Symmetric power functoriality for holomorphic modular forms, II

We establish the existence of the symmetric power liftings of all holomorphic Hecke eigenforms.


Introduction
Let F be a number field, and let π be a cuspidal automorphic representation of GL 2 (A F ). Langlands's functoriality principle predicts the existence, for any n ≥ 1, of an automorphic representation Sym n π of GL n+1 (A F ), characterized by the requirement that for any place v of F , the Langlands parameter of (Sym n π) v is the image of the Langlands parameter of π v under the nth symmetric power Sym n : GL 2 → GL n+1 of the standard representation of GL 2 .
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 [NT19], 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 n π under the assumption that there is no prime p such that π p is supercuspidal. (This includes the case that f has level SL 2 (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 2 (A Q ). Suppose that π is non-CM. Then for each integer n ≥ 1, Sym n π exists, as a regular algebraic, cuspidal automorphic representation of GL n+1 (A Q ).
In the 'missing' cases of π which are holomorphic limit of discrete series at ∞ or CM, the existence of Sym n π for all n is well known, although of course Sym n π is usually not cuspidal. The most difficult case of icosahedral weight one eigenforms ([Kim04, Theorem 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 Λ(Sym n 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 2 (F 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 2 (A Q ) the set sc(π) of primes p such that π p is supercuspidal. Fixing n ≥ 1, we prove the existence of Sym n π by induction on the cardinality of the set sc(π), the case |sc(π)| = 0 being exactly the main result of [NT19].
Our induction argument uses congruences between automorphic representations. If p is a prime and ι : Q p → C is an isomorphism, then there is an associated Galois representation r π,ι : Gal(Q/Q) → GL 2 (Q p ) and its mod p reduction r π,ι : Gal(Q/Q) → GL 2 (F p ).
If Sym n π ′ is known to exist, and the image of the representation Sym n r π ′ ,ι is sufficiently non-degenerate (for example, irreducible), then automorphy lifting theorems (such as those proved in [BLGGT14]) can be used to deduce the automorphy of Sym n r π,ι , hence the existence of Sym n π.
If p is a prime such that π p 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 n r π,ι is never irreducible, so the automorphy lifting theorems proved in [BLGGT14] 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 p is supercuspidal.) This approach might perhaps yield the existence of Sym n 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 π,ι and P is the universal pseudodeformation ring of the pseudocharacter associated to the symmetric power Sym n r π,ι ; the morphism P → R is the universal one classifying the pseudocharacter of Sym n of the universal deformation of r π,ι . A version of the Taylor-Wiles-Kisin patching argument upgrades this to a commutative diagram where P ∞ , R ∞ are 'patched deformation rings' and the vertical arrows are surjections. Since r π,ι is assumed to be irreducible, the arguments of [Kis09b,Kis09a] show that R ∞ 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 n r π ′ ,ι ; this in turn implies that Spec P ∞ is regular at the image of 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 n 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 : Q q → C such that r π,ιq has large image. We achieve this by introducing Steinberg type ramification at another auxiliary 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).
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.
Notation. If F is a perfect field, we generally fix an algebraic closure F /F and write G F for the absolute Galois group of F with respect to this choice. When the characteristic of F is not equal to p, we write ǫ : G F → Z × p for the p-adic cyclotomic character. We write ζ n ∈ F for a fixed choice of primitive n th root of unity (when this exists). If F is a number field, then we will also fix embeddings F → F v extending the map F → F v for each place v of F ; this choice determines a homomorphism G Fv → G F . When v is a finite place, we will write W Fv ⊂ G Fv for the Weil group, O Fv ⊂ F v for the valuation ring, ̟ v ∈ O Fv for a fixed choice of uniformizer, Frob v ∈ G Fv for a fixed choice of (geometric) Frobenius lift, k(v) = O Fv /(̟ v ) for the residue field, and q v = #k(v) for the cardinality of the residue field. If R is a ring and α ∈ R × , then we write ur α : W Fv → R × for the unramified character which sends Frob v to α. When v is a real place, we write c v ∈ G Fv for complex conjugation. If S is a finite set of finite places of F then we write F S /F for the maximal subextension of F unramified outside S and G F,S = Gal(F S /F ).
If p is a prime, then we call a coefficient field a finite extension E/Q p contained inside our fixed algebraic closure Q p , and write O for the valuation ring of E, ̟ ∈ O for a fixed choice of uniformizer, and k = O/(̟) for the residue field. We write C O for the category of complete Noetherian local O-algebras with residue field k. If G is a profinite group and ρ : G → GL n (Q p ) is a continuous representation, then we write ρ : G → GL n (F p ) for the associated semisimple residual representation (which is well-defined up to conjugacy). If F is a number field, v is a finite place of F , and ρ, ρ ′ : G Fv → GL n (O Q p ) are continuous representations, which are potentially crystalline if v|p, then 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 n ⊂ B n ⊂ GL n for the standard diagonal maximal torus and uppertriangular Borel subgroup. Let K be a non-archimedean characteristic 0 local field, and let Ω be an algebraically closed field of characteristic 0. If ρ : G K → GL n (Q p ) is a continuous 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 WD(ρ) F −ss for its Frobenius semisimplification. We use the cohomological normalisation of class field theory: it is the isomorphism Art K : K × → W ab K which sends uniformizers to geometric Frobenius elements. When Ω = C, we have the local Langlands correspondence rec K for GL n (K): a bijection between the sets of isomorphism classes of irreducible, admissible C[GL n (K)]-modules and Frobeniussemisimple Weil-Deligne representations over C of rank n. In general, we have the Tate normalisation of the local Langlands correspondence for GL n as described in [CT14, §2.1]. When Ω = C, we have rec T K (π) = rec K (π ⊗ | · | (1−n)/2 ). If F is a number field and χ : F × \A × F → C × is a Hecke character of type A 0 (equivalently: algebraic), then for any isomorphism ι : Q p → C there is a continuous character r χ,ι : G F → Q × p which is de Rham at the places v|p of F and such that for each finite place v of F , WD(r χ,ι ) p is a continuous character which is de Rham and unramified at all but finitely many places, then there exists a Hecke character χ : F × \A × F → C × of type A 0 such that r χ,ι = χ ′ . In this situation we abuse notation slightly by writing If F is a CM or totally real number field and π is an automorphic representation of GL n (A F ), we say that π is regular algebraic if π ∞ has the same infinitesimal character as an irreducible algebraic representation W of (Res F/Q GL n ) C .
If π is cuspidal, regular algebraic, and polarizable, in the sense of [BLGGT14], then for any isomorphism ι : Q p → C there exists a continuous, semisimple representation r π,ι : G F → GL n (Q p ) such that for each finite place v of F , WD(r π,ι | GF v ) F −ss ∼ = rec T Fv (ι −1 π v ) (see e.g. [Car14]). (When n = 1, this is compatible with our existing notation.) 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 has weight k if we can take W = (⊗ τ ∈Hom(F,C) Sym k−2 C 2 ) ∨ . (If F is totally real, then the cuspidal automorphic representations of weight k are those which are associated to cuspidal Hilbert modular forms of parallel weight k.) In this case the Hodge-Tate weights of r π,ι with respect to any embedding τ : F → Q p are {0, k − 1} and the character ǫ k−1 det r 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 , ). We write Iw v ⊂ GL n (O Fv ) for the standard Iwahori subgroup (elements which are upper-triangular modulo ̟ v ) and Iw v,1 ⊂ Iw v for the kernel of the natural map Iw v → (k(v) × ) n given by reduction modulo ̟ v , then projection to the diagonal. If U v ⊂ Iw v is a subgroup containing Iw v,1 , and 1 ≤ i ≤ n, then we define 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 p → C. Let n ≥ 1. Let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A F ) satisfying the following conditions: • π has weight 2 and is non-CM.
• For each place v|p of F , r π,ι | GF v is not ordinary, in the sense of [Tho16, §5.1]. Note that, together with the assumption that π has weight 2, this implies that r π,ι | GF v is potentially Barsotti-Tate. • Let Proj r π,ι : G F → PGL 2 (F p ) denote the projective representation associated to r π,ι . Then there exists a ≥ 1 such that p a > max(5, 2n − 1) and there is a sandwich up to conjugacy in PGL 2 (F p ). We impose the final condition to ensure that we can choose Taylor-Wiles primes such that the image of the corresponding Frobenius element under Sym n−1 r π,ι is regular semisimple.
The aim of this section is prove the following theorem: Theorem 2.1. Suppose that there exists another regular algebraic, cuspidal automorphic representation π ′ of GL 2 (A F ) such that the following conditions are satisfied: (1) π ′ has weight 2 and is non-CM.
(2) For each place v|p of F , r π ′ ,ι | GF v is not ordinary.
(4) For each place v ∤ p of F , π v is a character twist of the Steinberg representation if and only if π ′ v is. (5) Sym n−1 r π ′ ,ι is automorphic. Then Sym n−1 r π,ι is automorphic.
We begin with a preliminary reduction. Let E/Q p be a coefficient field. After possibly enlarging E, we can find conjugates r, r ′ of r π,ι , r π ′ ,ι respectively which take values in GL 2 (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. (7) det r π,ι = det r π ′ ,ι = ǫ −1 . (8) For each place v|p of F , π v and π ′ v are unramified. (9) For each finite place v ∤ p of F , π v and π ′ v are Iwahori-spherical. The number of places such that π v is ramified is even. (10) Let S p denote the set of p-adic places of F and Σ the set of places v such that π v is ramified. Let S = S p ∪ Σ. Then for each v ∈ S, r| GF v is trivial. For each v ∈ Σ, q v ≡ 1 mod p and π v , π ′ v are isomorphic to the Steinberg 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 = π K . Then Π is RACSDC (i.e. regular algebraic, conjugate self-dual, and cuspidal). We will show that the representation Sym n−1 r Π,ι is automorphic; this will imply Theorem 2.1, by soluble descent. We let Π ′ = π ′ K . Then Π ′ is also RACSDC, there is an isomorphism r Π,ι ∼ = r Π ′ ,ι , and Sym n−1 r Π ′ ,ι is automorphic. We write Π ′ n for the RACSDC automorphic representation of GL n (A K ) such that r Π ′ n ,ι ∼ = Sym n−1 r Π ′ ,ι . Recall that C O denotes the category of complete Noetherian local O-algebras with residue field k. If v is a place of F , we write R v ∈ C O for the object representing the functor Lift v : C O → Sets which associates to A ∈ C O the set of homomorphisms r : G Fv → GL 2 (A) lifting r| GF v (i.e. such that r mod m A = r) such that det r = ǫ −1 | GF v . We introduce certain quotients of R v :  If Q is a finite set of finite places of F , disjoint from S, then we write Def Q : C O → Sets for the functor which associates to A ∈ C O the set of 1 + M 2 (m A )-conjugacy classes of lifts r : G F → GL 2 (A) of r satisfying the following conditions: • r is unramified outside S ∪ Q, and det r = ǫ −1 .
• For each v ∈ S ∪ S ∞ , the homomorphism R v → A determined by r| GF v factors through the quotient R v → R v introduced above. Our assumption on the image of r implies that the functor Def Q is represented by an object R Q ∈ C O . If Q is empty then we write R = R ∅ . We also introduce some variants. Let R loc = ⊗ v∈S∪S∞ R v . Then R loc is an O-flat domain of Krull dimension 1 + 3[F : Q] + 3|S| (cf. [Kis09b,Lemma 3.4.12]). We write Def Q : C O → Sets for the functor of 1 + M 2 (m A )-conjugacy of tuples ( r, {A v } v∈S∪S∞ ), where r is as above and A v ∈ 1 + M 2 (m A ), and γ ∈ 1 + M 2 (m A ) acts by γ · ( r, v∈S∪S∞ is independent of the choice of representative for a given conjugacy class, and the universal property of R v determines a homomorphism R loc → R Q . The objects in this paragraph will only be used in the case p = 2. We write Def ′ Q : C O → Sets for the functor of 1 + M 2 (m A )-conjugacy classes of lifts r : G F → GL 2 (A) of r satisfying the following conditions: We write Def ′, Q for the functor of 1+M 2 (m A )-conjugacy classes of tuples ( r, {A v } v∈S∪S∞ ), where r is as above and A v ∈ 1 + M 2 (m A ). Then the functors Def ′ Q and Def ′, Q are represented by objects R ′ Q , R ′, Q ∈ C O and there is again a natural morphism R loc → R ′, Q . Let t = det Sym n−1 (r| GK ) and t ′ = det Sym n−1 (r ′ | GK ) denote the group determinants over O (in the sense of [Che14]) associated to these two symmetric power representations, and let t denote the group determinant over k which is their common reduction modulo ̟. We introduce the object P ∈ C O which is the quotient introduced in [NT20, §2.19]. Informally, P represents the functor of conjugate self-dual group determinants of G K,S lifting t which have similitude character ǫ 1−n and are semistable with Hodge-Tate weights in the interval [0, n− 1].
Lemma 2.2. Let A ∈ C O be Artinian, let v ∈ S p , and let r : Proof. It follows from the construction in [Kis09b,Kis09a]  Let A ∈ C O , and let r : G F → GL 2 (A) be a lift of r which determines a map R → A. Lemma 2.2 shows that the pseudocharacter associated to Sym n−1 ( r| GK ) satisfies condition (2.16.1) of [NT20]. In particular there are morphisms P → O associated to the pseudocharacters t, t ′ . Taking the pseudocharacter of the symmetric power of the universal deformation over R determines a morphism P → R in C O .
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, are distinct elements of k. 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.
Let r univ t,S∪Q introduced in [NT20, §2.19] corresponding to pseudodeformations t of t with the following additional properties: • For each v ∈ Q, t| GK v factors through the maximal Hausdorff abelian quo- -algebra, and again there is a universal morphism P Q → R Q . We remark that this need not be a morphism of O[∆ Q ]-algebras when n is odd, although it is when n is even.
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 , following 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 Σ. Fix a choice of maximal order O D and for each finite place v ∈ Σ, an 1 It is possible that our condition on the eigenvalues of Frobv 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 .
We define We introduce Hecke operators. If v ∈ Σ ∪ Q is a finite place of F , then the unramified Hecke operators T There is a unique maximal ideal m D ⊂ T univ D,Σ of residue field k such that for all finite places v ∈ S of F , the characteristic polynomial of r(Frob v ) equals is a Taylor-Wiles datum, then we write m D,Q for the maximal ideal of T univ,Q D,Σ∪Q generated by m D ∩ T univ D,Σ∪Q and the elements U Let m D ≥ 0 denote the p-adic valuation of the least common multiple of the exponents of the Sylow p-subgroups of the finite groups (1) The maximal ideals m D and m D,Q are in the support of H D (U 0 ) and H D (U 0 (Q)), respectively. 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 the same way as [KW09b, Lemma 9.1] (cf. also [Kis09a, Lemma 3.2.7]); note that since T 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 F . We recall that G comes equipped with isomorphisms ι w : if v ∈ Q and n is even, and We introduce Hecke operators for the group G. If v ∈ S ∪ Q is a place of F which splits v = ww c in K, then the unramified Hecke operators ι −1 w T (i) ̟ v (i = 1, . . . , n) act on the spaces H G (V 0 (Q)) and H G (V 1 (Q; N )). We write T univ G,S∪Q for the polynomial ring over O in the indeterminates T (i) w (where w is a place of K split over F , not lying above S ∪ Q and 1 ≤ i ≤ n), and T univ, There is a unique maximal ideal m G ⊂ T univ G,S of residue field k such that for all finite places w of K, split over F and not lying above S, the characteristic polynomial of Sym n−1 r(Frob w ) equals is a Taylor-Wiles datum, then we write m G,Q for the maximal ideal of T univ,Q G,S∪Q generated by m G ∩ T univ G,S∪Q and the elements The unitary group G comes with a determinant map det : (Q; N )) mG,Q . We will use this construction only in conjunction with the following lemma.
Lemma 2.4. Suppose that n is even and that p = 2. Suppose that Then there exists a unique character where the last map is z → z/z c . By Hilbert 90 we have a short exact sequence We need to check that θ χ is trivial on det V 1 (Q; N ). 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 identifying Since χ is quadratic, θ χ annihilates this subgroup, and we're done.
Let m G ≥ 0 denote the p-adic valuation of the least common multiple of the exponents of the Sylow p-subgroups of the finite groups G(F )∩tV 0 t −1 (t ∈ G(A ∞ F )).
Proposition 2.5. Let N ≥ 1 and let (Q, Q, (α v , β v ) v∈Q )) be a Taylor-Wiles datum of level N + m G . Then: (1) The maximal ideals m G , m G,Q are in the support of H G (V 0 ) and H G (V 0 (Q)), respectively.
. . , n) are the coefficients of the universal characteristic polynomial, defined as in [Che14, §1.10], then for any finite Proof. The first part may be deduced, as in [NT20, §4.3], from [Lab11, Théorème 5.4] (and the existence of Π ′ n ). The other parts are proved in a very similar way to the second and third parts of Proposition 2.3, as we now explain. We begin by constructing a more familiar set of objects.
As in the case of P Q , [ANT20, Proposition 2.5] again shows that for each v ∈ Q there are unique characters A is the restriction to G K v of the universal pseudocharacter over G K . We now claim that the following statements hold: To complete the proof of the first point, we need to explain why there is an isomorphism H G (V 0 (Q)) mG,Q ∼ = H G (V 0 ) mG . This follows from [NT20, Proposition 3.1]. The second part is proved in the same way as [NT20, Lemma 4.7]. The third part is proved using [NT20, Lemma 4.7] and [BC09, Proposition 1.5.1] (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 ′ Q → P Q and ∆ ′ Q → ∆ Q . The proof is complete on noting that trace induces an isomorphism (Q; N )) mG,Q and that the map We are now ready to prove Theorem 2.1. We will need to treat the cases p > 2 and p = 2 separately.
Proof of Theorem 2.1, case p > 2. Define H G = H G (V 0 ) mG and H D = H D (U 0 ) mD . 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 ∞ = O Y 1 , . . . , Y q , Z 1 , . . . , Z 4|S∪S∞|−1 . Then we can find the following data: (1) Complete Noetherian local W ∞ -algebras P ∞ , R ∞ equipped with isomor- 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. It is enough to show that p is in the support of H G as P -module. Indeed, this would imply (using [Lab11, Corollaire 5.3] and the irreducibility of Sym n−1 r| GK ) the existence of a RACSDC automorphic representation Π n of GL n (A K ) such that r Πn,ι ∼ = Sym n−1 (r| GK ). By descent (in the form of e.g. [BLGHT11, Lemma 1.5]), this would imply the sought-after automorphy of Sym n−1 r. Equivalently, we must show that if p ∞ is the pre-image of p under the morphism P ∞ → P , then p ∞ is in the support of H G,∞ as P ∞ -module. (Since Supp P H G = Supp P∞ H G,∞ ∩ Spec P , intersection taken inside Spec P ∞ .) The P ∞ -module H G,∞ is a Cohen-Macaulay module (i.e. it is finite and the dimension of its support is equal to its depth), since H G,∞ is a finite free W ∞module. Applying [Sta13, Tag 0BUS], it follows that each irreducible component of Supp P∞ H G,∞ has dimension q + 4|S ∪ S ∞ |. Similarly, we see that each irreducible component of Supp R∞ H D,∞ has dimension q + 4|S ∪ S ∞ |. Since R ∞ is a quotient of R loc X 1 , . . . , X g , a domain of Krull dimension q + 4|S ∪ S ∞ |, we see that R loc X 1 , . . . , X g → R ∞ is an isomorphism, that R ∞ is a domain, and that H D,∞ is a faithful R ∞ -module.
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 particular, p ∞ lies in Z, hence in Supp P∞ H G,∞ . This completes the proof of the theorem.
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 α n−i v β i−1 v (i = 1, . . . , n) are distinct does not cause any difficulty.
In either case we have established the claim; this completes the proof. Now we give the proof of Proposition 2.6. Let q = dim k H 1 (F S /F, ad 0 r(1)) and g = q+|S∪S ∞ |−1. For each N ≥ 1, fix a Taylor-Wiles datum (Q N , Q N , (α v , β v ) v∈QN ) of level N + max(m D , m G ) such that there exists a surjection R loc X 1 , . . . , X g → R QN of R loc -algebras. Fix v 0 ∈ S and define T = O {Z v,i,j } v∈S∪S∞,1≤i,j≤2 /(Z v0,1,1 ). We view T as an augmented O-algebra via the augmentation which sends each Z v,i,j to 0. Choose for each (Q; N )) mG,Q N . We fix a representative r univ : G F → GL 2 (R) for the universal deformation over R, and representatives r univ QN : G F → GL 2 (R QN ) for the universal deformations over R QN lifting r univ for each N ≥ 1. These choices determine ). Thus each ring R N , P N has a W ∞ -algebra structure, and there are isomorphisms 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 M ) M≥1 of integers N M ≥ M such that for each M ≥ 1 there is a commutative diagram of O-algebras where the morphisms from the back square of the cube to the front are isomorphisms, and there are commutative diagrams of modules compatible with the module structures arising from the previous commutative cube, and where the arrows from back to front are again isomorphisms. We define By passage to inverse limit, there is a diagram (of rings and modules) To complete the proof of the proposition, it remains to show that these objects have the following properties: • The morphism R loc X 1 , . . . , X g → R ∞ is surjective. is an isomorphism, or equivalently that the ideal aR ∞ of R ∞ is closed in the m R∞adic topology. This is true since R ∞ is a Noetherian ring. The same proof applies to the ring P ∞ .
Now we treat the case p = 2. Proposition 2.8. We can find an integer q ≥ |S ∪ S ∞ | − 2 with the following property: let Then we can find the following data: (1) Complete Noetherian local W ∞ -algebras P ∞ , R ∞ equipped with isomor- (2) A complete Noetherian local O-algebra R ′ ∞ and surjections R loc X 1 , . . . , X g → R ′ ∞ and R ′ ∞ → R ∞ in C O , where g = 2q + 1.
where G γ m acts on itself by the square of the identity. These objects have the following additional properties: 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 G as P -module, or equivalently that the pullback p ∞ ⊂ P ∞ of p is in the support of H G,∞ as P ∞ -module.
The P ∞ -module H G,∞ is a Cohen-Macaulay module, and each irreducible com- , showing that R inv ∞ has Krull dimension at least q + 4|S ∪ S ∞ |. We conclude that Spec R ′ ∞ has dimension at least q + 4|S ∪ S ∞ | + γ = dim R loc X 1 , . . . , X g . Since R loc X 1 , . . . , X g is a domain, it follows that the map R loc X 1 , . . . , 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 P∞ H G,∞ , by hypothesis. Similarly, let r, r ′ ⊂ 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 We now observe that the Zariski tangent space of the local ring P ∞,(p ′′ ∞ ) has dimension at most q + 4|S ∪ S ∞ | − 1. Indeed, translating by the element of G γ m [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 is a field. This again follows from [NT20, Theorem A, Proposition 2.21, Example 2.34]. It follows that P ∞,(p ′′ ∞ ) is a regular local ring of dimension 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 P∞ H G,∞ . We deduce that p ∞ ∈ Supp P∞ 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.
(2) There is a surjection R loc X 1 , . . . , X g → R ′, Q of R loc -algebras. (3) Let Θ Q 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 Proof. This is contained in [KW09b,Lemma 5.10], except that result specifies only that if v ∈ Q then the eigenvalues α v , β v ∈ k of r(Frob v ) are distinct. Here we require that the numbers α n−i v β i−1 v (i = 1, . . . , n) are distinct. However, reading the proof of loc. cit. we see that we can indeed choose v so that r(Frob v ) satisfies this stronger requirement (using of course our assumption that the projective image of r contains PSL 2 (F 2 a ) for some a ≥ 1 such that 2 a > 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 k H 1 (F S /F, ad r) − 2 and g = 2q + 1. For each N ≥ 1, fix a Taylor-Wiles datum (Q N , (Q; N )) mG,Q N . We fix a representative r univ : G F → GL 2 (R) for the universal deformation over R, and representatives r univ QN :  (V 1 (Q; N )) mG,Q N , which are trivial if n − 1 is even and which correspond to twisting by the quadratic characters χ| GK (resp. θ χ for χ ∈Θ QN [2](O)) when n − 1 is odd. We extend these to actions on H D,N and H G,N by completed tensor product with T . The morphism P N → R N is equivariant for these actions. A very similar 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.

Killing ramification
Our goal in this section is to prove the following theorem (Theorem A of the introduction): Theorem 3.1. Let n ≥ 1. Let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A Q ) which is non-CM. Then Sym n−1 π exists.
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 p is supercuspidal; the case where sc(π) is empty is exactly the main result of [NT19].
We begin with some preparatory definitions and results.
Definition 3.2. Let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A Q ). We define the semisimple conductor M π of π to be M π = l N ((rec Q l π l ) ss ) (where N denotes conductor). Lemma 3.3. Let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A Q ), let p be an odd prime, and let ι : Q p → C be an isomorphism. If r π,ι is reducible or dihedral 2 , then the prime-to-p part of its conductor divides M π .
Proof. If r π,ι is reducible or dihedral then its image has order prime to p, and for any prime l = p, r π,ι | GQ l is semisimple. This shows that the conductor of r π,ι | GQ l divides the conductor of (rec Q l π l ) ss . Lemma 3.4. Let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A Q ). Let p be a prime, let ι : Q p → C be an isomorphism, and suppose that Proj r π,ι (G Q ) contains a conjugate of PSL 2 (F p a ) for some p a > 5. Then we can find another regular algebraic, cuspidal automorphic representation π ′ of GL 2 (A Q ) with the following properties: (1) There is an isomorphism r π,ι ∼ = r π ′ ,ι .
(3) There is an isomorphism rec Qp π ′ p ∼ = ω 1 ⊕ ω 2 , where ω 1 , ω 2 : W Qp → C × are characters of conductor dividing p 3 . (4) For each prime l = p, π l is a twist of the Steinberg representation (resp. supercuspidal) if and only if π ′ l is. Proof. Let Σ be the set of primes l = p such that π l is a twist of the Steinberg representation, and let T be the set of primes l = p such that π l is supercuspidal. Let l 0 ≥ 5 be a prime such that l 0 ≡ 1 mod p, π l0 is unramified, and r π,ι (Frob l0 ) has distinct eigenvalues (such a prime exists because of our assumption on the image of Proj r π,ι ). Fix a coefficient field E containing a p 2 th root of unity. If l is a prime such that π l is supercuspidal, then we can find, after possibly enlarging is an open compact subgroup, we write Y (U ) for the locally symmetric space of level U (namely the object denoted X U Res D/Q Gm in [NT16, §3.1]). We regard M as an O[U ]-module by projection to l∈T GL 2 (Z l ). If U p = GL 2 (Z p ) and r ≥ 1 then we write U 0 (p r ) ⊂ U for the open compact subgroup with the same component at primes l = p and component at the prime p. We identify any pair of characters χ 1 , χ 2 : (Z/p r Z) × → O × with the character χ 1 ⊗ χ 2 : U 0 (p r ) → O × given by the formula ]-module. We use similar notation for k-and E-valued characters. Let δ = 0 if D is ramified at ∞ and δ = 1 otherwise. Let U = l U l be the open compact subgroup defined as follows: • U l0 = Iw l0,1 .
• If l ∈ Σ then π ′ l is a twist of the Steinberg representation.
We now show that (3.4.1) holds. We have also contains a copy of χ 2 ⊗ χ 1 with multiplicity one. The product of the eigenvalues of [U 0 (p 3 )α p,1 U 0 (p 3 )] on the χ 1 ⊗ χ 2 and χ 2 ⊗ χ 1 isotypic spaces of ι −1 π ′ p | U0(p 3 ) has p-adic valuation 1. We deduce that Using the exactness of H δ (Y (U 0 (p)), (?) ∨ ) mπ as a (contravariant) functor of smooth k[U 0 (p)]-modules, it is therefore enough to show that Ind U0(p) U0(p 3 ) χ 1 ⊗ χ 2 contains χ 1 ⊗ χ 2 and χ 2 ⊗ χ 1 as Jordan-Hölder factors with multiplicity at least 2 (or when χ 1 = χ 2 , that it contains χ 1 ⊗ χ 2 with multiplicity at least 3). This is true. Indeed, the semisimplification of a smooth k[Iw p ]-module (say finitedimensional as k-vector space) is determined by its restriction to the diagonal torus in Iw p , and we can then use Mackey's formula to show that if ψ : F × p → k × is a character then the semisimplification of Ind with multiplicity p + 2 if ψ = 1 and multiplicity p + 1 if ψ = 1. Lemma 3.5. Let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A Q ). Let p ≥ 3 be a prime, let ι : Q p → C be an isomorphism, and suppose that Proj r π,ι (G Q ) contains a conjugate of PSL 2 (F p a ) for some p a > 5. Then we can find another regular algebraic, cuspidal automorphic representation π ′ of GL 2 (A Q ) with the following properties: (1) There is an isomorphism r π,ι ∼ = r π ′ ,ι .
(2) There is an isomorphism rec Qq π q ∼ = Ind is a character such that χ| IQ q has order t. (In particular, q ∈ sc(π) and N (π q ) = q 2 .) (3) r is a primitive root modulo q. If M denotes the least common multiple of the prime-to-q part of M π and p∈sc(π)−{q} p 6 , then r ≡ 1 mod M . (4) π r 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 2 (F p ) of prime-to-p conductor dividing M q 2 , there exists a prime number s such that π s is an unramified twist of the Steinberg representation, ρ(Frob s ) 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 p → C such that r π,ι (G Q ) contains a conjugate of SL 2 (F p a ) for some p a > 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 q → C. We first claim that r π,ι is irreducible.
Since r ≡ 1 mod M (in particular, the characters χ i are unramified at r), we have (χ 1 /χ 2 )(Frob r ) = r i−1 . Since π r is an unramified twist of the Steinberg representation, we have (χ 1 /χ 2 )(Frob r ) = r or r −1 . Since r is a primitive root modulo q, this implies that 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 π,ι is irreducible.
We next claim that r π,ι is not dihedral. If r π,ι is dihedral, then Lemma 3.3 shows that the prime-to-q part of the conductor of r π,ι divides M π , so there exists a prime number s such that π s is an unramified twist of the Steinberg representation, r π,ι (Frob s ) is scalar, and s ≡ 1 mod 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 π,ι contains an element 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 2 (F q ) shows that the projective image of r π,ι contains PSL 2 (F q a ) (and is contained in PGL 2 (F q a )) for some a ≥ 1. If q a ≤ 2n − 1 then q 2a − 1 ≤ 4n(n − 1), so every element of PGL 2 (F q a ) of order prime to q has order at most 4n(n − 1). Since t/2 > 4n(n − 1), we see that we must have q a > 2n − 1. Now suppose that p = q, and fix an isomorphism ι : Q p → C. Then p = t and t ∤ q − 1, so r π,ι | GQ q is irreducible and its projective image contains elements of order t, and so r π,ι is irreducible and its projective image contains elements of order t. Using again the classification of finite subgroups of PGL 2 (F p ), we see that to complete the proof we just need to show that r π,ι is not dihedral. If it is dihedral then there exists a prime number s such that π s is an unramified twist of the Steinberg representation, r π,ι (Frob s ) is scalar, and s ≡ 1 mod 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.
Proof. Fix a prime t > max(10, 8n(n − 1), N (π)) such that t ≡ 1 mod 4 and there exists an isomorphism ι : Q t → C such that G = Proj r π,ι (G Q ) is conjugate either to PSL 2 (F t ) or PGL 2 (F t ). Since t > 5, the group PSL 2 (F t ) is simple. The condition t ≡ 1 mod 4 implies that −1 mod 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 ≡ −1 mod t, (q + 1) > 2t, and the image of Frob q in G is in the conjugacy class of complex conjugation.
Similarly, we can choose a prime r such that r is a primitive root modulo q, r π,ι (Frob r ) is scalar, and r splits in Q(ζ Mπ t ) and in Q(ζ p 6 ) for every p ∈ sc(π).
Indeed, the prime q is unramified in Q ker Proj rπ,ι (ζ Mπ t , {ζ p 6 } p∈sc(π) ) but totally ramified in Q(ζ q ), so 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 1 , . . . , p k } and let p k+1 = q. For each i = 1, . . . , k + 1, let ρ i,j (j ∈ X i ) be a set of representatives for the (finitely many) conjugacy classes of irreducible dihedral representations ρ : G Q → GL 2 (F pi ) of prime-to-p i conductor dividing lcm(M π q 2 , p∈sc(π) p 6 ). For any (i, j), the abelianization of the projective image of ρ i,j is isomorphic either to Z/2Z or (Z/2Z) 2 . In either case we claim that we can find a prime s i,j such that ρ i,j (Frob si,j ) is scalar, s i,j ≡ 1 mod p i , the image of Frob si,j in G is in the conjugacy class of complex conjugation, and s i,j ≡ −1 mod t.
Using Lemma 3.8 again, we're done if we can show that τ | E1(ζp i )∩E2(ζt) = c| E1(ζp i )∩E2(ζt) . Let E ab 2 denote the maximal abelian subfield of E 2 . It has degree 1 or 2 over Q (because of the form of the image of r π,ι ) and Gal(E 2 /E ab 2 ) is a non-abelian simple group. Thus the maximal soluble quotient of Gal(E 2 (ζ t )/Q) is Gal(E ab 2 (ζ t )/Q), which is in fact abelian. Since Gal(E 1 (ζ pi )/Q) is soluble, this shows that Gal(E 1 (ζ pi ) ∩ E 2 (ζ t )/Q) is abelian. Since t is coprime to qN (π), the prime t is unramified in E 1 (ζ pi ), while the quotient of [E ab 2 (ζ t ) : Q] by the ramification index of t is 1 or 2. We conclude that E 1 (ζ pi ) ∩ E 2 (ζ t ) is either trivial or quadratic. In particular, τ acts trivially on it. The element c also acts trivially on it, since it acts trivially on E ab 2 and also on the quadratic subfield of Q(ζ t ) (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 2 (A Q ) of weight 2 such that r π ′ ,ι ∼ = r π,ι , such that for each s ∈ {r, s i,j }, π ′ s is an unramified twist of the Steinberg representation, such that rec Qq π ′ q ∼ = Ind WQ q WQ q 2 χ for a character χ : W Q q 2 → C × such that χ| IQ q has order t, and such that for every other prime p, we have r π,ι | GQ p ∼ r π ′ ,ι | GQ p , with notation as in [BLGGT14,§1]. (The hypothesis '(ord)' of [Gee11, Proposition 3.1.5] is automatic in our situation.) In particular, we have M π ′ = q 2 M π . We see that π ′ is seasoned with respect to (q, t, r). To see that Sym n−1 π exists if and only if Sym n−1 π ′ does, apply e.g. [BLGGT14, Theorem 4.2.1]. The potential diagonalizability assumption is satisfied because r π,ι | GQ t , r π ′ ,ι | GQ t are both Fontaine-Laffaille, while the representations Sym n−1 r π,ι ∼ = Sym n−1 r π ′ ,ι are irreducible because the m th symmetric power of the standard representation of SL 2 (F t ) is irreducible whenever t > m. Proposition 3.10. Let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A Q ) of weight 2. Suppose that π is seasoned with respect to (q, t, r), and let p ∈ sc(π) satisfy p ≥ 5. Then we can find a regular algebraic, cuspidal automorphic representation π ′ of GL 2 (A Q ) with the following properties: (1) π ′ has weight 2 and is non-CM.
Proof. By Theorem 2.1, it's enough to find an isomorphism ι : Q p → C and a regular algebraic, cuspidal automorphic representation π ′ of GL 2 (A Q ) with the following properties: • The image of r π,ι contains a conjugate of SL 2 (F p a ) for some p a > 2n − 1.
• π ′ has weight 2 and is non-CM.
• If p = q then the conductor of π ′ p divides p 6 . 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 satisfying the first condition using Proposition 3.7; then the existence of a π ′ satisfying the above requirements is the content of Lemma 3.5.
Proof. If k ≥ 0, let (H k ) denote the hypothesis that the conclusion of the proposition holds when |sc(π)| ≤ k, and let (H ′ k ) 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 0 ) follows from the results of [NT19]. It therefore suffices to prove the implications (H k ) ⇒ (H ′ k+1 ) and (H ′ k ) ⇒ (H k ). The first implication follows immediately from Proposition 3.10. For the second, assume that (H ′ k ) holds and let π be a regular algebraic, cuspidal automorphic representation of GL 2 (A Q ) which is of weight 2 and non-CM, and such that |sc(π)| = k ≥ 1. By Proposition 3.9, we can find a regular algebraic, cuspidal automorphic representation π ′ of GL 2 (A Q ) which is seasoned with respect to (q, t, r), such that sc(π ′ ) = sc(π)∪{q}, and such that the existence of Sym n−1 π is equivalent to the existence of Sym n−1 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 2 (A Q ) which is seasoned with respect to (q, t, r), such that |sc(π ′′ )| = k, and such that the existence of Sym n−1 π ′′ implies that of Sym n−1 π ′ . The existence of Sym n−1 π ′′ follows from (H ′ k ), so we'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 automorphic representation of GL 2 (A Q ). We must show that Sym n−1 π exists. We first do this under the additional assumption that π has weight 2 and that 2 ∈ sc(π). We can assume that π 3 is supercuspidal. Fix a prime t > max(5, 4n(n − 1), N (π)) such that t ≡ 1 mod 4 and there exists an isomorphism ι t : Q t → C such that G = Proj r π,ιt (G Q ) is conjugate either to PSL 2 (F t ) or PGL 2 (F t ). Using the Chebotarev density theorem, we can find a prime q satisfying the following conditions: • The prime q satisfies q ≡ −1 mod t, q ≡ 1 mod 8, and q ≡ 1 mod l for every prime l < t. • The image of Frob q in G is in the conjugacy class of complex conjugation.
• There is an isomorphism rec Qq π ′ q ∼ = Ind WQ q W Q 2 q χ, where χ : W Q q 2 → C × is a character such that χ| IQ q has order t.
Theorem A.1. Let n ≥ 1. Let π be a cuspidal automorphic representation of GL 2 (A Q ) with π ∞ holomorphic limit of discrete series, or with π the automorphic induction of a Hecke character for a quadratic field. Then Sym n π exists.
Note that in these cases Sym n 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 π : G Q → GL 2 (C) with r π | WQ p ∼ = rec T Qp (π p ) for all primes p. The projective image of r π is a finite subgroup of PGL 2 (C), and is therefore dihedral or isomorphic to a copy of A 4 , S 4 or A 5 (moreover, each of these subgroups is unique up to conjugacy). We can then establish the automorphy of Sym n r π case by case, depending on the projective image. In the dihedral case, r π is induced from a character ψ of G K for K/Q quadratic, Sym n r π decomposes as a direct sum of characters and the inductions of characters from K to Q, and therefore Sym n r π is automorphic.
Twisting by a Dirichlet character, we can assume that r π (G Q ) = µ 2k Γ 1 (choose a prime p where π 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 n r π into irreducibles, it suffices to understand the decomposition of the representation Sym n C 2 of Γ 1 . See, for example, [Ste08, Appendix A] for the character tables of the binary polyhedral groups, or use [GAP20].
For the A 5 case, the irreducible representations of Γ 1 and their relationship to (symmetric powers of) the two Galois-conjugate irreducible two-dimensional representations are described in [Kim04,§5]. This allows automorphy of Sym n r π to be deduced from the automorphy of Sym m for m ≤ 4, together with tensor product functorialities GL 2 × GL 2 → GL 4 and GL 2 × GL 3 → GL 6 [Kim04, Theorem 6.4]. Now we turn to the A 4 case. Considering the character table of the binary tetrahedral 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 representations. Automorphy of Sym n r π therefore follows from automorphy of Sym 2 r π .
Finally, in the S 4 case, we consider the character table of the binary octahedral group. The irreducible representations of dimension > 1 are: • two faithful two-dimensional representations V 1 , V 2 , isomorphic up to twist, • a two-dimensional representation induced from a character of the normal index two subgroup, • two three-dimensional representations isomorphic to Sym 2 V 1 and its twist, • a four-dimensional representation isomorphic to Sym 3 V 1 .
So in this case automorphy of general symmetric powers follows from the automorphy of Sym m r π for m ≤ 3.
If π is an automorphic induction from a quadratic field K, as in the dihedral case, one can construct Sym n π as an isobaric direct sum of Hecke characters and automorphic inductions of Hecke characters for K.