induced action of G [2](O) on P lifts to H . ∞ G,∞ m m Once again, we show how Proposition 2.8 implies the theorem in this case before giving the proof of the proposition. Let p ⊂ P denote the kernel of the morphism P → O associated to t. It is again enough to show that p is in the support of H as P-module, or equivalently that the pullback p ⊂ P of p is in the support of H as P -module. ∞ ∞ G,∞ ∞ The P -module H is a Cohen-Macaulay module, and each irreducible compo- ∞ G,∞ nent of Supp H has dimension q + 4|S ∪ S |. Similarly, each irreducible component G,∞ ∞ of Supp H has dimension q + 4|S ∪ S |. D,∞ ∞ ∞ SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 137 inv  γ Let R ⊂ R denote the subring of invariants for the action of G (cf. [KW09b, ∞ ∞ m §2.4]). Then the morphism Sp → Sp is smooth of relative dimension γ (one can inv R R apply [KW09b, Proposition 2.5], which is used for a very similar purpose in the proof of [KW09b, Proposition 9.3]). On the other hand, Sp → Sp inv is a torsor for the R R ∞ ∞ γ inv group G [2], showing that R has Krull dimension at least q + 4|S ∪ S |.Wecon- m ∞ clude that Spec R has dimension at least q + 4|S ∪ S |+ γ = dim R X ,..., X . ∞ loc 1 g Since R X ,..., X  is a domain, it follows that the map R X ,..., X  → R is loc 1 g loc 1 g inv an isomorphism, that Spec R is irreducible, that Spec R is irreducible of dimension ∞ ∞ q + 4|S ∪ S |,and that G [2](O) acts transitively on the set of irreducible components of Spec R , all of which have dimension q + 4|S ∪ S |. Property (6) of Proposition 2.8 ∞ ∞ implies that Supp H is invariant under the action of G [2](O), so we conclude D,∞ R m 1 1 that H has full support in Spec(R ) (in fact, considering H [ ] over R [ ] we can D,∞ ∞ D,∞ ∞ 2 2 conclude that H is a faithful R -module). D,∞ ∞ Let p ⊂ P denote the kernel of the morphism P → O associated to t ,and let p denote its pullback to P .Then p ∈ Supp H , by hypothesis. Similarly, let r, r ⊂ ∞ G,∞ ∞ P R denote the kernels of the morphisms R → O associated to r, r respectively, and let r , r ⊂ R denote their pullbacks under the morphism R → R. Then p (resp. p )is ∞ ∞ ∞ ∞ ∞ ∞ the image of r (resp r ) under the map Spec R → Spec P .Let r ∈ Spec R denote ∞ ∞ ∞ ∞ ∞ ∞ a point which is in the G [2](O)-orbit of r and on the same irreducible component m ∞ of Spec R as r ,and let p denote the image of r in Spec P .Then p , p lie on ∞ ∞ ∞ ∞ ∞ ∞ ∞ a common irreducible component of Spec P . Since the action of G [2](O) extends to P and H and the morphism P → R is equivariant for this action, Supp H ∞ G,∞ ∞ ∞ G,∞ is invariant under G [2](O) and contains p . m ∞ We now observe that the Zariski tangent space of the local ring P  has dimen- ∞,(p ) sion at most q + 4|S ∪ S |− 1. Indeed, translating by the element of G [2](O) which takes p to p , it suffices to show that the Zariski tangent space of the local ring P ∞,(p ) ∞ ∞ ∞ has dimension at most q + 4|S ∪ S |− 1, or even that P /(Y ,..., Y , Z ,..., ∞ ∞,(p ) 1 q 1 Z ) = P is a field. This again follows from [NT20, Theorem A, Proposi- 4|S∪S |−1 (p ) tion 2.21, Example 2.34]. It follows that P is a regular local ring of dimension ∞,(p ) q + 4|S ∪ S |− 1 and that there is a unique irreducible component of P containing ∞ ∞ the point p , which has dimension q + 4|S ∪ S | and is contained in Supp H .We ∞ G,∞ ∞ P deduce that p ∈ Supp H , as required. ∞ ∞ The proof of Proposition 2.8 is again based on a patching argument. Here is the analogue of Lemma 2.7 in our case. Lemma 2.9. — Let q = dim H (F /F, ad r) − 2,and let g = 2q + 1.Then q ≥|S ∪ k S S |− 2 and for any N ≥ 1, we can find a Taylor–Wiles datum (Q, Q,(α ,β ) )) of level N ≥ 1 ∞ v v v∈Q with the following properties: (1) |Q|= q. (2) There is a surjection R X ,..., X  → R of R -algebras. loc 1 g loc Q 138 JAMES NEWTON, JACK A. THORNE (3) Let  denote the Galois group of the maximal abelian pro-2 extension of F which is unramified outside Q and (S ∪ S )-split. Then there is an isomorphism  /(2 ) ∞ Q N γ (Z/2 Z) ,where γ = 2 −|S ∪ S |+ q. Proof. — This is contained in [KW09b, Lemma 5.10], except that result specifies only that if v ∈ Q then the eigenvalues α ,β ∈ k of r(Frob ) are distinct. Here we require v v v n−i i−1 that the numbers α β (i = 1,..., n) are distinct. However, reading the proof of loc. v v cit. we see that we can indeed choose v so that r(Frob ) satisfies this stronger requirement (using of course our assumption that the projective image of r contains PSL (F a ) for 2 2 some a ≥ 1such that2 > max(5, 2n − 1), as we did in the proof of Lemma 2.7). Now we give the proof of Proposition 2.8.Let q = dim H (F /F, ad r) − 2and k S g = 2q + 1. For each N ≥ 1, fix a Taylor–Wiles datum (Q , Q ,(α ,β ) ) of level N N v v v∈Q N + max(m , m ) such that there exists a surjection R X ,..., X  → R of R - D G loc 1 g loc algebras. Define T = OZ ,..., Z . Choose for each N ≥ 1 a surjection 1 4(|S|+|S |)−1 OY ,..., Y  → O[ ]. Then we get a surjection W → T ⊗ O[ ]. 1 q Q ∞ O Q N N Define R = R ,R = R ,H = T ⊗ H (U (Q; N)) ,P = T ⊗ P , N D,N O D 1 m N O Q Q N Q D,Q N N N univ H = T ⊗ H (V (Q; N)) . We fix a representative r : G → GL (R) for the G,N O G 1 m F 2 G,Q univ universal deformation over R, and representatives r : G → GL (R ) for the uni- F 2 Q univ versal deformations over R lifting r for each N ≥ 1. These choices determine com- ∼ ∼ patible isomorphisms R = T ⊗ Rand R = T ⊗ R . Thus each ring R , P has O N O Q N N ∼ ∼ aW -algebra structure, and there are isomorphisms R ⊗ O = R, P ⊗ O = P. ∞ N W N W ∞ ∞ Moreover, Proposition 2.3 and Proposition 2.5 show that the modules H , H are G,N D,N finitefreeas T ⊗ O[ /(p )]-modules. Completed tensor product with T promotes O Q the morphism P → R to a morphism P → R . Q Q N N N N ˇ ˇ Let  = Sp denote the group functor A → Hom( , A ).Then Q Q Q N N N O[ ] acts on Sp  by twisting: if r is a lifting corresponding to a morphism R → Aand R N χ :  → A is a character, then r ⊗ χ is a lifting which determines another morphism R → A. Thereisamorphism δ : Sp  →  given by taking the determinant and N Q N R N −1 multiplying by ,and δ (1) = Sp . The induced action of  [2](O) on R lifts to Q N N R N an action on H (U (Q; N)) , given by twisting by quadratic characters as above D 1 m D,Q (see also [KW09b, §7.5]). Similarly we can define compatible actions of  [2] on P Q N and of  [2](O) on H (V (Q; N)) , which are trivial if n − 1 is even and which Q G 1 m N G,Q correspond to twisting by the quadratic characters χ | (resp. θ for χ ∈  [2](O)) G χ Q K N when n − 1 is odd. We extend these to actions on H and H by completed tensor D,N G,N product with T . The morphism P → R is equivariant for these actions. A very similar N N argument to the proof of [KW09b, Proposition 9.3] (with modifications as in the proof of Proposition 2.6 above) now shows how to use the above data to construct the objects required by the statement of Proposition 2.8.  SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 139 3. Killing ramification Our goal in this section is to prove the following theorem (Theorem A of the intro- duction): Theorem 3.1. — Let n ≥ 1.Let π be a regular algebraic, cuspidal automorphic representation n−1 of GL (A ) which is non-CM. Then Sym π exists. 2 Q We fix n, which we can assume to be ≥ 3. The proof of Theorem 3.1 will be roughly by induction on the cardinality of sc(π ), the set of primes p such that π is supercuspidal; the case where sc(π ) is empty is exactly the main result of [NT21]. We begin with some preparatory definitions and results. Definition 3.2. — Let π be a regular algebraic, cuspidal automorphic representation of ss GL (A ). We define the semisimple conductor M of π to be M = N((rec π ) ) (where 2 Q π π Q l l l N denotes conductor). Lemma 3.3. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ), 2 Q let p be an odd prime, and let ι : Q → C be an isomorphism. If r is reducible or dihedral ,thenthe π,ι prime-to-p part of its conductor divides M . Proof. — If r is reducible or dihedral then its image has order prime to p,and for π,ι any prime l = p, r | is semisimple. This shows that the conductor of r | divides π,ι G π,ι G Q Q l l ss the conductor of (rec π ) . Q l Lemma 3.4. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ). 2 Q Let p be a prime, let ι : Q → C be an isomorphism, and suppose that Proj r (G ) contains a con- π,ι Q jugate of PSL (F a ) for some p > 5. Then we can find another regular algebraic, cuspidal automorphic 2 p representation π of GL (A ) with the following properties: 2 Q (1) There is an isomorphism r = r . π,ι π ,ι (2) π has weight 2 and is not ι-ordinary. (3) There is an isomorphism rec π = ω ⊕ ω ,where ω ,ω : W → C are characters Q 1 2 1 2 Q p p of conductor dividing p . (4) For each prime l = p, π is a twist of the Steinberg representation (resp. supercuspidal) if and only if π is. Proof. — Let be the set of primes l = p such that π is a twist of the Steinberg representation, and let T be the set of primes l = p such that π is supercuspidal. Let l ≥ 5 be a prime such that l ≡ 1mod p, π is unramified, and r (Frob ) has distinct 0 0 l π,ι l 0 0 By dihedral, we mean that the representation is induced from an index two subgroup of G . Q 140 JAMES NEWTON, JACK A. THORNE eigenvalues (such a prime exists because of our assumption on the image of Proj r ). Fix π,ι a coefficient field E containing a p th root of unity. If l is a prime such that π is super- cuspidal, then we can find, after possibly enlarging E, an O[GL (Z )]-module M , finite 2 l l free as O-module, such that M ⊗ C is a type for the Bernstein component containing l O,ι π , in the sense of [BM02, Definition A.1.4.1]. We define M =⊗ M ,M = M ⊗ k, l l∈T l k O ∨ ∨ ∨ and M = M ⊗ E. We write M ,M and M for the O-, k-, and E-linear duals of these E O k E GL (Z )-modules, equipped with the dual group action. 2 l l∈T Let D denote the quaternion algebra over Q such that if l is a prime, then D is ramified at l if and only if l ∈ .(Thus D isramified at ∞ if and only if | | is odd.) Fix a maximal order O ⊂ D and for each prime l ∈ an identification O ⊗ Z = M (Z ). D D Z l 2 l If U = U ⊂ (O ⊗ Z) is an open compact subgroup, we write Y(U) for the locally l D Z symmetric space of level U (namely the object denoted X in [NT16, §3.1]). We Res G D/Q m regard M as an O[U]-module by projection to GL (Z ).If U = GL (Z ) and r ≥ 1 2 l p 2 p l∈T then we write U (p ) ⊂ U for the open compact subgroup with the same component at primes l = p and component ab ∈ GL (Z ) | c ≡ 0mod p 2 p cd r × × at the prime p. We identify any pair of characters χ ,χ : (Z/p Z) → O with the 1 2 r × character χ ⊗ χ : U (p ) → O given by the formula 1 2 0 ab r r χ ⊗ χ = χ (a mod p )χ (d mod p ). 1 2 1 2 cd We write M(χ ⊗ χ ) = M ⊗ O(χ ⊗ χ ),regardedas O[U (p )]-module. We use similar 1 2 O 1 2 0 notation for k- and E-valued characters. Let δ = 0ifDis ramified at ∞ and δ = 1 otherwise. Let U = U be the open compact subgroup defined as follows: • U = Iw . l l ,1 0 0 • If l ∈ ∪{l },then U = GL (Z ). 0 l 2 l × × • If l ∈ and π = St (χ ),then U = ker(χ ◦ det : (O ⊗ Z ) → C ). l 2 l l l D Z l Then U is neat, in the sense of [NT16, §3.1] (because of the choice of U ), and we can × × find characters χ , χ : F → k such that 1 2 δ ∨ H (Y(U (p)), M (χ ⊗ χ ) ) = 0, 0 k m 1 2 π univ −1 ∞ where m ⊂ T is the maximal ideal associated to ι π (notation for the Hecke D, ∪T∪{l ,p} algebra as in §2), cf. [BDJ10, Corollary 2.12]. 3 × × 3 Let χ ,χ : (Z/p Z) → O be lifts of χ , χ such that χ /χ has conductor p . 1 2 1 2 1 2 We will show that δ 3 ∨ 0<s<1 (3.4.1)H (Y(U (p )), M (χ ⊗ χ ) ) = 0, 0 E 1 2 π SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 141 3 3 where the superscript denotes the subspace where the Hecke operator [U (p )α U (p )] 0 p,1 0 acts with eigenvalues that have p-adic valuation 0 < s < 1. Assuming (3.4.1) holds, we can complete the proof of the lemma. Indeed, the Jacquet–Langlands correspondence then implies the existence of a regular algebraic, cuspidal automorphic representation π of GL (A ) satisfying the following conditions: 2 Q • Thereisanisomorphism r = r . π,ι π ,ι • π has weight 2. −1  3 3 • ι π | contains a copy of χ ⊗ χ on which [U (p )α U (p )] acts with 1 2 0 p,1 0 U (p ) p 0 eigenvalue of p-adic valuation 0 < s < 1. • If l ∈ then π is a twist of the Steinberg representation. • If l ∈ Tthen π | contains M ⊗ C, hence (by definition of a type) π is GL (Z ) l O,ι l 2 l l supercuspidal. • If l ∈ ∪ T ∪{p},then π is a principal series representation. By [BM02, A.2.4], there is an isomorphism rec π = ω ⊕ ω ,where ω ,ω : W → Q 1 2 1 2 Q p p p × × C are characters such that ω ◦ Art | = ιχ and ω ◦ Art | = ιχ .More- 1 Q 1 2 Q 2 p Z p Z p p −1 over, the space Hom 3 (χ ⊗ χ ,ι π ) is 1-dimensional. Since the Hecke operator U (p ) 1 2 0 p 3 3 [U (p )α U (p )] acts with eigenvalue of p-adic valuation 0 < s < 1, π is not ι-ordinary 0 p,1 0 and therefore satisfies our requirements (cf. [Ger19, Lemma 5.2]). We now show that (3.4.1) holds. We have δ 3 ∨ 0<s<1 δ 3 ∨ s=0 h (Y(U (p )), M (χ ⊗ χ ) ) + h (Y(U (p )), M (χ ⊗ χ ) ) 0 E 1 2 0 E 1 2 m m π π δ 3 ∨ s=1 + h (Y(U (p )), M (χ ⊗ χ ) ) 0 E 1 2 δ 3 ∨ = h (Y(U (p )), M (χ ⊗ χ ) ) , 0 E 1 2 m (where lowercase h denotes dimension of cohomology over k or E). For each π contribut- δ 3 ∨ −1 ing to H (Y(U (p )), M (χ ⊗ χ ) ), ι π | 3 also contains a copy of χ ⊗ χ with 0 E 1 2 U (p ) 2 1 p 0 3 3 multiplicity one. The product of the eigenvalues of [U (p )α U (p )] on the χ ⊗ χ 0 p,1 0 1 2 −1 and χ ⊗ χ isotypic spaces of ι π | 3 has p-adic valuation 1. We deduce that 2 1 U (p ) p 0 δ 3 ∨ s=1 δ 3 ∨ s=0 h (Y(U (p )), M (χ ⊗ χ ) ) = h (Y(U (p )), M (χ ⊗ χ ) ) . 0 E 1 2 0 E 2 1 m m π π Moreover, δ 3 ∨ s=0 δ ∨ ord h (Y(U (p )), M (χ ⊗ χ ) ) = h (Y(U (p)), M (χ ⊗ χ ) ) 0 E 1 2 0 k m 1 2 m π π and δ 3 ∨ s=0 δ ∨ ord h (Y(U (p )), M (χ ⊗ χ ) ) = h (Y(U (p)), M (χ ⊗ χ ) ) , 0 E 2 1 0 k m 2 1 m π π 142 JAMES NEWTON, JACK A. THORNE by Hida theory. It is therefore enough to show that δ 3 ∨ δ ∨ ord h (Y(U (p )), M (χ ⊗ χ ) ) > h (Y(U (p)), M (χ ⊗ χ ) ) 0 E 1 2 m 0 k π 1 2 m δ ∨ ord + h (Y(U (p)), M (χ ⊗ χ ) ) , 0 k 2 1 or even that δ 3 ∨ δ ∨ h (Y(U (p )), M (χ ⊗ χ ) ) > h (Y(U (p)), M (χ ⊗ χ ) ) 0 k m 0 k m 1 2 π 1 2 π δ ∨ + h (Y(U (p)), M (χ ⊗ χ ) ) . 0 k m 2 1 π δ ∨ Using the exactness of H (Y(U (p)), (?) ) as a (contravariant) functor of smooth 0 m U (p) k[U (p)]-modules, it is therefore enough to show that Ind χ ⊗ χ contains χ ⊗ χ 0 3 1 2 1 2 U (p ) and χ ⊗ χ as Jordan–Hölder factors with multiplicity at least 2 (or when χ = χ ,that 2 1 1 2 it contains χ ⊗ χ with multiplicity at least 3). 1 2 This is true. Indeed, the semisimplification of a smooth k[Iw ]-module (say finite- dimensional as k-vector space) is determined by its restriction to the diagonal torus in × × Iw , and we can then use Mackey’s formula to show that if ψ : F → k is a character −1 U (p) then the semisimplification of Ind χ ⊗ χ contains χ ψ ⊗ χ ψ with multiplicity 1 2 1 2 U (p ) p + 2if ψ = 1 and multiplicity p + 1if ψ = 1. Lemma 3.5. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ). 2 Q Let p ≥ 3 be a prime, let ι : Q → C be an isomorphism, and suppose that Proj r (G ) contains a π,ι Q conjugate of PSL (F a ) for some p > 5. Then we can find another regular algebraic, cuspidal automor- 2 p phic representation π of GL (A ) with the following properties: 2 Q (1) There is an isomorphism r = r . π,ι π ,ι (2) π hasweight2andisnot ι-ordinary. (3) There is an isomorphism rec π ω ⊕ ω ,where ω ,ω : W → C are characters Q 1 2 1 2 Q p p p 3  6 of conductor dividing p . In particular, N(π )|p . (4) For each prime l = p, r | ∼ r  | . In particular, N(π ) = N(π ). π,ι G π ,ι G l Q Q l l l Proof. — Lemma 3.4 implies the existence of a π satisfying requirements (1)–(3). We can appeal to [Gee11, Corollary 3.1.7] to replace it with a π also satisfying r | ∼ π,ι G r | for each prime l = p. By purity, r | and r | ‘strongly connect’ to each π ,ι G π,ι G π ,ι G Q Q Q l l l other in the terminology of [BLGGT14, §1.3]. Remark (6) of [BLGGT14, p. 524] implies that N(π ) = N(π ). Definition 3.6. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ) of weight 2 such that 2 ∈ sc(π ) and let q, t, r be prime numbers. We say that π is seasoned 2 Q with respect to (q, t, r) if the following properties hold: (1) t divides q + 1,t ∈ sc(π ),and t > max(10, 8n(n − 1)) and (q + 1)/t > 2. SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 143 (2) There is an isomorphism rec π Ind χ,where χ : W → C is a character Q q Q q W 2 such that χ | has order t. (In particular, q ∈ sc(π ) and N(π ) = q .) I q (3) r is a primitive root modulo q. If M denotes the least common multiple of the prime-to-q part of M and p ,then r ≡ 1 mod M. p∈sc(π )−{q} (4) π is an unramified twist of the Steinberg representation. (5) For each prime p ∈ sc(π ) and for each irreducible dihedral representation ρ : Gal(Q/Q) → GL (F ) of prime-to-p conductor dividing Mq , there exists a prime number s such that π 2 p s is an unramified twist of the Steinberg representation, ρ(Frob ) is scalar, and s ≡ 1 mod p. Proposition 3.7. — If π is seasoned with respect to (q, t, r) then for each prime p ∈ sc(π ) there exists an isomorphism ι : Q → C such that r (G ) contains a conjugate of SL (F a ) for some π,ι Q 2 p p > 2n − 1. Proof. — We split into cases depending on whether or not p = q. First suppose that p = q, and fix an isomorphism ι : Q → C.Wefirstclaim that r is irreducible. π,ι Otherwise, there’s an isomorphism r = χ ⊕ χ for characters χ : G → F of prime- π,ι Q 1 2 i q to-q conductor dividing M . Let ω : I → Q be the Teichmüller lift of the fundamental character of niveau 2. 2 Q q q a(q−1)b −1 Let b = (q + 1)/t.Then ι χ | = ω for some integer a ∈{1,..., t − 1} (this tame Q 2 character has niveau 2 since it extends to W and its order t does not divide q − 1). Write a(q − 1)b = i + (q + 1)j for some i ∈{1,..., q}.Then([Gee11, Theorem 4.6.1]) we i−1 1−i 1−i have (χ /χ )| =  or  . After relabelling, we can assume that (χ /χ )| =  . I I 1 2 Q 1 2 Q q q Since r ≡ 1 mod M (in particular, the characters χ are unramified at r), we have i−1 (χ /χ )(Frob ) = r . Since π is an unramified twist of the Steinberg representation, r r 1 2 −1 we have (χ /χ )(Frob ) = r or r . Since r is a primitive root modulo q, this implies that 1 2 one of i, i − 2 is divisible by q − 1. Since b divides i, i is among the numbers b, 2b,..., q + 1 − b. Since b > 2, we see that neither i nor i − 2 can be divisible by q − 1. This contradiction implies that r is π,ι irreducible. We next claim that r is not dihedral. If r is dihedral, then Lemma 3.3 shows π,ι π,ι that the prime-to-q part of the conductor of r divides M , so there exists a prime num- π,ι π ber s such that π is an unramified twist of the Steinberg representation, r (Frob ) is s π,ι s scalar, and s ≡ 1mod p. This is a contradiction. To finish the proof in the case p = q, we need to make a particular choice of ι. Such a choice fixes the value of a ∈{1,..., t − 1}; conversely, any a ∈{1,..., t − 1} can be obtained by making a suitable choice of ι. We choose a so that i = b. Invoking [Gee11, Theorem 4.6.1] once more, we see that the projective image of r contains an element π,ι 144 JAMES NEWTON, JACK A. THORNE of order in the set {(q + 1)/ gcd(q + 1, i + 1), (q + 1)/ gcd(q + 1, i − 1), (q − 1)/ gcd(q − 1, i − 1)}, therefore of order at least t/2. Since t/2 > 5, the classification of finite subgroups of PGL (F ) shows that the projective image of r contains PSL (F a ) (and is contained in 2 q π,ι 2 q a 2a PGL (F a ))for some a ≥ 1. If q ≤ 2n − 1then q − 1 ≤ 4n(n − 1), so every element of 2 q PGL (F ) of order prime to q has order at most 4n(n − 1). Since t/2 > 4n(n − 1), we see 2 q that we must have q > 2n − 1. Now suppose that p = q, and fix an isomorphism ι : Q → C.Then p = t and t  q − 1, so r | is irreducible and its projective image contains elements of order t, π,ι G and so r is irreducible and its projective image contains elements of order t. Using again π,ι the classification of finite subgroups of PGL (F ), we see that to complete the proof we 2 p just need to show that r is not dihedral. If it is dihedral then there exists a prime number π,ι s such that π is an unramified twist of the Steinberg representation, r (Frob ) is scalar, s π,ι s and s ≡ 1mod p. This is a contradiction. To prove the next proposition, we need to find primes with special properties, namely that their Frobenius elements act on the composita of certain field extensions in a prescribed way. Using the Chebotarev density theorem, we see that it is equivalent to exhibit Galois automorphisms acting in the correct way. In order to do so, it is helpful to recall the following lemma from basic Galois theory. Lemma 3.8. — Let E/K be a finite Galois extension, and let K /K, K /K be Galois subex- 1 2 tensions. Then the natural map Gal(K K /K) → Gal(K /K) × Gal(K /K) is an 1 2 1 Gal(K ∩K /K) 2 1 2 isomorphism. Proposition 3.9. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ) of weight 2 which is non-CM. Suppose that 2, 3 ∈ sc(π ). Then we can find a regular 2 Q algebraic, cuspidal automorphic representation π of GL (A ) with the following properties: 2 Q (1) There exist prime numbers (q, t, r) such that π is seasoned with respect to (q, t, r) and sc(π ) = sc(π ) ∪{q}. n−1 n−1 (2) Sym π exists if and only if Sym π does. Proof. — Fix a prime t > max(10, 8n(n − 1), N(π )) such that t ≡ 1mod 4 and there exists an isomorphism ι : Q → C such that G = Proj r (G ) is conjugate either π,ι Q to PSL (F ) or PGL (F ). Since t > 5, the group PSL (F ) is simple. The condition t ≡ 2 t 2 t 2 t 1 mod 4 implies that −1mod t is a square and that the image of complex conjugation c in G lies in [G, G]. Using the Chebotarev density theorem, we can therefore choose a prime q such that q ≡−1mod t, (q + 1)> 2t, and the image of Frob in G is in the conjugacy class of complex conjugation. SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 145 Similarly, we can choose a prime r such that r is a primitive root modulo q, r (Frob ) is scalar, and r splits in Q(ζ ) and in Q(ζ 6 ) for every p ∈ sc(π ). Indeed, π,ι r M t p ker Proj r π,ι the prime q is unramified in Q (ζ , {ζ } ) but totally ramified in Q(ζ ),so M t p p∈sc(π ) q the intersection of these two fields in Q is Q. We choose r so that it splits in the first field and is totally inert in the second. Let sc(π ) ={p ,..., p } and let p = q.For each i = 1,..., k + 1, let ρ 1 k k+1 i,j (j ∈ X ) be a set of representatives for the (finitely many) conjugacy classes of irre- ducible dihedral representations ρ : G → GL (F ) of prime-to-p conductor dividing Q 2 p i 2 6 lcm(M q , p ).For any (i, j), the abelianization of the projective image of ρ i,j p∈sc(π ) is isomorphic either to Z/2Z or (Z/2Z) . In either case we claim that we can find a prime s such that ρ (Frob ) is scalar, s ≡ 1mod p , the image of Frob in G is in the i,j s i,j i s i,j i,j i,j conjugacy class of complex conjugation, and s ≡−1mod t. i,j ker Proj ρ ker Proj r i,j π,ι To see this, let E = Q and E = Q . We want to show there is 1 2 σ ∈ Gal(E E (ζ ,ζ )/Q) such that σ | = 1, σ | = 1and σ | = c.First we find 1 2 p t E Q(ζ ) E (ζ ) i 1 p 2 t τ ∈ Gal(E (ζ )/Q) such that τ | = 1and τ | = 1. Gal(E /Q) is soluble and its 1 p E Q(ζ ) 1 i 1 p maximal abelian quotient is a quotient of (Z/2Z) ,so Gal(E ∩ Q(ζ )/Q) is a quotient 1 p 2 × 2 of (Z/2Z) . This shows that E ∩ Q(ζ ) is either trivial or quadratic. Since (F ) contains 1 p i p non-identity elements (because p ≥ 5, because p ∈ sc(π ))wecan finda τ with the desired i i property using Lemma 3.8. We can assume that τ acts trivially on the maximal abelian subextension of E (ζ ) of exponent 2. 1 p Using Lemma 3.8 again, we’re done if we can show that τ | = E (ζ )∩E (ζ ) 1 p 2 t ab c| .Let E denote the maximal abelian subfield of E .Ithas degree 1or2 E (ζ )∩E (ζ ) 2 1 p 2 t 2 ab over Q (because of the form of the image of r )and Gal(E /E ) is a non-abelian simple π,ι 2 ab group. Thus the maximal soluble quotient of Gal(E (ζ )/Q) is Gal(E (ζ )/Q),whichis 2 t t in fact abelian. Since Gal(E (ζ )/Q) is soluble, this shows that Gal(E (ζ ) ∩ E (ζ )/Q) 1 p 1 p 2 t i i is abelian. Since t is coprime to qN(π ), the prime t is unramified in E (ζ ), while the 1 p ab quotient of [E (ζ ) : Q] by the ramification index of t is 1 or 2. We conclude that E (ζ ) ∩ E (ζ ) is either trivial or quadratic. In particular, τ acts trivially on it. The 1 p 2 t ab element c also acts trivially on it, since it acts trivially on E and also on the quadratic subfield of Q(ζ ) (since t ≡ 1 mod 4). This completes the proof of the claim. To conclude the proof of the proposition, we apply [Gee11, Corollary 3.1.7]; it implies the existence of a regular algebraic, cuspidal automorphic representation π of GL (A ) of weight 2 such that r = r ,suchthatfor each s ∈{r, s }, π is an unram- 2 Q π ,ι π,ι i,j ified twist of the Steinberg representation, such that rec π Ind χ for a character q W χ : W → C such that χ | has order t, and such that for every other prime p,wehave Q I 2 Q r | ∼ r  | ,withnotationasin[BLGGT14, §1]. (The hypothesis ‘(ord)’of[Gee11, π,ι G π ,ι G Q Q p p Proposition 3.1.5] is automatic in our situation.) In particular, we have M = q M .We π π n−1 see that π is seasoned with respect to (q, t, r). To see that Sym π exists if and only if n−1 Sym π does, apply e.g. [BLGGT14, Theorem 4.2.1]. The potential diagonalizability 146 JAMES NEWTON, JACK A. THORNE assumption is satisfied because r | , r  | are both Fontaine–Laffaille, while the rep- π,ι G π ,ι G Q Q t t n−1 n−1 th resentations Sym r = Sym r are irreducible because the m symmetric power π,ι π ,ι of the standard representation of SL (F ) is irreducible whenever t > m. 2 t Proposition 3.10. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ) of weight 2. Suppose that π is seasoned with respect to (q, t, r),and let p ∈ sc(π ) sat- 2 Q isfy p ≥ 5. Then we can find a regular algebraic, cuspidal automorphic representation π of GL (A ) 2 Q with the following properties: (1) π hasweight2andisnon-CM. (2) sc(π ) = sc(π ) −{p}.If p = q, then π is seasoned with respect to (q, t, r). n−1  n−1 (3) If Sym π exists, then so does Sym π . Proof. — By Theorem 2.1, it’s enough to find an isomorphism ι : Q → C and a regular algebraic, cuspidal automorphic representation π of GL (A ) with the following 2 Q properties: • The image of r contains a conjugate of SL (F a ) for some p > 2n − 1. π,ι 2 p • π has weight 2 and is non-CM. • r  r . π ,ι π,ι • sc(π ) = sc(π ) −{p}. • For any prime l = p, r  | ∼ r | . π ,ι G π,ι G Q Q l l • π is not ι-ordinary. • If p = q then the conductor of π divides p . If p = q, then the last condition ensures that π is still seasoned with respect to (q, t, r) (more precisely, that conditions (3) and (5) in Definition 3.6 still hold). We choose ι sat- isfying the first condition using Proposition 3.7; then the existence of a π satisfying the above requirements is the content of Lemma 3.5. Proposition 3.11. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ). Suppose that π is of weight 2 and non-CM, and suppose that 2, 3 ∈ sc(π ).Then 2 Q n−1 Sym π exists. Proof. — If k ≥ 0, let (H ) denote the hypothesis that the conclusion of the proposi- tion holds when |sc(π )|≤ k,and let (H ) denote the hypothesis that the conclusion of the proposition holds when |sc(π )|≤ k and π is seasoned with respect to some tuple (q, t, r). As remarked above, (H ) follows from the results of [NT21]. It therefore suffices to prove the implications (H ) ⇒ (H ) and (H ) ⇒ (H ). k k k+1 k The first implication follows immediately from Proposition 3.10. For the second, assume that (H ) holds and let π be a regular algebraic, cuspidal automorphic represen- tation of GL (A ) which is of weight 2 and non-CM, and such that |sc(π )|= k ≥ 1. By 2 Q SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 147 Proposition 3.9, we can find a regular algebraic, cuspidal automorphic representation π of GL (A ) which is seasoned with respect to (q, t, r),suchthat sc(π ) = sc(π ) ∪{q},and 2 Q n−1 n−1 such that the existence of Sym π is equivalent to the existence of Sym π . Now choose a prime p ∈ sc(π ) (so p ∈ sc(π ) and p = q). Applying Proposition 3.10 with this choice of p gives another regular algebraic, cuspidal automorphic representation π of GL (A ) which is seasoned with respect to (q, t, r),suchthat |sc(π )|= k,and such 2 Q n−1  n−1  n−1 that the existence of Sym π implies that of Sym π . The existence of Sym π follows from (H ),sowe’re done. We can now give the proof of Theorem 3.1. Proof of Theorem 3.1.— Let π be a non-CM, regular algebraic, cuspidal automor- n−1 phic representation of GL (A ). We must show that Sym π exists. We first do this 2 Q under the additional assumption that π has weight 2 and that 2 ∈ sc(π ). We can assume that π is supercuspidal. Fix a prime t > max(5, 4n(n − 1), N(π )) such that t ≡ 1mod 4 and there exists an isomorphism ι : Q → C such that G = Proj r (G ) is conjugate t π,ι Q t t either to PSL (F ) or PGL (F ). Using the Chebotarev density theorem, we can find a 2 t 2 t prime q satisfying the following conditions: • The prime q satisfies q ≡−1mod t, q ≡ 1mod 8, and q ≡ 1mod l for every prime l < t. • The image of Frob in G is in the conjugacy class of complex conjugation. (Compare [KW09a, Lemma 8.2].) By [Gee11, Corollary 3.1.7], we can find another regular algebraic, cuspidal automorphic representation π of weight 2 satisfying the fol- lowing conditions: • r = r . π,ι π ,ι t t • If l = q is a prime, then r | ∼ r | . π,ι G π ,ι G t Q t Q l l • There is an isomorphism rec π Ind χ,where χ : W → C is a char- Q Q q W 2 2 q acter such that χ | has order t. n−1 n−1 Applying [BLGGT14, Theorem 4.2.1] to Sym r , we see that Sym π exists if and π,ι n−1 only if Sym π does. (The potential diagonalizability assumption is satisfied because n−1 r | , r  | are both Fontaine–Laffaille, while the representations Sym r π,ι G π ,ι G π,ι t Q t Q t t t n−1 th Sym r are irreducible because the m symmetric power of the standard representa- π ,ι tion of SL (F ) is irreducible whenever t > m.) 2 t Let ι : Q → C be an isomorphism. Then ([KW09a, Lemma 6.3]) there exists a ≥ 2 such that the image of Proj r is conjugate to PSL (F a ) or PGL (F a ).Infact, π,ι 2 3 2 3 a 2a we must have 3 > 2n − 1: otherwise t ≤ 3 − 1 ≤ 4n(n − 1), a contradiction to our assumption t > 4n(n − 1). Applying Lemma 3.4, we can find another regular algebraic, cuspidal automor- phic representation π of GL (A ) of weight 2 such that r  r  ,2, 3 ∈ sc(π ), 2 Q π ,ι π ,ι 148 JAMES NEWTON, JACK A. THORNE r  | is potentially crystalline and non-ordinary, and for each prime l = 3, π is a π ,ι G Q l twist of the Steinberg representation if and only if π is. Then Proposition 3.11 implies n−1 n−1 that Sym π exists. We can then invoke Theorem 2.1 to see that Sym π exists and n−1 hence Sym π exists. The next case to treat is when π has weight 2 but now 2 ∈ sc(π ). In this case we can n−1 repeat the same argument with 3 replaced by 2 to conclude the existence of Sym π . Finally we treat the general case where π has weight k for some k > 2and we make no assumption on sc(π ). In this case we can find a prime t > max(5, k(n + 1)) such that π is unramified and an isomorphism ι : Q → C such that the image of r t π,ι contains a conjugate of SL (F ). Applying Lemma 3.4 again, we can find another regu- 2 t lar algebraic, cuspidal automorphic representation π of GL (A ) of weight 2 such that 2 Q n−1 n−1 r  r and r  is potentially crystalline. Then Sym π exists and Sym r is ir- π ,ι π,ι π ,ι π,ι n−1 reducible. We can now apply [BLGGT14, Theorem 4.2.1] to conclude that Sym r π,ι n−1 n−1 is automorphic and therefore that Sym π exists. Note that Sym r  is the sym- π ,ι metric power of a 2-dimensional potentially diagonalizable representation ([GK14, Lemma 4.4.1]), and hence potentially diagonalizable, cf. the remark following [BLGG11, Definition 3.3.5]. Acknowledgements J.T.’s work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 714405). We thank Toby Gee and an anonymous referee for comments on earlier versions of this manuscript. 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. If material is not included in the article’s Creative Commons licence and your intended use SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 149 is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A: The case of weight one forms In this short appendix we record the automorphy of the symmetric power lifting for cuspidal Hecke eigenforms of weight 1, or with CM. The most difficult case is due to Kim [Kim04, Theorem 6.4]. Theorem A.1. — Let n ≥ 1.Let π be a cuspidal automorphic representation of GL (A ) with 2 Q π holomorphic limit of discrete series, or with π the automorphic induction of a Hecke character for a quadratic field. Then Sym π exists. Note that in these cases Sym π is usually not cuspidal. Proof. — First we assume that π is holomorphic limit of discrete series. Twisting by an algebraic Hecke character, we can assume that π is generated by a holomorphic weight 1 cuspidal Hecke eigenform. In particular, Deligne and Serre [DS74] constructed a continuous odd irreducible representation r : G → GL (C) with r | = rec (π ) π Q 2 π W p Q Q p p for all primes p. The projective image of r is a finite subgroup of PGL (C), and is there- π 2 fore dihedral or isomorphic to a copy of A , S or A (moreover, each of these subgroups 4 4 5 is unique up to conjugacy). We can then establish the automorphy of Sym r case by case, depending on the projective image. In the dihedral case, r is induced from a char- acter ψ of G for K/Q quadratic, Sym r decomposes as a direct sum of characters and K π the inductions of characters from K to Q, and therefore Sym r is automorphic. In the other cases, we denote the inverse image of Proj(r )(G ) in SL (C) by  . π Q 2 It is a binary polyhedral group. The image r (G ) is a subgroup of μ  ⊂ GL (C) for π Q 2k 2 some k (μ is the cyclic subgroup of the scalar matrices with order 2k). We have μ 2k 2k μ ×  /(−1, −I), so its irreducible representations are of the form ψ × σ ,with ψ a 2k character of μ , σ an irreducible representation of  ,and ψ(−1) = σ(−I). Twisting 2k by a Dirichlet character, we can assume that r (G ) = μ  (choose a prime p where π π Q 2k is unramified and which is 1 mod 2k, then twist by a Dirichlet character with conductor p and order 2k). Now to understand the decomposition of Sym r into irreducibles, it 2 1 suffices to understand the decomposition of the representation Sym C of  .See,for example, [Ste08, Appendix A] for the character tables of the binary polyhedral groups, or use [GAP20]. For the A case, the irreducible representations of  and their relationship to (symmetric powers of) the two Galois-conjugate irreducible two-dimensional representa- tions are described in [Kim04, §5]. This allows automorphy of Sym r to be deduced from the automorphy of Sym for m ≤ 4, together with tensor product functorialities GL × GL → GL and GL × GL → GL [Kim04, Theorem 6.4]. 2 2 4 2 3 6 150 JAMES NEWTON, JACK A. THORNE Now we turn to the A case. Considering the character table of the binary tetra- hedral group, we see that the irreducible representations of dimension > 1 comprise: three two-dimensional representations, isomorphic up to twist and a three-dimensional representation which is isomorphic to the symmetric square of the two-dimensional rep- n 2 resentations. Automorphy of Sym r therefore follows from automorphy of Sym r . π π Finally, in the S case, we consider the character table of the binary octahedral group. 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LABESSE, Changement de base CM et séries discrètes, in On the Stabilization of the Trace Formula,Stab. Trace Formula Shimura Var. Arith. Appl., vol. 1, pp. 429–470, Int. Press, Somerville, 2011. [NT16] J. NEWTON and J. A. THORNE, Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma, 4, e21 (2016). [NT20] J. NEWTON and J. A. THORNE, Adjoint Selmer groups of automorphic Galois representations of unitary type, J. Eur. Math. Soc. (2020), in press, arXiv:1912.11265. [NT21] J. NEWTON and J. A. THORNE, Symmetric power functoriality for holomorphic modular forms, Publ. Math. IHES (2021). https://doi.org/10.1007/s10240-021-00127-3. [Sta13] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2013. [Ste08] R. STEKOLSHCHIK, Notes on Coxeter Transformations and the McKay Correspondence, Springer Monographs in Math- ematics, Springer, Berlin, 2008. [Tho16] J. A. THORNE, Automorphy of some residually dihedral Galois representations, Math. Ann., 364 (2016), 589–648. J. N. Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK Current address: Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK 152 JAMES NEWTON, JACK A. THORNE 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 11 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, II

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SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II by JAMES NEWTON and JACK A. THORNE ABSTRACT Let f be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of the symmetric power lifting Sym f for every n ≥ 1. CONTENTS 1. Introduction ....................................................... 117 2. An automorphy lifting theorem for symmetric power representations ....................... 122 3. Killing ramification . .................................................. 139 Acknowledgements ..................................................... 148 Appendix A: The case of weight one forms ........................................ 149 References ......................................................... 150 1. Introduction Let F be a number field, and let π be a cuspidal automorphic representation of GL (A ). Langlands’s functoriality principle predicts the existence, for any n ≥ 1, of an 2 F automorphic representation Sym π of GL (A ), characterized by the requirement that n+1 F for any place v of F, the Langlands parameter of (Sym π) is the image of the Langlands parameter of π under the nth symmetric power Sym : GL → GL of the standard v 2 n+1 representation of GL . For a more detailed discussion of the context surrounding this problem, including known results by other authors, we refer the reader to the introduction of [NT21], of which this paper is a continuation. In that paper we studied the problem of symmetric power functoriality in the case that F = Q and π is regular algebraic (in which case π corresponds to a twist of a cuspidal Hecke eigenform f of weight k ≥ 2, cf. [Gel75,§3]). We established the existence of the symmetric power liftings Sym π under the assump- tion that there is no prime p such that π is supercuspidal. (This includes the case that f has level SL (Z).) In this paper we remove this assumption, proving the following theorem: Theorem A. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ). 2 Q Suppose that π is non-CM. Then for each integer n ≥ 1, Sym π exists, as a regular algebraic, cuspidal automorphic representation of GL (A ). n+1 Q In the ‘missing’ cases of π which are holomorphic limit of discrete series at ∞ or n n CM, the existence of Sym π for all n is well known, although of course Sym π is usually © IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2021 https://doi.org/10.1007/s10240-021-00126-4 118 JAMES NEWTON, JACK A. THORNE not cuspidal. The most difficult case of icosahedral weight one eigenforms ([Kim04,The- orem 6.4]) requires Kim and Shahidi’s results on tensor product and symmetric power functoriality. We provide some details in Appendix A. Using the modularity of elliptic curves over Q [BCDT01], we deduce the following corollary: Corollary B. — Let E be an elliptic curve over Q without complex multiplication. 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. Our strategy to prove Theorem A is inspired by the proof of Serre’s conjecture [KW09a]. There one takes as given Serre’s conjecture in the level 1 case (i.e. for every prime number p, the residual modularity of odd irreducible representations ρ : Gal(Q/Q) → GL (F ) 2 p unramified outside p), proved in [Kha06], and hopes to reduce the general case to this one by induction on the number of primes away from p at which ρ is ramified. Here we associate to any regular algebraic, cuspidal automorphic representation π of GL (A ) the set sc(π ) of primes p such that π is supercuspidal. Fixing n ≥ 1, we 2 Q p prove the existence of Sym π by induction on the cardinality of the set sc(π ),the case |sc(π )|= 0 being exactly the main result of [NT21]. Our induction argument uses congruences between automorphic representations. If p is a prime and ι : Q → C is an isomorphism, then there is an associated Galois representation r : Gal(Q/Q) → GL (Q ) π,ι 2 and its mod p reduction r : Gal(Q/Q) → GL (F ). π,ι 2 p If π is another regular algebraic, cuspidal automorphic representation of GL (A ),and 2 Q thereisanisomorphism r = r π,ι π ,ι (in other words, a congruence modulo p between π and π ), then passage to symmetric powers gives an isomorphism n n Sym r = Sym r . π,ι π ,ι n n If Sym π is known to exist, and the image of the representation Sym r  is sufficiently π ,ι non-degenerate (for example, irreducible), then automorphy lifting theorems (such as SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 119 those proved in [BLGGT14]) can be used to deduce the automorphy of Sym r ,hence π,ι the existence of Sym π . If p is a prime such that π is supercuspidal, it may be possible to choose π so that sc(π ) = sc(π ) −{p}, opening the way to an induction argument. This idea of ‘killing ramification’ plays a significant role in [KW09a]. The difficulty in applying this approach here is that if p ≤ n then the representation Sym r is never irreducible, so the automorphy lifting theorems proved in [BLGGT14] π,ι do not apply. (The automorphy lifting theorems for residually reducible representations proved in [ANT20] apply only for ordinary representations, a possibility which is ruled out if π is supercuspidal.) This approach might perhaps yield the existence of Sym π when p > n for every p ∈ sc(π ), but to get a result like Theorem A a new idea is required. Here we prove a new kind of automorphy lifting theorem, Theorem 2.1, specially tailored to the problem of symmetric powers (although we hope that these ideas will also be useful for other cases of Langlands functoriality). We consider the morphism P → R, where R is the universal deformation ring of the (supposed irreducible) representation r and P is the universal pseudodeformation ring of the pseudocharacter associated to π,ι the symmetric power Sym r ; the morphism P → R is the universal one classifying the π,ι pseudocharacter of Sym of the universal deformation of r . A version of the Taylor– π,ι Wiles–Kisin patching argument upgrades this to a commutative diagram P R ∞ ∞ P R, where P , R are ‘patched deformation rings’ and the vertical arrows are surjections. ∞ ∞ Since r is assumed to be irreducible, the arguments of [Kis09b, Kis09a] show that R π,ι ∞ is a domain which acts faithfully on a space of patched (rank-2) modular forms. The essential additional ingredient is the main result of [NT20], which shows that Spec P is regular (of dimension 0) at the point corresponding to the pseudocharacter of the representation Sym r ; this in turn implies that Spec P is regular at the image of π ,ι ∞ this point in Spec P , and allows us to deduce that the image of Spec R → Spec P is ∞ ∞ ∞ contained in the support of a space of patched (rank-(n + 1)) modular forms, leading to a proof of Theorem 2.1. Our a priori knowledge about the ring P obviates the need to kill the dual Selmer group of Sym r . π,ι To actually prove Theorem A, we combine Theorem 2.1 with a modified version of the ‘killing ramification’ technique of [KW09a], based on a variation of the notion of ‘good dihedral’ representation introduced in that paper. This is not quite routine since we need our ‘good dihedral’ automorphic representations π to have the property that, if q is the good dihedral prime, then there is an isomorphism ι : Q → C such that r has q π,ι q q 120 JAMES NEWTON, JACK A. THORNE large image. We achieve this by introducing Steinberg type ramification at another auxil- iary prime r, which is acceptable since the presence of r does not affect the set sc(π ).We call an automorphic representation π that comes equipped with the requisite auxiliary primes ‘seasoned’ (see Definition 3.6). Notation. If F is a perfect field, we generally fix an algebraic closure F/F and write G for the absolute Galois group of F with respect to this choice. When the characteristic of F is not equal to p, we write  : G → Z for the p-adic cyclotomic character. We th 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 v v each place v of F; this choice determines a homomorphism G → G . When v is a F F finite place, we will write W ⊂ G for the Weil group, O ⊂ F for the valuation ring, F F F v v v v ∈ O for a fixed choice of uniformizer, Frob ∈ G for a fixed choice of (geometric) v F v F v v Frobenius lift, k(v) = O /( ) for the residue field, and q = #k(v) for the cardinality F v v × × of the residue field. If R is a ring and α ∈ R , then we write ur : W → R for the α F unramified character which sends Frob to α. When v is a real place, we write c ∈ G v v F for complex conjugation. If S is a finite set of finite places of F then we write F /Ffor 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. We write C for the category of complete Noetherian local O-algebras with residue field k.If G is a profinite group and ρ : G → GL (Q ) is a continuous representation, then we write ρ : G → GL (F ) for the associated semisimple residual representation (which is well- n p defined up to conjugacy). If F is a number field, v is a finite place of F, and ρ, ρ : G → GL (O ) are continuous representations, which are potentially crystalline if v|p,then n Q we use the notation ρ ∼ ρ established in [BLGGT14, §1] (which indicates that these two representations define points on a common component of a suitable deformation ring). 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 corre- spondence rec for GL (K): a bijection between the sets of isomorphism classes of ir- K n reducible, admissible C[GL (K)]-modules and Frobenius-semisimple Weil–Deligne rep- resentations over C of rank n. In general, we have the Tate normalisation of the local Langlands correspondence for GL as described in [CT14, §2.1]. When = C,wehave T (1−n)/2 rec (π ) = rec (π ⊗|·| ). K SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 121 × × 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 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 n F ∞ as an irreducible algebraic representation W of (Res GL ) . F/Q n C If π is cuspidal, regular algebraic, and polarizable, in the sense of [BLGGT14], then for any isomorphism ι : Q → C there exists a continuous, semisimple represen- F−ss tation r : G → GL (Q ) such that for each finite place v of F, WD(r | ) = π,ι F n π,ι G p Fv T −1 rec (ι π ) (see e.g. [Car14]). (When n = 1, this is compatible with our existing nota- tion.) We use the convention that the Hodge–Tate weight of the cyclotomic character is −1. We use special terminology in the case n = 2: if k ≥ 2 is an integer, we say that π k−2 2 ∨ has weight k if we can take W = (⊗ Sym C ) . (If F is totally real, then the τ ∈Hom(F,C) cuspidal automorphic representations of weight k are those which are associated to cusp- idal Hilbert modular forms of parallel weight k.) In this case the Hodge–Tate weights of k−1 r with respect to any embedding τ : F → Q are {0, k − 1} and the character  det r π,ι π,ι has finite order. If F is a number field, G is a reductive group over F, v is a finite place of F, and U is an open compact subgroup of G(F ), then we write H(G(F ), U ) for the convo- v v v v lution algebra of compactly supported U -biinvariant functions f : G(F ) → Z (convo- v v lution defined with respect to the Haar measure on G(F ) which gives U volume 1). v v Then H(G(F ), U ) is a free Z-module, with basis given by the characteristic functions v v [U g U ] of double cosets for g ∈ U \G(F )/U . v v v v v v v If 1 ≤ i ≤ n,let α = diag( ,..., , 1,..., 1) ∈ GL (F ) (where there are i ,i v v n v occurrences of  on the diagonal). We define (i) T =[GL (O )α GL (O )]∈ H(GL (F ), GL (O )). n F  ,i n F n v n F v v v v v We write Iw ⊂ GL (O ) for the standard Iwahori subgroup (elements which are upper- v n F × n triangular modulo  )and Iw ⊂ Iw for the kernel of the natural map Iw → (k(v) ) v v,1 v v given by reduction modulo  , then projection to the diagonal. If U ⊂ Iw is a subgroup v v v containing Iw ,and 1 ≤ i ≤ n,thenwedefine v,1 (i) U =[U α U ]∈ H(GL (F ), U ). v  ,i v n v v v 122 JAMES NEWTON, JACK A. THORNE 2. An automorphy lifting theorem for symmetric power representations Let p be a prime and let F be a totally real field. Fix an isomorphism ι : Q → C. Let n ≥ 1. Let π be a regular algebraic, cuspidal automorphic representation of GL (A ) 2 F satisfying the following conditions: • π has weight 2 and is non-CM. • For each place v|p of F, r | is not ordinary, in the sense of [Tho16, §5.1]. π,ι G Note that, together with the assumption that π has weight 2, this implies that r | is potentially Barsotti–Tate. π,ι G • Let Proj r : G → PGL (F ) denote the projective representation associated π,ι F 2 p to r . Then there exists a ≥ 1such that p > max(5, 2n − 1) and there is a π,ι sandwich a a PSL (F ) ⊂ Proj r (G ) ⊂ PGL (F ), 2 p π,ι F 2 p up to conjugacy in PGL (F ). 2 p We impose the final condition to ensure that we can choose Taylor–Wiles primes such n−1 that the image of the corresponding Frobenius element under Sym r is regular π,ι semisimple. The aim of this section is prove the following theorem: Theorem 2.1. — Suppose that there exists another regular algebraic, cuspidal automorphic rep- resentation π of GL (A ) such that the following conditions are satisfied: 2 F (1) π hasweight2andisnon-CM. (2) For each place v|pof F,r  | is not ordinary. π ,ι G (3) There is an isomorphism r  r . π ,ι π,ι (4) For each place v  pof F, π is a character twist of the Steinberg representation if and only if π is. n−1 (5) Sym r  is automorphic. π ,ι n−1 Then Sym r is automorphic. π,ι We begin with a preliminary reduction. Let E/Q be a coefficient field. After pos- sibly enlarging E, we can find conjugates r, r of r , r respectively which take values in π,ι π ,ι GL (O). We can also assume that the eigenvalues of each element in the image of r lie in k. After passage to a soluble totally real extension, we can assume that the following additional conditions are satisfied: (6) [F : Q] is even. −1 (7) det r = det r  =  . π,ι π ,ι (8) For each place v|p of F, π and π are unramified. v SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 123 (9) For each finite place v  p of F, π and π are Iwahori-spherical. The number of places such that π is ramified is even. (10) Let S denote the set of p-adic places of F and the set of places v such that π is ramified. Let S = S ∪ .Thenfor each v ∈ S, r| is trivial. v p G For each v ∈ , q ≡ 1mod p and π ,π are isomorphic to the Steinberg v v representation (not just up to twist — note that condition (7) already implies that any such twist is by a quadratic character). (11) There exists an everywhere unramified CM quadratic extension K/F, with each place v ∈ S split in K. Let  = π .Then  is RACSDC (i.e. regular algebraic, conjugate self-dual, and cus- n−1 pidal). We will show that the representation Sym r is automorphic; this will imply ,ι Theorem 2.1, by soluble descent. We let  = π .Then  is also RACSDC, there is an n−1 isomorphism r = r , and Sym r is automorphic. We write  for the RACSDC ,ι  ,ι  ,ι n−1 automorphic representation of GL (A ) such that r = Sym r . n K  ,ι  ,ι Recall that C denotes the category of complete Noetherian local O-algebras with residue field k.If v is a place of F, we write R ∈ C for the object representing the functor Lift : C → Sets which associates to A ∈ C the set of homomorphisms v O O −1 r : G → GL (A) lifting r| (i.e. such that r mod m = r)suchthat det r =  | .We F 2 G A G v F F v v introduce certain quotients of R : • If v ∈ S , the smallest reduced O-torsion-free quotient R of R such that if p v F : R → Q is a homomorphism such that the pushforward of the universal v p lifting to Q is crystalline of Hodge–Tate weights {0, 1} (with respect to any em- bedding F → Q ) and is not ordinary, then f factors through R .By[Kis09a, v v Corollary 2.3.13], R is a domain of dimension 4 +[F : Q ]. v v p • If v ∈ , the smallest reduced O-torsion-free quotient R of R such that if f : R → Q is a homomorphism such that the pushforward of the universal v p −1 lifting to Q is an extension of  by the trivial character, then f factors through R .By[Kis09b, Corollary 2.6.7], R is a domain of dimension 4. v v −1, • If v ∈ S (the set of infinite places of F), the quotient R of R denoted R ∞ v in [Kis09a, Proposition 2.5.6]. Then R is a domain of dimension 3. If Q is a finite set of finite places of F, disjoint from S, then we write Def : C → Sets Q O for the functor which associates to A ∈ C the set of 1 + M (m )-conjugacy classes of lifts O 2 A r : G → GL (A) of r satisfying the following conditions: F 2 −1 •  r is unramified outside S ∪ Q, and det r =  . • For each v ∈ S ∪ S , the homomorphism R → A determined by r| factors ∞ G v v through the quotient R → R introduced above. Our assumption on the image of r implies that the functor Def is represented by an ob- ject R ∈ C . If Q is empty then we write R = R . We also introduce some variants. Let Q O ∅ 124 JAMES NEWTON, JACK A. THORNE R = ⊗ R .Then R is an O-flat domain of Krull dimension 1 + 3[F : Q]+ 3|S| loc v∈S∪S v loc (cf. [Kis09b, Lemma 3.4.12]). We write Def : C → Sets for the functor of 1 + M (m )- O 2 A conjugacy of tuples ( r, {A } ),where  r is as above and A ∈ 1 + M (m ),and v v∈S∪S v 2 A −1 γ ∈ 1 + M (m ) acts by γ · ( r, {A } ) = (γ rγ , {γ A } ). This functor is rep- 2 A v v∈S∪S v v∈S∪S ∞ ∞ −1 resented by an object denoted R ∈ C . The tuple of representations (A  r| A ) O G v v∈S∪S F ∞ Q v v is independent of the choice of representative for a given conjugacy class, and the univer- sal property of R determines a homomorphism R → R . loc v Q The objects in this paragraph will only be used in the case p = 2. We write Def : C → Sets for the functor of 1 + M (m )-conjugacy classes of lifts r : G → GL (A) of r O 2 A F 2 satisfying the following conditions: −1 •  r is unramified outside S ∪ Q, and if v ∈ Sthendet r| =  | . G G F F v v • For each v ∈ S ∪ S , the homomorphism R → A determined by r| factors ∞ G v Fv through the quotient R → R introduced above. We write Def for the functor of 1 + M (m )-conjugacy classes of tuples ( r, {A } ), 2 A v v∈S∪S Q ∞ where r is as above and A ∈ 1 + M (m ). Then the functors Def and Def are repre- v 2 A Q Q , , sented by objects R ,R ∈ C and there is again a natural morphism R → R . O loc Q Q Q n−1  n−1 Let t = det Sym (r| ) and t = det Sym (r | ) denote the group determi- G G K K nants over O (in the sense of [Che14]) associated to these two symmetric power represen- tations, and let t denote the group determinant over k which is their common reduction [0,n−1] modulo  . We introduce the object P ∈ C which is the quotient R of R intro- O S t,S duced in [NT20, §2.19]. Informally, P represents the functor of conjugate self-dual group 1−n determinants of G lifting t which have similitude character  and are semistable K,S with Hodge–Tate weights in the interval [0, n − 1]. Lemma 2.2. — Let A ∈ C be Artinian, let v ∈ S ,and let  r : G → GL (A) be a O p F 2 −1 lift of r of determinant  | such that the associated homomorphism R → A factors through F v n n−1 R → R .View A as an A[G ]-module via the representation Sym  r : G → GL (A) (we v F F n v v v n−1 2 n equip Sym A with its standard ordered basis, cf. [BLGG11, Definition 3.3.1]). Then A is iso- morphic, as Z [G ]-module, to a subquotient of a lattice in a crystalline (in particular, semistable) p F Q [G ]-module with all Hodge–Tate weights in the interval [0, n − 1]. p F Proof. — It follows from the construction in [Kis09b, Kis09a] that the dual of A is isomorphic, as Z [G ]-module, to the generic fibre of a finite flat group scheme p F over O .By[BBM82, Théorème 3.1.1], there exists a lattice L in a crystalline Q [G ]- F p F v v representation with Hodge–Tate weights in the interval [0, 1] such that A is isomorphic, n−1 2 as Z [G ]-module, to a quotient of L. It follows that Sym A is isomorphic, as Z [G ]- p F p F v v n−1 n−1 module, to a quotient of Sym L. The proof is complete on noting that Sym Lis Z Z p p n−1 a lattice in (Sym L)[1/p], a crystalline Q [G ]-representation with all Hodge–Tate p F Z v weights in the interval [0, n − 1].  SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 125 Let A ∈ C ,and let r : G → GL (A) be a lift of r which determines a map R → A. O F 2 n−1 Lemma 2.2 shows that the pseudocharacter associated to Sym ( r| ) satisfies condition (2.16.1) of [NT20]. In particular there are morphisms P → O associated to the pseu- docharacters t, t . Taking the pseudocharacter of the symmetric power of the universal deformation over R determines a morphism P → Rin C . We will study this morphism using the Taylor–Wiles method. In this paper we call a Taylor–Wiles datum of level N ≥ 1 a tuple (Q, Q,(α ,β ) ),where: v v v∈Q • Q is a finite set of places of F, split in K, such that for each v ∈ Q, q ≡ 1mod p . • For each v ∈ Q, we have fixed a factorisation v =  v v in K such that Q ={ v | v ∈ Q}. • For each v ∈ Q, α ,β ∈ k are eigenvalues of r(Frob ). We require that v v v n−1 n−2 n−1 α ,α β ,...,β are distinct elements of k. v v v We note that this last condition is stronger than the one typically appearing in applications to automorphy of 2-dimensional Galois representations and is specially adapted to our purposes here. univ Let r : G → GL (R ) denote a representative of the universal deformation. F 2 Q univ If v ∈ Qthen r | is conjugate (in GL (R )) to a unique representation of the G 2 Q Q v form A ⊕ B ,where A mod m = ur and B mod m = ur . The characters v v v R α v R β Q v Q v × univ × A , B : G → R are independent of the choice of r . We write  = k(v) (p) v v F Q v Q Q v∈Q (i.e. product of maximal p-power quotients of k(v) ). The collection of characters A ◦ Art | (v ∈ Q) determine a homomorphism O[ ]→ R with the property that v F O Q Q the natural map R → R factors through a canonical isomorphism R ⊗ O R. Q Q O[ ] [0,n−1] We write P ∈ C for the quotient of the quotient R of R introduced in Q O S∪Q t,S∪Q [NT20, §2.19] corresponding to pseudodeformations t of t with the following additional properties: • For each v ∈ Q, t| factors through the maximal Hausdorff abelian quo- ab 1 tient G → G . By [ANT20, Proposition 2.5] (cf. [NT20, Lemma 4.28], in [ANT20] we generalise [BC09, Proposition 1.5.1] to arbitrary characteristic), n−i i−1 there are unique characters χ ,...,χ lifting ur such that t| is the v,1 v,n G α β K v v  v pseudocharacter associated to χ ⊕ ··· ⊕ χ . v,1 v,n n+1−2i • Let m =n/2 and let i ∈{1,..., n}.If n = 2m is even, then χ | = χ | . v,i I I K v,m K v  v (n+1−2i)/2 If n = 2m + 1 is odd, then χ | = χ | . v,i I I K v,m K v  v The characters χ (v ∈ Q) give P the structure of O[ ]-algebra, and again there v,m Q Q is a universal morphism P → R . We remark that this need not be a morphism of Q Q O[ ]-algebras when n is odd, although it is when n is even. It is possible that our condition on the eigenvalues of Frob necessarily entails that t is abelian at  v.However,we haven’t verified this and it doesn’t cost us anything to build this in to the definition of P . Q 126 JAMES NEWTON, JACK A. THORNE To carry out the patching argument, we need to introduce spaces of automorphic forms. We first discuss automorphic forms on a definite quaternion algebra over F, follow- ing the set-up of [Kis09a, §3.1] and [KW09b, §7]. Let D be a definite quaternion algebra over F, ramified precisely at the infinite places and at the places of . Fix a choice of max- imal order O and for each finite place v ∈ , an identification O ⊗ O = M (O ). D D O F 2 F F v v ∞ × If U = U ⊂ (D ⊗ A ) is an open subgroup, then we write H (U) for the set of v F D v F ∞ × × ∞ × ∞ × functions f : (D ⊗ A ) → O such that for all γ ∈ D , z ∈ (A ) , g ∈ (D ⊗ A ) , O O F F F F F and u ∈ U, we have f (γ gzu) = f (g). We define × × U = (O ⊗ O ) × (D ⊗ F ) . 0 D O F F v F v v∈ v∈ If (Q, Q, {α ,β } ) is a Taylor–Wiles datum of level N ≥ 1, then we define U (Q; N) = v v v∈Q 1 U (Q; N) ⊂ U (Q) = U (Q) by U (Q; N) = U (Q) = U if v ∈ Q, and 1 v 0 0 v 1 v 0 v 0,v v v U (Q) = Iw and 0 v v ab −1 × N U (Q; N) = ∈ Iw : ad → 1 ∈ k(v) (p)/(p ) 1 v v cd if v ∈ Q. Thus U (Q; N) ⊂ U (Q) is a normal subgroup with quotient 1 0 U (Q)/U (Q; N) =  /(p ). 0 1 Q We introduce Hecke operators. If v ∈ ∪ Q is a finite place of F, then the un- (1) (2) ramified Hecke operators T , T act on H (U (Q)) and H (U (Q; N)).If v ∈ Q D 0 D 1 v v (1) univ then the operator U acts on H (U (Q)) and H (U (Q; N)). We write T for D 0 D 1 D, ∪Q univ,Q (1) (2) the polynomial ring over O in the indeterminates T , T (v ∈ ∪ Q) and T = v v D, ∪Q univ (1) T [{U } ]. v∈Q D, ∪Q univ There is a unique maximal ideal m ⊂ T of residue field k such that for all D, 2 (1) finite places v ∈ S of F, the characteristic polynomial of r(Frob ) equals X − T X + (2) (1) (2) q T mod m and for each v ∈ S ,T ∈ m and T − 1 ∈ m .If (Q, Q,(α ,β ) ) v D p D D v v v∈Q v v v univ,Q is a Taylor–Wiles datum, then we write m for the maximal ideal of T generated D,Q D, ∪Q univ (1) by m ∩ T and the elements U − α (v ∈ Q). D v D, ∪Q × ∞ × × If χ : F \(A ) / det U (Q; N) → O is a quadratic character and f ∈ H (U (Q; N)), then we define f ⊗ χ ∈ H (U (Q; N)) by the formula (f ⊗ χ)(g) = D 1 D 1 χ(det(g))f (g). We observe that if p = 2and f ∈ H (U (Q; N)) then f ⊗ χ ∈ D 1 m D,Q H (U (Q; N)) . D 1 m D,Q Let m ≥ 0denote the p-adic valuation of the least common multiple of the × ∞ × −1 × exponents of the Sylow p-subgroups of the finite groups F \(U(A ) ∩ t D t) for ∞ × t ∈ (D ⊗ A ) ; this number is finite, see [KW09b, §7.2]. F SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 127 Proposition 2.3. — Let N ≥ 1 and let (Q, Q,(α ,β ) ) be a Taylor–Wiles datum of v v v∈Q level N + m . (1) The maximal ideals m and m are in the support of H (U ) and H (U (Q)), D D,Q D 0 D 0 respectively. univ N N (2) H (U (Q; N)) is a T ⊗ O[ /(p )]-module free as O[ /(p )]- D 1 m O Q Q D,Q D, ∪Q module, and there is an isomorphism H (U (Q; N)) ⊗ O = H (U ) D 1 m D 0 m O[ /(p )] D,Q Q D univ of T -modules. D, ∪Q (3) There exists a structure on H (U (Q; N)) of R -module such that for any repre- D 1 m Q D,Q univ sentative r of the universal deformation of r and for each finite place v ∈ S ∪ Q of F, univ (1) univ (2) tr r (Frob ) acts as T and det r (Frob ) acts as q T . Moreover, the O[ ]- v v v Q Q v Q v module structure induced by the map O[ ]→ R agrees with the one in the second part Q Q of the lemma. Proof. — The first part is a consequence of the Jacquet–Langlands correspondence (and the existence of π ). The second part is [KW09b, Corollary 7.5]. The third part is proved in thesameway as [KW09b, Lemma 9.1] (cf. also [Kis09a, Lemma 3.2.7]); note (1) that since T ∈ m for each v ∈ S , only automorphic representations which are non- D p ordinary at each v ∈ S can contribute to H (U (Q; N)) . p D 1 m We next discuss automorphic forms on a definite unitary group of rank n.We therefore fix a unitary group G over F, split by K/F, as in [NT20, §4.1], together with an extension of G to a reductive group scheme over O . We recall that G comes equipped with isomorphisms ι : G → Res GL for each place v ofFwhich splits v = w O O /O n F K F v w v ww in K. Moreover, for each place v  ∞ of F, G is quasi-split, while for each place v|∞ of F, G(F ) is compact. If V = V ⊂ G(A ) is an open compact subgroup, then we write H (V) v + G v F for the set of functions f : G(F)\G(A )/V → O.Wedefine V = V by choosing 0 0,v −1 V = G(O ) if v ∈ and V = ι Iw if v ∈ .If (Q, Q, {α ,β } ) is a Taylor– 0,v F 0,v  v v v v∈Q v  v Wiles datum of level N ≥ 1, then we define V (Q; N) = V (Q; N) ⊂ V (Q) = 1 1 v 0 −1 V (Q) by V (Q; N) = V (Q) = V if v ∈ Q, and V (Q) = ι Iw and 0 v 1 v 0 v 0,v 0 v  v −1 n+1−2i × N V (Q; N) = ι {(a ) ∈ Iw | a → 1 ∈ k( v) (p)/(p )} 1 v ij  v v ii i=1 if v ∈ Qand n is even, and (n+1−2i)/2 −1 × N V (Q; N) = ι {(a ) ∈ Iw | a → 1 ∈ k( v) (p)/(p )} 1 v ij  v v ii i=1 if v ∈ Qand n is odd. Thus V (Q; N) ⊂ V (Q) is a normal subgroup with quotient 1 0 V (Q)/V (Q; N)  /(p ). 0 1 Q 128 JAMES NEWTON, JACK A. THORNE We introduce Hecke operators for the group G. If v ∈ S ∪ Q is a place of F which c −1 (i) splits v = ww in K, then the unramified Hecke operators ι T (i = 1,... n)act on the w w −1 (i) spaces H (V (Q)) and H (V (Q; N)).If v ∈ Q then the operators ι U (i = 1,..., n) G 0 G 1 univ act on the spaces H (V (Q)) and H (V (Q; N)). We write T for the polynomial G 0 G 1 G,S∪Q (i) ring over O in the indeterminates T (where w is a place of K split over F, not lying univ,Q i=1,...,n univ (i) above S ∪ Qand 1 ≤ i ≤ n), and T = T [{U } ].Thus H (V (Q)) and G 0 G,S∪Q G,S∪Q  v∈Q univ,Q H (V (Q; N)) are T -modules. G 1 G,S∪Q univ There is a unique maximal ideal m ⊂ T of residue field k such that for all G,S finite places w of K, split over F and not lying above S, the characteristic polynomial n−1 i i(i−1)/2 (i) n−i of Sym r(Frob ) equals (−1) q T X mod m .If (Q, Q,(α ,β ) ) is a w G v v v∈Q i=0 w w univ,Q Taylor–Wiles datum, then we write m for the maximal ideal of T generated by G,Q G,S∪Q univ m ∩ T and the elements G,S∪Q i(1−i)/2 (i) n−j j−1 U − q α β v v j=1 for v ∈ Q, i = 1,..., n. The unitary group G comes with a determinant map det : G → U ,where U = 1 1 ker(N : Res G → G ).If K/F K/F m m ∞ × θ : U (F)\U (A )/ det V (Q; N) → O 1 1 1 is a character and f ∈ H (V (Q; N)) then we define f ⊗ θ ∈ H (V (Q; N)) by the G 1 G 1 formula (f ⊗ θ)(g) = θ(det(g))f (g).If f ∈ H (V (Q; N)) and θ is trivial, then G 1 m G,Q f ⊗ θ ∈ H (V (Q; N)) . We will use this construction only in conjunction with the G 1 m G,Q following lemma. Lemma 2.4. — Suppose that n is even and that p = 2. Suppose that × ∞ × × χ : F \(A ) / det U (Q; N) → O is a quadratic character. Then there exists a unique character ∞ × θ : U (F)\U (A )/ det V (Q; N) → O χ 1 1 1 ∞ × c c such that for all z ∈ (A ) we have θ (z/z ) = χ(zz ). Proof. — There is a short exact sequence of F-groups 0 → G → Res G → U → 0, m K/F m 1 where the last map is z → z/z . By Hilbert 90 we have a short exact sequence × ∞ × × ∞ × ∞ 0 → F \(A ) → K \(A ) → U (F)\U (A ) → 0. 1 1 F K F SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 129 ∞ × This shows that there is a unique character θ : U (F)\U (A ) → O such that χ 1 1 c c ∞ × θ (z/z ) = χ(zz ) for all z ∈ (A ) . We need to check that θ is trivial on det V (Q; N). χ χ 1 This can be checked locally at each finite place of F. At places v ∈ Q it follows from the fact that χ is unramified at v.If v ∈ Q then we see, using that n is even and identify- × × ing U (F ) with F ,that det V (Q; N) is contained in the subgroup of O with square 1 v 1 v v F image in k( v) . Since χ is quadratic, θ annihilates this subgroup, and we’re done. Let m ≥ 0denote the p-adic valuation of the least common multiple of the expo- −1 ∞ nents of the Sylow p-subgroups of the finite groups G(F) ∩ tV t (t ∈ G(A )). Proposition 2.5. — Let N ≥ 1 and let (Q, Q,(α ,β ) )) be a Taylor–Wiles datum of v v v∈Q level N + m .Then: (1) The maximal ideals m , m are in the support of H (V ) and H (V (Q)),respec- G G,Q G 0 G 0 tively. univ N N (2) H (V (Q; N)) is a T ⊗ O[ /(p )]-module free as O[ /(p )]- G 1 m O Q Q G,Q G,S∪Q module, and there is an isomorphism H (V (Q; N)) ⊗ O = H (V ) G 1 m O[ /(p )] G 0 m G,Q Q G univ of T -modules. G,S∪Q univ (3) There exists a structure on H (V (Q; N)) of P -module such that if  : G → G 1 m Q K G,Q i P (i = 1,..., n) are the coefficients of the universal characteristic polynomial, defined as in [Che14, §1.10], then for any finite place v ∈ S ∪ Q of F which splits v = ww in univ i(i−1)/2 (i) K,  (Frob ) acts on H (V (Q; N)) as q T . Moreover, the O[ ]- w G 1 m Q i G,Q w w module structure on H (V (Q; N)) induced by the map O[ ]→ P agrees with G 1 m Q Q G,Q the one in the second part of the lemma. Proof. — The first part may be deduced, as in [NT20, §4.3], from [Lab11, Théorème 5.4] (and the existence of  ). The other parts are proved in a very sim- ilar way to the second and third parts of Proposition 2.3, as we now explain. We × n begin by constructing a more familiar set of objects. Let  = k(v) (p) .Let Q v∈Q V (Q; N) = V (Q; N) ⊂ V (Q) be the subgroup defined by V (Q; N) = V (Q) v 0 v 0 v 1 v 1 1 if v ∈ Qand −1 × N V (Q; N) = ι {(a ) ∈ Iw |∀i = 1,..., n, a → 1 ∈ k( v) (p)/(p )} v ij  v ii 1  v if v ∈ Q. Thus V (Q; N) ⊂ V (Q) is a normal subgroup with quotient V (Q)/ 0 0 V (Q; N)  /(p ). We write P ∈ C for the quotient of the ring R introduced O S∪Q 1 Q Q in [NT20, §2.19] corresponding to pseudodeformations whose restriction to G factors ab through G for each v ∈ Q. As in the case of P ,[ANT20, Proposition 2.5] again shows (i)  × that for each v ∈ Q there are unique characters A : G → (P ) (i = 1,..., n)such v Q (i) (1) (n) that A mod m = ur n−i i−1 and det(A ⊕ ··· ⊕ A ) is the restriction to G of the P K v α β v v  v Q v v universal pseudocharacter over G . We now claim that the following statements hold: K 130 JAMES NEWTON, JACK A. THORNE univ  N  N • H (V (Q; N)) is a T ⊗ O[ /(p )]-module free as O[ /(p )]- G m O 1 G,Q G,S∪Q Q Q module, and there is an isomorphism H (V (Q; N)) ⊗ N O G m O[ /(p )] 1 G,Q univ H (V ) of T -modules. G 0 m G G,S∪Q • There is a unique structure on H (V (Q; N)) of P -module such that for G m 1 G,Q Q c univ any place v ∈ S ∪ QofFwhich splits v = ww in K,  (Frob ) acts on i(i−1)/2 (i) H (V (Q; N)) as q T . G m G,Q 1 w w • The two induced O[ ]-module structures on H (V (Q; N)) (one by the G m G,Q Q 1 isomorphism V (Q)/V (Q; N) =  /(p ), the other by the map  → P 1 Q Q Q (i) associated to the tuple of characters A ◦ Art | )are thesame. v  v O To prove the first point, we need to explain why H (V (Q; N)) is an O[ /(p )]- G m 1 G,Q Q module with the claimed coinvariants. The action of  /(p ) is induced by the ac- tion of V (Q) via the isomorphism V (Q)/V (Q; N) =  /(p ) (which therefore 0 0 1 Q univ commutes with the action of T ). The freeness follows because V (Q)/V (Q; N) G,S∪Q 1 acts freely on the quotient G(F)\G(A )/V (Q; N), because V (Q; N) contains all F 1 1 the p-torsion elements of V (Q); compare the proof of [KW09b, Lemma 7.4] and [BHKT19, Lemma 8.18]. The freeness of this action implies that there is an isomor- univ phism H (V (Q; N)) ⊗ O = H (V (Q)) of T -modules. To com- G m O[ /(p )] G 0 m 1 G,Q G,Q G,S∪Q plete the proof of the first point, we need to explain why there is an isomorphism H (V (Q)) = H (V ) . This follows from [NT20, Proposition 3.1]. The second G 0 m G 0 m G,Q G part is proved in the same way as [NT20, Lemma 4.7]. The third part is proved using [NT20, Lemma 4.7] and [ANT20, Proposition 2.5] (in particular, the uniqueness of the decomposition of residually multiplicity-free pseudocharacters). We now need to explain why the above claims imply the properties in the statement of the proposition. There are canonical quotient morphisms P → P and →  . The proof is complete on noting that trace induces an isomorphism H (V (Q; N)) ⊗ N O[ /(p )] = H (V (Q; N)) and that the map G m O[ /(p )] Q G 1 m 1 G,Q G,Q P → P factors through an isomorphism P ⊗ O[ ] P . Q O[ ] Q Q Q Q We are now ready to prove Theorem 2.1. We will need to treat the cases p > 2and p = 2separately. Proof of Theorem 2.1,case p > 2.— Define H = H (V ) and H = H (U ) . G G 0 m D D 0 m G D The proof of the theorem will be based on the following proposition: Proposition 2.6. — We can find an integer q ≥ 0 with the following property: let W = OY ,..., Y , Z ,..., Z . Then we can find the following data: 1 q 1 4|S∪S |−1 (1) Complete Noetherian local W -algebras P , R equipped with isomorphisms P ⊗ ∞ ∞ ∞ ∞ W ∼ ∼ O P and R ⊗ O R in C . = = ∞ W O (2) A surjection R X ,..., X  → R in C , where g = q +|S ∪ S |− 1. loc 1 g ∞ O ∞ SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 131 (3) A P -module H and an R -module H such that both H , H are finite ∞ G,∞ ∞ D,∞ G,∞ D,∞ free W -modules, complete with isomorphisms H ⊗ O H (as P-module) and ∞ G,∞ W G H ⊗ O H (as R-module). D,∞ W D (4) A morphism P → R of O-algebras making the diagram ∞ ∞ P R ∞ ∞ P R commute. Before giving the proof of Proposition 2.6, we show how it implies the theorem. Let p ⊂ P denote the kernel of the morphism P → O associated to t.Itisenoughto show that p is in the support of H as P-module. Indeed, this would imply (using [Lab11, n−1 Corollaire 5.3] and the irreducibility of Sym r| ) the existence of a RACSDC au- n−1 tomorphic representation  of GL (A ) such that r Sym (r| ). By descent n n K  ,ι G n K (in the form of e.g. [BLGHT11, Lemma 1.5]), this would imply the sought-after auto- n−1 morphy of Sym r. Equivalently, we must show that if p is the pre-image of p un- der the morphism P → P, then p is in the support of H as P -module. (Since ∞ ∞ G,∞ ∞ Supp H = Supp H ∩ Spec P, intersection taken inside Spec P .) G G,∞ ∞ P P The P -module H is a Cohen–Macaulay module (i.e. it is finite and the di- ∞ G,∞ mension of its support is equal to its depth), since H is a finite free W -module. Ap- G,∞ ∞ plying [Sta13, Tag 0BUS], it follows that each irreducible component of Supp H G,∞ has dimension q + 4|S ∪ S |. Similarly, we see that each irreducible component of Supp H has dimension q + 4|S ∪ S |. Since R is a quotient of R X ,..., X , D,∞ ∞ ∞ loc 1 g a domain of Krull dimension q + 4|S ∪ S |, we see that R X ,..., X  → R is an ∞ loc 1 g ∞ isomorphism, that R is a domain, and that H is a faithful R -module. ∞ D,∞ ∞ Let p ⊂ P denote the kernel of the morphism P → O associated to t ,and let p denote the pre-image of p under the morphism P → P. Then p ∈ Supp H ,by ∞ G,∞ ∞ P hypothesis, and therefore dim(P  ) ≥ q + 4|S ∪ S |− 1. We claim that the Zariski ∞,(p ) ∞ tangent space to the local ring P  has dimension at most q + 4|S ∪ S |− 1. Indeed, ∞,(p ) ∞ it suffices to note that the quotient P  /(Y ,..., Y , Z ,..., Z ) = P  equals ∞,(p ) 1 q 1 4|S∪S |−1 (p ) its residue field E, by the vanishing of the adjoint Bloch–Kato Selmer group of r  (i.e. ,ι by [NT20, Theorem A, Proposition 2.21, Example 2.34]). We deduce that P is ∞,(p ) a regular local ring of dimension q + 4|S ∪ S |− 1, so there is a unique irreducible component Z of Spec P containing the point p , which has dimension q + 4|S ∪ S | ∞ ∞ and is contained in Supp H . G,∞ Since Spec R is irreducible and the image of the morphism Spec R → Spec P ∞ ∞ ∞ contains p , we find that the morphism Spec R → Spec P factors through Z. In par- ∞ ∞ ticular, p lies in Z, hence in Supp H . This completes the proof of the theorem. ∞ G,∞ ∞ 132 JAMES NEWTON, JACK A. THORNE The proof of Proposition 2.6 is based on a patching argument. We first prove a lemma which shows that there are enough Taylor–Wiles data. The argument is very similar (and essentially identical in the case n = 2) to the proof of [DDT97,Theorem 2.49]. We spell out the details here just to show that the condition that the numbers n−i i−1 α β (i = 1,..., n) are distinct does not cause any difficulty. v v 1 0 Lemma 2.7. — Let q = dim H (F /F, ad r(1)),and let g = q +|S ∪ S |− 1. Then k S ∞ for any N ≥ 1, we can find a Taylor–Wiles datum (Q, Q,(α ,β ) ) of level N ≥ 1 such that v v v∈Q there is a surjection R X ,..., X  → R of R -algebras. loc 1 g loc Proof. — Let (Q, Q,(α ,β ) ) be a Taylor–Wiles datum. A standard computa- v v v∈Q tion (compare e.g. [Kis09b, Proposition 3.2.5], [Tho16, Proposition 5.10]), shows that there is a surjection R X ,..., X  → R where g = λ +|Q|+|S ∪ S |− 1and λ loc 1 g Q ∞ Q is the dimension of the group ⎛ ⎞ 1 0 1 0 ⎝ ⎠ ker H (F /F, ad r (1)) → H (F , ad r(1)) . S v v∈Q We therefore need to show that for any N ≥ 1, we can find a Taylor–Wiles datum of level Nsuch that |Q|= q and λ = 0. By induction, and the Chebotarev density theorem, it is enough to show the following claim: 1 0 • Let [φ]∈ H (F /F, ad r(1)) be non-zero. Then for any N ≥ 1, there exists σ ∈ G such that φ(σ ) ∈ (σ − 1) ad r(1) and the eigenvalues α, β of r(σ ) satisfy F(ζ ) (α/β) = 1for i = 1,..., n − 1. Let L/F denote the extension cut out by Proj r. Recall our assumption that  = Gal(L/F) is conjugate in PGL (F ) either to PSL (F a ) or PGL (F a ) for some p > max(5, 2n − 1). 2 p 2 p 2 p In either case [, ] PSL (F a ) is a non-abelian simple group that acts absolutely irre- 2 p 0 0 ducibly on ad .Moreover, [CPS75, Table (4.5)] shows that H (PSL (F a ), ad ) = 0. The 2 p inflation-restriction exact sequence implies that H (L(ζ N )/F, ad r(1)) = 0. Another ap- plication of inflation-restriction shows that Res [φ]= 0; we may identify this restric- L(ζ )/F 0 0 tion with a non-zero G -equivariant homomorphism f : G → ad r(1). Since ad r F L(ζ ) is absolutely irreducible as a k[PSL (F )]-module, f (G ) spans ad r(1) as k-vector 2 p L(ζ ) space. To prove the claim above, choose any σ ∈ G such that the eigenvalues α, β ∈ k F(ζ ) of r(σ ) satisfy (α/β) = 1for i = 1,..., n − 1. (This is possible since if t generates F a,then 2 a t has multiplicative order ≥ (p − 1)/2 > n − 1and Proj r(G ) contains an element F(ζ ) −1 2 which is conjugate to diag(t, t ) = diag(t , 1).) If φ(σ ) ∈ (σ − 1) ad r(1) then we’re done. Suppose instead that φ(σ ) ∈ (σ − 0 0 1) ad r(1). Since f (G ) spans ad r(1),wecan find τ ∈ G such that f (τ ) ∈ (σ − L(ζ ) L(ζ ) N N p p SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 133 1) ad r(1).Then τσ ∈ G ,Proj r(τ σ ) = Proj r(σ ), and the cocycle relation shows F(ζ ) that φ(τ σ ) = φ(τ ) + φ(σ ) ∈ (τ σ − 1) ad r(1). In either case we have established the claim; this completes the proof. 1 0 Now we give the proof of Proposition 2.6.Let q = dim H (F /F, ad r(1)) and k S g = q +|S ∪ S |− 1. For each N ≥ 1, fix a Taylor–Wiles datum (Q , Q ,(α ,β ) ) ∞ N N v v v∈Q of level N + max(m , m ) such that there exists a surjection R X ,..., X  → R D G loc 1 g of R -algebras. Fix v ∈ Sand define T = O{Z } /(Z ).Weview T loc 0 v,i,j v∈S∪S ,1≤i,j≤2 v ,1,1 ∞ 0 as an augmented O-algebra via the augmentation which sends each Z to 0. Choose v,i,j for each N ≥ 1 a surjection OY ,..., Y  → O[ /(p )]. Then we get surjections 1 q Q N N W → T ⊗ O[ /(p )]→ O[ /(p )]. ∞ O Q Q N N Define R = R ,H = T ⊗ H (U (Q; N)) ,P = T ⊗ P ,H = N D,N O D 1 m N O Q G,N Q D,Q N univ T ⊗ H(V (Q; N)) . We fix a representative r : G → GL (R) for the universal O 1 m F 2 G,Q univ deformation over R, and representatives r : G → GL (R ) for the universal defor- F 2 Q univ mations over R lifting r for each N ≥ 1. These choices determine isomorphisms ∼ ∼ R = T ⊗ Rand R = T ⊗ R , which classify the universal S ∪ S -framed liftings O N O Q ∞ univ univ (r , {1 + (Z )} ) (resp. (r , {1 + (Z )} )). Thus each ring R , P has v,i,j v∈S∪S v,i,j v∈S∪S N N ∞ Q ∞ ∼ ∼ aW -algebra structure, and there are isomorphisms R ⊗ O R, P ⊗ O P. = = ∞ N W N W ∞ ∞ Moreover, Proposition 2.3 and Proposition 2.5 show that the modules H , H are G,N D,N finite free as T ⊗ O[ /(p )]-modules. Finally, completed tensor product with T pro- O Q motes the morphism P → R to a morphism P → R . Q Q N N N N To patch these objects together we now carry out a diagonalisation argument along very similar lines to the proof of [Kis09b, Proposition 3.3.1]. By [NT20, Lemma 2.16], we can find an integer g ≥ 0and for eachN ≥ 1 a surjection OX ,..., X  → P of 0 1 g N O-algebras. Let a ⊂ W denote the kernel of the augmentation W → O,and forany ∞ ∞ N ≥ 1let a denote the kernel of the map W → O[ /(p )]. Choose a sequence N ∞ Q (b ) of open ideals of W satisfying the following conditions: N N≥1 ∞ • For each N ≥ 1, b ⊂ b . N+1 N • For each N ≥ 1, a ⊂ b . N N •∩ b = 0. N≥1 N Let s = max(dim H [1/p], dim H [1/p]).Let r = length (W /b ) . Then the se- E D E G N ∞ N quence (r ) is non-decreasing and N N≥1 length H /(b ) = length H /(b ) ≤ r , D,N N D,N N N R W N ∞ so H /(b ) has a natural structure of R /m -module. Similarly, H /(b ) has a D,N N N G,N N natural structure of P /m -module. Thus for every pair of integers N ≥ M ≥ 1we have, N 134 JAMES NEWTON, JACK A. THORNE by passage to quotient from the data constructed above, a diagram (of rings and modules) H /(b ) H /(b ) D,N M D M r r M M R X ,..., X ⊗ W R /m R/m loc 1 g O ∞ N R R r r M M OX ,..., X ⊗ W P /m P/m 1 g O ∞ N 0 P P H /(b ) H /(b ) G,N M G M where the horizontal arrows are all surjective. Keeping M fixed, the cardinalities of the rings and modules appearing in this diagram (excepting those in the first column) are uniformly bounded as N varies. By the pigeonhole principle, we can therefore find an increasing sequence (N ) of integers N ≥ Msuch that for each M ≥ 1 there is a M M≥1 M commutative diagram of O-algebras r r M M R X ,..., X ⊗ W R /m R/m loc 1 g O ∞ N M+1 R R M+1 r r M M R /m R/m M R R r r M M OX ,..., X ⊗ W P /m P/m 1 g O ∞ N 0 M+1 P P M+1 r r M M P /m P/m , M P P where the morphisms from the back square of the cube to the front are isomorphisms, and there are commutative diagrams of modules H /(b ) H /(b ) D,N M D M M+1 H /(b ) H /(b ) D,N M D M H /(b ) H /(b ) G,N M G M M+1 H /(b ) H /(b ), G,N M G M M SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 135 compatible with the module structures arising from the previous commutative cube, and where the arrows from back to front are again isomorphisms. We define r r M M R = lim R /m and P = lim P /m . ∞ N ∞ N M R M P N N M M ← − ← − M M and similarly H = lim H /(b ) and H = lim H /(b ). D,∞ D,N M G,∞ G,N M M M ← − ← − M M By passage to inverse limit, there is a diagram (of rings and modules) H H D,∞ D R X ,..., X ⊗ W R R loc 1 g O ∞ ∞ OX ,..., X ⊗ W P P 1 g O ∞ ∞ H H . G,∞ G To complete the proof of the proposition, it remains to show that these objects have the following properties: • The morphism R X ,..., X  → R is surjective. loc 1 g ∞ • The modules H ,H are finite free over W and the morphism H → D,∞ G,∞ ∞ D,∞ H (resp. H → H ) factors over an isomorphism H /(a) = H (resp. D G,∞ G D,∞ D H /(a) H ). G,∞ G • The morphism R → R (resp. P → P) factors over an isomorphism R /(a) ∞ ∞ ∞ R (resp. P /(a) = P). The first point holds because R X ,..., X  is a complete local ring and each map loc 1 g R X ,..., X  → R /m is surjective. The second follows from e.g. Nakayama’s loc 1 g N M R lemma and the freeness of the modules H /b ,H /b . For the third point, we D,N M G,N M M M r r M M recall that R /(a) = R, and consequently R /(m , a) = R/m . We therefore need N N M M R R to show that the natural map r r M M lim R /m /(a) → lim R /(m , a) N N M R M R N N M M ← − ← − M M is an isomorphism, or equivalently that the ideal aR of R is closed in the m -adic ∞ ∞ R topology. This is true since R is a Noetherian ring. The same proof applies to the ring P . ∞ 136 JAMES NEWTON, JACK A. THORNE Now we treat the case p = 2. Proof of Theorem 2.1,case p = 2.— Define H = H(V ) and H = H (U ) . G 0 m D D 0 m G D The proof of the theorem in this case will be based on the following proposition, incor- porating ideas from the 2-adic patching argument given in [KW09b, Kis09a]. If A ∈ C , we follow [KW09b, §2.1] in writing Sp for the functor Sp : C → Sets represented A A by A. We write G : C → Groups for the functor which sends A ∈ C to the group m O O × × ker(A → (A/m ) ). Proposition 2.8. — We can find an integer q ≥|S ∪ S |− 2 with the following property: let W = OY ,..., Y , Z ,..., Z  and let γ = 2 −|S ∪ S |+ q. Then we can find the ∞ 1 q 1 4|S∪S |−1 ∞ following data: (1) Complete Noetherian local W -algebras P , R equipped with isomorphisms P ⊗ ∞ ∞ ∞ ∞ W ∼ ∼ O P and R ⊗ O R in C . = = ∞ W O (2) A complete Noetherian local O-algebra R and surjections R X ,..., X  → R loc 1 g ∞ ∞ and R → R in C , where g = 2q + 1. ∞ O (3) A P -module H and an R -module H such that both H , H are finite ∞ G,∞ ∞ D,∞ G,∞ D,∞ free W -modules, together with isomorphisms H ⊗ O = H (as P-module) and ∞ G,∞ W G H ⊗ O H (as R-module). D,∞ W D (4) A morphism P → R of O-algebras making the diagram ∞ ∞ P R ∞ ∞ P R commute. γ γ γ (5) A free action of G on Sp and a G -equivariant morphism δ : Sp → G ,where m R m R m ∞ ∞ G acts on itself by the square of the identity. These objects have the following additional properties: −1 γ (6) We have δ (1) = Sp . The induced action of G [2](O) on R lifts to H . ∞ D,∞ R m (7) There exists an action of G [2] on Sp such that the morphism Sp → Sp is m P R P ∞ ∞ ∞ γ γ G [2]-equivariant, and the induced action of G [2](O) on P lifts to H . ∞ G,∞ m m Once again, we show how Proposition 2.8 implies the theorem in this case before giving the proof of the proposition. Let p ⊂ P denote the kernel of the morphism P → O associated to t. It is again enough to show that p is in the support of H as P-module, or equivalently that the pullback p ⊂ P of p is in the support of H as P -module. ∞ ∞ G,∞ ∞ The P -module H is a Cohen-Macaulay module, and each irreducible compo- ∞ G,∞ nent of Supp H has dimension q + 4|S ∪ S |. Similarly, each irreducible component G,∞ ∞ of Supp H has dimension q + 4|S ∪ S |. D,∞ ∞ ∞ SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 137 inv  γ Let R ⊂ R denote the subring of invariants for the action of G (cf. [KW09b, ∞ ∞ m §2.4]). Then the morphism Sp → Sp is smooth of relative dimension γ (one can inv R R apply [KW09b, Proposition 2.5], which is used for a very similar purpose in the proof of [KW09b, Proposition 9.3]). On the other hand, Sp → Sp inv is a torsor for the R R ∞ ∞ γ inv group G [2], showing that R has Krull dimension at least q + 4|S ∪ S |.Wecon- m ∞ clude that Spec R has dimension at least q + 4|S ∪ S |+ γ = dim R X ,..., X . ∞ loc 1 g Since R X ,..., X  is a domain, it follows that the map R X ,..., X  → R is loc 1 g loc 1 g inv an isomorphism, that Spec R is irreducible, that Spec R is irreducible of dimension ∞ ∞ q + 4|S ∪ S |,and that G [2](O) acts transitively on the set of irreducible components of Spec R , all of which have dimension q + 4|S ∪ S |. Property (6) of Proposition 2.8 ∞ ∞ implies that Supp H is invariant under the action of G [2](O), so we conclude D,∞ R m 1 1 that H has full support in Spec(R ) (in fact, considering H [ ] over R [ ] we can D,∞ ∞ D,∞ ∞ 2 2 conclude that H is a faithful R -module). D,∞ ∞ Let p ⊂ P denote the kernel of the morphism P → O associated to t ,and let p denote its pullback to P .Then p ∈ Supp H , by hypothesis. Similarly, let r, r ⊂ ∞ G,∞ ∞ P R denote the kernels of the morphisms R → O associated to r, r respectively, and let r , r ⊂ R denote their pullbacks under the morphism R → R. Then p (resp. p )is ∞ ∞ ∞ ∞ ∞ ∞ the image of r (resp r ) under the map Spec R → Spec P .Let r ∈ Spec R denote ∞ ∞ ∞ ∞ ∞ ∞ a point which is in the G [2](O)-orbit of r and on the same irreducible component m ∞ of Spec R as r ,and let p denote the image of r in Spec P .Then p , p lie on ∞ ∞ ∞ ∞ ∞ ∞ ∞ a common irreducible component of Spec P . Since the action of G [2](O) extends to P and H and the morphism P → R is equivariant for this action, Supp H ∞ G,∞ ∞ ∞ G,∞ is invariant under G [2](O) and contains p . m ∞ We now observe that the Zariski tangent space of the local ring P  has dimen- ∞,(p ) sion at most q + 4|S ∪ S |− 1. Indeed, translating by the element of G [2](O) which takes p to p , it suffices to show that the Zariski tangent space of the local ring P ∞,(p ) ∞ ∞ ∞ has dimension at most q + 4|S ∪ S |− 1, or even that P /(Y ,..., Y , Z ,..., ∞ ∞,(p ) 1 q 1 Z ) = P is a field. This again follows from [NT20, Theorem A, Proposi- 4|S∪S |−1 (p ) tion 2.21, Example 2.34]. It follows that P is a regular local ring of dimension ∞,(p ) q + 4|S ∪ S |− 1 and that there is a unique irreducible component of P containing ∞ ∞ the point p , which has dimension q + 4|S ∪ S | and is contained in Supp H .We ∞ G,∞ ∞ P deduce that p ∈ Supp H , as required. ∞ ∞ The proof of Proposition 2.8 is again based on a patching argument. Here is the analogue of Lemma 2.7 in our case. Lemma 2.9. — Let q = dim H (F /F, ad r) − 2,and let g = 2q + 1.Then q ≥|S ∪ k S S |− 2 and for any N ≥ 1, we can find a Taylor–Wiles datum (Q, Q,(α ,β ) )) of level N ≥ 1 ∞ v v v∈Q with the following properties: (1) |Q|= q. (2) There is a surjection R X ,..., X  → R of R -algebras. loc 1 g loc Q 138 JAMES NEWTON, JACK A. THORNE (3) Let  denote the Galois group of the maximal abelian pro-2 extension of F which is unramified outside Q and (S ∪ S )-split. Then there is an isomorphism  /(2 ) ∞ Q N γ (Z/2 Z) ,where γ = 2 −|S ∪ S |+ q. Proof. — This is contained in [KW09b, Lemma 5.10], except that result specifies only that if v ∈ Q then the eigenvalues α ,β ∈ k of r(Frob ) are distinct. Here we require v v v n−i i−1 that the numbers α β (i = 1,..., n) are distinct. However, reading the proof of loc. v v cit. we see that we can indeed choose v so that r(Frob ) satisfies this stronger requirement (using of course our assumption that the projective image of r contains PSL (F a ) for 2 2 some a ≥ 1such that2 > max(5, 2n − 1), as we did in the proof of Lemma 2.7). Now we give the proof of Proposition 2.8.Let q = dim H (F /F, ad r) − 2and k S g = 2q + 1. For each N ≥ 1, fix a Taylor–Wiles datum (Q , Q ,(α ,β ) ) of level N N v v v∈Q N + max(m , m ) such that there exists a surjection R X ,..., X  → R of R - D G loc 1 g loc algebras. Define T = OZ ,..., Z . Choose for each N ≥ 1 a surjection 1 4(|S|+|S |)−1 OY ,..., Y  → O[ ]. Then we get a surjection W → T ⊗ O[ ]. 1 q Q ∞ O Q N N Define R = R ,R = R ,H = T ⊗ H (U (Q; N)) ,P = T ⊗ P , N D,N O D 1 m N O Q Q N Q D,Q N N N univ H = T ⊗ H (V (Q; N)) . We fix a representative r : G → GL (R) for the G,N O G 1 m F 2 G,Q univ universal deformation over R, and representatives r : G → GL (R ) for the uni- F 2 Q univ versal deformations over R lifting r for each N ≥ 1. These choices determine com- ∼ ∼ patible isomorphisms R = T ⊗ Rand R = T ⊗ R . Thus each ring R , P has O N O Q N N ∼ ∼ aW -algebra structure, and there are isomorphisms R ⊗ O = R, P ⊗ O = P. ∞ N W N W ∞ ∞ Moreover, Proposition 2.3 and Proposition 2.5 show that the modules H , H are G,N D,N finitefreeas T ⊗ O[ /(p )]-modules. Completed tensor product with T promotes O Q the morphism P → R to a morphism P → R . Q Q N N N N ˇ ˇ Let  = Sp denote the group functor A → Hom( , A ).Then Q Q Q N N N O[ ] acts on Sp  by twisting: if r is a lifting corresponding to a morphism R → Aand R N χ :  → A is a character, then r ⊗ χ is a lifting which determines another morphism R → A. Thereisamorphism δ : Sp  →  given by taking the determinant and N Q N R N −1 multiplying by ,and δ (1) = Sp . The induced action of  [2](O) on R lifts to Q N N R N an action on H (U (Q; N)) , given by twisting by quadratic characters as above D 1 m D,Q (see also [KW09b, §7.5]). Similarly we can define compatible actions of  [2] on P Q N and of  [2](O) on H (V (Q; N)) , which are trivial if n − 1 is even and which Q G 1 m N G,Q correspond to twisting by the quadratic characters χ | (resp. θ for χ ∈  [2](O)) G χ Q K N when n − 1 is odd. We extend these to actions on H and H by completed tensor D,N G,N product with T . The morphism P → R is equivariant for these actions. A very similar N N argument to the proof of [KW09b, Proposition 9.3] (with modifications as in the proof of Proposition 2.6 above) now shows how to use the above data to construct the objects required by the statement of Proposition 2.8.  SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 139 3. Killing ramification Our goal in this section is to prove the following theorem (Theorem A of the intro- duction): Theorem 3.1. — Let n ≥ 1.Let π be a regular algebraic, cuspidal automorphic representation n−1 of GL (A ) which is non-CM. Then Sym π exists. 2 Q We fix n, which we can assume to be ≥ 3. The proof of Theorem 3.1 will be roughly by induction on the cardinality of sc(π ), the set of primes p such that π is supercuspidal; the case where sc(π ) is empty is exactly the main result of [NT21]. We begin with some preparatory definitions and results. Definition 3.2. — Let π be a regular algebraic, cuspidal automorphic representation of ss GL (A ). We define the semisimple conductor M of π to be M = N((rec π ) ) (where 2 Q π π Q l l l N denotes conductor). Lemma 3.3. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ), 2 Q let p be an odd prime, and let ι : Q → C be an isomorphism. If r is reducible or dihedral ,thenthe π,ι prime-to-p part of its conductor divides M . Proof. — If r is reducible or dihedral then its image has order prime to p,and for π,ι any prime l = p, r | is semisimple. This shows that the conductor of r | divides π,ι G π,ι G Q Q l l ss the conductor of (rec π ) . Q l Lemma 3.4. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ). 2 Q Let p be a prime, let ι : Q → C be an isomorphism, and suppose that Proj r (G ) contains a con- π,ι Q jugate of PSL (F a ) for some p > 5. Then we can find another regular algebraic, cuspidal automorphic 2 p representation π of GL (A ) with the following properties: 2 Q (1) There is an isomorphism r = r . π,ι π ,ι (2) π has weight 2 and is not ι-ordinary. (3) There is an isomorphism rec π = ω ⊕ ω ,where ω ,ω : W → C are characters Q 1 2 1 2 Q p p of conductor dividing p . (4) For each prime l = p, π is a twist of the Steinberg representation (resp. supercuspidal) if and only if π is. Proof. — Let be the set of primes l = p such that π is a twist of the Steinberg representation, and let T be the set of primes l = p such that π is supercuspidal. Let l ≥ 5 be a prime such that l ≡ 1mod p, π is unramified, and r (Frob ) has distinct 0 0 l π,ι l 0 0 By dihedral, we mean that the representation is induced from an index two subgroup of G . Q 140 JAMES NEWTON, JACK A. THORNE eigenvalues (such a prime exists because of our assumption on the image of Proj r ). Fix π,ι a coefficient field E containing a p th root of unity. If l is a prime such that π is super- cuspidal, then we can find, after possibly enlarging E, an O[GL (Z )]-module M , finite 2 l l free as O-module, such that M ⊗ C is a type for the Bernstein component containing l O,ι π , in the sense of [BM02, Definition A.1.4.1]. We define M =⊗ M ,M = M ⊗ k, l l∈T l k O ∨ ∨ ∨ and M = M ⊗ E. We write M ,M and M for the O-, k-, and E-linear duals of these E O k E GL (Z )-modules, equipped with the dual group action. 2 l l∈T Let D denote the quaternion algebra over Q such that if l is a prime, then D is ramified at l if and only if l ∈ .(Thus D isramified at ∞ if and only if | | is odd.) Fix a maximal order O ⊂ D and for each prime l ∈ an identification O ⊗ Z = M (Z ). D D Z l 2 l If U = U ⊂ (O ⊗ Z) is an open compact subgroup, we write Y(U) for the locally l D Z symmetric space of level U (namely the object denoted X in [NT16, §3.1]). We Res G D/Q m regard M as an O[U]-module by projection to GL (Z ).If U = GL (Z ) and r ≥ 1 2 l p 2 p l∈T then we write U (p ) ⊂ U for the open compact subgroup with the same component at primes l = p and component ab ∈ GL (Z ) | c ≡ 0mod p 2 p cd r × × at the prime p. We identify any pair of characters χ ,χ : (Z/p Z) → O with the 1 2 r × character χ ⊗ χ : U (p ) → O given by the formula 1 2 0 ab r r χ ⊗ χ = χ (a mod p )χ (d mod p ). 1 2 1 2 cd We write M(χ ⊗ χ ) = M ⊗ O(χ ⊗ χ ),regardedas O[U (p )]-module. We use similar 1 2 O 1 2 0 notation for k- and E-valued characters. Let δ = 0ifDis ramified at ∞ and δ = 1 otherwise. Let U = U be the open compact subgroup defined as follows: • U = Iw . l l ,1 0 0 • If l ∈ ∪{l },then U = GL (Z ). 0 l 2 l × × • If l ∈ and π = St (χ ),then U = ker(χ ◦ det : (O ⊗ Z ) → C ). l 2 l l l D Z l Then U is neat, in the sense of [NT16, §3.1] (because of the choice of U ), and we can × × find characters χ , χ : F → k such that 1 2 δ ∨ H (Y(U (p)), M (χ ⊗ χ ) ) = 0, 0 k m 1 2 π univ −1 ∞ where m ⊂ T is the maximal ideal associated to ι π (notation for the Hecke D, ∪T∪{l ,p} algebra as in §2), cf. [BDJ10, Corollary 2.12]. 3 × × 3 Let χ ,χ : (Z/p Z) → O be lifts of χ , χ such that χ /χ has conductor p . 1 2 1 2 1 2 We will show that δ 3 ∨ 0<s<1 (3.4.1)H (Y(U (p )), M (χ ⊗ χ ) ) = 0, 0 E 1 2 π SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 141 3 3 where the superscript denotes the subspace where the Hecke operator [U (p )α U (p )] 0 p,1 0 acts with eigenvalues that have p-adic valuation 0 < s < 1. Assuming (3.4.1) holds, we can complete the proof of the lemma. Indeed, the Jacquet–Langlands correspondence then implies the existence of a regular algebraic, cuspidal automorphic representation π of GL (A ) satisfying the following conditions: 2 Q • Thereisanisomorphism r = r . π,ι π ,ι • π has weight 2. −1  3 3 • ι π | contains a copy of χ ⊗ χ on which [U (p )α U (p )] acts with 1 2 0 p,1 0 U (p ) p 0 eigenvalue of p-adic valuation 0 < s < 1. • If l ∈ then π is a twist of the Steinberg representation. • If l ∈ Tthen π | contains M ⊗ C, hence (by definition of a type) π is GL (Z ) l O,ι l 2 l l supercuspidal. • If l ∈ ∪ T ∪{p},then π is a principal series representation. By [BM02, A.2.4], there is an isomorphism rec π = ω ⊕ ω ,where ω ,ω : W → Q 1 2 1 2 Q p p p × × C are characters such that ω ◦ Art | = ιχ and ω ◦ Art | = ιχ .More- 1 Q 1 2 Q 2 p Z p Z p p −1 over, the space Hom 3 (χ ⊗ χ ,ι π ) is 1-dimensional. Since the Hecke operator U (p ) 1 2 0 p 3 3 [U (p )α U (p )] acts with eigenvalue of p-adic valuation 0 < s < 1, π is not ι-ordinary 0 p,1 0 and therefore satisfies our requirements (cf. [Ger19, Lemma 5.2]). We now show that (3.4.1) holds. We have δ 3 ∨ 0<s<1 δ 3 ∨ s=0 h (Y(U (p )), M (χ ⊗ χ ) ) + h (Y(U (p )), M (χ ⊗ χ ) ) 0 E 1 2 0 E 1 2 m m π π δ 3 ∨ s=1 + h (Y(U (p )), M (χ ⊗ χ ) ) 0 E 1 2 δ 3 ∨ = h (Y(U (p )), M (χ ⊗ χ ) ) , 0 E 1 2 m (where lowercase h denotes dimension of cohomology over k or E). For each π contribut- δ 3 ∨ −1 ing to H (Y(U (p )), M (χ ⊗ χ ) ), ι π | 3 also contains a copy of χ ⊗ χ with 0 E 1 2 U (p ) 2 1 p 0 3 3 multiplicity one. The product of the eigenvalues of [U (p )α U (p )] on the χ ⊗ χ 0 p,1 0 1 2 −1 and χ ⊗ χ isotypic spaces of ι π | 3 has p-adic valuation 1. We deduce that 2 1 U (p ) p 0 δ 3 ∨ s=1 δ 3 ∨ s=0 h (Y(U (p )), M (χ ⊗ χ ) ) = h (Y(U (p )), M (χ ⊗ χ ) ) . 0 E 1 2 0 E 2 1 m m π π Moreover, δ 3 ∨ s=0 δ ∨ ord h (Y(U (p )), M (χ ⊗ χ ) ) = h (Y(U (p)), M (χ ⊗ χ ) ) 0 E 1 2 0 k m 1 2 m π π and δ 3 ∨ s=0 δ ∨ ord h (Y(U (p )), M (χ ⊗ χ ) ) = h (Y(U (p)), M (χ ⊗ χ ) ) , 0 E 2 1 0 k m 2 1 m π π 142 JAMES NEWTON, JACK A. THORNE by Hida theory. It is therefore enough to show that δ 3 ∨ δ ∨ ord h (Y(U (p )), M (χ ⊗ χ ) ) > h (Y(U (p)), M (χ ⊗ χ ) ) 0 E 1 2 m 0 k π 1 2 m δ ∨ ord + h (Y(U (p)), M (χ ⊗ χ ) ) , 0 k 2 1 or even that δ 3 ∨ δ ∨ h (Y(U (p )), M (χ ⊗ χ ) ) > h (Y(U (p)), M (χ ⊗ χ ) ) 0 k m 0 k m 1 2 π 1 2 π δ ∨ + h (Y(U (p)), M (χ ⊗ χ ) ) . 0 k m 2 1 π δ ∨ Using the exactness of H (Y(U (p)), (?) ) as a (contravariant) functor of smooth 0 m U (p) k[U (p)]-modules, it is therefore enough to show that Ind χ ⊗ χ contains χ ⊗ χ 0 3 1 2 1 2 U (p ) and χ ⊗ χ as Jordan–Hölder factors with multiplicity at least 2 (or when χ = χ ,that 2 1 1 2 it contains χ ⊗ χ with multiplicity at least 3). 1 2 This is true. Indeed, the semisimplification of a smooth k[Iw ]-module (say finite- dimensional as k-vector space) is determined by its restriction to the diagonal torus in × × Iw , and we can then use Mackey’s formula to show that if ψ : F → k is a character −1 U (p) then the semisimplification of Ind χ ⊗ χ contains χ ψ ⊗ χ ψ with multiplicity 1 2 1 2 U (p ) p + 2if ψ = 1 and multiplicity p + 1if ψ = 1. Lemma 3.5. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ). 2 Q Let p ≥ 3 be a prime, let ι : Q → C be an isomorphism, and suppose that Proj r (G ) contains a π,ι Q conjugate of PSL (F a ) for some p > 5. Then we can find another regular algebraic, cuspidal automor- 2 p phic representation π of GL (A ) with the following properties: 2 Q (1) There is an isomorphism r = r . π,ι π ,ι (2) π hasweight2andisnot ι-ordinary. (3) There is an isomorphism rec π ω ⊕ ω ,where ω ,ω : W → C are characters Q 1 2 1 2 Q p p p 3  6 of conductor dividing p . In particular, N(π )|p . (4) For each prime l = p, r | ∼ r  | . In particular, N(π ) = N(π ). π,ι G π ,ι G l Q Q l l l Proof. — Lemma 3.4 implies the existence of a π satisfying requirements (1)–(3). We can appeal to [Gee11, Corollary 3.1.7] to replace it with a π also satisfying r | ∼ π,ι G r | for each prime l = p. By purity, r | and r | ‘strongly connect’ to each π ,ι G π,ι G π ,ι G Q Q Q l l l other in the terminology of [BLGGT14, §1.3]. Remark (6) of [BLGGT14, p. 524] implies that N(π ) = N(π ). Definition 3.6. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ) of weight 2 such that 2 ∈ sc(π ) and let q, t, r be prime numbers. We say that π is seasoned 2 Q with respect to (q, t, r) if the following properties hold: (1) t divides q + 1,t ∈ sc(π ),and t > max(10, 8n(n − 1)) and (q + 1)/t > 2. SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 143 (2) There is an isomorphism rec π Ind χ,where χ : W → C is a character Q q Q q W 2 such that χ | has order t. (In particular, q ∈ sc(π ) and N(π ) = q .) I q (3) r is a primitive root modulo q. If M denotes the least common multiple of the prime-to-q part of M and p ,then r ≡ 1 mod M. p∈sc(π )−{q} (4) π is an unramified twist of the Steinberg representation. (5) For each prime p ∈ sc(π ) and for each irreducible dihedral representation ρ : Gal(Q/Q) → GL (F ) of prime-to-p conductor dividing Mq , there exists a prime number s such that π 2 p s is an unramified twist of the Steinberg representation, ρ(Frob ) is scalar, and s ≡ 1 mod p. Proposition 3.7. — If π is seasoned with respect to (q, t, r) then for each prime p ∈ sc(π ) there exists an isomorphism ι : Q → C such that r (G ) contains a conjugate of SL (F a ) for some π,ι Q 2 p p > 2n − 1. Proof. — We split into cases depending on whether or not p = q. First suppose that p = q, and fix an isomorphism ι : Q → C.Wefirstclaim that r is irreducible. π,ι Otherwise, there’s an isomorphism r = χ ⊕ χ for characters χ : G → F of prime- π,ι Q 1 2 i q to-q conductor dividing M . Let ω : I → Q be the Teichmüller lift of the fundamental character of niveau 2. 2 Q q q a(q−1)b −1 Let b = (q + 1)/t.Then ι χ | = ω for some integer a ∈{1,..., t − 1} (this tame Q 2 character has niveau 2 since it extends to W and its order t does not divide q − 1). Write a(q − 1)b = i + (q + 1)j for some i ∈{1,..., q}.Then([Gee11, Theorem 4.6.1]) we i−1 1−i 1−i have (χ /χ )| =  or  . After relabelling, we can assume that (χ /χ )| =  . I I 1 2 Q 1 2 Q q q Since r ≡ 1 mod M (in particular, the characters χ are unramified at r), we have i−1 (χ /χ )(Frob ) = r . Since π is an unramified twist of the Steinberg representation, r r 1 2 −1 we have (χ /χ )(Frob ) = r or r . Since r is a primitive root modulo q, this implies that 1 2 one of i, i − 2 is divisible by q − 1. Since b divides i, i is among the numbers b, 2b,..., q + 1 − b. Since b > 2, we see that neither i nor i − 2 can be divisible by q − 1. This contradiction implies that r is π,ι irreducible. We next claim that r is not dihedral. If r is dihedral, then Lemma 3.3 shows π,ι π,ι that the prime-to-q part of the conductor of r divides M , so there exists a prime num- π,ι π ber s such that π is an unramified twist of the Steinberg representation, r (Frob ) is s π,ι s scalar, and s ≡ 1mod p. This is a contradiction. To finish the proof in the case p = q, we need to make a particular choice of ι. Such a choice fixes the value of a ∈{1,..., t − 1}; conversely, any a ∈{1,..., t − 1} can be obtained by making a suitable choice of ι. We choose a so that i = b. Invoking [Gee11, Theorem 4.6.1] once more, we see that the projective image of r contains an element π,ι 144 JAMES NEWTON, JACK A. THORNE of order in the set {(q + 1)/ gcd(q + 1, i + 1), (q + 1)/ gcd(q + 1, i − 1), (q − 1)/ gcd(q − 1, i − 1)}, therefore of order at least t/2. Since t/2 > 5, the classification of finite subgroups of PGL (F ) shows that the projective image of r contains PSL (F a ) (and is contained in 2 q π,ι 2 q a 2a PGL (F a ))for some a ≥ 1. If q ≤ 2n − 1then q − 1 ≤ 4n(n − 1), so every element of 2 q PGL (F ) of order prime to q has order at most 4n(n − 1). Since t/2 > 4n(n − 1), we see 2 q that we must have q > 2n − 1. Now suppose that p = q, and fix an isomorphism ι : Q → C.Then p = t and t  q − 1, so r | is irreducible and its projective image contains elements of order t, π,ι G and so r is irreducible and its projective image contains elements of order t. Using again π,ι the classification of finite subgroups of PGL (F ), we see that to complete the proof we 2 p just need to show that r is not dihedral. If it is dihedral then there exists a prime number π,ι s such that π is an unramified twist of the Steinberg representation, r (Frob ) is scalar, s π,ι s and s ≡ 1mod p. This is a contradiction. To prove the next proposition, we need to find primes with special properties, namely that their Frobenius elements act on the composita of certain field extensions in a prescribed way. Using the Chebotarev density theorem, we see that it is equivalent to exhibit Galois automorphisms acting in the correct way. In order to do so, it is helpful to recall the following lemma from basic Galois theory. Lemma 3.8. — Let E/K be a finite Galois extension, and let K /K, K /K be Galois subex- 1 2 tensions. Then the natural map Gal(K K /K) → Gal(K /K) × Gal(K /K) is an 1 2 1 Gal(K ∩K /K) 2 1 2 isomorphism. Proposition 3.9. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ) of weight 2 which is non-CM. Suppose that 2, 3 ∈ sc(π ). Then we can find a regular 2 Q algebraic, cuspidal automorphic representation π of GL (A ) with the following properties: 2 Q (1) There exist prime numbers (q, t, r) such that π is seasoned with respect to (q, t, r) and sc(π ) = sc(π ) ∪{q}. n−1 n−1 (2) Sym π exists if and only if Sym π does. Proof. — Fix a prime t > max(10, 8n(n − 1), N(π )) such that t ≡ 1mod 4 and there exists an isomorphism ι : Q → C such that G = Proj r (G ) is conjugate either π,ι Q to PSL (F ) or PGL (F ). Since t > 5, the group PSL (F ) is simple. The condition t ≡ 2 t 2 t 2 t 1 mod 4 implies that −1mod t is a square and that the image of complex conjugation c in G lies in [G, G]. Using the Chebotarev density theorem, we can therefore choose a prime q such that q ≡−1mod t, (q + 1)> 2t, and the image of Frob in G is in the conjugacy class of complex conjugation. SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 145 Similarly, we can choose a prime r such that r is a primitive root modulo q, r (Frob ) is scalar, and r splits in Q(ζ ) and in Q(ζ 6 ) for every p ∈ sc(π ). Indeed, π,ι r M t p ker Proj r π,ι the prime q is unramified in Q (ζ , {ζ } ) but totally ramified in Q(ζ ),so M t p p∈sc(π ) q the intersection of these two fields in Q is Q. We choose r so that it splits in the first field and is totally inert in the second. Let sc(π ) ={p ,..., p } and let p = q.For each i = 1,..., k + 1, let ρ 1 k k+1 i,j (j ∈ X ) be a set of representatives for the (finitely many) conjugacy classes of irre- ducible dihedral representations ρ : G → GL (F ) of prime-to-p conductor dividing Q 2 p i 2 6 lcm(M q , p ).For any (i, j), the abelianization of the projective image of ρ i,j p∈sc(π ) is isomorphic either to Z/2Z or (Z/2Z) . In either case we claim that we can find a prime s such that ρ (Frob ) is scalar, s ≡ 1mod p , the image of Frob in G is in the i,j s i,j i s i,j i,j i,j conjugacy class of complex conjugation, and s ≡−1mod t. i,j ker Proj ρ ker Proj r i,j π,ι To see this, let E = Q and E = Q . We want to show there is 1 2 σ ∈ Gal(E E (ζ ,ζ )/Q) such that σ | = 1, σ | = 1and σ | = c.First we find 1 2 p t E Q(ζ ) E (ζ ) i 1 p 2 t τ ∈ Gal(E (ζ )/Q) such that τ | = 1and τ | = 1. Gal(E /Q) is soluble and its 1 p E Q(ζ ) 1 i 1 p maximal abelian quotient is a quotient of (Z/2Z) ,so Gal(E ∩ Q(ζ )/Q) is a quotient 1 p 2 × 2 of (Z/2Z) . This shows that E ∩ Q(ζ ) is either trivial or quadratic. Since (F ) contains 1 p i p non-identity elements (because p ≥ 5, because p ∈ sc(π ))wecan finda τ with the desired i i property using Lemma 3.8. We can assume that τ acts trivially on the maximal abelian subextension of E (ζ ) of exponent 2. 1 p Using Lemma 3.8 again, we’re done if we can show that τ | = E (ζ )∩E (ζ ) 1 p 2 t ab c| .Let E denote the maximal abelian subfield of E .Ithas degree 1or2 E (ζ )∩E (ζ ) 2 1 p 2 t 2 ab over Q (because of the form of the image of r )and Gal(E /E ) is a non-abelian simple π,ι 2 ab group. Thus the maximal soluble quotient of Gal(E (ζ )/Q) is Gal(E (ζ )/Q),whichis 2 t t in fact abelian. Since Gal(E (ζ )/Q) is soluble, this shows that Gal(E (ζ ) ∩ E (ζ )/Q) 1 p 1 p 2 t i i is abelian. Since t is coprime to qN(π ), the prime t is unramified in E (ζ ), while the 1 p ab quotient of [E (ζ ) : Q] by the ramification index of t is 1 or 2. We conclude that E (ζ ) ∩ E (ζ ) is either trivial or quadratic. In particular, τ acts trivially on it. The 1 p 2 t ab element c also acts trivially on it, since it acts trivially on E and also on the quadratic subfield of Q(ζ ) (since t ≡ 1 mod 4). This completes the proof of the claim. To conclude the proof of the proposition, we apply [Gee11, Corollary 3.1.7]; it implies the existence of a regular algebraic, cuspidal automorphic representation π of GL (A ) of weight 2 such that r = r ,suchthatfor each s ∈{r, s }, π is an unram- 2 Q π ,ι π,ι i,j ified twist of the Steinberg representation, such that rec π Ind χ for a character q W χ : W → C such that χ | has order t, and such that for every other prime p,wehave Q I 2 Q r | ∼ r  | ,withnotationasin[BLGGT14, §1]. (The hypothesis ‘(ord)’of[Gee11, π,ι G π ,ι G Q Q p p Proposition 3.1.5] is automatic in our situation.) In particular, we have M = q M .We π π n−1 see that π is seasoned with respect to (q, t, r). To see that Sym π exists if and only if n−1 Sym π does, apply e.g. [BLGGT14, Theorem 4.2.1]. The potential diagonalizability 146 JAMES NEWTON, JACK A. THORNE assumption is satisfied because r | , r  | are both Fontaine–Laffaille, while the rep- π,ι G π ,ι G Q Q t t n−1 n−1 th resentations Sym r = Sym r are irreducible because the m symmetric power π,ι π ,ι of the standard representation of SL (F ) is irreducible whenever t > m. 2 t Proposition 3.10. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ) of weight 2. Suppose that π is seasoned with respect to (q, t, r),and let p ∈ sc(π ) sat- 2 Q isfy p ≥ 5. Then we can find a regular algebraic, cuspidal automorphic representation π of GL (A ) 2 Q with the following properties: (1) π hasweight2andisnon-CM. (2) sc(π ) = sc(π ) −{p}.If p = q, then π is seasoned with respect to (q, t, r). n−1  n−1 (3) If Sym π exists, then so does Sym π . Proof. — By Theorem 2.1, it’s enough to find an isomorphism ι : Q → C and a regular algebraic, cuspidal automorphic representation π of GL (A ) with the following 2 Q properties: • The image of r contains a conjugate of SL (F a ) for some p > 2n − 1. π,ι 2 p • π has weight 2 and is non-CM. • r  r . π ,ι π,ι • sc(π ) = sc(π ) −{p}. • For any prime l = p, r  | ∼ r | . π ,ι G π,ι G Q Q l l • π is not ι-ordinary. • If p = q then the conductor of π divides p . If p = q, then the last condition ensures that π is still seasoned with respect to (q, t, r) (more precisely, that conditions (3) and (5) in Definition 3.6 still hold). We choose ι sat- isfying the first condition using Proposition 3.7; then the existence of a π satisfying the above requirements is the content of Lemma 3.5. Proposition 3.11. — Let π be a regular algebraic, cuspidal automorphic representation of GL (A ). Suppose that π is of weight 2 and non-CM, and suppose that 2, 3 ∈ sc(π ).Then 2 Q n−1 Sym π exists. Proof. — If k ≥ 0, let (H ) denote the hypothesis that the conclusion of the proposi- tion holds when |sc(π )|≤ k,and let (H ) denote the hypothesis that the conclusion of the proposition holds when |sc(π )|≤ k and π is seasoned with respect to some tuple (q, t, r). As remarked above, (H ) follows from the results of [NT21]. It therefore suffices to prove the implications (H ) ⇒ (H ) and (H ) ⇒ (H ). k k k+1 k The first implication follows immediately from Proposition 3.10. For the second, assume that (H ) holds and let π be a regular algebraic, cuspidal automorphic represen- tation of GL (A ) which is of weight 2 and non-CM, and such that |sc(π )|= k ≥ 1. By 2 Q SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 147 Proposition 3.9, we can find a regular algebraic, cuspidal automorphic representation π of GL (A ) which is seasoned with respect to (q, t, r),suchthat sc(π ) = sc(π ) ∪{q},and 2 Q n−1 n−1 such that the existence of Sym π is equivalent to the existence of Sym π . Now choose a prime p ∈ sc(π ) (so p ∈ sc(π ) and p = q). Applying Proposition 3.10 with this choice of p gives another regular algebraic, cuspidal automorphic representation π of GL (A ) which is seasoned with respect to (q, t, r),suchthat |sc(π )|= k,and such 2 Q n−1  n−1  n−1 that the existence of Sym π implies that of Sym π . The existence of Sym π follows from (H ),sowe’re done. We can now give the proof of Theorem 3.1. Proof of Theorem 3.1.— Let π be a non-CM, regular algebraic, cuspidal automor- n−1 phic representation of GL (A ). We must show that Sym π exists. We first do this 2 Q under the additional assumption that π has weight 2 and that 2 ∈ sc(π ). We can assume that π is supercuspidal. Fix a prime t > max(5, 4n(n − 1), N(π )) such that t ≡ 1mod 4 and there exists an isomorphism ι : Q → C such that G = Proj r (G ) is conjugate t π,ι Q t t either to PSL (F ) or PGL (F ). Using the Chebotarev density theorem, we can find a 2 t 2 t prime q satisfying the following conditions: • The prime q satisfies q ≡−1mod t, q ≡ 1mod 8, and q ≡ 1mod l for every prime l < t. • The image of Frob in G is in the conjugacy class of complex conjugation. (Compare [KW09a, Lemma 8.2].) By [Gee11, Corollary 3.1.7], we can find another regular algebraic, cuspidal automorphic representation π of weight 2 satisfying the fol- lowing conditions: • r = r . π,ι π ,ι t t • If l = q is a prime, then r | ∼ r | . π,ι G π ,ι G t Q t Q l l • There is an isomorphism rec π Ind χ,where χ : W → C is a char- Q Q q W 2 2 q acter such that χ | has order t. n−1 n−1 Applying [BLGGT14, Theorem 4.2.1] to Sym r , we see that Sym π exists if and π,ι n−1 only if Sym π does. (The potential diagonalizability assumption is satisfied because n−1 r | , r  | are both Fontaine–Laffaille, while the representations Sym r π,ι G π ,ι G π,ι t Q t Q t t t n−1 th Sym r are irreducible because the m symmetric power of the standard representa- π ,ι tion of SL (F ) is irreducible whenever t > m.) 2 t Let ι : Q → C be an isomorphism. Then ([KW09a, Lemma 6.3]) there exists a ≥ 2 such that the image of Proj r is conjugate to PSL (F a ) or PGL (F a ).Infact, π,ι 2 3 2 3 a 2a we must have 3 > 2n − 1: otherwise t ≤ 3 − 1 ≤ 4n(n − 1), a contradiction to our assumption t > 4n(n − 1). Applying Lemma 3.4, we can find another regular algebraic, cuspidal automor- phic representation π of GL (A ) of weight 2 such that r  r  ,2, 3 ∈ sc(π ), 2 Q π ,ι π ,ι 148 JAMES NEWTON, JACK A. THORNE r  | is potentially crystalline and non-ordinary, and for each prime l = 3, π is a π ,ι G Q l twist of the Steinberg representation if and only if π is. Then Proposition 3.11 implies n−1 n−1 that Sym π exists. We can then invoke Theorem 2.1 to see that Sym π exists and n−1 hence Sym π exists. The next case to treat is when π has weight 2 but now 2 ∈ sc(π ). In this case we can n−1 repeat the same argument with 3 replaced by 2 to conclude the existence of Sym π . Finally we treat the general case where π has weight k for some k > 2and we make no assumption on sc(π ). In this case we can find a prime t > max(5, k(n + 1)) such that π is unramified and an isomorphism ι : Q → C such that the image of r t π,ι contains a conjugate of SL (F ). Applying Lemma 3.4 again, we can find another regu- 2 t lar algebraic, cuspidal automorphic representation π of GL (A ) of weight 2 such that 2 Q n−1 n−1 r  r and r  is potentially crystalline. Then Sym π exists and Sym r is ir- π ,ι π,ι π ,ι π,ι n−1 reducible. We can now apply [BLGGT14, Theorem 4.2.1] to conclude that Sym r π,ι n−1 n−1 is automorphic and therefore that Sym π exists. Note that Sym r  is the sym- π ,ι metric power of a 2-dimensional potentially diagonalizable representation ([GK14, Lemma 4.4.1]), and hence potentially diagonalizable, cf. the remark following [BLGG11, Definition 3.3.5]. Acknowledgements J.T.’s work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 714405). We thank Toby Gee and an anonymous referee for comments on earlier versions of this manuscript. 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. If material is not included in the article’s Creative Commons licence and your intended use SYMMETRIC POWER FUNCTORIALITY FOR HOLOMORPHIC MODULAR FORMS, II 149 is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A: The case of weight one forms In this short appendix we record the automorphy of the symmetric power lifting for cuspidal Hecke eigenforms of weight 1, or with CM. The most difficult case is due to Kim [Kim04, Theorem 6.4]. Theorem A.1. — Let n ≥ 1.Let π be a cuspidal automorphic representation of GL (A ) with 2 Q π holomorphic limit of discrete series, or with π the automorphic induction of a Hecke character for a quadratic field. Then Sym π exists. Note that in these cases Sym π is usually not cuspidal. Proof. — First we assume that π is holomorphic limit of discrete series. Twisting by an algebraic Hecke character, we can assume that π is generated by a holomorphic weight 1 cuspidal Hecke eigenform. In particular, Deligne and Serre [DS74] constructed a continuous odd irreducible representation r : G → GL (C) with r | = rec (π ) π Q 2 π W p Q Q p p for all primes p. The projective image of r is a finite subgroup of PGL (C), and is there- π 2 fore dihedral or isomorphic to a copy of A , S or A (moreover, each of these subgroups 4 4 5 is unique up to conjugacy). We can then establish the automorphy of Sym r case by case, depending on the projective image. In the dihedral case, r is induced from a char- acter ψ of G for K/Q quadratic, Sym r decomposes as a direct sum of characters and K π the inductions of characters from K to Q, and therefore Sym r is automorphic. In the other cases, we denote the inverse image of Proj(r )(G ) in SL (C) by  . π Q 2 It is a binary polyhedral group. The image r (G ) is a subgroup of μ  ⊂ GL (C) for π Q 2k 2 some k (μ is the cyclic subgroup of the scalar matrices with order 2k). We have μ 2k 2k μ ×  /(−1, −I), so its irreducible representations are of the form ψ × σ ,with ψ a 2k character of μ , σ an irreducible representation of  ,and ψ(−1) = σ(−I). Twisting 2k by a Dirichlet character, we can assume that r (G ) = μ  (choose a prime p where π π Q 2k is unramified and which is 1 mod 2k, then twist by a Dirichlet character with conductor p and order 2k). Now to understand the decomposition of Sym r into irreducibles, it 2 1 suffices to understand the decomposition of the representation Sym C of  .See,for example, [Ste08, Appendix A] for the character tables of the binary polyhedral groups, or use [GAP20]. For the A case, the irreducible representations of  and their relationship to (symmetric powers of) the two Galois-conjugate irreducible two-dimensional representa- tions are described in [Kim04, §5]. This allows automorphy of Sym r to be deduced from the automorphy of Sym for m ≤ 4, together with tensor product functorialities GL × GL → GL and GL × GL → GL [Kim04, Theorem 6.4]. 2 2 4 2 3 6 150 JAMES NEWTON, JACK A. THORNE Now we turn to the A case. Considering the character table of the binary tetra- hedral group, we see that the irreducible representations of dimension > 1 comprise: three two-dimensional representations, isomorphic up to twist and a three-dimensional representation which is isomorphic to the symmetric square of the two-dimensional rep- n 2 resentations. Automorphy of Sym r therefore follows from automorphy of Sym r . π π Finally, in the S case, we consider the character table of the binary octahedral group. 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THORNE, Automorphy of some residually dihedral Galois representations, Math. Ann., 364 (2016), 589–648. J. N. Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK Current address: Mathematical Institute, Woodstock Road, Oxford OX2 6GG, UK 152 JAMES NEWTON, JACK A. THORNE 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 11 octobre 2021.

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