The Sato-Tate Conjecture

for Modular Forms of Weight 3

Toby Gee^{1}

Received: August 26, 2009 Revised: December 15, 2009

Communicated by Don Blasius

Abstract. We prove a natural analogue of the Sato-Tate conjecture for modular forms of weight 2 or 3 whose associated automorphic representations are a twist of the Steinberg representation at some finite place.

2000 Mathematics Subject Classification: 11F33.

Contents

1. Introduction 772 2. Notation and assumptions 775 3. Modular forms 779 4. Potential automorphy in weight 0 781 5. Changing weight 785 6. Potential automorphy 794 7. The Sato-Tate Conjecture 797 References 799

1The author was partially supported by NSF grant DMS-0841491

1. Introduction

The Sato-Tate conjecture is a conjecture about the distribution of the number of points on an elliptic curve over finite fields. Specifically, if E is an elliptic curve over ℚ without CM, then for each prime l such that E has good reduction at l we set

a_{l} := 1 + l − #E(𝔽_{l}).

Then the Sato-Tate conjecture states that the quantities cos^{−1}(a_{l}/2\sqrt{l}) are equidistributed with respect to the measure

\frac{2}{π} sin^{2} θdθ

on [0, π]. Alternatively, by the Weil bounds for E, the polynomial

X^{2} − a_{l}X + l = (X − α_{l}l^{1/2})(X − β_{l}l^{1/2})

satisfies |α_{l}| = |β_{l}| = 1, and there is a well-defined conjugacy class x_{E,l} in SU(2), the conjugacy class of the matrix

(\begin{matrix}α_{l} & 0 \\ 0 & β_{l} \end{matrix}) .

The Sato-Tate conjecture is then equivalent to the statement that the classes x_{E,l} are equidistributed with respect to the Haar measure on SU(2). Tate observed that the conjecture would follow from properties of the symmetric power L-functions of E, specifically that these L-functions (suitably normalised) should have nonvanishing analytic continuation to the region ℜs ≥ 1. This would follow (given the modularity of elliptic curves) from the Langlands conjectures (specifically, it would be a consequence of the symmetric power functoriality from GL_{2} to GL_{n} for all n). Unfortunately, proving this functoriality appears to be well beyond the reach of current techniques. However, Harris, Shepherd-Baron and Taylor observed that the required analytic properties would follow from a proof of the potential automorphy of the symmetric power L-functions (that is, the automorphy of the L-functions after base change to some extension of ℚ), and were able to use Taylor’s potential automorphy techniques to prove the Sato-Tate conjecture for all elliptic curves E with nonintegral j-invariant (see [HSBT09]). There are various possible generalisations of the Sato-Tate conjecture; if one wishes to be maximally ambitious, one could consider equidistribution results for the Satake parameters of rather general automorphic representations (see for example section 2 of [Lan79]). Again, such results appear to be well beyond the range of current technology. There is, however, one special case that does seem to be reasonable to attack, which is the case of Hilbert cuspidal eigenforms of regular weight. In this paper, we prove a natural generalisation of the Sato-Tate conjecture for modular newforms (over ℚ) of weight 2 or 3, subject to the natural analogue of the condition that an elliptic curve has non-integral j-invariant. We note that previously the only modular forms for which the conjecture was known were those corresponding to elliptic curves; in particular, there were no examples of weight 3 modular forms for which the conjecture was known. After this paper was made available, the conjecture was proved for all modular forms of weight at least 2 in [BLGHT09], by rather different methods. Our approach is similar to that of [HSBT09], and we are fortunate in being able to quote many of their results. Indeed, it is straightforward to check that Tate’s argument shows that the conjecture would follow from the potential automorphy of the symmetric powers of the l-adic Galois representations associated to a modular form. One might then hope to prove this potential automorphy in the style of [HSBT09]; one would proceed by realising the symmetric powers of the mod l Galois representation geometrically in such a way that their potential automorphy may be established, and then deduce the potential automorphy of the l-adic representations by means of the modularity lifting theorems of [CHT08] and [Tay08]. It turns out that this simple strategy encounters some significant obstacles. First and foremost, it is an unavoidable limitation of the known potential automorphy methods that they can only deduce that a mod l Galois representation is automorphic of minimal weight (which we refer to as “weight 0”). However, the symmetric powers of the Galois representations corresponding to modular forms of weight greater than 2 are never automorphic of minimal weight, so one has no hope of directly proving their potential automorphy in the fashion outlined above without some additional argument. If, for example, one knew the weight part of Serre’s conjecture for GL_{n} (or even for unitary groups) one would be able to deduce the required results, but this appears to be an extremely difficult problem in general. There is, however, one case in which the analysis of the Serre weights is rather easier, which is the case that the l-adic Galois representations are ordinary. It is this observation that we exploit in this paper. In general, it is anticipated that for a given newform f of weight k ≥ 2, there is a density one set of primes l such that there is an ordinary l-adic Galois representation corresponding to f. Unfortunately, if k > 3 then it is not even known that there is an infinite set of such primes; this is the reason for our restriction to k = 2 or 3. In these cases, one may use the Ramanujan conjecture and Serre’s form of the Cebotarev density theorem (see [Ser81]) to prove that the set of l which are “ordinary” in this sense has density one, via an argument that is presumably well-known to the experts (although we have not been able to find the precise argument that we use in the literature). We note that it is important for us to be able to choose l arbitrarily large in certain arguments (in order to satisfy the hypotheses of the automorphy lifting theorems of [Tay08]), so it does not appear to be possible to apply our methods to any modular forms of weight greater than 3. Similarly, we cannot prove anything for Hilbert modular forms of parallel weight 3 over any field other than ℚ. We now outline our arguments in more detail, and explain exactly what we prove. The early sections of the paper are devoted to proving the required potential automorphy results. In section 2 we recall some basic definitions and results from [CHT08] on the existence of Galois representations attached to regular automorphic representations of GL_{n} over totally real and CM fields, subject to suitable self-duality hypotheses and to the existence of finite places at which the representations are square integrable. Section 3 recalls some standard results on the Galois representations attached to modular forms, and proves the result mentioned above on the existence of a density one set of primes for which there is an ordinary Galois representation. In section 4 we prove the potential automorphy in weight 0 of the symmetric powers of the residual Galois representations associated to a modular form, under the hypotheses that the residual Galois representation is ordinary and irreducible, and the automorphic representation corresponding to the modular form is an unramified twist of the Steinberg representation at some finite place. The latter condition arises because of restrictions of our knowledge as to when there are Galois representations associated to automorphic representations on unitary groups, and it is anticipated that it will be possible to remove it in the near future. That would then allow us to prove our main theorems for any modular forms of weights 2 or 3 which are not of CM type. (Note added in proof: such results are now available, cf. [Shi09], [CH09], [Gue09], and it is thus an easy exercise to deduce our main results without any Steinberg assumption.) One approach to proving the potential automorphy result in weight 0 would be to mimic the proofs for elliptic curves in [HSBT09]. In fact we can do better than this, and are able to directly utilise their results. We are reduced to proving that after making a quadratic base change and twisting, the mod l representation attached to our modular form is, after a further base change, congruent to a mod l representation arising from a certain Hilbert-Blumenthal abelian variety. This is essentially proved in [Tay02], and we only need to make minor changes to the proofs in [Tay02] in order to deduce the properties we need. We can then directly apply one of the main results of [HSBT09] to deduce the automorphy of the even-dimensional symmetric powers of the Hilbert-Blumenthal abelian variety, and after twisting back we deduce the required potential automorphy of our residual representations. Note that apart from resulting in rather clean proofs, the advantage of making an initial congruence to a Galois representation attached to an abelian variety and then using the potential automorphy of the symmetric powers of this abelian variety is that we are able to obtain local-global compatibility at all finite places (including those dividing the residue characteristic). This compatibility is not yet available for automorphic representations on unitary groups in general, and is needed in our subsequent arguments. In particular, it tells us that the automorphic representations of weight 0 which correspond to the symmetric powers of the l-adic representations coming from our Hilbert-Blumenthal abelian variety are ordinary at l. In section 5 we exploit this ordinarity to deduce that the even-dimensional symmetric powers of the mod l representations are potentially automorphic of the “correct” weight. This is a basic consequence of Hida theory for unitary groups, but we are not aware of any reference that proves the precise result we need. Accordingly, we provide a proof in the style of the arguments of [Tay88].

There is nothing original in this section, and as the arguments are somewhat technical the reader may wish to skip it on a first reading. The results of the preceding sections are combined in section 6 to establish the required potential automorphy results for l-adic (rather than mod l) representations. This essentially comes down to checking the hypotheses of the modularity lifting theorem that we wish to apply from [Tay08], which follow from the analogous arguments in [HSBT09] together with the conditions that we have imposed in our potential automorphy arguments. It is here that we need the freedom to choose l to be arbitrarily large, which results in our restriction to weights 2 and 3. Finally, in section 7 we deduce the form of the Sato-Tate conjecture mentioned above. As in [HSBT09] we have only proved the potential automorphy of the even-dimensional symmetric powers of the l-adic representations associated to our modular form, and we deduce the required analytic properties for the Lfunctions attached to odd-dimensional symmetric powers via an argument with Rankin-Selberg convolutions exactly analogous to that of [HSBT09]. In fact, we need to prove the same results for the L-functions of certain twists of our representations by finite-order characters, but this is no more difficult. We now describe the form of the final result, which is slightly different from that for elliptic curves, because our modular forms may have non-trivial nebentypus (and indeed are required to do so if they have weight 3). Suppose that the newform f has level N, nebentypus χ_{f} and weight k; then the image of χ_{f} is precisely the m-th roots of unity for some m. Then if p ∤ N is a prime, we know that if

X^{2} − a_{p}X + p^{k−1}χ_{f}(p) = (X − α_{p}p^{(k−1)/2})(X − β_{p}p^{(k−1)/2})

where a_{p} is the eigenvalue of f for the Hecke operator T_{p}, then the matrix

(\begin{matrix}α_{p} & 0 \\ 0 & β_{p} \end{matrix})

defines a conjugacy class x_{f,p} in U(2)_{m}, the subgroup of U(2) of matrices with determinant an m-th root of unity. Then our main result is

Theorem. If f has weight 2 or 3 and the associated automorphic representation is a twist of the Steinberg representation at some finite place, then the conjugacy classes x_{f,p} are equidistributed with respect to the Haar measure on U(2)_{m} (normalised so that U(2)_{m} has measure 1).

One can make this more concrete by restricting to primes p such that χ_{f}(p) is a specific m-th root of unity; see the remarks at the end of section 7. We would like to thank Thomas Barnet-Lamb, David Geraghty and Richard Taylor for various helpful discussions during the writing of this paper.

2. Notation and assumptions

We let ϵ denote the l-adic cyclotomic character, regarded as a character of the absolute Galois group of a number field or of a completion of a number field at a finite place. We sometimes use the same notation for the mod l cyclotomic character; it will always be clear from the context which we are referring to. We denote Tate twists in the usual way, i.e. ρ(n) := ρ ⊗ ϵ^{n}. We write K̄ for a separable closure of a field K. If x is a finite place of a number field F, we will write I_{x} for the inertia subgroup of Gal(\overline{F}_{x}/F_{x}). We fix an algebraic closure \overline{ℚ} of ℚ, and regard all finite extensions of ℚ as being subfields of \overline{ℚ}. We also fix algebraic closures \overline{ℚ}_{p} of ℚ_{p} for all primes p, and embeddings \overline{ℚ} ↪ \overline{ℚ}_{p}. We need several incarnations of the local Langlands correspondence. Let K be a finite extension of ℚ_{p}, and l ≠ p a prime. We have a canonical isomorphism

Art_{K} : K^{×} → W_{K}^{ab}

normalised so that geometric Frobenius elements correspond to uniformisers. Let Irr(GL_{n}(K)) denote the set of isomorphism classes of irreducible admissible representations of GL_{n}(K) over ℂ, and let WDRep_{n}(W_{K}) denote the set of isomorphism classes of n-dimensional Frobenius semi-simple complex Weil-Deligne representations of the Weil group W_{K} of K. The main result of [HT01] is that there is a family of bijections

rec_{K} : Irr(GL_{n}(K)) → WDRep_{n}(W_{K})

satisfying a number of properties that specify them uniquely (see the introduction to [HT01] for a complete list). Among these properties are:

• If π ∈ Irr(GL_{1}(K)) then rec_{K}(π) = π ∘ Art_{K}^{−1}.
• rec_{K}(π^{∨}) = rec_{K}(π)^{∨}.

• If χ_{1}, …, χ_{n} ∈ Irr(GL_{1}(K)) are such that the normalised induction

n-Ind(χ_{1}, …, χ_{n}) is irreducible, then
rec_{K}(n-Ind(χ_{1}, …, χ_{n})) = ⊕_{i=1}^{n} rec_{K}(χ_{i}).

We will often just write rec for rec_{K} when the choice of K is clear from the context. After choosing an isomorphism ι : \overline{ℚ}_{l} → ℂ one obtains bijections rec_{l} from the set of isomorphism classes of irreducible admissible representations of GL_{n}(K) over \overline{ℚ}_{l} to the set of isomorphism classes of n-dimensional Frobenius semi-simple Weil-Deligne \overline{ℚ}_{l}-representations of W_{K}. We then define r_{l}(π) to be the l-adic representation of Gal(\overline{K}/K) associated to rec_{l}(π^{∨} ⊗ | · |^{(1−n)/2}) whenever this exists (that is, whenever the eigenvalues of rec_{l}(π^{∨}⊗|·|^{(1−n)/2})(ϕ) are l-adic units, where ϕ is a Frobenius element). We will, of course, only use this notation where it makes sense. It is useful to note that

r_{l}(π)^{∨}(1 − n) = r_{l}(π^{∨}).

Let M denote a CM field with maximal totally real subfield F (by “CM field” we always mean “imaginary CM field”). We denote the nontrivial element of Gal(M/F) by c. Following [CHT08] we define a RACSDC (regular, algebraic, conjugate self dual, cuspidal) automorphic representation of GL_{n}(𝔸_{M}) to be a cuspidal automorphic representation π such that

• π^{∨} ≅ π^{c}, and • π_{∞} has the same infinitesimal character as some irreducible algebraic

representation of Res_{M/ℚ} GL_{n}.

We say that a ∈ (ℤ^{n})^{Hom(M,ℂ)} is a weight if • a_{τ,1} ≥ ⋯ ≥ a_{τ,n} for all τ ∈ Hom(M, ℂ), and

• a_{τc,i} = −a_{τ,n+1−i}.

For any weight a we may form an irreducible algebraic representation W_{a} of GL_{n}^{Hom(M,ℂ)}, the tensor product over τ of the irreducible algebraic representations of GL_{n} with highest weight a_{τ}. We say that π has weight a if it has the same infinitesimal character as W_{a}^{∨}; note that any RACSDC automorphic representation has some weight. Let S be a non-empty finite set of finite places of M. For each v ∈ S, choose an irreducible square integrable representation ρ_{v} of GL_{n}(M_{v}) (in this paper, we will in fact only need to consider the case where each ρ_{v} is the Steinberg representation). We say that an RACSDC automorphic representation π has type {ρ_{v}}_{v∈S} if for each v ∈ S, π_{v} is an unramified twist of ρ_{v}^{∨}. There is a compatible family of Galois representations associated to such a representation in the following fashion.

Proposition 2.1. Let ι : \overline{ℚ}_{l} ⟶^{∼} ℂ. Suppose that π is an RACSDC automorphic representation of GL_{n}(𝔸_{M}) of type {ρ_{v}}_{v∈S} for some nonempty set of finite places S. Then there is a continuous semisimple representation

r_{l,ι}(π) : Gal(\overline{M}/M) → GL_{n}(\overline{ℚ}_{l})

such that (1) For each finite place v ∤ l of M, we have

r_{l,ι}(π)|_{Gal(\overline{M}_{v}/M_{v})}^{ss} = r_{l}(ι^{−1}π_{v})^{∨}(1 − n)^{ss}.

(2) r_{l,ι}(π)^{c} = r_{l,ι}(π)^{∨}ϵ^{1−n}.

Proof. This follows from Proposition 4.2.1 of [CHT08] (which in fact also gives information on r_{l,ι}|_{Gal(\overline{M}_{v}/M_{v})} for places v|l). □

The representation r_{l,ι}(π) may be conjugated to be valued in the ring of integers of a finite extension of ℚ_{l}, and we may reduce it modulo the maximal ideal of this ring of integers and semisimplify to obtain a well-defined continuous representation

r̄_{l,ι}(π) : Gal(\overline{M}/M) → GL_{n}(\overline{𝔽}_{l}).

Let a ∈ (ℤ^{n})^{Hom(M,\overline{ℚ}_{l})}, and let ι : \overline{ℚ}_{l} ⟶^{∼} ℂ. Define ι_{∗}a ∈ (ℤ^{n})^{Hom(M,ℂ)} by (ι_{∗}a)_{ιτ,i} = a_{τ,i}. Now let ρ_{v} be a discrete series representation of GL_{n}(M_{v}) over \overline{ℚ}_{l} for each v ∈ S. If r : Gal(\overline{M}/M) → GL_{n}(\overline{ℚ}_{l}), we say that r is automorphic of weight a and type {ρ_{v}}_{v∈S} if r ≅ r_{l,ι}(π) for some RACSDC automorphic representation π of weight ι_{∗}a and type {ιρ_{v}}_{v∈S}. Similarly, if r̄: Gal(\overline{M}/M) → GL_{n}(\overline{𝔽}_{l}), we say that r̄ is automorphic of weight a and type {ρ_{v}}_{v∈S} if r̄ ≅ r̄_{l,ι}(π) for some RACSDC automorphic representation π with π_{l} unramified, of weight ι_{∗}a and type {ιρ_{v}}_{v∈S}.

We now consider automorphic representations of GL_{n}(𝔸_{F}). We say that a cuspidal automorphic representation π of GL_{n}(𝔸_{F}) is RAESDC (regular, algebraic, essentially self dual, cuspidal) if

• π^{∨} ≅ χπ for some character χ : F^{×}\𝔸_{F}^{×} → ℂ^{×} with χ_{v}(−1) independent of v|∞, and

• π_{∞} has the same infinitesimal character as some irreducible algebraic representation of Res_{F/ℚ} GL_{n}.

We say that a ∈ (ℤ^{n})^{Hom(F,ℂ)} is a weight if

a_{τ,1} ≥ ⋯ ≥ a_{τ,n}

for all τ ∈ Hom(F, ℂ). For any weight a we may form an irreducible algebraic representation W_{a} of GL_{n}^{Hom(F,ℂ)}, the tensor product over τ of the irreducible algebraic representations of GL_{n} with highest weight a_{τ}. We say that an RAESDC automorphic representation π has weight a if it has the same infinitesimal character as W_{a}^{∨}. In this case, by the classification of algebraic characters over a totally real field, we must have a_{τ,i} + a_{τ,n+1−i} = w_{a} for some w_{a} independent of τ. Let S be a non-empty finite set of finite places of F. For each v ∈ S, choose an irreducible square integrable representation ρ_{v} of GL_{n}(M_{v}). We say that an RAESDC automorphic representation π has type {ρ_{v}}_{v∈S} if for each v ∈ S, π_{v} is an unramified twist of ρ_{v}^{∨}. Again, there is a compatible family of Galois representations associated to such a representation in the following fashion.

Proposition 2.2. Let ι : \overline{ℚ}_{l} ⟶^{∼} ℂ. Suppose that π is an RAESDC automorphic representation of GL_{n}(𝔸_{F}), of type {ρ_{v}}_{v∈S} for some nonempty set of finite places S, with π^{∨} ≅ χπ. Then there is a continuous semisimple representation

r_{l,ι}(π) : Gal(\overline{F} /F) → GL_{n}(\overline{ℚ}_{l})

such that (1) For each finite place v ∤ l of F, we have

r_{l,ι}(π)|_{Gal(\overline{F}_{v}/F_{v})}^{ss} = r_{l}(ι^{−1}π_{v})^{∨}(1 − n)^{ss}.

(2) r_{l,ι}(π)^{∨} = r_{l,ι}(χ)ϵ^{n−1}r_{l,ι}(π). Here r_{l,ι}(χ) is the l-adic Galois representation associated to χ via ι (see Lemma 4.1.3 of [CHT08]).

Proof. This is Proposition 4.3.1 of [CHT08] (which again obtains a stronger result, giving information on r_{l,ι}|_{Gal(\overline{F}_{v}/F_{v})} for places v|l). □

Again, the representation r_{l,ι}(π) may be conjugated to be valued in the ring of integers of a finite extension of ℚ_{l}, and we may reduce it modulo the maximal ideal of this ring of integers and semisimplify to obtain a well-defined continuous representation

r̄_{l,ι}(π) : Gal(\overline{F} /F) → GL_{n}(\overline{𝔽}_{l}).

Let a ∈ (ℤ^{n})^{Hom(F,\overline{ℚ}_{l})}, and let ι : \overline{ℚ}_{l} ⟶^{∼} ℂ. Define ι_{∗}a ∈ (ℤ^{n})^{Hom(F,ℂ)} by (ι_{∗}a)_{ιτ,i} = a_{τ,i}. Let ρ_{v} be a discrete series representation of GL_{n}(M_{v}) over \overline{ℚ}_{l} for each v ∈ S. If r : Gal(\overline{F} /F) → GL_{n}(\overline{ℚ}_{l}), we say that r is automorphic of weight a and type {ρ_{v}}_{v∈S} if r ≅ r_{l,ι}(π) for some RAESDC automorphic representation π of weight ι_{∗}a and type \{ιρ_{v}\}_{v∈S}. Similarly, if r̄: Gal(\overline{F} /F) → GL_{n}(\overline{𝔽}_{l}), we say that r̄ is automorphic of weight a and type \{ρ_{v}\}_{v∈S} if r̄ ≅ r̄_{l,ι}(π) for some RAESDC automorphic representation π with π_{l} unramified, of weight ι_{∗}a and type {ιρ_{v}}_{v∈S}. As in [HSBT09] we denote the Steinberg representation of GL_{n}(K), K a nonarchimedean local field, by Sp_{n}(1).

3. Modular forms

3.1. Let f be a cuspidal newform of level Γ_{1}(N), nebentypus χ_{f}, and weight k ≥ 2. Suppose that for each prime p ∤ N we have T_{p}f = a_{p}f. Then each a_{p} is an algebraic integer, and the set {a_{p}} generates a number field K_{f} with ring of integers O_{f}. We will view K_{f} as a subfield of ℂ. It is known that K_{f} contains the image of χ_{f}. For each place λ|l of O_{f} there is a continuous representation

ρ_{f,λ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(K_{f,λ})

which is determined up to isomorphism by the property that for all p ∤ Nl, ρ_{f,λ}|_{Gal(\overline{ℚ}_{p}/ℚ_{p})} is unramified, and the characteristic polynomial of ρ_{f,λ}(Frob_{p}) is X^{2} − a_{p}X + p^{k−1}χ_{f}(p) (where Frob_{p} is a choice of a geometric Frobenius element at p). Assume from now on that f is not of CM type.

Definition 3.1. Let λ be a prime of O_{f} lying over a rational prime l. Then we say that f is ordinary at λ if λ ∤ a_{l}. We say that f is ordinary at l if it is ordinary at λ for some λ|l.

Lemma 3.2. If k = 2 or 3, then the set of primes l such that f is ordinary at l has density one.

Proof. The proof is based on an argument of Wiles (see the final lemma of [Wil88]). Let S be the finite set of primes which either divide N or which are ramified in O_{f}. Suppose that f is not ordinary at p ∉ S. By definition we have that λ|a_{p} for each prime λ of O_{f} lying over p. Since p is unramified in O_{f},

(p) = ∏_{λ|p} λ, so p|a_{p}. Write a_{p} = pb_{p} with b_{p} ∈ O_{f}. Since p ∤ N, the Weil bounds (that is, the Ramanujan-Petersson conjecture) tell us that for each embedding ι : K_{f} ↪ ℂ we have |ι(a_{p})| ≤ 2p^{(k−1)/2}. Since k ≤ 3, this implies that |ι(b_{p})| ≤ 2 for all ι. Let T be the set of y ∈ O_{f} such that |ι(y)| ≤ 2 for all ι. This is a finite set, because one can bound the absolute values of the coefficients of the characteristic polynomial of such a y. From the above analysis, it is sufficient to prove that for each y ∈ T, the set of primes p for which a_{p} = py has density zero. However, by Corollaire 1 to Théorème 15 of [Ser81], the number of primes p ≤ x for which a_{p} = py is O(x/(log x)^{5/4−δ}) for any δ > 0, which immediately shows that the density of such primes is zero, as required. □

The following result is well known, and follows from, for example, [Sch90] and Theorem 2 of [Wil88].

Lemma 3.3. If f is ordinary at a place λ|l of O_{f}, and l ∤ N, then the Galois representation ρ_{f,λ} is crystalline, and furthermore it is ordinary; that is,

ρ_{f,λ}|_{Gal(\overline{ℚ}_{l}/ℚ_{l})} ≅ (\begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2}ϵ^{1−k} \end{matrix})

where ψ_{1} and ψ_{2} are unramified characters of finite order. In addition, ψ_{1} takes Frob_{l} to the unit root of X^{2} − a_{l}X + χ_{f}(l)l^{k−1}.

3.2. Let \overline{ρ}_{f,λ} denote the semisimplification of the reduction mod λ of ρ_{f,λ}; this makes sense because ρ_{f,λ} may be conjugated to take values in GL_{2}(O_{f,λ}), and it is independent of the choice of lattice. It is valued in GL_{2}(k_{f,λ}), where k_{f,λ} is the residue field of K_{f,λ}.

Definition 3.4. We say that \overline{ρ}_{f,λ} has large image if

SL_{2}(k) ⊂ \overline{ρ}_{f,λ}(Gal(\overline{ℚ}/ℚ)) ⊂ k_{f,λ}^{×} GL_{2}(k)

for some subfield k of k_{f,λ}.

We will need to know that the residual Galois representations \overline{ρ}_{f,λ} frequently have large image. The following result is essentially due to Ribet (see [Rib75], which treats the case N = 1; for a concrete reference, which also proves the corresponding result for Hilbert modular forms, see [Dim05]).

Lemma 3.5. For all but finitely many primes λ of O_{f}, \overline{ρ}_{f,λ} has large image.

3.3. We let π(f) be the automorphic representation of GL_{2}(𝔸_{ℚ}) corresponding to f, normalised so that π(f) is RAESDC of weight (k − 2, 0) (it is essentially self dual because

π(f)^{∨} ≅ χπ(f)

where χ = | · |^{k−2}χ_{f}^{−1}). Let λ|l be a place of O_{f}, and choose an isomorphism ι : \overline{ℚ}_{l} ⟶^{∼} ℂ and a compatible embedding K_{f,λ} ↪ \overline{ℚ}_{l}; that is, an embedding such that the diagram

\xymatrix{K_{f} \ar[r] \ar[d] & ℂ \\ K_{f,λ} \ar[r] & _{\overline{ℚ}_{l}} \ar[u]_{ι}}

commutes. Assume that π_{f,v} is square integrable for some finite place v. Then by Proposition 2.2 there is a Galois representation

r_{l,ι}(π(f)) : Gal(\overline{ℚ}/ℚ) → GL_{2}(\overline{ℚ}_{l})

associated to π_{f}, and it follows from the definitions that

r_{l,ι}(π(f)) ≅ ρ_{f,λ} ⊗_{K_{f,λ}} \overline{ℚ}_{l}.

Definition 3.6. We say that f is Steinberg at a prime q if π(f)_{q} is an unramified twist of the Steinberg representation.

Definition 3.7. We say that f is potentially Steinberg at a prime q if π(f)_{q} is a (possibly ramified) twist of the Steinberg representation.

Note that if f is (potentially) Steinberg at q for some q then it is not CM. Note also that if f is potentially Steinberg at q then there is a Dirichlet character θ such that f ⊗ θ is Steinberg at q.

4. Potential automorphy in weight 0

4.1. Let l be an odd prime, and let f be a modular form of weight 2 ≤ k < l and level N, l ∤ N. Assume that f is Steinberg at q. Suppose that λ|l is a place of O_{f} such that f is ordinary at λ. Assume that \overline{ρ}_{f,λ} is absolutely irreducible. By Lemma 3.3 we have

\overline{ρ}_{f,λ}|_{Gal(\overline{ℚ}_{l}/ℚ_{l})} ≅ (\begin{matrix}\overline{ψ}_{1} & ∗ \\ 0 & \overline{ψ}_{2}ϵ^{1−k} \end{matrix})

where \overline{ψ}_{1} and \overline{ψ}_{2} are unramified characters. We wish to prove that the symmetric powers of \overline{ρ}_{f,λ} are potentially automorphic of some weight. To do so, we use a potential modularity argument to realise \overline{ρ}_{f,λ} geometrically, and then appeal to the results of [HSBT09]. The potential modularity result that we need is almost proved in [Tay02]; the one missing ingredient is that we wish to preserve the condition of being Steinberg at q. This is, however, easily arranged, and rather than repeating all of the arguments of [Tay02], we simply indicate the modifications required. We begin by recalling some definitions from [Tay02]. Let N be a totally real field. Then an N-HBAV over a field K is a triple (A, i, j) where

• A/K is an abelian variety of dimension [N : ℚ],

• i : O_{N} ↪ End(A/K), and

• j : O_{N}^{+} ⟶^{∼} P(A, i) is an isomorphism of ordered invertible O_{N}-

modules. For the definitions of ordered invertible O_{N}-modules and of O_{N}^{+} and P(A, i), see page 133 of [Tay02]. Choose a totally real quadratic field F in which l is inert and q is unramified and which is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ, a finite extension k/k_{f,λ} and a character \overline{θ} : Gal(\overline{F} /F) → k^{×} which is unramified at q such that

det \overline{ρ}_{f,λ}|_{Gal(\overline{F} /F)} = ϵ^{−1}\overline{θ}^{−2}

and (\overline{ρ}_{f,λ}|_{Gal(\overline{F} /F)} ⊗ \overline{θ})(Frob_{w}) has eigenvalues 1, #k(w), where w|q is a place of F. This is possible as the obstruction to taking a square root of a character is in the 2-part of the Brauer group, and because any class in the Brauer group of a local field splits over an unramified extension. Let \overline{ρ} = \overline{ρ}_{f,λ}|_{Gal(\overline{F} /F)} ⊗ \overline{θ} :

Gal(\overline{F} /F) → GL_{2}(k), so that det \overline{ρ} = ϵ^{−1}. If x is the place of F lying over l, then we may write (for some character \overline{χ}_{x} of Gal(\overline{F_{x}}/F_{x}))

\overline{ρ}|_{Gal(\overline{F_{x}}/F_{x})} ≅ (\begin{matrix}\overline{χ}_{x}^{−1} & ∗ \\ 0 & \overline{χ}_{x}ϵ^{−1} \end{matrix})

with \overline{χ}_{x}^{2}|_{I_{x}} = ϵ^{2−k}.

Theorem 4.1. There is a finite totally real Galois extension E/F which is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ and in which the unique prime of F dividing l splits completely, a totally real field N, a place λ′|l of N, a place v_{q}|q of E, and an N-HBAV (A, i, j)/E with potentially good reduction at all places dividing l such that

• the representation of Gal(Ē/E) on A[λ′] is equivalent to (\overline{ρ}|_{Gal(Ē/E)})^{∨},

• at each place x|l of E, the action of Gal(\overline{E_{x}}/E_{x}) on T_{λ′}A ⊗ ℚ_{l} is of the form

(\begin{matrix}χ_{x}^{−1}ϵ & ∗ \\ 0 & χ_{x} \end{matrix})

with χ_{x} a tamely ramified lift of \overline{χ}_{x}, and

• A has multiplicative reduction at v_{q}.

Proof. As remarked above, this is essentially proved in [Tay02]. Indeed, if k > 2 then with the exception of the fact that E can be chosen to be linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ, and the claim that A can be chosen to have multiplicative reduction at some place over q, the result is obtained on page 136 of [Tay02] (the existence of A with A[λ′] equivalent to (\overline{ρ}|_{Gal(Ē/E)})^{∨} is established in the second paragraph on that page, and the form of the action of Gal(\overline{E_{x}}/E_{x}) for x|l follows from Lemma 1.5 of loc. cit. ). We now indicate the modifications needed to the arguments of [Tay02] to obtain the slight strengthening that we require. Suppose firstly that k > 2. Rather than employing the theorem of Moret-Bailly stated as Theorem G of [Tay02], we use the variant given in Proposition 2.1 of [HSBT09]. This immediately allows us to assume that E is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ, so we only need to ensure that A has multiplicative reduction at some place dividing q. Let X be the moduli space defined in the first paragraph of page 136 of [Tay02]. Let v be a place of F lying over q. It is enough to check that there is a non-empty open subset Ω_{v} of X(F_{v}) such that for each point of Ω_{v}, the corresponding N-HBAV has multiplicative reduction. Let Ω_{v} denote the set of all points of X(F_{v}) such that the corresponding N-HBAV has multiplicative reduction; this is an open subset of X(F_{v}), and it is non-empty (by the assumptions on θ̄ at places of F dividing q, and the assumption that π(f) is an unramified twist of the Steinberg representation, we see that \overline{ρ}(Frob_{v}) has eigenvalues 1 and #k(v), and is congruent to a Galois representation attached to an unramified twist of a Steinberg representation, so any N-HBAV with multiplicative reduction suffices), as required.

If k = 2, then the only additional argument needed is one to ensure that if \overline{χ}_{x}^{2} = 1, then the abelian variety can be chosen to have good reduction rather than multiplicative reduction. This follows easily from the fact that \overline{ρ}|_{Gal(\overline{F_{x}}/F_{x})} is finite flat (cf. the proof of Theorem 2.1 of [KW08], which establishes a very similar result). □

Let M be a totally real field, and let (A, i, j)/M be an N-HBAV. Fix an embedding N ⊂ ℝ. We recall some definitions from section 4 of [HSBT09]. For each finite place v of M there is a two dimensional Weil-Deligne representation WD_{v}(A, i) defined over \overline{N} such that if 𝔭 is a place of N of residue characteristic p different from the residue characteristic of v, we have

WD(H^{1}(A × \overline{M}, ℚ_{p})|_{Gal(\overline{M}_{v}/M_{v})} ⊗_{N_{p}} \overline{N}_{𝔭}) ≅ WD_{v}(A, i) ⊗\overline{N} \overline{N}_{𝔭}.

Definition 4.2. We say that Sym^{m} A is automorphic of type {ρ_{v}}_{v∈S} if there is an RAESDC representation π of GL_{m+1}(𝔸_{M}) of weight 0 and type {ρ_{v}}_{v∈S} such that for all finite places v of M,

rec(π_{v})| Art_{M_{v}}^{−1} |^{−m/2} = Sym^{m} WD_{v}(A, i).

Theorem 4.3. Let E, A be as in the statement of Theorem 4.1. Let N be a finite set of even positive integers. Then there is a finite Galois totally real extension F′/E and a place w_{q}|q of F′ such that

• for any n ∈ N, Sym^{n−1} A is automorphic over F′ of weight 0 and type {Sp_{n}(1)}_{{w_{q}}},

• The primes of E dividing l are unramified in F′, and

• F′ is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ.

Proof. This is essentially Theorem 4.1 of [HSBT09]. In particular, the proof in [HSBT09] establishes that there is a Galois totally real extension F′/E, and a place w_{q} of F′ lying over q such that for any n ∈ N, Sym^{n−1} A is automorphic over F′ of weight 0 and type {Sp_{n}(1)}_{{w_{q}}}. Note that the l used in their argument is not the l used here. To complete the proof, we need to establish that it is possible to obtain an F′ in which l is unramified, and which is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ. The latter point causes no difficulty, but the first point requires some minor modifications of the arguments of [HSBT09]. We now outline the necessary changes. To aid comparison to [HSBT09], for the rest of this proof we will refer to our l as s; all references to l will be to primes of that name in the proofs of various theorems in [HSBT09]. We begin by choosing a finite solvable totally real extension L of E, linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ, such that the base change of A to L has good reduction at all places dividing s. Choose a prime l as in the proof of Theorem 4.1 of [HSBT09]. We then apply a slight modification of Theorem 4.2 of loc.cit., with the conclusion strengthened to include the hypothesis that s is unramified in F′. To prove this, in the proof of Theorem 4.2 of loc.cit., note that F_{1} = E. Choose all auxiliary primes not to divide s. Rather than constructing a moduli space X_{W} over E, construct the analogous space over L, and consider the restriction of scalars Y = Res_{L/E}(X_{W}). Applying Proposition 2.1 of [HSBT09] to Y, rather than X_{W}, we may find a finite totally real Galois extension F^{(1)}/E in which s is unramified, such that Y has an F^{(1)}- point. Furthermore, we may assume that F^{(1)} is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ. Note that an F^{(1)}-point of Y corresponds to an F^{(1)}L-point of X_{W}. We now make a similar modification to the proof of Theorem 3.1 of [HSBT09], replacing the schemes T_{W_{i}} over F with Res_{LF/F} T_{W_{i}}. We conclude that there is a finite Galois totally real extension F′/E in which s is unramified, which is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ, such that for any n ∈ N, Sym^{n−1} A is automorphic over F′L of weight 0 and type {Sp_{n}(1)}_{{w_{q}}}. Since the extension F′L/F′ is solvable, it follows from solvable base change (e.g. Lemma 1.3 of [BLGHT09]) that in fact for any n ∈ N, Sym^{n−1} A is automorphic over F′ of weight 0 and type {Sp_{n}(1)}_{{w_{q}}}, as required. □

We may now twist \overline{ρ} by \overline{θ}^{−1} in order to deduce results about \overline{ρ}_{f,λ}. Let N and λ be as in the statement of Theorem 4.1. Fix an embedding N_{λ′} ↪ \overline{ℚ}_{l}. Let θ be the Teichmüller lift of \overline{θ}, and let ρ_{n} denote the action of Gal(Ē/E) on

Sym^{n−1}(H^{1}(A × \overline{E}, ℚ_{l}) ⊗_{N_{l}} N_{λ′} ⊗ θ^{−1}) ⊗_{N′_{λ}} \overline{ℚ}_{l}.

By construction, ρ_{n} is a lift of Sym^{n−1} \overline{ρ}_{f,λ}|_{Gal(\overline{E}/E)} ⊗_{k_{f,λ}} 𝔽̄_{l} (where the embedding k_{f,λ} ↪ \overline{𝔽}_{l} is determined by the embedding k ↪ \overline{𝔽}_{l} induced by the embedding N_{λ′} ↪ \overline{ℚ}_{l}). Note also that (again by construction) at each place x|l of E,

ρ_{2}|_{Gal(\overline{E_{x}}/E_{x})} ≅ (\begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2}ω^{2−k}ϵ^{−1} \end{matrix})

with ψ_{1}, ψ_{2} unramified lifts of \overline{ψ}_{1}|_{Gal(\overline{E_{x}}/E_{x})} and \overline{ψ}_{2}|_{Gal(\overline{E_{x}}/E_{x})} respectively, and ω the Teichmüller lift of ϵ.

Corollary 4.4. Let N be a finite set of even positive integers. Then there is a Galois totally real extension F′/E and a place w_{q}|q of F′ such that

• for any n ∈ N, ρ_{n}|_{Gal(\overline{ℚ}/F′)} is automorphic of weight 0 and type {Sp_{n}(1)}_{{w_{q}}},

• every prime of E dividing l is unramified in F′ (so that l is unramified in F′), and

• F′ is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ.

Let ι : \overline{ℚ}_{l} ⟶^{∼} ℂ, and for n ∈ N let π_{n} be the RAESDC representation of GL_{n}(𝔸_{F′}) with r_{l,ι}(π_{n}) ≅ ρ_{n}|_{Gal(\overline{ℚ}_{l}/F′)}. If k = 2 then π_{n,x} is unramified for each x|l, and if k > 2 then for each place x|l of F′, π_{n,x} is a principal series representation n-Ind_{B_{n}(F′_{x})}^{GL_{n}(F′_{x})}(χ_{1}, …, χ_{n}) with ι^{−1}χ_{i} ∘ Art_{F′_{x}}^{−1} |_{I_{x}} = ω^{(i−1)(2−k)} and v_{l}(ι^{−1}χ_{i}(l)) = [F′_{x} : ℚ_{l}] (i − 1 + \frac{1−n}{2}), where v_{l} is the l-adic valuation on \overline{ℚ}_{l} with v_{l}(l) = 1.

Proof. This is a straightforward consequence of Theorem 4.3. The only part that needs to be checked is the assertion about the form of π_{n,x} for x|l when k > 2. Without loss of generality, we may assume that 2 ∈ N. Note firstly that any principal series representation of the given form is irreducible, so that we need only check that

ι^{−1} rec(π_{n,x}) = ⨁_{i=1}^{n} ω^{(i−1)(2−k)}α_{i},

where α_{i} is an unramified character with v_{l}(α_{i}(l)) = [F′_{x} : ℚ_{l}] (i − 1 + \frac{1−n}{2}). By Definition 4.2 and Theorem 4.3 we see that rec(π_{n,x}) = Sym^{n−1} rec(π_{2,x}), so it suffices to establish the result in the case n = 2, or rather (because of the compatibility of rec with twisting) it suffices to check the corresponding result for WD_{v}(A, i) at places v|l. This is now an immediate consequence of localglobal compaitibility, and follows at once from, for example, Lemma B.4.1 of [CDT99], together with the computations of the Weil-Deligne representations associated to characters in section B.2 of loc. cit.

□

5. Changing weight

5.1. We now explain how to deduce from the results of the previous section that Sym^{n} \overline{ρ}_{f,λ} is potentially automorphic of the correct weight (that is, the weight of the conjectural automorphic representation corresponding to Sym^{n} ρ_{f,λ}), rather than potentially automorphic of weight 0. We accomplish this as a basic consequence of Hida theory; note that we simply need a congruence, rather than a result about families, and the result follows from a straightforward combinatorial argument. This result is certainly known to the experts, but as we have been unable to find a reference which provides the precise result we need, we present a proof in the spirit of the arguments of [Tay88].

5.2. For each n-tuple of integers a = (a_{1}, …, a_{n}) with a_{1} ≥ ⋯ ≥ a_{n} there is an irreducible representation of the algebraic group GL_{n} defined over ℚ_{l}, with highest weight (with respect to the Borel subgroup of upper-triangular matrices) given by

diag(t_{1}, …, t_{n}) ↦ ∏_{i=1}^{n} t_{i}^{a_{i}}.

We will need an explicit model of this representation, for which we follow section 2 of [Che04]. Let K be an algebraic extension of ℚ_{l}, N the subgroup of GL_{n}(K) consisting of upper triangular unipotent matrices, \overline{N} the subgroup of lower triangular unipotent matrices, and T the subgroup of diagonal matrices. Let R := K[GL_{n}] = K[{X_{i,j}}_{1≤i,j≤n}, det(X_{i,j})^{−1}]. We have commuting natural actions of GL_{n}(K) on R by left and right multiplication. For an element g ∈ GL_{n}(K) we denote these actions by g_{l} and g_{r} respectively, so that if we let M denote the matrix (X_{i,j})_{i,j} ∈ M_{n}(R), we have

(g_{l}.X)_{i,j} = g^{−1}M

and

(g_{r}.X)_{i,j} = Mg.

If (t_{1}, …, t_{n}) ∈ ℤ^{n}, we say that an element f ∈ R is of left weight t (respectively of right weight t) if for all d ∈ T we have d_{l}f = t^{−1}(d)f (respectively d_{r}f = t(d)f) where

t(diag(x_{1}, …, x_{n})) = ∏_{i=1}^{n} x_{i}^{t_{i}}.

For each 1 ≤ i ≤ n and each i-tuple j = (j_{1}, …, j_{i}), 1 ≤ j_{1} < ⋯ < j_{i} ≤ n, we let Y_{i,j} be the minor of order i of M obtained by taking the entries from the first i rows and columns j_{1},…,j_{i}. Let R\overline{N} denote the subalgebra of R of elements fixed by the g_{l}-action of \overline{N}; it is easy to check that Y_{i,j} ∈ R\overline{N}. Because T normalises \overline{N} it acts on R\overline{N} on the left, and we let R\overline{N} ; t be the sub K-vector space of elements of left weight t; this has a natural action of GL_{n}(K) induced by g_{r}.

Proposition 5.1. Suppose that t_{1} ≥ ⋯ ≥ t_{n}. Then R\overline{N} ; t is a model of the irreducible algebraic representation of GL_{n}(K) of highest weight t. Furthermore, it is generated as a K-vector space by the monomials in Y_{i,j} of left weight t, and a highest weight vector is given by the unique monomial in Y_{i,j} of left and right weight t.

Proof. This follows from Proposition 2.2.1 of [Che04]. □

Assume that in fact t_{1} ≥ ⋯ ≥ t_{n} ≥ 0, and let X_{t} denote the free O_{K}-module with basis the monomials in Y_{i,j} of left weight t. By Proposition 5.1, X_{t} is a GL_{n}(O_{K})-stable lattice in R\overline{N} ; t. Let T^{+} be the submonoid of T consisting of elements of the form

diag(l^{b_{1}}, …, l^{b_{n}})

with b_{1} ≥ ⋯ ≥ b_{n} ≥ 0; then X_{t} is certainly also stable under the action of T^{+}. Let α = diag(l^{b_{1}}, …, l^{b_{n}}) ∈ T^{+}. We wish to determine the action of α on X_{t}.

Lemma 5.2. If Y ∈ X_{t} is a monomial in the Y_{i,j}, then α(Y) ⊂ l^{∑_{i=1}^{n} b_{i}t_{n+1−i}}X_{t}. If in fact b_{1} > ⋯ > b_{n} then α(Y) ⊂ l^{1+∑_{i=1}^{n} b_{i}t_{n+1−i}}X_{t} unless Y is the unique lowest weight vector.

Proof. If Y has (right) weight (v_{1}, …, v_{n}), then α(Y) = l^{∑_{i=1}^{n} b_{i}v_{i}}Y. The unique lowest weight vector has weight (t_{n}, …, t_{1}), so it suffices to prove that for any other Y of weight (v_{1}, …, v_{n}) which occurs in R\overline{N} ; t, the quantity ∑_{i=1}^{n} b_{i}v_{i} is at least as large, and is strictly greater if b_{1} > ⋯ > b_{n}. However, by standard weight theory we know that we may obtain (v_{1}, …, v_{n}) from (t_{n}, …, t_{1}) by successively adding vectors of the form (0, …, 1, 0 …, 0, −1, 0, …, 0), and it is clear that the addition of such a vector does not decrease the sum, and in fact increases it if b_{1} > ⋯ > b_{n}, as required. □

We define a new action of T^{+} on X_{t}, which we denote by ·_{twist}, by multiplying the natural action of diag(l^{b_{1}}, …, l^{b_{n}}) by l^{− ∑_{i=1}^{n} b_{i}t_{n+1−i}}; this is legitimate by Lemma 5.2.

5.3. Fix for the rest of this section a choice of isomorphism ι : \overline{ℚ}_{l} ⟶^{∼} ℂ. Assume for the rest of this section that F′ is a totally real field in which each l is unramified, and π′ is an RAESDC representation of GL_{n}(𝔸_{F′}) of weight 0 and type {Sp_{n}(1)}_{{w_{q}}} for some place w_{q}|q of F′, with (π′)^{∨} = χπ′. Suppose furthermore that there is an integer k > 2 such that

• for each place x|l, π′_{x} is a principal series n-Ind_{B_{n}(F_{x})}^{GL_{n}(F_{x})}(χ_{1}, …, χ_{n}) with v_{l}(ι^{−1}χ_{i}(l)) = [F′_{x} : ℚ_{l}] (i − 1 + \frac{1−n}{2}) and ι^{−1}χ_{i} ∘ Art_{F_{x}}^{−1} |_{I_{x}} =

ω^{(i−1)(2−k)}.

(See Corollary 4.4 for an example of such a representation.) We transfer to a unitary group, following section 3.3 of [CHT08]. Firstly, we make a quartic totally real Galois extension F/F′, linearly disjoint from \overline{ℚ}^{ker \overline{r}_{l,ι}(π)} over ℚ, such that w_{q} and all primes dividing l split in F. Let S(B) be the set of places of F lying over w_{q}. Let E be a imaginary quadratic field in which l and q split, such that E is linearly disjoint from \overline{ℚ}^{ker \overline{r}_{l,ι}(π)} over ℚ. Let M = F E. Let c denote the nontrivial element of Gal(M/F). Let S_{l} denote the places of F dividing l, and let S̃_{l} denote a set of places of M dividing l such that the natural map S̃_{l} → S_{l} is a bijection. If v|l is a place of F then we write ṽ for the corresponding place in S̃_{l}.

Lemma 5.3. There is a finite order character ϕ : M^{×}\𝔸_{M}^{×} → ℂ^{×} such that • ϕ ∘ N_{M/F} = χ ∘ N_{M/F}, and

• ϕ is unramified at all places lying over S(B) and at all places in S̃_{l}.

Proof. By Lemma 4.1.1 of [CHT08] (or more properly its proof, which shows that the character produced may be arranged to have finite order) there is a finite order character ψ : M^{×}\𝔸_{M}^{×} → ℂ^{×} such that for each v ∈ S_{l}, ψ|_{Mṽ}^{×} = 1 and ψ|_{M_{cṽ}}^{×} = χ|_{F_{v}^{×}}, and such that ψ is unramified at each place in S(B). It now suffices to prove the result for the character χ(ψ|_{𝔸_{F}^{×}})^{−1}, which is unramified at S(B) ∪ S_{l}, and the result now follows from Lemma 4.1.4 of [CHT08]. □

Now let π = π′_{M} ⊗ ϕ, which is an RACSDC representation of GL_{n}(𝔸_{M}), satisfying:

• π has weight 0.

• π has type {Sp_{n}(1)}_{w|w_{q}}.

• for each place x ∈ S̃_{l}, π_{x} is a principal series n-Ind_{B_{n}(M_{x})}^{GL_{n}(M_{x})}(χ_{1}, …, χ_{n}) with v_{l}(ι^{−1}χ_{i}(l)) = [F′_{x} : ℚ_{l}] (i − 1 + \frac{1−n}{2}) and ι^{−1}χ_{i} ∘ Art_{M_{x}}^{−1} |_{I_{x}} = ω^{(i−1)(2−k)} with k > 2.

5.4. Choose a division algebra B with centre M such that

• B splits at all places not dividing a place in S(B).

• If w is a place of M lying over a place in S(B), then B_{w} is a division algebra.

• dim_{M} B = n^{2}.

• B^{op} ≅ B ⊗_{M,c} M.

For any involution ‡ on B with ‡|_{M} = c, we may define a reductive algebraic group G_{‡}/F by

G_{‡}(R) = {g ∈ B ⊗_{F} R : g^{‡⊗1}g = 1}

for any F -algebra R. Because [F : ℚ] is divisible by 4 and #S(B) is even, we may (by the argument used to prove Lemma 1.7.1 of [HT01]) choose ‡ such that

• If v ∉ S(B) is a finite place of F then G_{‡}(F_{v}) is quasi-split, and

• If v|∞, G_{‡}(F_{v}) ≅ U(n).

Fix such a choice of ‡, and write G for G_{‡}. We wish to work with algebraic modular forms on G; in order to do so, we need an integral model for G. We obtain such a model by fixing an order O_{B} in B such that O_{B}^{‡} = O_{B} and O_{B,w} is a maximal order for all primes w which are split over M (see section 3.3 of [CHT08] for a proof that such an order exists). We now regard G as an algebraic group over O_{F}, by defining

G(R) = {g ∈ O_{B} ⊗_{O_{F}} R : g^{‡⊗1}g = 1}

for all O_{F} -algebras R. We may identify G with GL_{n} at places not in S(B) which split in M in the following way. Let v ∉ S(B) be a place of F which splits in M. Choose an isomorphism i_{v} : O_{B,v} ⟶^{∼} M_{n}(O_{M_{v}}) such that i_{v}(x^{‡}) = ^{t}i_{v}(x)^{c} (where ^{t} denotes matrix transposition). Choosing a prime w|v of M gives an isomorphism

i_{w} : G(F_{v}) ⟶^{∼} GL_{n}(M_{w})
i_{v}^{−1}(x,^{t}x^{−c}) ↦ x.

This identification satisfies i_{w}G(O_{F,v}) = GL_{n}(O_{M,w}). Similarly, if v ∈ S(B) then v splits in M, and if w|v then we obtain an isomorphism

i_{w} : G(F_{v}) ⟶^{∼} B_{w}^{×}

with i_{w}G(O_{F,v}) = O_{B,w}^{×}. Now let K = \overline{ℚ}_{l}. Write O for the ring of integers of K, and k for the residue field \overline{𝔽}_{l}. Let I_{l} = Hom(F, K), and let Ĩ_{l} be the subset of elements of Hom(M, K) such that the induced place of M is in S̃_{l}. Let a ∈ (ℤ^{n})^{Hom(M,K)}; we assume that

• a_{τ,1} ≥ ⋯ ≥ a_{τ,n} ≥ 0 if τ ∈ Ĩ_{l}, and
• a_{τc,i} = −a_{τ,n+1−i}.

Consider the constructions of section 5.2 applied to our choice of K. Then we have an O-module

Y_{a} = ⊗_{τ∈Ĩ_{l}}X_{a_{τ}}

which has a natural action of G(O_{F,l}), where g ∈ G(O_{F,l}) acts on X_{a_{τ}} by τ(i_{τ} g_{τ}). From now on, if v|l is a place of F, we will identify G(O_{F_{v}}) with GL_{n}(O_{Mṽ}) via the map iṽ without comment. We say that an open compact subgroup U ⊂ G(A_{F}^{∞}) is sufficiently small if for some place v of F the projection of U to G(F_{v}) contains no nontrivial elements of finite order. Assume from now on that U is sufficiently small, and in addition that we may write U = ∏_{v} U_{v}, U_{v} ⊂ G(O_{F_{v}}), such that

• if v ∈ S(B) and w|v is a place of M, then i_{w}(U_{v}) = O_{B,w}^{×}, and

• if v|l then U_{v} is the Iwahori subgroup of matrices which are upper-

triangular mod l. If v|l, let U′_{v} denote the pro-l subgroup of U_{v} corresponding to the group of matrices which are (upper-triangular) unipotent mod l, and let

χ_{v} : U_{v}/U′_{v} → O^{×}

be a character. Let χ = ⊗χ_{v} : ∏_{v|l} U_{v} → O^{×}, and write

Y_{a,χ} = Y_{a} ⊗_{O} χ,

a ∏_{v|l} U_{v}-module. Let A be an O-algebra. Then we define the space of algebraic modular forms

S_{a,χ}(U, A)

to be the space of functions

f : G(F)\G(𝔸_{F}^{∞}) → A ⊗_{O} Y_{a,χ}

satisfying

f(gu) = u^{−1}f(g)

for all u ∈ U, g ∈ G(𝔸_{F}^{∞}), where the action of U on A ⊗_{O} Y_{a,χ} is inherited from the action of ∏_{v|l} U_{v} on Y_{a,χ}. Note that because U is sufficiently small we have

S_{a,χ}(U, A) = S_{a,χ}(U, O) ⊗_{O} A.

More generally, if V is any U′′-module with U′′ a sufficiently small compact open subgroup, we define the space of algebraic modular forms

S(U′′, V)

to be the space of functions

f : G(F)\G(𝔸_{F}^{∞}) → V

satisfying

f(gu) = u^{−1}f(g)

for all u ∈ U′′, g ∈ G(𝔸_{F}^{∞}). Let T_{l}^{+} denote the monoid of elements of G(𝔸_{F}^{∞}) which are trivial outside of places dividing l, and at places dividing l correspond to matrices diag(l^{b_{1}}, …, l^{b_{n}}) with b_{1} ≥ ⋯ ≥ b_{n} ≥ 0. In addition to the action of U on Y_{a,χ}, we can also allow T_{l}^{+} to act. We define the action of T_{l}^{+} via the action ·_{twist} on X_{t} defined above. This gives us an action of the monoid ⟨U, T_{l}^{+}⟩ on Y_{a,χ}. Now suppose that g is an element of G(𝔸_{F}^{∞}) with either g_{l} ∈ G(O_{F,l}) or g ∈ T_{l}^{+}; then we write

UgU = ∐_{i} g_{i}U,

a finite union of cosets, and define a linear map

[UgU] : S_{a,χ}(U, A) → S_{a,χ}(U, A)

by

([UgU]f)(h) = ∑_{i} g_{i}f(hg_{i}).

We now introduce some notation for Hecke algebras. Let v be a place of F which splits in M, and suppose that v ∉ S(B) and that U_{v} = G(O_{F_{v}}) (so, in particular v ∤ l). Suppose that w|v is a place of M, so that we may regard G(O_{F_{v}}) as GL_{n}(O_{M_{w}}) via i_{w}. Then we let T_{w}^{(j)}, 1 ≤ j ≤ n denote the Hecke operator given by

[U diag(ϖ_{w}, …, ϖ_{w}, 1, …, 1)U]

where ϖ_{w} is a uniformiser of M_{w}, and there are j occurrences of it in this matrix. We let 𝕋_{a,χ}(U, A) denote the commutative A-subalgebra of End(S_{a,χ}(U, A)) generated by the operators T_{w}^{(j)} and (T_{w}^{(n)})^{−1} for all w, j as above. Note that 𝕋_{a,χ}(U, A) commutes with [UgU] for all g ∈ T_{l}^{+}. More generally, let 𝕋(U) denote the polynomial ring over O in the formal variables T_{w}^{(j)} and (T_{w}^{n})^{−1}, which we may think of as acting on S_{a,χ}(U, A) via the obvious map 𝕋(U) → 𝕋_{a,χ}(U, A). We also wish to consider the Hecke operator U_{l} = [UuU], where u ∈ T_{l}^{+} has u_{v} = diag(l^{n−1}, …, l, 1) for each v|l. As usual, we can define a Hida idempotent

e_{l} = lim_{n→∞} U_{l}^{n!},

which has the property that U_{l} is invertible on e_{l}S_{a,χ}(U, O) and is topologically nilpotent on (1 − e_{l})S_{a,χ}(U, O). We write

S_{a,χ}^{ord}(U, A) := e_{l}S_{a,χ}(U, A).

Let a ∈ (ℤ^{n})^{Hom(M,K)} be a weight, and let χ_{a} = ⊗_{v|l}χ_{a,v}, where χ_{a,v} : U_{v}/U′_{v} ≅ ((O_{M_{v}}/𝔪_{M_{v}})^{×})^{n} → O^{×} is given by the character (x_{1}, …, x_{n}) ↦ ∏_{τ} ∏_{i} τ(x̃_{i})^{a_{ṽ,n+1−i}}, where x̃_{i} is the Teichmüller lift of x_{i}, and the product is over the embeddings τ ∈ Ĩ_{l} which give rise to v. The main lemma we require is the following.

Lemma 5.4. Let a be a weight. Then there is a 𝕋(U)-equivariant isomorphism

S_{a,χ}^{ord}(U, k) → S_{0,χχ_{a}}^{ord}(U, k).

Proof. Note firstly that there is a natural projection map j from Y_{a,χ} to the O-module given by the tensor product Z_{a,χ} of the lowest weight vectors. This is a map of ∏_{v|l} U_{v}-modules, and by Lemma 5.2 we see that j induces an isomorphism

u ·_{twist}Y_{a,χ} ⊗_{O} k → u ·_{twist}Z_{a,χ} ⊗_{O} k.

Note also that by definition we have an isomorphism of ⟨U, T_{l}^{+}⟩-modules Z_{a,χ} → Y_{0,χχ_{a}}. It thus suffices to prove that the induced map

j : S_{a,χ}^{ord}(U, k) → S^{ord}(U, Z_{a,χ} ⊗_{O} k) (= S_{0,χχ_{a}}^{ord}(U, k))

is an isomorphism. We claim that there is a diagram

\xymatrix{S_{a,χ}(U, k) \ar[r]^{j} & S(U, Z_{a,χ} ⊗_{O} k) \ar[r]^{u·_{twist}} & S(U ∩ uUu^{−1}, u ·_{twist}Z_{a,χ} ⊗_{O} k) \ar[d]^{j^{−1}} \\  & S_{a,χ}(U ∩ uUu^{−1}, k) \ar[ul]^{cor} & S(U ∩ uUu^{−1}, u ·_{twist} Y_{a,χ} ⊗_{O} k) \ar[l]_{i}}

such that the maps

cor ∘i ∘ j^{−1} ∘ u ·_{twist} ∘ j : S_{a,χ}(U, k) → S_{a,χ}(U, k)

and

j ∘ cor ∘i ∘ j^{−1} ∘ u ·_{twist} : S(U, Z_{a,χ} ⊗_{O} k) → S(U, Z_{a,χ} ⊗_{O} k)

are both given by U_{l}. Since U_{l} is an isomorphism on S_{a,χ}^{ord}(U, k), the result will follow. In fact, the construction of the diagram is rather straightforward. The maps j, j^{−1} are just the natural maps on the coefficients (note that both are maps of U-modules). The map u ·_{twist} is given by

(u ·_{twist}f)(h) = u ·_{twist} f(hu).

The map i is given by the inclusion of U-modules u·_{twist}Y_{a,χ}⊗_{O} k ↪ Y_{a,χ}⊗_{O} k. Finally, the map cor is defined in the following fashion. We may write

U = ∐ u_{i}(U ∩ uUu^{−1}),

and we define

(cor f)(h) = ∑ u_{i}f(hu_{i}).

The claims regarding the compositions of these maps follow immediately from the observation that

UuU = ∐ u_{i}uU.
□

5.5. We now recall some results on tamely ramified principal series representations of GL_{n} from [Roc98]. Let L be a finite extension of ℚ_{p} for some p, and let π_{L} be an irreducible smooth complex representation of GL_{n}(L). Let I denote the Iwahori subgroup of GL_{n}(O_{L}) consisting of matrices which are upper-triangular mod 𝔪_{L}, and let I_{1} denote its Sylow pro-l subgroup. Let l be the residue field of L, and let ϖ_{L} denote a uniformiser of L. Then there is a natural isomorphism I/I_{1} ≅ (l^{×})^{n}. If χ = (χ_{1}, …, χ_{n}) : (l^{×})^{n} → ℂ^{×} is a character, then we let π_{L}^{I,χ} denote the space of vectors in π_{L} which are fixed by I_{1} and transform by χ under the action of I/I_{1}. The space π_{L}^{I,χ} has a natural action of the Hecke algebra H(I, χ) of compactly supported χ^{−1}-spherical functions on GL_{n}(L). We consider the commutative subalgebra 𝕋(I, χ) of H(I, χ) generated by double cosets [IαI] where α = diag(ϖ_{L}^{b_{1}}, …, ϖ_{L}^{b_{n}}) with b_{1} ≥ ⋯ ≥ b_{n} ≥ 0. If χ : (O_{L}^{×})^{n} → ℂ^{×} is tamely ramified, then we let π_{L}^{I,χ} denote π_{L}^{I,χ̄}, where χ̄ is the character (l^{×})^{n} → ℂ^{×} determined by χ. Let δ denote the modulus character of GL_{n}(L), so that

δ(diag(a_{1}, …, a_{n})) = |a_{1}|^{n−1}|a_{2}|^{n−3} … |a_{n}|^{1−n}

where | · | denotes the usual norm on L.

Proposition 5.5. (1) If π_{L}^{I} ≠ 0 then π is a subquotient of an unramified

principal series representation.

(2) If π_{L}^{I_{1}} ≠ 0 then π is a subquotient of a tamely ramified principal series representation. More precisely, if π_{L}^{I,χ} ≠ 0 then π_{L} is a subquotient of a tamely ramified principal series representation n-Ind_{B_{n}(L)}^{GL_{n}(L)}(χ′_{1}, …, χ′_{n}) with χ′_{i} extending χ_{i} for each i.

(3) If π_{L} = n-Ind_{B_{n}(L)}^{GL_{n}(L)}(χ) with χ tamely ramified, then
π_{L}^{I,χ} ≅ ⊕_{w}χδ^{−1/2}

as a 𝕋(I, χ)-module, where the sum is over the elements w of the Weyl group of GL_{n} with χ^{w} = χ; that is, the double coset [IαI] acts via (χδ^{−1/2})(α) on π_{L}^{I,χ}.

Proof. The first two parts follow from Lemma 3.1.6 of [CHT08] and its proof. All three parts follow at once from Theorem 7.7 and Remark 7.8 of [Roc98] (which are valid for GL_{n} without any restrictions on L - see the proof of Lemma 3.1.6 of [CHT08]), together with the standard calculation of the Jacquet module of a principal series representation, for which see for example Theorem 6.3.5 of [Cas95] (although note that there is a missing factor of δ^{1/2} (or rather δ_{Ω}^{1/2} in the notation of loc. cit.) in the formula given there). □

5.6. Keep our running assumptions on π. Suppose that U = ∏_{v} U_{v} is a sufficiently small subgroup of G(𝔸_{F}). Assume further that U has been chosen such that if v ∉ S(B), v = ww^{c} splits completely in M, and U_{v} is a maximal compact subgroup of G(F_{v}), then π_{w} is unramified. Recall that we have fixed an isomorphism ι : \overline{ℚ}_{l} ⟶^{∼} ℂ. There is a maximal ideal 𝔪_{ι,π} of 𝕋(U) determined by π in the following fashion. For each place v = ww^{c} as above the Hecke operators T_{w}^{(i)} act via scalars α_{w,i} on (π_{w})^{GL_{n}(O_{Mw})}. The α_{w,i} are all algebraic integers, so that ι^{−1}(α_{w,i}) ∈ O. Then 𝔪_{ι,π} is the maximal ideal of 𝕋(U) containing all the T_{w}^{(i)} − ι^{−1}(α_{w,i}). Let σ_{k} ∈ (ℤ^{n})^{Hom(M,K)} be the weight determined by (σ_{k})_{τ,i} = (k − 2)(n − i) for each τ ∈ Ĩ_{l}.

Lemma 5.6. Suppose that π is a RACSDC representation of GL_{n}(𝔸_{M}) of weight 0 and type {Sp_{n}(1)}_{S(B)}. Suppose that for each place x ∈ S̃_{l}, π_{x} is a principal series n-Ind_{B_{n}(M_{x})}^{GL_{n}(M_{x})}(χ_{x,1}, …, χ_{x,n}) with ι^{−1}χ_{x,i} ∘ Art_{F_{x}}^{−1} |_{I_{x}} = ω^{(i−1)(2−k)}. Then there is a sufficiently small compact open subgroup U of G(𝔸_{F}) such that U satisfies the requirements above (in particular, U = ∏_{v} U_{v} where U_{v} is an Iwahori subgroup of GL_{n}(F_{v}) for each v|l) and S_{0,χ_{σ_{k}}} (U, O)_{𝔪_{ι,π}} ≠ 0. If we assume furthermore that v_{l}(ι^{−1}χ_{x,i}(l)) = [M_{x} : ℚ_{l}] (i − 1 + \frac{1−n}{2}) for all i (and all x ∈ S̃_{l}) then S_{0,χ_{σ_{k}}}^{ord} (U, O)_{𝔪_{ι,π}} ≠ 0.

Proof. This is a consequence of Proposition 3.3.2 of [CHT08]. The only issues are at places dividing l and places in S(B). For the latter, it is enough to note that under the Jacquet-Langlands correspondence, Sp_{n}(1) corresponds to the trivial representation. For the first part, we also need to check that at each place x ∈ S̃_{l}, π_{x}^{I_{x},χ_{x}} ≠ 0, where I_{x} is the standard Iwahori subgroup of GL_{n}(M_{x}), and χ_{x} = (χ_{x,1}, …, χ_{x,n}). This follows at once from Proposition 5.5. For the second part, we must check in addition that if the Hecke operator [I_{x}u_{x}I_{x}] (where u_{x} = diag(l^{n−1}, …, 1)) acts via the scalar α_{x} on π_{x}^{I_{x},χ_{x}}, then ι^{−1}(α_{x}) is an l-adic unit. This is straightforward; by Proposition 5.5(3), α_{x} = χ_{x}(u)δ^{−1/2}(u). Thus

v_{l}(ι^{−1}(α_{x})) = v_{l}(ι^{−1}(χ_{x}(u)δ^{−1/2}(u)))

= ∑_{i=1}^{n}(n − i)v_{l}(ι^{−1}χ_{x,i}(l)) + ∑_{i=1}^{n}(n − i)v_{l}((l^{−[M_{x}:ℚ_{l}]})^{−(n+1−2i)/2)})

= ∑_{i=1}^{n}(n − i)([M_{x} : ℚ_{l}] (i − 1 + \frac{1 − n}{2}))+

+ ∑_{i=1}^{n}[M_{x} : ℚ_{l}](n − i)(n + 1 − 2i)/2

= \frac{[M_{x} : ℚ_{l}]}{2} ∑_{i=1}^{n}(n − i)((2i − 1 − n) + (n + 1 − 2i))

= 0,

as required. □

Lemma 5.7. Keep (all) the assumptions of Lemma 5.6. Then there is an RACSDC representation π′′ of GL_{n}(𝔸_{M}) of weight ι_{∗}σ_{k}, type {Sp_{n}(1)}_{{S(B)}} and with π′′_{l} unramified such that \overline{r}_{l,ι}(π′′) ≅ r̄_{l,ι}(π).

Proof. This is essentially a consequence of Lemma 5.6, Lemma 5.4, and Proposition 5.5, together with Proposition 3.3.2 of [CHT08]. Indeed, Lemma 5.4 and Lemma 5.6 show that S_{σ_{k},1}^{ord}(U, O)_{𝔪_{ι,π}} ≠ 0, which by Proposition 5.5(1) and Proposition 3.3.2 of [CHT08] gives us a π′′ satisfying all the properties we claim, except that we only know that for each x|l, π′′_{x} is a subquotient of an unramified principal series representation. We claim that this unramified principal series is irreducible, so that π′′_{x} is unramified. To see this, note that the fact that we know that S_{σ_{k},1}^{ord}(U, O)_{𝔪_{ι,π}} ≠ 0 (rather than merely S_{σ_{k},1}(U, O)_{𝔪_{ι,π}} ≠ 0) means that we can choose π′′ so that for each x ∈ S̃_{l}, π′′_{x} is a subquotient of an unramified principal series representation n-Ind_{B_{n}(M_{x})}^{GL_{n}(M_{x})}(χ_{x,1}, …, χ_{x,n}) with

v_{l}(ι^{−1}χ_{x,i}(l)) = [M_{x} : ℚ_{l}] ((i − 1)(k − 1) + (1 − n)/2)

(this follows from the comparison of the Hecke actions on (π′′_{x})^{I_{x}} and S_{σ_{k},1}(U, O), noting that the latter action is defined in terms of ·_{twist}). Now, if the principal series n-Ind_{B_{n}(M_{x})}^{GL_{n}(M_{x})}(χ_{x,1}, …, χ_{x,n}) were reducible, there would be i, j with χ_{x,i} = χ_{x,j}|·|, so that χ_{x,i}(l)l^{[M_{x}:ℚ_{l}]} = χ_{x,j}(l), which is a contradiction because k > 2. The result follows. □

Combining Corollary 4.4 with Lemma 5.7, we obtain

Proposition 5.8. Let l be an odd prime, and let f be a modular form of weight 2 ≤ k < l and level coprime to l. Assume that f is Steinberg at q, and that for some place λ|l of O_{f}, f is ordinary at λ and \overline{ρ}_{f,λ} is absolutely irreducible. Fix an embedding K_{f,λ} ↪ \overline{ℚ}_{l}. Let N be a finite set of even positive integers. Then there is a Galois totally real extension F/ℚ and a quadratic imaginary field E, together with a place w_{q}|q of M = F E such that if we choose a set S̃_{l} of places of M consisting of one place above each place of F dividing l, and define σ_{k} ∈ (ℤ^{n})^{Hom(M,K)} by (σ_{k})_{τ,i} = (k − 2)(n − i), then

• for each n ∈ N, there is a character ϕ̄_{n} : Gal(\overline{M}/M) → \overline{𝔽}_{l}^{×} which is unramified at all places in S̃_{l}, which satisfies

ϕ̄_{n}ϕ̄_{n}^{c} = (ϵ det \overline{ρ}_{f,λ} ⊗ \overline{𝔽}_{l})^{1−n}|_{Gal(\overline{M}/M)}

and (Sym^{n−1} \overline{ρ}_{f,λ} ⊗ \overline{𝔽}_{l})|_{Gal(\overline{M}/M)} ⊗ ϕ̄_{n} is automorphic of weight σ_{k} and type {Sp_{n}(1)}_{{w_{q}}}.

• l is unramified in M.

• M is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ.

6. Potential automorphy

6.1. Assume as before that f is a cuspidal newform of level Γ_{1}(N), weight k ≥ 2, and nebentypus χ_{f}. Let π(f) be the RAESDC representation of GL_{2}(𝔸_{ℚ}) corresponding to f. We will think of χ_{f} as an automorphic representation of GL_{1}(𝔸_{ℚ}), and write χ_{f} = ⊗_{p}χ_{f,p}. We now define what we mean by the claim that the symmetric powers of f are potentially automorphic. If F is a totally real field and v|p is a place of F, we write rec(π_{f,p})|_{F_{v}} for the restriction of the Weil-Deligne representation rec(π_{f,p}) to the Weil group of F_{v}. Then we say that Sym^{n−1} f is potentially automorphic over F if there is an RAESDC representation π_{n} of GL_{n}(𝔸_{F}) such that for all primes p and all places v|p of F we have

rec(π_{n,v}) = Sym^{n−1}(rec(π_{f,p})|_{F_{v}}).

By a standard argument (see for example section 4 of [HSBT09]) this is equivalent to asking that Sym^{n−1} ρ_{f,λ}|_{Gal(\overline{F} /F)} be automorphic for some place (equivalently for all places) λ of K_{f}. Similarly, we may speak of Sym^{n−1} f being potentially automorphic of a specific weight and type. We then define (for each n ≥ 1 and each integer a) the L-series

L(χ_{f}^{a} ⊗ Sym^{n−1} f, s) = ∏_{p} L((χ_{f,p}^{a} ∘ Art_{ℚ_{p}}^{−1}) ⊗ Sym^{n−1} rec(π_{f,p}), s + (1 − n)/2).

We now normalise the L-functions of RAESDC automorphic representations to agree with those of their corresponding Galois representations. Specifically, if π is an RAESDC representation of GL_{n}(𝔸_{F}), we define

L(π, s) = ∏_{v∤∞} L(π_{v}, s + (1 − n)/2).

If π is square integrable at some finite place, then for each isomorphism ι : \overline{ℚ}_{l} ⟶^{∼} ℂ there is a Galois representation r_{l,ι}(π), and by definition we have

L(π, s) = ∏_{v∤∞} L(π_{v} ⊗ (| · | ∘ det)^{(1−n)/2}, s)

= ∏_{v∤∞} L(rec(π_{v} ⊗ (| · | ∘ det)^{(1−n)/2}), s)

= ∏_{v∤∞} L(r_{l}(ι^{−1}π_{v})^{∨}(1 − n), s)

= L(r_{l,ι}(π), s).

Theorem 6.1. Suppose that f is a cuspidal newform of level Γ_{1}(N) and weight k = 2 or 3. Suppose that f is Steinberg at q. Let N be a finite set of even positive integers. Then there is a Galois totally real field F such that for any n ∈ N and any subfield F′ ⊂ F with F/F′ soluble, Sym^{n−1} f is automorphic over F′.

Proof. By Lemma 3.2 and Lemma 3.5 we may choose a prime l > 3 and a place λ of O_{f} lying over l such that

• l ∤ N.

• f is ordinary at λ.

• l > max(2n + 1)_{n∈N}.

• \overline{ρ}_{f,λ} has large image.

By Corollary 5.8 there is an embedding K_{f,λ} ↪ \overline{ℚ}_{l}, a Galois totally real extension F/ℚ and a quadratic imaginary field E, together with a place w_{q}|q of M = F E such that if we choose a set S̃_{l} of places of M consisting of one place above each place of F dividing l, and define σ_{k} ∈ (ℤ^{n})^{Hom(M,K)} by (σ_{k})_{τ,i} = (k − 2)(n − i), then

• for each n ∈ N, there is a character ϕ̄_{n} : Gal(\overline{M}/M) → \overline{𝔽}_{l}^{×} which is unramified at all places in S̃_{l} and satisfies

ϕ̄_{n}ϕ̄_{n}^{c} = (ϵ det \overline{ρ}_{f,λ} ⊗ \overline{𝔽}_{l})^{1−n}|_{Gal(\overline{M}/M)},

and (Sym^{n−1} \overline{ρ}_{f,λ} ⊗ \overline{𝔽}_{l})|_{Gal(\overline{M}/M)} ⊗ ϕ̄_{n} is automorphic of weight σ_{k} and type {Sp_{n}(1)}_{{w_{q}}}.

• l is unramified in M.

• M is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ. Fix n ∈ N, and let ρ := Sym^{n−1} ρ_{f,λ}|_{Gal(\overline{F} /F)} ⊗ \overline{ℚ}_{l}. There is a crystalline character χ : Gal(\overline{F} /F) → O_{\overline{ℚ}_{l}}^{×} which is unramified above q such that

ρ^{∨} ≅ ρχϵ^{n−1};

in fact,

χ = (ϵ det ρ_{f,λ} ⊗ O_{\overline{ℚ}_{l}})^{1−n}|_{Gal(\overline{F} /F)}.

By Lemma 4.1.6 of [CHT08] we can choose an algebraic character

ψ : Gal(\overline{M}/M) → O_{\overline{ℚ}_{l}}^{×}

such that • χ|_{Gal(\overline{M}/M)} = ψψ^{c},

• ψ is crystalline,

• ψ is unramified at each place in S̃_{l}.

• ψ is unramified above q,

• ψ̄ = ϕ̄_{n}. Then ρ′ = ρ|_{Gal(\overline{M}/M)}ψ satisfies

(ρ′)^{c} ≅ (ρ′)^{∨}ϵ^{1−n}.

We claim that ρ′ is automorphic of weight σ_{k}, level prime to l and type {Sp_{n}(1)}_{{w_{q}}}. This follows from Theorem 5.2 of [Tay08]; we now check the hypotheses of that theorem. Certainly ρ̄′ ≅ (Sym^{n−1} \overline{ρ}_{f,λ} ⊗ \overline{𝔽}_{l})|_{Gal(\overline{M}/M)} ⊗ ϕ̄_{n} is automorphic of weight σ_{k}, level prime to l and type {Sp_{n}(1)}_{{w_{q}}}. The only non-trivial conditions to check are that:

• \overline{M}^{ker ad \overline{ρ}′} does not contain M(ζ_{l}), and

• The image \overline{ρ}′(Gal(\overline{M}/M(ζ_{l}))) is big in the sense of Definition 2.5.1 of [CHT08].

These both follow from the assumption that \overline{ρ}_{f,λ} has large image, the fact that

M is linearly disjoint from \overline{ℚ}^{ker(\overline{ρ}_{f,λ})} over ℚ, Corollary 2.5.4 of [CHT08], and the fact that PSL_{2}(k) is simple if k is a finite field of cardinality greater than 3. It follows from Lemma 4.3.3 of [CHT08] that ρ is automorphic. Then from Lemma 4.3.2 of [CHT08] we see that for each F′ with F/F′ soluble, Sym^{n−1} ρ_{f,λ}|_{Gal(\overline{F} /F′)} is automorphic, as required. □

Corollary 6.2. Suppose that f is a cuspidal newform of level Γ_{1}(N) and weight k = 2 or 3. Suppose that f is potentially Steinberg at q. Let N be a finite set of even positive integers. Then there is a Galois totally real field F such that for any n ∈ N and any subfield F′ ⊂ F with F/F′ soluble, Sym^{n−1} f is automorphic over F′.

Proof. Let θ be a Dirichlet character such that f′ = f ⊗ θ is Steinberg at q. The result then follows from Theorem 6.1 applied to f′. □

7. The Sato-Tate Conjecture

7.1. Let f be a cuspidal newform of level Γ_{1}(N), nebentypus χ_{f}, and weight k ≥ 2. Suppose that χ_{f} has order m, so that the image of χ_{f} is precisely the group μ_{m} of m-th roots of unity. Let U(2)_{m} be the subgroup of U(2) consisting of matrices with determinant in μ_{m}. For each prime l ∤ N, if we write

X^{2} − a_{l}X + l^{k−1}χ_{f}(l) = (X − α_{l}l^{(k−1)/2})(X − β_{l}l^{(k−1)/2})

then (by the Ramanujan conjecture) the matrix

(\begin{matrix}α_{l} & 0 \\ 0 & β_{l} \end{matrix})

defines a conjugacy class x_{f,l} in U(2)_{m}. A natural generalisation of the Sato-Tate conjecture is

Conjecture 7.1. If f is not of CM type, then the conjugacy classes x_{f,l} are equidistributed with respect to the Haar measure on U(2)_{m} (normalised so that U(2)_{m} has measure 1).

The group U(2)_{m} is compact, and its irreducible representations are given by det^{a} ⊗ Sym^{b} ℂ^{2} for 0 ≤ a < m and b ≥ 0. By the corollary to Theorem 2 of section I.A.2 of [Ser68] (noting the different normalisations of L-functions in force there), Conjecture 7.1 follows if one knows that for each b ≥ 1, the functions L(χ_{f}^{a}⊗Sym^{b} f, s) are holomorphic and non-zero for ℜs ≥ 1+b(k−1)/2 (the required results for b = 0 are classical).

Theorem 7.2. Suppose that f is a cuspidal newform of level Γ_{1}(N), character χ_{f}, and weight k = 2 or 3. Suppose that χ_{f} has order m. Suppose also that f is potentially Steinberg at q for some prime q. Then for all integers 0 ≤ a < m, b ≥ 1 the function L(χ_{f}^{a}⊗Sym^{b} f, s) has meromorphic continuation to the whole complex plane, satisfies the expected functional equation, and is holomorphic and nonzero in ℜs ≥ 1 + b(k − 1)/2.

Proof. The argument is very similar to the proof of Theorem 4.2 of [HSBT09]. We argue by induction on b; suppose that b is odd, and the result is known for all 1 ≤ b′ < b. We will deduce the result for b and for b + 1 simultaneously. Apply Corollary 6.2 with N = {2, b + 1}. Let F be as in the conclusion of Corollary 6.2. By Brauer’s theorem, we may write

1 = Σ_{j}a_{j} Ind_{Gal(F/F_{j})}^{Gal(F/ℚ)} χ_{j}

where F ⊃ F_{j} with F/F_{j} soluble, χ_{j} a character Gal(F/F_{j}) → ℂ^{×}, and a_{j} ∈ ℤ. Then for each j, Sym^{b} f is automorphic over F_{j}, corresponding to an RAESDC representation π_{j} of GL_{b+1}(𝔸_{F_{j}}). In addition, f is automorphic over F_{j}, corresponding to an RAESDC representation σ_{j} of GL_{2}(𝔸_{F_{j}}). Then we have

L(χ_{f}^{a} ⊗ Sym^{b} f, s) = ∏_{j} L(π_{j} ⊗ (χ_{j} ∘ Art_{F_{j}}) ⊗ (χ_{f}^{a} ∘ N_{F_{j}/ℚ}), s)^{a_{j}},

L(χ_{f}^{a} ⊗ Sym^{2} f, s) = ∏_{j} L((Sym^{2} σ_{j}) ⊗ (χ_{j} ∘ Art_{F_{j}}) ⊗ (χ_{f}^{a} ∘ N_{F_{j}/ℚ}), s)^{a_{j}},

and

L(χ_{f}^{a} ⊗ Sym^{b+1} f, s)L(χ_{f}^{a+1} ⊗ Sym^{b−1} f, s − k + 1) =

= ∏_{j} L((π_{j} ⊗ (χ_{j} ∘ Art_{F_{j}}) ⊗ (χ_{f}^{a} ∘ N_{F_{j}/ℚ})) × σ_{j}, s + b(k − 1)/2)^{a_{j}}.

The result then follows from the main results of [CPS04] and [GJ78] (in the case b = 1) together with Theorem 5.1 of [Sha81]. □

Corollary 7.3. Suppose that f is a cuspidal newform of level Γ_{1}(N) and weight k = 2 or 3. Suppose also that f is potentially Steinberg at q for some prime q. Then Conjecture 7.1 holds for f.

Finally, we note that one can make this result more concrete, as one can easily explicitly determine the Haar measure on U(2)_{m} from that of its finite index subgroup SU(2). One finds that (as already follows from Dirchlet’s theorem) the classes x_{f,l} are equidistributed by determinant, and that furthermore the classes with fixed determinant are equidistributed with respect to the natural analogue of the usual Sato-Tate measure. That is, suppose that ζ ∈ μ_{m}, and fix a square root ζ^{1/2} of ζ. Then any conjugacy class x_{f,l} in U(2)_{m} with determinant ζ contains a representative of the form

(\begin{matrix}ζ^{1/2}e^{iθ_{l}} & 0 \\ 0 & ζ^{1/2}e^{−iθ_{l}} \end{matrix})

with θ_{l} ∈ [0, π], and the θ_{l} are equidistributed with respect to the measure \frac{2}{π} sin^{2} θdθ.

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Toby Gee

Department of Mathematics Harvard University gee@math.harvard.edu