Potential automorphy of odd-dimensional symmetric powers of elliptic curves, and applications

Michael Harris

UFR de Mathématiques, Université Paris 7, 2 Pl. Jussieu, 75251 Paris cedex 05, FRANCE

harris@math.jussieu.fr

To Yuri Ivanovich Manin

Introduction

The present article was motivated by a question raised independently by Barry Mazur and Nick Katz. The articles [CHT,HST,T] contain a proof of the Sato-Tate conjecture for an elliptic curve E over a totally real field whose j-invariant j(E) is not an algebraic integer. The Sato-Tate conjecture for E is an assertion about the equidistribution of Frobenius angles of E, or equivalently about the number of points |E(𝔽_{p})| on E modulo p as p varies. The precise statement of the conjecture, which is supposed to hold for any elliptic curve without complex multiplication, is recalled in §5. Now suppose E and E′ are two elliptic curves without complex multiplication, and suppose E and E′ are not isogenous. The question posed by Mazur and Katz is roughly the following: are the distributions of the Frobenius angles of E and E′, or equivalently of the numbers p + 1 − |E(𝔽_{p})| and p + 1 − |E′(𝔽_{p})|, independent?

The Sato-Tate conjecture, in the cases considered in [CHT,HST,T], is a consequence of facts proved there about L-functions of symmetric powers of the Galois representation on the Tate module T_{ℓ}(E) of E, following a strategy elaborated by Serre in [S]. These facts in turn follow from one of the main theorems of [CHT,HST,T], namely that, if n is even, the (n − 1)st symmetric power of T_{ℓ}(E) is potentially automorphic, in that it is associated to a cuspidal automorphic representation of GL(n) over some totally real Galois extension of the original base field. The restriction to even n is inherent in the approach to potential modularity developed in [HST], which applies only to even-dimensional representations. The necessary properties of all symmetric power L-functions follow from this result for even-dimensional symmetric powers, together with basic facts about Rankin-Selberg L-functions proved by Jacquet-Shalika-Piatetski-Shapiro and Shahidi. In a similar way, the Mazur-Katz question can be resolved affirmatively if we can prove potential automorphy for all symmetric power L-functions of E and E′ over the same field.

The main purpose of the present article is to explain how to prove potential automorphy for odd-dimensional symmetric power L-functions, thus providing a response to the question of Mazur and Katz. The principal innovation is a tensor product trick that converts an odd-dimensional representation to an even-dimensional representation. Briefly, in §§2 and 3 one tensors with a two-dimensional representation. One has to choose a two-dimensional representation of the right kind, which is not difficult. The challenge is then to recover the odd-dimensional symmetric power unencumbered by the extraneous two-dimensional factor; this is the subject of §4. I say “explain how to prove” rather than “prove” because the proofs of the main results of this article make use of stronger modularity theorems than those proved in [CHT] and [T], and are thus conditional. I explain in §1 how I expect these modularity results to result from a strengthening of known theorems associating compatible families of ℓ-adic Galois representations to certain kinds of automorphic representations. These stronger theorems, stated as Expected Theorems 1.2 and 1.4, are the subject of work in progress by participants in the Paris automorphic forms seminar, and described in [H]. This work has progressed to a point where it seems legitimate to admit these Expected Theorems. Nevertheless, the present article should be viewed as a promissory note which will not be negotiable until the project outlined in [H] has been completed.

One can of course generalize the question and ask whether the distributions of the Frobenius angles of n pairwise non-isogenous elliptic curves without complex multiplication are independent. For n ≥ 3 this seems completely inaccessible by current techniques in automorphic forms.

The article concludes with some speculations regarding additional applications of the tensor product trick.

I met Yuri Ivanovich Manin briefly near the beginning of my career. Later, as a National Academy of Sciences exchange fellow I had the remarkable good fortune of spending a year as his guest in Moscow, and as a (mostly passive but deeply appreciative) participant in his seminar at Moscow State University during what may well have been its final year, and saw first-hand what the Moscow mathematical community owed to his insight and personality. The influence of Yuri Ivanovich on my own work is pervasive, and the present work is no exception: though it is not apparent in what follows, the article [HST], on which all the results presented here are based, can be read as an extended meditation on the Gauss-Manin connection as applied to a particular family of Calabi-Yau varieties. It is an honor to dedicate this article to Yuri Ivanovich Manin.

I thank Barry Mazur and Nick Katz for raising the question that led to this paper. Apart from the tensor product trick, practically all the ideas in this paper are contained in [CHT], [HST], and [T]; I thank my coauthors – Laurent Clozel, Nick Shepherd-Barron, and Richard Taylor – for their collaboration over many years. I thank Richard Taylor specifically for his help with the proof of the crucial Lemma 4.2. Finally, I thank the referee for a careful reading, and for helping me to clarify a number of important points.

1. Reciprocity for n-dimensional Galois representations

All finite-dimensional representations of Galois groups are assumed to be continuous. When E is a number field, contained in a fixed algebraic closure \overline{ℚ} of ℚ, we let Γ_{E} denote Gal(\overline{ℚ}/E). Let ρ be a (finite-dimensional) ℓ-adic representation of Γ_{E}. Say ρ is pure of weight w if for all but finitely many primes v of E the restriction ρ_{v} of ρ to the decomposition group Γ_{v} is unramified and if the eigenvalues of ρ_{v}(F rob_{v}) are all algebraic numbers whose absolute values equal qv^{\frac{w}{2}} ; here q_{v} is the order of the residue field k_{v} at v and F rob_{v} is geometric Frobenius. If ρ is pure of weight w, the normalized L-function of ρ is

L^{norm}(s, ρ) = L(s + \frac{w}{2} , ρ);

here we assume we have a way to define the local Euler factors at primes dividing ℓ (for example, ρ belongs to a compatible system of λ-adic representations). Then L^{norm}(s, ρ) converges absolutely for Re(s) > 1.

Let F be a CM field, F^{+} ⊂ F its maximal totally real subfield, so that [F : F^{+}] ≤ 2. Let c ∈ Gal(F/F^{+}) be complex conjugation; by transport of structure it acts on automorphic representations of GL(n, F). The following theorem is the basis for many of the recent results on reciprocity for Galois representations of dimension > 2. For the purposes of the following theorem, a unitary Harish-Chandra module σ for GL(n, ℂ) will be called “cohomological” if σ ⊗ || det ||^{\frac{n−1}{2}} || is cohomological in the usual sense, i.e. if there is a finitedimensional irreducible representation W of GL(n, ℂ) such that

H^{•}(Lie(GL(n, ℂ)), U(n); σ ⊗ || det ||^{\frac{n−1}{2}} || ⊗ W) ≠ 0.

The half-integral twist is required by the unitarity.

Theorem 1.1 ([C2], [Ko], [HT], [TY]). In what follows, Π denotes a cuspidal automorphic representation of GL(n, F), and {ρ_{•,λ}} denotes a compatible family of n-dimensional λ-adic representations of Γ_{F}.

There is an arrow Π ↦ {ρ_{Π,λ}}, where λ runs through non-archimedean completions of a certain number field E(Π), under the following hypotheses:

\{ \begin{matrix}(1) The factor Π_{∞} \\ is cohomological \\ (2) Π ∘ c ≅ Π^{∨} \\ (3) ∃v_{0}, Π_{v_{0}} \\ discrete series \end{matrix}\} ⇒ \{ \begin{matrix}(a) ρ = ρ_{Π,λ} geometric, \\ HT regular \\ (b) ρ ⊗ ρ ∘ c → ℚ_{ℓ}(1 − n) \\ (c) & local condition \\ at v_{0} \end{matrix}\}

This correspondence has the following properties:

(i) For any finite place v prime to the residue characteristic ℓ of λ,

[ρ_{Π,λ} |_{W D_{v}}]^{F rob−ss} = L(Π_{v} ⊗ | • |v^{\frac{1−n}{2}} ).

Here W D_{v} is the local Weil-Deligne group at v, L is the normalized local Langlands correspondence, and F rob − ss denotes Frobenius semisimplification;

(ii)The representation ρ_{Π,λ} |_{G_{v}} is potentially semistable, in Fontaine’s sense, for any v dividing ℓ, and the Hodge-Tate weights at v are explicitly determined by the infinitesimal character of the Harish-Chandra module Π_{∞}.

The local Langlands correspondence is given the unitary normalization. This means that the correspondence identifies L(s, Π) and L^{norm}(s, ρ_{Π,λ}), so that the functional equations always exchange values at s and 1 − s.

The term “geometric” is used in the sense of Fontaine-Mazur: each ρ_{Π,λ} is unramified outside a finite set of places of F, in addition to the potential semistability mentioned in the statement of the theorem. The condition “HT regular” (Hodge-Tate) means that the Hodge-Tate weights at v have multiplicity at most one.

For the local condition (c), we can take the condition that the representation of the decomposition group at v_{0} is indecomposable as long as v_{0} is prime to the residue characteristic of λ, or equivalently that this representation of the decomposition group at v_{0} corresponds to a discrete series representation of GL(n, F_{v_{0}}). The conditions on both sides of the diagram match: (1) ↔ (a),

(2) ↔ (b), (3) ↔ (c).

When Π is a base change of a representation Π^{+} of GL(n, F^{+}), condition (2) just means that Π is self-dual.

In what follows, we will admit the following extension of Theorem 1.1:

Expected Theorem 1.2. The assertions of Theorem 1.1 remain true provided Π satisfies conditions (1) and (2); then ρ_{Π,λ} satisfies conditions (a) and (b), as well as (i) and (ii).

Here “Expected Theorem” means something more than conjecture. The claim of Expected Theorem 1.2 is a very special case of the general Langlands conjectures, in the version for Galois representations developed in Clozel’s article [C1]. This specific case is the subject of work in progress on the part of participants in the Paris automorphic forms seminar and others, and an outline of the various steps in the proof can be found in [H]. There are quite a lot of intermediate steps, the most difficult of which involve analysis of the stable trace formula, twisted or not, but I would ”expect” that they will all have been verified, and the theorem completely proved, by 2010 at the latest.

I single out one of the intermediate steps. By a unitary group over F^{+} I will mean the group of automorphisms of a vector space V/F preserving a nondegenerate hermitian form. The group is denoted U(V), the hermitian form being understood. We will consider a hermitian vector space V of dimension n with the following properties:

(1.3.1) For every real place σ of F^{+}, the local group U(V_{σ}) is compact (the

hermitian form is totally definite); (1.3.2) For every finite place v of F^{+}, the local group U(V_{v}) is quasi-split and

split over an unramified extension.

We write G_{0} = U(V). Such a unitary group always exists when n is odd, provided F/F^{+} is everywhere unramified. When n is even, there is a sign obstruction that can be removed by replacing F^{+} by a totally real quadratic extension. Such restrictions are harmless for applications (see [H]).

We let K = ∏_{v} K_{v} ⊂ G_{0}(A^{f}) be an open compact level subgroup. Hypothesis (1.3.2) guarantees that G_{0}(F_{v}^{+}) contains a hyperspecial maximal compact subgroup for all finite v. If v is split then any maximal compact subgroup is hyperspecial and conjugate to GL(n, O_{v}). We assume

Hypothesis 1.3.3. K_{v} is hyperspecial maximal compact for all v that remain inert in F.

For any ring R, let

M_{K}(G_{0}, R) = C(G_{0}(F^{+})\G_{0}(A)/G_{0}(ℝ) · K, R),

where for any topological space X, C(X, R) means the R-module of continuous functions from X to R, the latter endowed with the discrete topology. The Hecke algebra H_{K}(R) of double cosets of K in G_{0}(A^{f}) with coefficients in R acts on M_{K}(G_{0}, R). It contains a subring H_{K}^{hyp}(R) generated by the double cosets of K_{v} in G_{0}(F_{v}^{+}) where v runs over primes that split in F at which K_{v} is hyperspecial maximal compact. We denote by 𝕋_{K}(R) the image of H_{K}^{hyp}(R) in End_{R}(M_{K}(G_{0}, R)). The algebra 𝕋_{K}(R) is reduced if R is a semisimple algebra flat over ℤ (cf. [CHT], Corollary 2.3.3).

We can also consider M(G_{0}, R), the direct limit of M_{K}(G_{0}, R) over all K, including those not satisfying (1.3.3). This is a representation of G_{0}(A^{f}) and decomposes as a sum of irreducible representations when R is an algebraically closed field of characteristic zero. Let π ⊂ M(G_{0}, ℂ) be an irreducible summand. Write π = π_{∞} ⊗ π_{f}, π_{f} = ⊗′_{v}π_{v}, the restricted tensor product over finite primes v of F^{+} of representations of G_{0}(F_{v}^{+}). With our hypotheses π_{∞} is the trivial representation of G_{0,ℝ} = ∏_{σ} U(V_{σ}), where σ is as in (1.3.1). Suppose π^{K} ≠ {0} for some K satisfying (1.3.3). Then for every finite v one can define the local base change Π_{v} = BC_{F_{v}/F_{v}^{+}}π_{v}, a representation of G_{0}(F_{v}^{+})→^{∼}⟶ ∏_{w|v} GL(n, F_{w}). If v is inert, then by (1.3.3) we know that π_{v} is an unramified representation, and so is Π_{v}. If not, then π_{v} is a representation of GL(n, F_{v}^{+}) and Π_{v}→^{∼}⟶π_{v} ⊗π_{v}^{∨}, with the appropriate normalization. Thus

Lemma 1.3.4. The local factor π_{v} is uniquely determined by Π_{v}.

Expected Theorem 1.4. There is a cohomological representation Π_{∞} of GL(n, F_{∞}) = GL(n, F ⊗_{ℚ} ℝ) such that the formal base change

Π = BC_{F/F^{+}}π = Π_{∞} ⊗ ⨂_{v}^{′} Π_{v}

is an automorphic representation of GL(n, F). Moreover, there is a partition n = a_{1} + a_{2} + ⋯ + a_{r} and an automorphic representation ⨂_{j} Π_{j} of the group ∏_{j} GL(a_{j}, A_{F}) such that each Π_{j} is in the discrete automorphic spectrum of GL(a_{j}, F) and Π, as a representation of GL(n, A_{F}), is parabolically induced from the inflation of ⨂_{j} Π_{j} to the standard parabolic subgroup P (A) ⊂ GL(n, A_{F}) associated to the partition. Moreover, each Π_{j} satisfies conditions (1) and (2) of Theorem 1.1, where “cohomological” is understood as in loc. cit..

We say π is F/F^{+}-cuspidal if Π is cuspidal, in which case it follows from the classification of generic cohomological representations that Π_{∞} is necessarily tempered and is uniquely determined by the condition that π_{∞} is trivial. This representation is denoted Π_{∞,0}, or Π_{∞,0}(n, F) when this is necessary.

Fix a prime ℓ and let O be the ring of integers in a finite extension of ℚ_{ℓ}. The ring 𝕋_{K}(O) is semilocal and 𝕋_{K}(O ⊗ ℚ̄_{ℓ}) is a product of fields. Let 𝔪 ⊂ 𝕋_{K}(O) be a maximal ideal and let I ⊂ 𝕋_{K}(O ⊗ ℚ̄_{ℓ}) be any prime ideal whose intersection with 𝕋_{K}(O) is contained in 𝔪. Then I determines an irreducible G_{0}(A^{f})-summand π_{0} of M(G_{0}, ℚ̄_{ℓ}), or equivalently of M(G_{0}, ℂ), if one identifies the algebraic closures of ℚ in ℂ and in ℚ̄_{ℓ}, as in [HT], p. 20. More precisely, I only determines π_{0,v} locally at v for which K_{v} is hyperspecial, but this includes all inert primes. Let S be the set at which K_{v} is not hyperspecial. By Expected Theorem 1.4, I determines a collection of cohomological automorphic representations Π of GL(n, F) which are isomorphic outside the finite set S. By strong multiplicity one, Π is in fact unique; we denote it Π_{I}. Then π is unique by Lemma 1.3.4.

Combining Expected Theorems 1.4 and 1.2, we thus obtain an n-dimensional representation ρ_{Π,ℓ} = ρ_{I,ℓ} of Γ_{F}. We say I is Eisenstein at ℓ if the reduction mod ℓ, denoted ρ̄_{Π,ℓ}, is not absolutely irreducible. It follows in the usual way from property (i) of the correspondence between ρ_{Π,ℓ} and Π that if I is Eisenstein at ℓ then every prime ideal of 𝕋_{K}(O ⊗ ℚ̄_{ℓ}) lying above 𝔪 is Eisenstein. In that case we say 𝔪 is Eisenstein at ℓ.

Lemma 1.5. Admit Expected Theorems 1.4 and 1.2. Fix a prime ℓ, and suppose 𝔪 ⊂ 𝕋_{K}(O) is not an Eisenstein ideal at ℓ. Then any prime ideal I ⊂ 𝕋_{K}(O ⊗ ℚ̄_{ℓ}) lying above 𝔪 has the property that Π_{I} is cuspidal (in that case we say 𝔪 and I are F/F^{+} − cuspidal).

Sketch of proof. This follows from the classification of automorphic representations of GL(n) and from properties of base change. Suppose I corresponds to π ⊂ M(G_{0}, ℂ). If the base change Π of π belongs to the discrete spectrum of GL(n, F) – that is, if the partition in Expected Theorem 1.4 is a singleton – then it is either cuspidal or in the non-tempered discrete spectrum. In the latter case, the Moeglin-Waldspurger classification implies that n factors as ab, with a > 1, b > 1, and Π is the Speh representation attached to a cuspidal automorphic representation Π_{1} of GL(b, F). Clozel has checked in [C3] that Π_{1} satisfies properties (1) and (2) of Expected Theorem 1.2 for GL(b), hence is associated to a b-dimensional ℓ-adic representation ρ_{1} of Γ_{F}. It follows from condition (i) of Expected Theorem 1.2 that the semisimple representation ρ_{Π,ℓ} decomposes as a sum of a constituents, each of which is an abelian twist of ρ_{1}.

Suppose Π does not belong to the discrete spectrum of GL(n, F). Then π is endoscopic, hence is associated to a partition n = ∑ a_{j}, with each a_{j} > 0, and an automorphic representation ⊗_{j}Π_{a_{j}} in the discrete spectrum of ∏_{j} GL(a_{j}, F), such that each Π_{a_{j}}, satisfies properties (1) and (2) of Expected Theorem 1.2. Then ρ_{Π,ℓ} decomposes as a sum of r > 1 pieces of dimensions a_{j}. ⊓ ⊔

I recall the main modularity lifting theorem of [T], whose proof builds on and completes the main results of [CHT].

1.6. Modularity lifting theorem. Let ℓ > n be a prime unramified in F^{+} (resp. and such that every divisor of ℓ in F^{+} splits in F) and let

r : Γ_{F^{+}} → GL(n, ℚ̄_{ℓ}) (resp. r : Γ_{F} → GL(n, ℚ̄_{ℓ}))

be a continuous irreducible representation satisfying the following properties:

(a) r ramifies at only finitely many primes, is crystalline at all primes dividing ℓ, and is Hodge-Tate regular;

(b) r ≃ r^{∨}(1 − n) · χ (resp. r^{c} ≃ r^{∨}(1 − n)) where (1 − n) is the Tate twist and χ is a character whose value is constant on all complex conjugations (resp. c denotes complex conjugation);

(c) At some finite place v not dividing ℓ, r_{v} corresponds to a square-integrable representation of GL(n, F_{v}^{+}) under the local Langlands correspondence, and satisfies the final “minimality” hypotheses of [CHT, 4.3.4 (5)] or [T, 5.2 (5)]

In addition, we assume that r̄

(d) has “big” image in the sense of Definition 3.1 below;

(e) is absolutely irreducible;

(f) is of the form ρ̄_{Π,ℓ} for some cuspidal automorphic representation Π of GL(n, F^{+}) satisfying conditions (1)-(3) of Theorem 1.1.

Then r is of the form ρ_{Π′,ℓ} for some cuspidal automorphic representation Π′ of GL(n, F^{+}) satisfying conditions (i)-(iii) of Theorem 1.1.

I have not written out the last part of hypothesis (c) in detail, because it will be dropped in the remainder of the article. More precisely, if we admit Expected Theorems 1.2 and 1.4, then we obtain the 1.7. Expected modularity lifting theorem. Let ℓ and r be as in 1.6, but we no longer assume condition (c), and in (f) we drop condition (3). Then r is of the form ρ_{Π′,ℓ} for some cuspidal automorphic representation Π′ of GL(n, F^{+}) satisfying properties (i)-(ii) of Expected Theorem 1.2.

I Expect the arguments of [CHT] and [T] will apply without change to yield this Expected Theorem. The various rings of deformations of r̄ are defined exactly as in [CHT] and [T], and the target Hecke algebra is the localization of 𝕋_{K}(O) at the maximal ideal 𝔪 associated to r̄. Hypothesis (e) guarantees that 𝔪 is not Eisenstein at ℓ, and the remaining arguments should go through unchanged. However, there are many steps to the proof of Theorem 1.6, and it is necessary to check carefully that, as I believe, Expected Theorems 1.2 and 1.4 eliminate all dependence on the local condition (c).

In the remainder of the paper, I draw consequences from 1.7.

2. Potential modularity of a Galois representation

Let F and F^{+} be as in §1. The article [HST] develops a method for proving that certain n-dimensional ℓ-adic representations ρ of Γ_{F^{+}} (resp. Γ_{F}) that look like they arise from automorphic representations via the correspondence of Theorem 1.1, are potentially automorphic in the following sense: there exists a totally real Galois extension F′/F^{+} such that ρ |_{Γ_{F′}} (resp. ρ |_{Γ_{F ·F′}}) does correspond to a cuspidal automorphic representation of GL(n, F′) (resp. GL(n, F · F′)). The relevant result is Theorem 3.1 of [HST], which is in turn based on Theorem 1.6. Although the latter theorem is valid for representations of arbitrary dimension n, Theorem 3.1 only applies to an even-dimensional representation of Γ_{F^{+}} endowed with an alternating form that is preserved by Γ_{F^{+}} up to a multiplier.

If we admit the expected results of §1, then Theorem 3.1 of [HST] admits the following simplification. The constant C(n_{i}) is a positive number introduced in [HST], Corollary 1.11; its precise definition is irrelevant to the applications. For a finite prime w, G_{w} denotes a decomposition group, I_{w} ⊂ G_{w} the inertia group.

Theorem 2.1 [HST]. Assume the Expected Theorems of §1. Let F^{+}/F^{0,+} be a Galois extension of totally real fields and let n_{1}, …, n_{r} be even positive integers. Suppose that ℓ > max{C(n_{i}), n_{i}} is a prime which is unramified in F and which splits in ℚ(ζ_{n_{i}+1}), i = 1, …, r. Let L be a finite set of primes of F^{+} not containing primes above ℓ and let M be a finite extension of F.

Suppose that for i = 1, …, r

r_{i} : Γ_{F^{+}} → GSp(n_{i}, ℤ_{ℓ})

is a continuous representation with the following properties.

• r_{i} has multiplier ω_{ℓ}^{1−n_{i}}, where ω_{ℓ} is the ℓ-adic cyclotomic character.

• r_{i} ramifies at only finitely many primes.

• The image r̄_{i}(Γ_{F^{+}(ζ_{ℓ})}) is big, in the sense of Definition 3.1 below, where ζ_{ℓ} is a primitive ℓ-th root of 1.

• The fixed field F_{i} of ker ad(r̄_{i}) ⊂ Γ_{F^{+}} does not contain F (ζ_{ℓ}).

• r_{i} is unramified at all primes in L.

• If w | ℓ is a prime of F then r_{i} |_{G_{w}} is crystalline with Hodge-Tate weights 0, 1, …, n_{i} − 1, with the conventions of [HST]. Moreover,

r̄_{i} |_{I_{w}}≃ ⨁_{j=0}^{n_{i}−1} ω_{ℓ}^{−j}.

Then there is a totally real field F^{′,+}/F^{+}, Galois over F_{0}^{+} and linearly disjoint from the compositum of the F_{i} with M over F^{+}, with the property that each r_{i,F^{′,+}} = r_{i} |_{Γ_{F}^{′,+}} corresponds to an automorphic representation Π_{i} of GL(n_{i}, F^{′,+}). If F′/F^{′,+} is a CM quadratic extension, then the base change Π_{i,F′} has archimedean constituent isomorphic to Π_{∞,0}(n_{i}, F′) (cf. the remarks after 1.4). Finally, all primes of L and all primes of F dividing ℓ are unramified in F′.

Apart from a few slight changes in notation, this theorem is practically identical to Theorem 3.1 of [HST]. There is no field M in [HST] but the proof yields an F^{′,+} linearly disjoint over F^{+} from any fixed extension. Only condition (7) of Theorem 3.1 of [HST], corresponding to condition (3) of Theorem 1.1, has been eliminated. The proof is identical but simpler: references to Theorem 1.6 are replaced by references to Expected Theorem 1.7, and all arguments involving the primes q and q′ in [HST] are no longer necessary.

Let E be an elliptic curve over F^{+}, and let ρ_{E,ℓ} : Γ_{F^{+}} → GL(2, ℚ_{ℓ}) denote the representation on H^{1}(E\overline{ℚ}, ℚ_{ℓ}), i.e. the dual of the ℓ-adic Tate module. For n ≥ 1 let

ρ_{E,ℓ}^{n} = Sym^{n−1}ρ_{E,ℓ} : Γ_{F^{+}} → GL(n, ℚ_{ℓ}).

We will always assume E has no complex multiplication. Then ρ_{E,ℓ}^{n} is irreducible by a theorem of Serre, for all n, and for almost all ℓ > n, Im(ρ̄) contains the image of SL(2, 𝔽_{ℓ}) under the symmetric power representation, and hence is absolutely irreducible. When F^{+} = ℚ it was proved in the series of papers initiated by Wiles and Taylor-Wiles and completed by Breuil, Conrad, Diamond, and Taylor that L(s, ρ_{E,ℓ}) is automorphic, which in this case means is attached to a classical new form of weight 2. The prototype for Theorem 2.1 is the theorem proved by Taylor in [T02], which shows that L(s, ρ_{E,ℓ}) is potentially automorphic for any F^{+}.

In [HST] and [T], Theorem 2.1 is notably applied to show that L(s, ρ_{E,ℓ}^{n}) is potentially automorphic for any even n, provided E has non-integral jinvariant. One can hardly hope to apply Theorem 2.1 as such when n is odd, given that symplectic groups are only attached to even integers. Moreover, when n is odd ρ_{E,ℓ}^{n} has an orthogonal polarization rather than a symplectic polarization. These two related flaws – the oddness of n and the orthogonality of ρ^{n} – can be cured simultaneously by tensoring ρ_{E,ℓ}^{n} by a two-dimensional representation τ : Γ_{F^{+}} → GL(2, ℤ_{ℓ}). Such a representation is necessarily symplectic, with multiplier det τ. We suppose τ has determinant ω_{ℓ}^{−n}. In order to preserve the hypotheses of Theorem 2.1, specifically 2.1 (6), we need to assume

Hypothesis 2.2. If w | ℓ is a prime of F then τ |_{G_{w}} is crystalline with Hodge-Tate weights 0, n, with the conventions of [HST]. Moreover,

τ̄ |_{I_{w}}≃ 1 ⊕ ω_{ℓ}^{−n}.

For example, let f be a classical new form of weight n + 1 for Γ_{0}(N), for some integer N. Associated to f is a number field ℚ(f), generated by the Fourier coefficients of f, and a compatible system of 2-dimensional λ-adic representations τ_{f,λ} of Γ_{ℚ} as λ varies over the primes of ℚ(f). We choose a prime ℓ that splits completely in ℚ(f) and such that ℓ ∤ N. Fix λ dividing ℓ, and write τ = τ_{f,λ}. Then τ takes values in GL(2, ℤ_{ℓ}); since f has trivial nebentypus, the determinant of τ is indeed ω_{ℓ}^{−n}. The hypothesis regarding τ̄ |_{I_{w}} is in general a serious restriction, but we will find explicit examples.

(2.3)

We say the residual representation τ̄ is “big enough” if its image contains a non-commutative subgroup of the normalizer N(T) of a maximal torus T ⊂ SL(2, 𝔽_{ℓ}), and more specifically that Im(τ̄) contains an element h of T with distinct eigenvalues that acts trivially on the cyclotomic field ℚ(ζ_{ℓ}), and an element w of order 2 that does not commute with h.

Here are the representations we will use. Let L be an imaginary quadratic field not contained in F, and let η_{L} be the corresponding quadratic Dirichlet character, viewed as an idèle class character of ℚ. Let χ be a Hecke character of (the idèles of) L whose restriction to the idèles of ℚ is the product |•|_{A}^{−n}·η_{L}, where |•|_{A} is the idèle norm. Choose an isomorphism L_{∞} = L⊗_{ℚ}ℝ→^{∼}⟶ℂ, let z be the corresponding coordinate function on L_{∞}, and assume χ_{∞}(z) = z^{−n}. Then χ is an algebraic Hecke character and is associated to a compatible system of ℓ-adic characters χ_{λ} : Γ_{L} → ℚ(χ)_{λ}^{+}, where ℚ(χ) is the field of coefficients of χ, a finite extension of ℚ, and λ runs through the places of ℚ(χ). Choose a prime ℓ > 2n + 1 that splits in L and in ℚ(χ); then for any λ dividing ℓ we can view χ_{λ} as a ℚ_{ℓ}^{×}-valued character of Γ_{L}; we write χ_{ℓ} = χ_{λ}.

For such an ℓ, we define the monomial induced representation

τ_{ℓ} = Ind_{Γ_{L}}^{Γ_{ℚ}}χ_{ℓ} : Γ_{ℚ} → GL(2, ℚ_{ℓ}).

Lemma 2.4. Under the above hypotheses, τ̄_{ℓ} is “big enough” in the sense of (2.3).

Proof. Let v, v′ be the two primes of L dividing ℓ. Let k(v), k(v′) denote the corresponding residue fields; then the product of the respective Teichmüller characters defines an inclusion

k(v)^{×} × k(v′)^{×} ↪ O_{v}^{×} × O_{v′}^{×} ⊂ A_{L}^{×}.

Let b and b′ denote generators of the images of the cyclic groups k(v)^{×} and k(v′)^{×} in O_{v}^{×} and O_{v′}^{×}, respectively, and let s(b), s(b′) denote their images in Gal(L^{ab}/L) under the reciprocity map. Our hypotheses imply that s(b) acts on τ̄ with eigenvalues b^{n}, 1, and likewise for s(b′). Moreover, the trivial eigenspace for s(b) is the non-trivial eigenspace for s(b′), and vice versa. We identify b and b′ with roots of unity in ℚ_{ℓ}^{×}. The element h = τ̄(s(b) · s(b′)^{−1}) then belongs to SL(2, 𝔽_{ℓ}) and has eigenvalues b^{±n}. Since ℓ − 1 > 2n, these eigenvalues are distinct; in particular, h does not commute with the image w of complex conjugation in Im(τ̄). Since the action of O_{v}^{×} × O_{v′}^{×} on ℚ(ζ_{ℓ}) factors through the norm to ℤ_{ℓ}^{×}, we see that h acts trivially on ℚ(ζ_{ℓ}). This completes the proof of Lemma 2.4. ⊓ ⊔

Corollary 2.5. Let τ = τ_{f,λ} be as in the preceding paragraph and satisfy Hypothesis 2.2. We continue to admit the Expected Theorems of §1. We use the same notation for τ |_{Γ_{F^{+}}} . Let E be an elliptic curve over F. Let n be an odd positive integer, and suppose there is a prime ℓ > 2n + 1, unramified in F^{+}, such that

(i) E has good ordinary reduction at all primes w dividing ℓ (ii)ℓ does not divide the conductor N of f, and

τ̄ |_{I_{w}}≃ 1 ⊕ ω_{ℓ}^{−n}.

(iii)ℓ splits in ℚ(ζ_{2i+1}), i = 1, …, n − 1; in particular, ℓ ≡ 1 (mod n) (take

i = \frac{n−1}{2}). (iv)\frac{ℓ−1}{n} > 2.

Suppose τ̄ is “big enough” in the sense of (2.3). Let M be an arbitrary extension of F^{+}. For every integer i ≤ n, let r_{i} = ρ_{E,ℓ}^{2i}; let r_{τ} = ρ_{E,ℓ}^{n} ⊗τ. Then there is a totally real Galois extension F^{′,+}/F^{+}, linearly disjoint from M over F^{+}, with the property that for i = 1, …, n, r_{i,F^{′,+}} = r_{i} |_{Γ_{F}^{′,+}} corresponds to an automorphic representation Π_{i} of GL(2i, F^{′,+}), and such that r_{τ} corresponds to an automorphic representation Π_{τ} of GL(2n, F^{′,+}). If F′/F^{′,+} is a CM quadratic extension, then the base change Π_{i,F′} (resp. Π_{τ,F′}) has archimedean constituent isomorphic to Π_{∞,0}(2i, F′) (resp. Π_{∞,0}(2n, F′)).

Proof. Under our hypotheses on ℓ and the image of τ̄, r̄_{τ} is absolutely irreducible. We first prove the corollary under the hypothesis that for each w dividing ℓ,

ρ̄_{E,ℓ} |_{I_{w}}≃ 1 ⊕ ω_{ℓ}^{−1}.  (2.5.1)

Conditions (1), (2), and (6) of Theorem 2.1 are clearly satisfied. Since ℓ is unramified in F^{+}, F^{+} and ℚ(ζ_{ℓ}) are linearly disjoint over ℚ. By condition (1), the intersection of Im(r̄_{i}) with the center of GSp(n_{i}, 𝔽_{ℓ}) maps onto the subgroup of Gal(ℚ(ζ_{ℓ})/ℚ) generated by 2(1 − n_{i})-th powers, which implies condition (4) for all r_{i}, and for r_{τ} as well.

Condition (5) is irrelevant. It remains to verify condition (3). For r̄_{i}, i = 1, …, n, this is Lemma 3.2 of [HST]; the case of r̄_{τ} is Lemma 3.2.

This completes the proof under hypothesis (2.5.1). We reduce to this case as in the proof of Theorem 3.3 of [HST], replacing ℓ by a second prime ℓ′ > 2n+1 also split in L and in ℚ(χ), τ̄_{ℓ} by τ̄_{ℓ′}, and E by a curve E′ (unfortunately denoted E in [loc. cit.]) such that ρ_{E′,ℓ}→^{∼}⟶ρ_{E,ℓ} but ρ_{E′,ℓ′} satisfies hypothesis (2.5.1) at ℓ′. ⊓ ⊔

Remark 2.6. Recall that the results of this section are all conditional on the Expected Theorems of §1. In particular, we are not assuming that E has potentially multiplicative reduction at some place. If we do assume j(E) is not integral, then the automorphic representations Π_{i} are constructed unconditionally in [CHT, HST, T]. However, the local condition on j(E) does not suffice to impose a strong enough local condition on r_{τ} (corresponding to a discrete series representation on the automorphic side).

3. A lemma about certain residual representations

Definition 3.1. Let V/𝔽̄_{ℓ} be a finite dimensional vector space. Let ad^{0}(V) ⊂ ad(V) = Hom(V, V) be the subspace of trace 0 endomorphisms. A subgroup Δ ⊂ GL(V) is big if the following hold:

(a) H^{i}(Δ, ad^{0}V) = (0) for i = 0, 1.

(b) For all irreducible 𝔽̄_{ℓ}[Δ]-submodules W ⊂ Hom(V, V) we can find h ∈ Δ and α ∈ 𝔽̄_{ℓ} with the following properties. The α-generalized eigenspace V_{h,α} of h on V is one-dimensional. Let

π_{h,α} : V → V_{h,α}; i_{h,α} : V_{h,α} ↪ V

denote, respectively, the h-equivariant projection and the h-equivariant inclusions of the indicated spaces. Then π_{h,α} ∘ W ∘ i_{h,α} ≠ (0).

Remark. It is not the case that if Δ contains a subgroup Δ′ which is big in the above sense, then Δ itself is necessarily big (bigger than big is not necessarily big). This is because condition (a) is not preserved under passage to a bigger group. To check condition (b), on the other hand, it clearly suffices to show that it holds for some subgroup Δ′ ⊂ Δ.

Lemma 3.2. Let F^{+}, τ, ρ^{n} = ρ_{E,ℓ}^{n}, and r_{τ} be as in the statement of Corollary 2.5, with τ as in the proof of Lemma 2.4. Suppose ℓ > 4n−1. Then r̄_{τ} (Γ_{F^{+}(ζ_{ℓ})}) is big in the sense of Definition 3.1.

Proof. We begin by establishing notation. Write ρ̄^{n} = ρ̄_{E,ℓ}^{n}. We define ad^{0} as in Definition 3.1, and write

ad ρ̄^{n} = ad^{0} ρ̄^{n} ⊕ 1, ad τ̄ = ad^{0} τ̄ ⊕ 1,

where 1 denotes the trivial representation. Then

ad^{0} r̄_{τ} = ad^{0} ρ̄^{n} ⊗ ad^{0} τ̄ ⊕ ad^{0} ρ̄^{n} ⊕ ad^{0} τ̄.  (3.2.1)

Let Δ = r̄_{τ} (Γ_{F^{+}(ζ_{ℓ})}), and define

Δ̃ = ρ̄^{n}(Γ_{F^{+}(ζ_{ℓ})}) × τ̄((Γ_{F^{+}(ζ_{ℓ})}).

The tensor product defines an exact sequence

1 → C → Δ̃ → Δ → 1

where the kernel C maps injectively to the center of GL(2, 𝔽_{ℓ}), viewed as the group of linear transformations of the space of τ̄. In particular, the order of C is prime to ℓ. (In fact, as the referee pointed out, under our hypotheses one checks easily that C is trivial, but this makes no difference in the sequel.) Finally, let Δ_{ρ} denote the image of ρ̄^{n}(Γ_{F^{+}(ζ_{ℓ})}) ⊂ Δ̃ in Δ. This is a normal subgroup of Δ isomorphic to the simple finite group P SL(2, 𝔽_{ℓ}), since n is odd and ℓ > 2n + 1. Moreover, Δ_{τ} := Δ/Δ_{ρ} is of order prime to ℓ, since the image of τ̄ is contained in the normalizer of a maximal torus. It follows from the inflation-restriction sequence that

H^{1}(Δ, W)→^{∼}⟶H^{0}(Δ_{τ} , H^{1}(Δ_{ρ}, W))  (3.2.2)

for any summand W of (3.2.1). Proof of (a): We first note that Δ acts irreducibly on r̄_{τ}, hence H^{0}(Δ, ad^{0} r̄_{τ}) = (0). We apply (3.2.2) to show that H^{1}(Δ, W) = 0 for each summand W of (3.2.1). Indeed, it suffices to show that H^{1}(P SL(2, 𝔽_{ℓ}), W) = 0 for each W. But as a representation of P SL(2, 𝔽_{ℓ}), each W is a direct sum of copies of i-dimensional symmetric powers Sym^{i−1} of the standard representation P SL(2, 𝔽_{ℓ}), where i runs through (odd) integers at most equal to 2n − 1. Since ℓ > 2n + 1, it is well known that H^{1}(P SL(2, 𝔽_{ℓ}), Sym^{i−1}) = 0 for i ≤ 2n − 1. Proof of (b): Let b denote a generator of the cyclic group k(v)^{×} ≃ μ_{ℓ−1}, as in the proof of Lemma 2.4. As remarked above, if Δ′ ⊂ Δ is a subgroup that satisfies 3.1(b) for a given summand W of (3.2.1), then Δ also satistifes this property for the given W. We may thus assume Im(τ̄) is contained in the normalizer of a maximal torus T in SL(2, 𝔽_{ℓ}) and contains an element t_{0} ∈ T with the two distinct eigenvalues b^{n}, b^{−n} on τ̄ , with corresponding eigenvectors v_{1} and v_{2}, as well as the element w ∉ T with eigenvalues 1 and −1. With appropriate normalizations, we can assume the corresponding eigenvectors are v_{1} + v_{2} and v_{1} − v_{2}, respectively. Write k = 𝔽_{ℓ}. We can write

ad τ̄ = k_{+} ⊕ k_{−} ⊕ U,

where U = Ind_{Γ_{L}·F^{+}}^{Γ_{F^{+}}} χ̄_{ℓ}/χ̄_{ℓ}^{c} and k_{+} and k_{−} are representations of Γ_{F^{+}} that factor through Gal(L · F^{+}/F^{+}), with the non-trivial element acting by the indicated sign. We thus have

Hom(r̄_{τ} , r̄_{τ}) = ad ρ̄^{n} ⊗ [k_{+} ⊕ k_{−} ⊕ U].  (3.2.3)

For i, j ∈ {1, 2}), let p_{i,j} ∈ End(τ̄) be the endomorphism that takes v_{i} to v_{j} and vanishes on v_{k} if k ≠ i. Then k_{+} (resp. k_{−}) is spanned by p_{1,1} +p_{2,2} (resp. p_{1,1} − p_{2,2}, whereas U is spanned by p_{1,2} and p_{2,1}.

Let t ∈ Im(ρ̄^{n}) be the image of an element of a split maximal torus of SL(2, 𝔽_{ℓ}), with n distinct eigenvalues under ρ̄^{n}, as in the proof of Lemma 3.2 of [HST]. More precisely, we can take t to be the diagonal element diag(b, b^{−1}), so that ρ̄^{n}(t) has eigenvalues b^{n−1}, b^{n−3}, …, b^{1−n}. The formula (3.2.3) expresses Hom(r̄_{τ} , r̄_{τ}) as a sum of four copies of ad ρ̄^{n}, as representation of t. Let h_{0}, resp h_{w} denote the image in Δ of (t, t_{0}) ∈ Δ̃, resp. the image of (t, w). Since ℓ > 4n − 1, b^{i} ≠ 1 for any i < 4n − 1, hence no ratio of eigenvalues of ρ̄^{n}(t) equals a ratio of eigenvalues of τ̄(t_{0}), nor of τ̄(w). It follows that all the generalized eigenspaces of h_{0} and h_{w} in r̄_{τ} are of dimension 1. It was shown in the proof of Lemma 3.2 of [HST] that ad ρ̄^{n} satisfies 3.1(b), with Δ replaced by Im(ρ̄^{n}) and with t playing the role of h; here we use the hypothesis that ℓ > 2n + 1. It thus follows that, if W is an irreducible summand of ad ρ̄^{n} ⊗[k_{+} ⊕k_{−}], then 3.1(b) is satisfied for this W with h = h_{0}. On the other hand, if W is an irreducible summand of ad ρ̄^{n} ⊗ U, then 3.1(b) is satisfied for this W with h = h_{w}. Indeed, it suffices to observe that the element (p_{1,2} + p_{2,1}) ∈ U takes the eigenvector v_{1} + v_{2} to itself. ⊓ ⊔

4. Removing τ

We fix an odd number n as above. The hypotheses of the earlier sections remain in force; in particular, we admit the Expected Theorems of §1.

Corollary 4.1. Let F^{+}(ρ̄_{E,ℓ}) denote the splitting field of ρ̄_{E,ℓ} and M = L · F^{+}(ρ̄_{E,ℓ}). Then there is a totally real Galois extension F^{′,+}/F^{+}, linearly disjoint from M over F^{+}, with the property that for i = 0, …, n, r_{i,F^{′,+}} = r_{i} |_{Γ_{F}^{′,+}} corresponds to a cuspidal automorphic representation Π_{i} of GL(2i, F^{′,+}), and such that r_{τ,F^{′,+}} = r_{τ} |_{Γ_{F}^{′,+}} corresponds to a cuspidal automorphic representation Π_{τ} of GL(2n, F^{′,+}).

Let E be a number field, and let ρ be an ℓ-adic representation of Γ_{E}. We assume ρ to be pure of some weight w, as in §1; thus we can define L^{norm}(s, ρ). We say L(s, ρ) (or L^{norm}(s, ρ) is invertible if it extends to a meromorphic function on ℂ and if L^{norm}(s, ρ) has no zeroes for Res ≥ 1 and no poles for Res ≥ 1 except for a possible pole at s = 1.

Let L′ = L · F^{′,+}, and let c ∈ Gal(L′/F^{′,+}) denote complex conjugation The proof of the following lemma was devised with a great deal of help from Richard Taylor.

Lemma 4.2. The representation Π_{τ} of GL(2n, F^{′,+}) is isomorphic to the automorphic induction from L′ of some cuspidal automorphic representation Π_{1}(τ) of GL(n, L′). After possibly replacing Π_{1}(τ) by its Galois conjugate Π_{1}(τ)^{c}, the tensor product Π_{1}(τ) ⊗ χ is isomorphic to its c-conjugate, hence descends to a cuspidal automorphic representation π of GL(n, F^{′,+}).

Proof. Let η_{L′} be the quadratic character of F^{′,+} associated to the extension

L′. By construction, τ_{ℓ} |_{Γ_{F^{′,+}}} ⊗η_{L′} →^{∼}⟶τ_{ℓ} |_{Γ_{F^{′,+}}} , hence

Π_{τ} ⊗ η_{L′} →^{∼}⟶Π_{τ} .  (4.2.1)

It follows from [AC, Chapter 3, Theorem 4.2 (b)] that there exists a cuspidal automorphic representation Π_{1}(τ) of GL(n, L′) such that Π_{τ} is isomorphic to the automorphic induction of Π_{1}(τ) from L′ to F^{′,+}. This means in particular that

Π_{1}(τ)^{c} ≄ Π_{1}(τ),

and

L(s, ρ_{τ,L′}) = L(s, Π_{τ,L′}) = L(s, Π_{1}(τ))L(s, Π_{1}(τ)^{c})  (4.2.2)

It follows from Corollary 2.5 that L(s, ρ_{E,F}^{m}), and more generally L(s, ρ_{E,F}^{m} ⊗ ξ_{ℓ}), is entire for all even m ≤ 2n, F = F^{′,+} or F = L′, when ξ_{ℓ} is the ℓ-adic Galois avatar of an algebraic Hecke character ξ of A_{F}^{×}. The proof of Theorem 4.2 of [HST] shows that L(s, ρ_{E,F}^{m} ⊗ ξ_{ℓ}) is invertible for all m ≤ 2n and for all algebraic Hecke characters ξ, for F = F^{′,+} or F = L′. Of course L(s, ρ_{τ,F^{′,+}}) is also entire and L(s, ρ_{τ,L′}) is entire unless n = 1, which gives rise to the only possible pole at s = 1.

Consider the automorphic L-function

L(s) = L(s, Π_{τ,L′} × Π_{τ,L′}^{∨} ⊗ (χ/χ^{c}))  (4.2.3)

Comparing this to (4.2.2) we find that

L(s) = L(s, [ρ_{E,L′}^{n} ⊗ (χ ⊕ χ^{c})] ⊗ [ρ_{E,L′}^{n,∨} ⊗ (χ^{−1} ⊕ χ^{c,−1})] ⊗ [χ/χ^{c}])

= L(s, (ρ_{E,L′}^{n} ⊗ ρ_{E,L′}^{n,∨}) ⊗ [(χ/χ^{c}) ⊕ (χ/χ^{c}) ⊕ (χ/χ^{c})^{2}]) · L(s, ρ_{E,L′}^{n} ⊗ ρ_{E,L′}^{n,∨}) (4.2.4)

Writing

ρ_{E,L′}^{n} ⊗ ρ_{E,L′}^{n,∨} = ⊕_{i=0}^{n−1}ρ_{E,L′}^{2i+1} ⊗ ω_{ℓ}^{−i}

we see that the first factor of the last line of (4.2.4) is a product of invertible L-functions without poles at s = 1; the final factor of the last line has a simple pole at s = 1. Thus L(s) has a simple pole at s = 1. But L(s) is an automorphic L-function for GL(n) × GL(n). We rewrite L(s) using (4.2.2):

L(s) = L(s, Π_{1}(τ) × Π_{1}(τ)^{∨} ⊗ (χ/χ^{c})) · L(s, Π_{1}(τ)^{c} × Π_{1}(τ)^{c,∨} ⊗ (χ/χ^{c})) ·L(s, Π_{1}(τ) × Π_{1}(τ)^{c,∨} ⊗ (χ/χ^{c})) · L(s, Π_{1}(τ)^{c} × Π_{1}(τ)^{∨} ⊗ (χ/χ^{c}))

Applying the Jacquet-Shalika classification theorem we see that exactly one of the factors has a simple pole, and in that case the factor is necessarily of the form L(s, Π × Π^{∨}). Since (χ/χ^{c})_{∞} is a character of infinite order, neither of the first two factors can have a pole; we must therefore either have

Π_{1}(τ)^{∨}→^{∼}⟶Π_{1}(τ)^{c,∨} ⊗ (χ/χ^{c})

or

Π_{1}(τ)^{c,∨}→^{∼}⟶Π_{1}(τ)^{∨} ⊗ (χ/χ^{c}).

In other words, up to exchanging Π_{1}(τ) with Π_{1}(τ)^{c}, we have

Π_{1}(τ) ⊗ χ→^{∼}⟶(Π_{1}(τ) ⊗ χ)^{c};

hence Π_{1}(τ)⊗χ descends to a cuspidal automorphic representation of GL(n, F^{′,+}). This completes the proof. ⊓ ⊔

Now (Π_{τ})_{∞} is cohomological, hence (Π_{τ})_{L′,∞} is also cohomological. But (Π_{τ})_{L′,∞} is represented as a subquotient of the representation of GL(2n, L′_{∞}) induced from the representation (Π_{1}(τ))_{∞} ⊗ (Π_{1}(τ)^{c})_{∞} of the Levi factor GL(n, L′_{∞})×GL(n, L′_{∞}) of the relevant maximal parabolic, and it follows that Π_{1}(τ)_{∞} is also cohomological. Thus Π_{1}(τ) satisfies condition (1) of Expected Theorem 1.2.

On the other hand, ρ_{τ} has a symplectic polarization with multiplier ω_{ℓ}^{1−2n}, by construction. It follows that the associated automorphic representation Π_{τ} is self dual. This property is preserved under base change to L′. It thus follows from (4.2.2) and the Jacquet-Shalika classification theorem that

{Π_{1}(τ), Π_{1}(τ)^{c}} = {Π_{1}(τ)^{∨}, Π_{1}(τ)^{c,∨}}

as sets. Thus either (a) Π_{1}(τ) satisfies condition (2) of Expected Theorem 1.2,

or (b) Π_{1}(τ)→^{∼}⟶Π_{1}(τ)^{∨}.

Assume (a) holds. Then Π_{1}(τ) satisfies both conditions of Expected Theorem 1.2, hence is associated to an n-dimensional Galois representation ρ_{1}(τ) of Γ_{L′}. It follows from (4.2.2) that

ρ_{1}(τ) ⊕ ρ_{1}(τ)^{c}→^{∼}⟶ρ_{E,ℓ}^{n} ⊗ χ_{ℓ} ⊕ ρ_{E,ℓ}^{n} ⊗ χ_{ℓ}^{c}  (4.3)

Since ρ_{E,ℓ}^{n} is irreducible, it follows that it must be equal to either ρ_{1}(τ)⊗χ_{ℓ} or ρ_{1}(τ) ⊗ χ_{ℓ}^{c}. In either case, ρ_{E,ℓ}^{n} is associated to a cuspidal automorphic representation Π_{n,L′} of GL(n, L′). Since ρ_{E,ℓ}^{n} descends to a representation of Γ_{F^{′,+}}, Π_{n,L′} descends to a cuspidal automorphic representation Π_{n} of GL(n, F^{′,+}). Thus ρ_{E,ℓ}^{n} is automorphic over F^{′,+}.

Assume (b) holds. Then the central character ξ of Π_{1}(τ) is also self-dual, i.e. ξ = ξ^{−1}. It follows from Lemma . that

Π_{1}(τ) ⊗ χ→^{∼}⟶Π_{1}(τ)^{c} ⊗ χ^{c}.

Combining this with (b), we have

ξ · χ ∘ det = ξ^{c} · χ^{c} ∘ det;

This implies that χ/χ^{c} is self-dual, ie.

(χ^{c})^{2} = χ^{2}.

But this is already false for the archimedean components. Thus (b) is impossible.

We have thus proved that ρ_{E,ℓ}^{n} is automorphic over F^{′,+}. More generally, Theorem 2.1 allows us to add new r_{τ} ’s of different dimensions 2n_{i}, with n_{i} odd, to the list in Corollary 2.5. By adding ρ_{E,ℓ}^{2i+1} ⊗ τ_{i} to the list – we are free to vary the 2-dimensional τ_{i} if we like, we thus obtain our main theorem:

Theorem 4.4. Assume the Expected Theorems of §1. Let F^{+} be a totally real field, and let E be an elliptic curve over F^{+}. Let n be a positive integer. Then there is a finite totally real Galois extension F^{′,+}/F and, for each positive integer i ≤ n, a cuspidal automorphic representation Π_{i} of GL(i, F^{′,+}), satisfying conditions (a) and (b) of Expected Theorem 1.2, such that

ρ_{E,ℓ}^{i} |_{Γ_{F^{′,+}}} = ρ_{Π_{i},ℓ}.

In particular, if i > 1, L^{norm}(s, ρ_{E,ℓ,F^{′,+}}^{i}) = L(s, Π_{i}) is an entire function.

I repeat that we are not assuming that E have non-integral j-invariant; however, all statements are conditional on the Expected Theorems of §1.

5. Applications and generalizations

We continue to admit the Expected Theorems of §1. Let E and E′ be two elliptic curves over F^{+} without complex multiplication. Assume E and E′ are not isogenous. It then follows from Faltings’ isogeny theorem that ρ_{E,ℓ} and ρ_{E′,ℓ} are not isomorphic as representations of Γ_{F^{+}} for all ℓ. Since the traces of F rob_{v}, for primes v of good reduction for E and E′, are integers that determine ρ_{E,ℓ} and ρ_{E′,ℓ} up to isomorphism, it follows that ρ̄_{E,ℓ} and ρ̄_{E′,ℓ} are not isomorphic for sufficiently large ℓ. By Serre’s theorem, if ℓ is sufficiently large, Im(ρ̄_{E,ℓ}) = Im(ρ̄_{E′,ℓ}) = GL(2, 𝔽_{ℓ}).

Let m, m′ be two positive integers. Applying Theorem 2.1, we obtain the analogue of Corollary 2.5 for the collection of representations r_{i} = ρ_{E,ℓ}^{2i}, r′_{j} = ρ_{E′,ℓ}^{2j}, 1 ≤ i ≤ m, 1 ≤ j ≤ m′, together with r_{τ} = ρ_{E,ℓ}^{m} ⊗ τ and r′_{τ′} = ρ^{m′} ⊗ τ′ if m or m′ is odd, provided τ̄ and τ̄′ are “big enough”. Bearing in mind the results of §3, we have the following statement:

Proposition 5.1. Let L be as in §3, and define

τ_{ℓ} = Ind_{Γ_{L}}^{Γ_{ℚ}}χ_{ℓ}; τ′_{ℓ} = Ind_{Γ_{L}}^{Γ_{ℚ}}χ′_{ℓ}

where χ (resp. χ′) is a Hecke character with χ_{∞}(z) = z^{−m}, (resp. χ′_{∞}(z) = z^{−m′}) if m (resp. m′) is odd. We assume

χ |_{A_{ℚ}^{×}}= | • |_{A}^{−m} · η_{L}; χ′ |_{A_{ℚ}^{×}}= | • |_{A}^{−m′} · η_{L}

Suppose there is a prime ℓ > sup(2m + 1, 2m′ + 1), unramified in F^{+}, such that

(i) E has good ordinary reduction at all primes w dividing ℓ (ii) ℓ splits in L; (iii) ℓ splits in ℚ(ζ_{2i+1}), i = 1, …, sup(m − 1, m′ − 1); in particular, ℓ ≡ 1

(mod mm′). (iv) \frac{ℓ−1}{m} > 2, \frac{ℓ−1}{m′} > 2.

Let M be an arbitrary extension of F^{+}. Define r_{i}, r′_{j} r_{τ} and r_{τ′} as above. Then there is a totally real Galois extension F^{′,+}/F^{+}, linearly disjoint from M over F^{+}, with the property that for i = 1, …, m, (resp. j = 1, …, m′) r_{i,F^{′,+}} = r_{i} |_{Γ_{F}^{′,+}} (resp. r′_{j,F^{′,+}}) corresponds to an automorphic representation Π_{i} of GL(2i, F^{′,+}) ((resp. Π′_{j} of GL(2j, F^{′,+})) and such that r_{τ} (resp. r′_{τ′}) corresponds to an automorphic representation Π_{τ} of GL(2m, F^{′,+}) (resp. Π′_{τ′} of GL(2m′, F^{′,+}). If F′/F^{′,+} is a CM quadratic extension, then the base change Π_{i,F′} (resp. Π_{τ,F′}) has archimedean constituent isomorphic to Π_{∞,0}(2i, F′) (resp. Π_{∞,0}(2n, F′)), and likewise for Π′_{j,F′}, Π′_{τ′,F′}.

The discussion of §4 applies to both Π_{τ} and Π′_{τ}, and we obtain the following strengthening of Theorem 4.4.

Theorem 5.2. Assume the Expected Theorems of §1. Let F^{+} be a totally real field, let E and E′ be elliptic curves over F^{+}, and assume E and E′ do not become isogenous over an abelian extension of F^{+}. Let m and m′ be positive integers. Then there is a finite totally real Galois extension F^{′,+}/F and, for each positive integer i ≤ m, (resp. j ≤ m′) a cuspidal automorphic representation Π_{i} of GL(i, F^{′,+}) (resp. Π′_{j} of GL(j, F^{′,+})) satisfying conditions (a) and (b) of Expected Theorem 1.2, such that

ρ_{E,ℓ}^{i} |_{Γ_{F^{′,+}}} = ρ_{Π_{i},ℓ}; ρ_{E′,ℓ}^{j} |_{Γ_{F^{′,+}}} = ρ_{Π′_{j},ℓ}

In particular, if m · m′ > 1,

L^{norm}(s, ρ_{E,ℓ,F^{′,+}}^{m} ⊗ ρ_{E′,ℓ,F^{′,+}}^{m′}) = L(s, Π_{m} × Π′_{m′})

is an entire function.

The Rankin-Selberg L-function has no poles if m ≠ m′; if m = m′ it has a pole if and only if Π′_{m′} →^{∼}⟶ Π_{m}^{∨}. which implies that the corresponding Galois representations ρ_{E,ℓ}^{m} and ρ_{E′,ℓ}^{m} are isomorphic. The kernel of the map of the standard 2-dimensional representation of GL(2) to its mth symmetric power is finite and contained in the center. If ρ_{E,ℓ}^{m}→^{∼}⟶ρ_{E′,ℓ}^{m}, it thus follows that the corresponding adjoint representations ad ρ_{E,ℓ} and ad ρ_{E′,ℓ} are isomorphic, hence that there exists an abelian character η, necessarily finite, such that ρ_{E′,ℓ}→^{∼}⟶ρ_{E,ℓ} ⊗ η. Thus ρ_{E′,ℓ} and ρ_{E,ℓ} become isomorphic over a finite extension of F^{+}, hence E and E′ are isogenous by Faltings’ theorem.

Using Brauer’s theorem, as in the proof of Theorem 4.2 of [HST], we then obtain:

Theorem 5.3. Assume the Expected Theorems of §1. Let F^{+} be a totally real field, let E and E′ be elliptic curves over F^{+}, and assume E and E′ do not become isogenous over an abelian extension of F^{+}. Let m and m′ be positive integers. Then the L-function L(s, ρ_{E,ℓ}^{m} ⊗ ρ_{E′,ℓ}^{m′}) is invertible and satisfies the expected functional equation.

Proof. This is obtained from Theorem 5.2 by applying Brauer’s theorem, as in the proof of Theorem 4.2 of [HST]. It suffices to mention that the nonvanishing of the Rankin-Selberg L-function along the line Re(s) = 1 of a pair of cuspidal automorphic representations (with unitary central characters) is due in general to Shahidi [Sh]. ⊓ ⊔

Finally, here is the precise statement of the question of Mazur and Katz mentioned in the introduction, together with the affirmative response. Recall the notation k_{v} and q_{v} of §1.

Theorem 5.4. Assume the Expected Theorems of §1. Let F^{+} be a totally real field, let E and E′ be elliptic curves over F^{+}, and assume E and E′ do not become isogenous over an abelian extension of F^{+}. For any prime v of F^{+} where E and E′ both have good reduction, we let

|E(k_{v})| = (1 − qv^{\frac{1}{2}} e^{iϕ_{v}})(1 − qv^{\frac{1}{2}} e^{−iϕ_{v}})

|E′(k_{v})| = (1 − qv^{\frac{1}{2}} e^{iψ_{v}})(1 − qv^{\frac{1}{2}} e^{−iψ_{v}})

where ϕ_{v}, ψ_{v} ∈ [0, π].

Then the pairs (ϕ_{v}, ψ_{v}) ∈ [0, π] × [0, π] are uniformly distributed with respect to the measure

\frac{4}{π^{2}} sin^{2}ϕ sin^{2}ψ dϕdψ.

Proof. Theorem 5.4 follows directly from Theorem 5.3 by the argument in [S], Appendix to §I. ⊓ ⊔

6. Concluding remarks

The author and Richard Taylor have independently noticed that tensoring with an induced representation from a Hecke character may be useful in other situations. For example, let f be an elliptic modular form of weight k, ρ_{f,ℓ} : Γ_{ℚ} → GL(2, ℚ̄_{ℓ}) the associated two-dimensional Galois representation, and let ρ_{f,ℓ}^{n} = Sym^{n−1}ρ_{f,ℓ}. There is no hope of applying the potential modularity technique of [HST] to ρ_{f,ℓ}^{n} if k > 2: the series of Hodge-Tate weights at ℓ has gaps for all n, and Griffiths transversality implies it is impossible to obtain families of positive dimension of motives with such Hodge-Tate numbers. However, if k is odd, one can choose a Hecke character χ of an abelian CM extension L/ℚ of degree k −1 with infinity type so chosen that ρ_{f,ℓ}^{n} ⊗Ind_{Γ_{L}}^{Γ_{ℚ}}χ_{ℓ} has an unbroken series of Hodge-Tate weights. (If k is even one takes L of degree 2(k − 1).)

Two serious obstacles remain. In the first place, the constructions in §4 require that we know in advance that the symmetric power L-functions are invertible, and this information is not available a priori for k > 2. In the second place, the arguments for finding rational points on moduli spaces over number fields unramified at ℓ break down in higher weights.

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