MODULARITY OF THE RANKIN-SELBERG L-SERIES, AND MULTIPLICITY ONE FOR SL(2)

DINAKAR RAMAKRISHNAN

Contents

1. Introduction 1

2. Notations and Preliminaries 5

3. Construction of ⊠ : A(GL(2)) × A(GL(2)) → A(GL(4)) 8 3.1. Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product L-functions: local factors and holomorphy 15 3.4. Boundedness in vertical strips 18 3.5. Modularity in the good case 30 3.6. A descent criterion 32 3.7. Modularity in the general case 35

4. Applications 37 4.1. A multiplicity one theorem for SL(2) 37 4.2. Some new functional equations 40 4.3. Root numbers and representations of orthogonal type 42 4.4. Triple product L-functions revisited 44 4.5. The Tate conjecture for 4-fold products of modular curves 47

Bibliography 52

1. Introduction

Let f, g be primitive cusp forms, holomorphic or otherwise, on the upper half plane H of levels N, M repectively, with (unitarily normalized) L-functions

L(s, f) = ∑_{n≥1} \frac{a_{n}}{n^{s}} = ∏_{p prime} [(1 − α_{p}p^{−s})(1 − β_{p}p^{−s})]^{−1}

and

L(s, g) = ∑_{n≥1} \frac{b_{n}}{n^{s}} = ∏_{p prime} [(1 − α′_{p}p^{−s})(1 − β′_{p}p^{−s})]^{−1}.

When p does not divide N (resp. M), the inverse roots α_{p}, β_{p} (resp. α′_{p}, β′_{p}) are non-zero with sum a_{p} (resp. b_{p}). For every p prime to NM, set

L_{p}(s, f × g) = [(1 − α_{p}α′_{p}p^{−s})(1 − α_{p}β′_{p}p^{−s})(1 − β_{p}α′_{p}p^{−s})(1 − β_{p}β′_{p}p^{−s})]^{−1}.

Let L^{∗}(s, f × g) denote the (incomplete Euler) product of L_{p}(s, f × g) over all p not dividing NM. This is closely related to the convolution L-series ∑_{n≥1} a_{n}b_{n}n^{−s},

whose miraculous properties were first studied by Rankin and Selberg.

Modularity of the Rankin-Selberg L-series 3

A fundamental question, first raised by Langlands, is to know whether this Rankin-Selberg product is modular, i.e., if there exists an automorphic form f ⊠ g on GL(4)/ℚ whose standard L-function equals L^{∗}(s, f ×g) after removing the ramified and archimedean factors. The first main result of this paper is to answer it in the affirmative, in fact with the base field ℚ replaced by any number field F (see Theorem M, section 3). Our proof uses a mixture of converse theorems, base change and descent, and it also appeals to the local regularity properties of Eisenstein series and the scalar products of their truncations. Briefly, experts have long suspected that this result should follow from the converse theorem for GL(4) requiring only twists by GL(2) and GL(1), which has recently been published by Cogdell and Piatetski-Shapiro ([CoPS]). While this is morally true, three difficulties crop up when one tries to implement this, and new ideas are required to surmount them. The first difficulty arises because, given cuspidal automorphic representations π, π′ on GL(2)/F, one needs to have a definition of a candidate for π ⊠ π′ as an admissible representation of GL(4, 𝔸_{F}) (which is needed before we can test its analytic properties to show modularity). If one admits the local Langlands correspondence, then the local candidates π_{v} ⊠ π′_{v} can be defined as the admissible representations of GL(4, F_{v}) corresponding to σ_{v} ⊗ σ′_{v}, where σ_{v} (resp. σ′_{v}) is the 2-dimensional representation of the Weil-Deligne group W′_{F_{v}} associated to π_{v} (resp. π′_{v}) by Kutzko ([Ku]). (Since the work on this paper was completed, a preprint of M. Harris and R.L. Taylor has appeared, establishing the local correspondence for GL(n). But we feel that it will be satisfying not to to have to appeal to it here; global arguments should always be able to circumvent fine local difficulties.) We get around this problem by appealing to the base change results of Arthur and Clozel ([AC]). To be precise, we first make use of the fact that every supercuspidal representation becomes a principal series representation after a finite normal solvable base change, so that a candidate for π ⊠ π′ can be defined over suitable (infinite families of) global, solvable extensions K/F. Then, after proving modularity upstairs, we perform a simultaneous descent via an inductive argument in cyclic layers. At the end, we get as a byproduct the definition of π_{v} ⊠ π′_{v} at any v (and any F). Once one has the admissible Π on GL(4)/K, one needs, for modularity via the converse theorem, good analytic information on L(s, Π×η), for any cuspidal η on GL(m)/K, for m ≤ 2. The m = 1 case is easy by the Rankin-Selberg theory as extended to general K by Jacquet ([J]). But the m = 2 case is subtle and leads to problems. One of them, which is the second difficulty, is caused by not knowing the equality, at every place, with each other and with L(s, Π×η), of the three candidates for the triple product L-function L(s, π × π′ × η). The first is defined formally as an Euler product of {L(s, σ_{v} ⊗ σ′_{v} ⊗ τ_{v})}, where τ_{v} is the 2-dimensional of W′_{F_{v}} associated to η_{v}; the second candidate is defined by the integral representation of Garrett ([G]), as generalized by Piatetski-Shapiro and Rallis ([PS-R2]); and the third is given by the machinery of Langlands-Shahidi ([La2], [Sh1]). Though the unramified local factors are known to be the same in all cases, something close to an equality is essential as no candidate has all the desired properties, and they have complementary strengths. (One simply cannot avoid dealing with the bad places!) By a careful analysis and synthesis of known results due to Ikeda and others, and by using in addition some global arguments involving the works of Langlands and Tunnell on the Artin conjecture, we manage to prove at the end that the first candidate has all the desired properties under some local restrictions which can be achieved by solvable base change. So base change is used for yet another reason! (At the archimedean places, we make use of the work of Ikeda ([Ik2]) on the computation of the triple product L-factors for unramified representations.) We prove later the equality, at each place v, of L(s, π_{v} × π′_{v} × η_{v}) (resp. ε(s, π_{v} × π′_{v} × η_{v})) with the GL(4)×GL(2)- factor L(s, (π_{v} ⊠π′_{v})×τ_{v}) (resp. ε(s, (π_{v} ⊠π′_{v})×η_{v})). The third and final difficulty is the question of boundedness in vertical strips of the triple product L-function. This is absolutely crucial for applying the converse theorem, and this does not seem to be elementary as, for example, in the case of the standard L-function of GL(2). Our method is to use Arthur’s truncation ([A1]) of the non-cuspidal Eisenstein series E(g, s) on GSp(6)/F which occurs in the integral representation of L(s, π × π′ × η), and reduce the problem, via local regularity of eigenfunctions of the Laplacian, to the norms of the truncated Eisenstein series, making use of the fact that the intervening constant terms are in our case expressible in terms of abelian L-functions. The reader is referred to section 3.1 for a fuller discussion of the strategy of proof of the various parts. The details of proof occupy the sections 3.2- 3.7.

As a consequence of Theorem M, we settle a conjecture of Labesse and Langlands ([LL], [Lab1]) asserting that the space of cusp forms on SL(2) has multiplicity one (see Theorem 4.1.1). To see what this means concretely, consider f, g as above with trivial characters such that

a_{p}^{2} = b_{p}^{2}

for almost all p. Then multiplicity one (in this context) implies that there exists a quadratic Dirichlet character χ such that (for almost all p)

a_{p} = b_{p}χ(p).

(In fact, if N, M are in addition square-free, χ must be trivial.) When we started on this project in 1994, we were able to settle quickly the case when f and g are holomorphic, but then learnt that this case had been known to various people including D. Blasius and J.-P. Serre; here the idea is to make use of the associated ℓ-adic Galois representations (see [K], pp.90-91, and [Ra2], where there is also a mod ℓ analog and a density result). But this method does not work for Maass forms, and the starting point for this paper was our realization in Fall 1994 that both cases could be tackled simultaneously if one knew of the existence of f ⊠g. We first managed to prove multiplicity one for SL(2) over ℚ in Fall 96 by some special tricks and a weaker form of Theorem M. The proof given here works over arbitrary F, but is shorter, partly because some the earlier arguments over ℚ have been transplanted to the proof of Theorem M. If there is any creativity in this paper, it is perhaps foremost in the application of Theorem M to this problem, though it is not the most technically difficult part. It should also be remarked that it has been expected for some time now that the Labesse-Langlands conjecture should follow from the adjoint square lifting from SL(2) to PGL(3), more precisely from a careful comparison of the stable trace formula for SL(2) and the twisted trace formula for PGL(3) relative to g → ^{t}g^{−1}. This approach has been considered by Y. Flicker in a series of papers culminating in [F]. But we ignore the question of whether this program has been completed, as our approach is totally different and (hopefully) has independent interest. We would also like to note in passing that, as shown by D. Blasius ([B]), multiplicity one fails for SL(n), for every n > 2.

A corollary of Theorem M is the deduction of the standard analytic properties of 4-fold convolutions L(s, f_{1} × f_{2} × f_{3} × f_{4}) for quadruples (f_{1}, f_{2}, f_{3}, f_{4}) of primitive cusp forms. This is done by appealing, in addition, to the higher Rankin-Selberg theory for GL(4)×GL(4) due to Jacquet, Piatetski-Shapiro and Shalika ([JPSS1]), and Shahidi ([Sh 1,3,5]). We show, moreover, that L(s, sym^{2}(f_{1}) × f_{2} × f_{3}) and L(s, sym^{3}(f_{1}) × f_{2}) have meromorphic continuations and functional equations (see Theorem 4.2.1) and are non-zero in {ℜ(s) ≥ 1} without any pole there except possibly at s = 1.

By using the existence of ⊠ on GL(2)×GL(2) in conjunction with a global argument we prove an identity, even at the ramified places v, equating the Rankin-Selberg L and ε-factors of (π_{1,v}⊠π_{2,v}, π_{3,v}⊠π_{4,v}) with those of σ_{1,v}⊗σ_{2,v}⊗σ_{3,v}⊗σ_{4,v}, where σ_{j,v} denotes, for each j ≤ 4, the 2-dimensional representation of W′_{F_{v}} associated to π_{j,v} (cf. [Ku]). This gives a little bit of information (see Proposition 4.3.3) on the local Langlands correspondence for GL(4). Moreover, we use the identities and the method of [PR] to prove that when π_{j} has trivial central character for every j, the global root number W ((π_{1} ⊠ π_{2}) × (π_{3} ⊠ π_{4})) is 1 (see Theorem 4.3.4).

We then turn the tables and prove in section 4.4, that as a consequence of Theorem M, the three candidates for the triple product L-functions on GL(2) all agree at all places. (Recall that we needed to know that they were very nearly the same to get Theorem M in the first place.) This may be of independent interest.

The final application of our main result is the proof of the Tate conjecture for 4-fold products V of modular curves, asserting in particular that the order of pole at s = 2 of the L-function over any solvable (normal) number field K of the Galois module W_{ℓ} := H_{et}^{4}(V\overline{ℚ}, ℚ_{ℓ}) equals the rank of the group of K-rational codimension 2 Tate cycles on V\overline{ℚ} (see Theorem 4.5.1). Moreover we show, in line with the works of Ribet ([Ri1]) and V.K. Murty ([Mu]) on the Jacobians of modular curves, that the latter number can also be computed with the Tate cycles replaced by the algebraic cycles modulo homological equivalence if the level of at least one of the curves is square-free. We refer to chapter 5 for a precise statement.

We would like to express our gratitude to Ilya Piatetski-Shapiro for his continued interest in this project, and for kindly writing down, with J. Cogdell, the form of the converse theorem for GL(4) which we need ([CoPS]). Thanks are also due to T. Ikeda for writing down his calculations of the archimedean factors of the triple product L-functions ([Ik2]), to S. Rallis for useful remarks on these L-functions, to F. Shahidi for explaining his approach to the same via Langlands’s theory of Eisenstein series and for commenting on an earlier version, to my colleague T. Wolff for helpful conversations on an analytic lemma we use in section 3.4, and to many others, including H. Jacquet, R.P. Langlands, J. Rogawski and P. Sarnak, who have shown encouragement and interest. Special thanks must go to J. Cogdell for reading the earlier and the revised versions thoroughly and making crucial remarks. Part of the technical typing of this paper was done by Cherie Galvez, whom we thank. Finally, we would like to express our appreciation to the following: the National Science Foundation for support through the grants DMS-9501151 and DMS-9801328, Université Paris-sud, Orsay, where we spent a fruitful month during September 1996, the DePrima Mathematics House in Sea Ranch, CA, for inviting us to visit and work there during August 1996 and 1998, MATSCIENCE, India, for hospitality in Feb. 98, and - last, but not the least - the MSRI, Berkeley, for (twice) providing the right climate to work in; this project was started (in 1994) and essentially ended there.

2. Notations and Preliminaries

2.1. Let \overline{ℚ} denote the algebraic closure of ℚ in ℂ. For any subfield K of \overline{ℚ}, let Gal(\overline{ℚ}/K) denote the Galois group of \overline{ℚ} over K, together with the profinite topology. For any number field F with ring of integers 𝔒_{F}, let Σ(F) (resp. Σ_{∞}(F), resp. Σ_{0}(F)) denote the places (resp. archimedean places, resp. finite places) of F. At each v, let F_{v} denote the completion of F relative to v, and if moreover v is finite, let 𝔒_{v}, 𝔓_{v}, 𝔽_{v}, |.|_{v} and Nv = q_{v} respectively denote the ring of integers of F_{v}, the maximal ideal, the residue field, the normalized absolute value and the norm of v. When F = ℚ, there is a unique archimedean place ∞ given by the canonical imbedding of ℚ in ℝ, and every finite place v corresponds to a rational prime number p, in which case we will write ℤ_{p} instead of O_{v}. Denote by 𝔸_{F} the ring of adeles of F, which is the restricted direct product of {F_{v}} relative to {O_{v}}, equipped with its usual locally compact topology. Let I_{F} denote the group of ideles of F, and C_{F} the idele class group I_{F} /F^{∗}.

We will fix a non-trivial unitary character ψ of 𝔸_{F} which is trivial on F, and let Ψ_{v} the v-component of Ψ. Various quantities, such as the ε-factors, will depend on this choice, which we will suppress in our notation.

In this paper we will systematically use the powerful language of automorphic representations, though at relevant places we will indicate briefly the essence of what we do in the classical language. Given any primitive cusp form f on the upper half plane of level N, character ω, there exists a (unique) cuspidal automorphic representation π = π(f) of GL(2, 𝔸_{ℚ}) of conductor N and central character ω such that, at every prime p, the p-Euler factor of L(s, f) agrees with the L-factor L(s, π_{p}) of the p-component π_{p}. We refer the reader to the expository monograph [Ge] to understand how to go back and forth between the two approaches.

2.2. For every algebraic group G over F, let G(𝔸_{F}) denote the restricted direct product ∏′_{v} G(F_{v}), endowed with the usual locally compact topology. By a cuspidal representation of G(𝔸_{F}) = G(F_{∞}) × G(𝔸_{F,0}), we will always mean an irreducible, unitary, cuspidal automorphic representation. Such a representation is in particular a restricted tensor product π = ⊗′_{v}π_{v} = π_{∞} ⊗ π_{0}, where each π_{v} is an (irreducible) admissible representation of G(F_{v}) for v finite, and an admissible (LieG_{v}, K_{v})−module for v archimedean, with K_{v} denoting a compact modulo center subgroup of G(F_{v}); π_{0} (resp. π_{∞}) is the restricted tensor product of π_{v} over all finite (resp. archimedan) places v. By definition, π_{v} must be unramified at almost all v.

For any irreducible, automorphic representation π of GL(n, 𝔸_{F}), let L(s, π) = L(s, π_{∞})L(s, π_{0}) denote the associated standard L−function ([J]) of π; it has an Euler product expansion

(2.2.1)  L(s, π) = ∏_{v} L(s, π_{v}),

convergent in a right-half plane. If v is an archimedean place, then one knows (cf. [La1]) how to associate a semisimple n−dimensional ℂ−representation σ(π_{v}) of the Weil group W_{F_{v}}, and L(π_{v}, s) identifies with L(σ_{v}, s). On the other hand, if v is a finite place where π_{v} is unramified, there is a corresponding semisimple (Langlands) conjugacy class A_{v}(π) in GL(n, ℂ) such that

(2.2.2)  L(s, π_{v}) = det(1 − A_{v}T)^{−1}|_{T =q_{v}^{−s}}.

We may find a diagonal representative diag(α_{1,v}(π), ..., α_{n,v}(π)) for A_{v}(π), which is unique up to permutation of the diagonal entries. Let [α_{1,v}(π), ..., α_{n,v}(π)] denote the resulting unordered n−tuple. Since W_{F,v}^{ab} ≃ F_{v}^{∗}, A_{v}(π) clearly defines an abelian n−dimesional representation σ(π_{v}) of W_{F,v}. One has

Theorem 2.2.3 ([JS1]) Let n ≥ 1, and π a non-trivial cuspidal representation of GL(n, 𝔸_{F}). Then L(s, π) is entire. Moreover, for any finite set S of places of F, the incomplete L−function L^{S}(s, π) = ∏_{v∉S} L(s, π_{v}) is holomorphic in ℜ(s) > 0.

When n = 1, such a π is simply a unitary idele class character and this result is due to Hecke. Also, when π is trivial, L(s, π) = ζ_{F} (s).

By the theory of Eisenstein series, one has a sum operation ⊞ ([La3]), which results in the following

Theorem 2.2.4 ([JS2]) Given any m−tuple of cuspidal representations π_{1}, ..., π_{m} of GL(n_{1}, 𝔸_{F}), ..., GL(n_{m}, 𝔸_{F}) respectively, there exists a unitary, irreducible, automorphic representation π_{1} ⊞ ... ⊞ π_{m} of GLn, 𝔸_{F}), n = n_{1} + ... + n_{m}, which is unique up to equivalence, such that for any finite set S of places,

L^{S}(s, ⊞_{j=1}^{m}π_{j}) = ∏_{j=1}^{m} L^{S}(s, π_{j}).

Call such a (Langlands) sum π ≃ ⊞_{j=1}^{m}π_{j}, with each π_{j} cuspidal, an isobaric representation. Denote by ram(π) the finite set of finite places where π is ramified, and let 𝔑(π) be its conductor ([JPSS2]).

For every integer n ≥ 1, set:

(2.2.5)  A(n, F) = {π : isobaric representation of GL(n, 𝔸_{F})}/≃,

and

A_{0}(n, F) = {π ∈ A(n, F)| π cuspidal}.

Put A(F) = ∪_{n≥1}A(n, F) and A_{0}(F) = ∪_{n≥1}A_{0}(n, F).

Remark. One can also define the analogs of A(n, F) for local fields F, where the “cuspidal” subset A_{0}(n, F) consists of essentially square-integrable representations of GL(n, F). See [La3] and [Ra1] for details.

Let π, π′ be isobaric automorphic representations in A(n, F), A(n′, F) respectively. Then there exist an associated Euler product L(s, π × π′) ([JS2,4], [JPSS], [COPS2]; [Sh1,3]), which converges in {ℜ(s) > 1}, and admits a meromorphic continuation to the whole s−plane with a functional equation. When v is archimedean or a finite place outside ram(π), one has

(2.2.6)  L_{v}(s, π × π′) = L(s, σ(π_{v}) ⊗ σ(π′_{v})).

When n = 1, L(s, π × π′) = L(s, ππ′), and when n = 2 and F = ℚ, this function is the Rankin-Selberg L−function, extended to arbitrary global fields by Jacquet ([J2]).

Theorem 2.2.7 (Jacquet-Shalika [JS2]) Let π ∈ 𝔄_{0}(n, F), π′ ∈ 𝔄_{0}(n′, F), and S a finite set of places. Then L^{S}(s, π × π′) is holomorphic in {ℜ(s) > 0} unless π ≃ π′^{∨}, in which case it has a unique pole (of order 1) at s = 1.

We will also need the general construction of automorphic L-functions attached to any π ∈ A(n, F). Let S be any finite set of places containing the archimedean and ramified places (for π). If r is any algebraic representation of GL(n, ℂ), which is the connected component of the L-group of GL(n), we put

L^{S}(s, π, r) = ∏_{v∉S} L(s, π, r),

where

L(s, π, r) = det(I − (Nv)^{−s}r(A_{v}(π)))^{−1}.

Of particular interest is when r is the exterior square Λ^{2}, or the symmetric kth power Sym^{k}, for some k ≥ 1, of the standard represenation. When r is simply the standard representation, L^{S}(s, π, r) is evidently just L^{S}(s, π).

One knows (cf. [JS1], [BuF], [Sh1]) that L^{S}(s, π, Λ^{2}) converges absolutely in ℜ(s) > 1 and admits a meromorphic continuation with a functional equation of the usual type relating s and 1 − s. One also knows, by [BuG], [Sh1], the analogous properties of L^{S}(s, π, sym^{2}). For n = 2, one knows more by [GJ], namely that there is an (isobaric) automorphic representation sym^{2}(π) such that

L^{S}(s, sym^{2}(π)) = L^{S}(s, π, sym^{2}).

For n = 2, one also knows the meromorphic continuation and functional equation (cf. [Sh2]) of L^{S}(s, π, sym^{k}) for all k ≤ 5.

2.3. Base change and automorphic induction.

We will now review the results of Arthur and Clozel [AC] in a form which is suitable for the applications found in this article. See also [Lab2].

In [AC], chapter 3 (sec. 3 - 6), one finds a construction, for any cyclic extension K/F with [K : F] a prime ℓ, of maps

b_{K/F} : A(n, F) → A(n, K), π → π_{K} (base change)

and

I_{K/F} : A(n, K) → A(nℓ, F), π → I(π) (automorphic induction),

such that at every place v of F which is finite and unramified for the representations and K/F (or archimedean) and a place w of K above v, we have (respectively)

(3.1)  res_{K_{w}}^{F_{v}}(σ(π_{v})) ≃ σ((π_{K})_{w}),

and

σ(I(π)_{v}) ≃ ind_{K_{w}}^{F_{v}}(π_{w}).

Given any local field E, if β an unramified representation of GL(r, E) (or if E is archimedean), we write σ(β) to signify the associated r-dimensional representation of W′_{E} (or W_{E}).

There are also local analogs of these maps, customarily called “local base change” and “local automorphic induction”. The former is constructed and discussed in great detail in chapter 1 of [AC]. The later is discussed briefly in [Cℓ1] and the relevant assertions are consequences of the results of [AC]. An alternate construction of the local automorphic induction for essentially tempered representations, which works also in characteristic p, is given in the paper [HH] of Henniart and Herb. The properties we will need are summarized as follows:

Proposition 2.3.1 Let K/F be a cyclic extension of number fields or local fields of degree ℓ, a prime. Let θ be a generator of Gal(K/F), and let χ denote the character of the idele class group (resp. multiplicative group) of F in the global (resp. local) case associated to K. Then

1. The image of b_{K/F} consists precisely of those β ∈ A(n, K) such that β ≃ β∘θ.

2. The image of I_{K/F} consists precisely of those π ∈ A(nℓ, F) such that π ≃ π ⊗ χ.

3. For every π ∈ A(m, F) and β ∈ A(n, K), we have the adjointness property:

L(s, π × I(β)) = L(s, π_{K} × β),

and

ε(s, π × I(β))ε(s, 1_{K})^{nm} = ε(s, π_{K} × β) ∏_{j=0}^{ℓ−1} ε(s, χ^{j})^{nm}.

4. Suppose β is cuspidal (resp. supercuspidal) in A(n, K), for K global (resp.

local). Then

I(β)_{K} ≃ ⊞_{j=0}^{ℓ−1} β ∘ θ^{j}.

Moreover, I(β) is cuspidal (resp. supercuspidal) iff β is not isomorphic to β ∘ θ.

5. Suppose π is cuspidal (resp. supercuspidal) in A(n, F), for F global (resp.

local). Then

I(π_{K}) ≃ ⊞_{j=0}^{ℓ−1} π ⊗ χ^{j}.

Moreover, π_{K} is cuspidal (resp. supercuspidal) iff π is not isomorphic to π ⊗ χ.

For a proof of 1., 2. and the second half of 5., see [AC], [Cℓ1] and [HH]. For the remaining assertions, see [PR], pages 7,8, where this Proposition is stated as Prop. 3.1, and where A(n, F) is denoted Isob(n, F).

3. Construction of ⊠ : A(GL(2)) × A(GL(2)) → A(GL(4))

The object of this chapter is to prove the following

Theorem M Let π, π′ be in A(2, F). Then we have:

Existence: There exists an isobaric automorphic representation π ⊠ π′ of GL(4, 𝔸_{F}) satisfying (at every finite place v)

(L_{v})  L(s, (π ⊠ π′)_{v}) = L(s, π_{v} × π′_{v}),

and

(ε_{v})  ε(s, (π ⊠ π′)_{v}) = ε(s, π_{v} × π′_{v}).

We also have

(L_{∞})  L(s, (π ⊠ π′)_{∞}) = L(s, π_{∞} × π′_{∞}).

Cuspidality Criterion: Suppose π, π′ are both cuspidal.

If neither of them is associated to a character of a quadratic extension, then π ⊠ π′ is cuspidal iff we have

(C) π′ is not equivalent to π ⊗ χ, for any idele class character χ of F.

If π′ = I_{K}^{F}(μ), for a character μ of a quadratic extension K, then π ⊠ π′ is cuspidal iff the base change π_{K} is cuspidal and not isomorphic to π_{K} ⊗ (μ ∘ θ)μ^{−1}, where θ denotes the non-trivial automorphism of K/F.

We will call an isobaric representation π ⊠ π′ satisfying (L_{v}) and (ε_{v}), for all v, a strong lifting to GL(4)/F, or the automorphic tensor product, of the pair (π, π′). If it only satisfies (L_{v}) for almost all v, and not necessarily (ε_{v}) or (L_{∞}), we will call it a weak lifting.

We will briefly explain the arithmetic motivation for the cuspidality criterion above. Suppose π, π′ correspond to irreducible, continuous 2-dimensional representations σ, σ′ over ℂ of Gal(\overline{ℚ}/F). Then, since the operation ⊠ on A(2, F)×A(2, F) is supposed to correspond to the tensor product on the Galois side, π ⊠ π′ should be cuspidal iff σ ⊗ σ′ is irreducible. When σ, σ′ are non-dihedral, reducibility happens iff one is a twist of the other by a character. That the criterion should be the same for all non-dihedral cuspidal pairs (π, π′) is motivated by the hope (see [La3], [Cl2], and [Ra1]) that there is a group L_{F}, whose connected component L_{F}^{0} is pro-reductive with L_{F} /L_{F}^{0} ≃ Gal(\overline{ℚ}/F), and whose irreducible n-dimensional representations parametrize cuspidal automorphic representations of GL(n, 𝔸_{F}).

3.1. Relevant objects and the strategy. We begin with a simple

Lemma 3.1.1. Theorem M holds in the following three special cases:

(I) At least one of {π, π′} is not cuspidal. (II) At least one of {π, π′} is automorphically induced by a character μ of (the idele class group of) a quadratic extension K of F. (III) π′ is a twist of π, i.e., there exists a character χ of C_{F} such that

π′ ≃ π ⊗ χ.

Proof. Suppose π′ is not cuspidal. Then as we have seen in section 2, there exist idele class characters μ_{1}, μ_{2} of F such that π′ = μ_{1} ⊞ μ_{2}. In this case, we set

π ⊠ π′ = (π ⊠ μ_{1}) ⊞ (π ⊠ μ_{2}),

which is not cuspidal. In this case one knows ([JS2]) that the identities (L_{v}), (ε_{v}), and (L_{∞}) are all satisfied everywhere. Thus we are done in case (I).

Suppose π′ is of the form I_{K}^{F}(μ), for a character μ of C_{K}, for some K. Then we set

π ⊠ π′ = I_{K}^{F}(π_{K} ⊗ μ),

where π_{k} denotes the base change of π to K. The identity (L_{v}) at the unramified places v is a direct consequence of the identities of Proposition 2.3.1. The fact that (L_{v}) holds at every place, and that (L_{∞}) and (ε_{v}) also hold, will be shown as part of a general assertion in the next section (see Proposition 3.2.1).

It is left to consider (III). In this case, we set

π ⊠ π′ = (sym^{2}(π) ⊗ χ) ⊞ ωχ,

where ω is the central character of π. The asserted identities are then consequences of the work of Gelbart and Jacquet ([GJ]) and the strong multiplicity one theorem ([JS2]).

We will say that (π, π′) is of general type if we are not in either of these three special cases.

Given any pair (Π, η) of irreducible, generic, admissible representations of GL(n, 𝔸_{F}), GL(m, 𝔸_{F}) respectively, which are not necessarily automorphic, we may set

(3.1.2)  L(s, Π × η) = ∏_{v} L(s, Π_{v} × η_{v})

and

ε(s, Π × η) = ∏_{v} ε(s, Π_{v} × η_{v}).

The following result is a crucial ingredient of our approach.

Theorem 3.1.3 (Cogdell - Piatetski-Shapiro ([CoPS]). Let T be a fixed finite set of finite places of F. Let Π be an irreducible unitary, admissible, generic representation of GL(4, 𝔸_{F}) which satisfies the following: For every η ∈ A_{0}(n, F), n ≤ 2, with η_{v} unramified at every v in T, we have:

(MC)  L(s, Π × η) and L(s, Π^{∨} × η^{∨}) converge absolutely in large ℜ(s),

and they admit meromorphic continuations to the whole s-plane.

(E)  L(s, Π × η) and L(s, Π^{∨} × η^{∨}) are entire.

There is a functional equation

(F E)  L(1 − s, Π^{∨} × η^{∨}) = ε(s, Π × η)L(s, Π × η).

(BV)  L(s, Π × η) is bounded in vertical strips.

Then Π is nearly automorphic, i.e., there exists an automorphic representation Π_{1} of GL(4, 𝔸_{F}) such that Π_{v} ≃ Π_{1,v} for almost all v.

Consequently, Theorem M can be attacked for (π, π′) of general type if we can solve the following two problems (for a finite set T of finite places):

(P1) Define an irreducible, generic admissible representation Π of GL(4, 𝔸_{F}) such that we have at every place v,

(3.1.4)  L(s, Π_{v}) = L(s, π_{v} × π′_{v})

and

ε(s, Π_{v}) = ε(s, π_{v} × π′_{v}).

(P2) Given any η ∈ A_{0}(m, F), m ≤ 2, with η_{v} unramified at every v in T, the hypotheses (MC), (E), (FE) and (BV) hold for L(s, Π × η) and L(s, Π^{∨} × η^{∨}).

(P1) cannot be solved as posed without knowing the local Langlands correspondence for GL(4), which in any case we would like to avoid using. We can, however, define a candidate Π_{v} at almost all places, in particular at the archimedean places and at finite places v where π_{v} and π′_{v} are not both supercuspidal. (Even the supercuspidal cases can be dealt with in odd residual characteristics.) See section 3.7. Given any π, we can find,using [Ku], a chain F = K_{0} ⊂ K_{1} ⊂ … ⊂ K_{m} = K with each K_{j}/K_{j−1} cyclic of prime degree, in fact an infinite family X = X(π) of such chains, such that the base change π_{K} of π to K, which exists by [AC], has the property that at any place w of K, π_{K,w} is not supercuspidal. Hence (P1) can be solved over K, and if we can also solve (P2) over a “sufficiently large” subset Y of X, we can construct π⊠π′ over each K in Y, and then try to find a common descent to F satisfying the desired properties (listed in Theorem M). The part dealing with descent follows an approach taken for GL(2) in our earlier joint work [BR] with D. Blasius. (There is a mistake in [BR], but it does not affect the relevant section there discussing the descent criterion; in any case, we give a complete argument here for GL(n) - see Proposition 3.6.1) The best possible situation will be one in which almost every finite place v of F splits completely in some K ∈ Y. There are complications with finding a common descent if (π_{K}, π′_{K}) is not of general type for too many K. For this and other reasons, it will be best for us to use an inductive argument in cyclic layers of prime degree. We refer to sections 3.6 and 3.7 for details.

After replacing F by a suitable larger field, let us suppose that we have solved (P1), and that (π, π′) is of general type with π_{v} not supercuspidal. Fix any T, η as above. If m = 1, then all the hypotheses of Theorem 3.1.3 are just known assertions of the Rankin-Selberg theory. So we may assume that m = 2. (P2) can be solved if we can find a “triple product L-function” L(s, π × π′ × η) which satisfies all these hypotheses and satisfies (at every v)

(3.1.5)  L(s, π_{v} × π′ × η_{v}) = L(s, Π_{v} × η_{v}).

There are in fact three candidates for such an L-function, but unfortunately, none of them possesses apriori all the desired properties. The first one is in some sense the most natural one. To define it, let σ_{v} (resp. σ′_{v}) denote, at each v, the 2-dimensional representation of W′_{F_{v}} associated to π_{v} (resp. π′_{v}) by the local Langlands correspondence for GL(2) ([Ku]). (When v is archimedean, W′_{F_{v}} signifies just usual the Weil group by W′_{F_{v}}.) Put

(3.1.6)  L(s, π × π′ × η) = ∏_{v} L(s, σ_{v} ⊗ σ′_{v} ⊗ τ_{v}),

where τ_{v} denotes, at each v, the representation of W′_{F_{v}} attached to η_{v}. We do not know apriori any of the desired properties for this Euler product.

One defines the epsilon factor ε(π × π′ × η) as the product over all v of ε(s, σ_{v} ⊗ σ′_{v} ⊗ τ_{v}); these local factors will also be denoted ε(s, π_{v} × π′_{v} × η_{v}).

The second candidate, which we denote by L_{1}(s, π × π′ × η), is the one defined by Piatetski-Shapiro and Rallis in [PSR2], generalizing the construction of Garrett ([G]) over ℚ when all three representations correspond to holomorphic modular forms. (see also Ikeda ([Ik1]). This is defined by using an integral representation, and is known to satisfy (MC) and (FE). It also satisfies (E) by the arguments of Ikeda (see Theorem 2.7 of [Ik2], and also the discussion following Theorem 3.3.11 of this paper). But one does not yet have (BV), which is problematic, which we propose to rectify in this paper. One knows that at the finite unramified places ([PSR1]) and the infinite places ([Ik3]) that

(3.1.9)  L_{1}(s, π_{v} × π′_{v} × η_{v}) = L(s, π_{v} × π′_{v} × η_{v}).

See section 3.3 for a more complete discussion.

The third candidate, which we denote by L_{2}(s, π × π′ × η), was constructed in the work of Shahidi (cf. [Sh1], page 582), developing the ideas of Langlands in the monograph ([La2]). (See also section 0 of [PSR2] for an exposition.) To be precise, let G denote the group Spin(4, 4). It has a parabolic P = MU with Levi component M which is a quotient (over F) of GL(1)× SL(2)× SL(2)× SL(2) by {1, −1}, where −1 signifies H_{1}H_{3}H_{4}(−1), with the H_{j} denoting the standard simple coroots. Take the representation β := ω ⊗ π_{1} ⊗ π′_{1} ⊗ τ_{1} on M(𝔸_{F}), with π_{1} (resp. π′_{1}, resp. τ_{1}) being any irreducible component of the restriction of π (resp. π′, resp. τ) to SL(2, 𝔸_{F}) and ω being the product of the central characters of π, π′ and τ. Extend β in the usual way to P (𝔸_{F}) by letting U(𝔸_{F}) act trivially. Then the Langlands Eisenstein series on G(𝔸_{F}) associated to the representation of G(𝔸_{F}) induced by β defines, and gives analytic information on, L_{2}(s, π × π′ × η). One knows that this L-function satisfies (MC) and (FE) (cf. [Sh1]), and also (E) by the very recent work of H. Kim and F. Shahidi([K-Sh]). If v is archimedean or unramified or tempered, we have

(3.1.8)  L_{2}(s, π_{v} × π′_{v} × η_{v}) = L(s, π_{v} × π_{v} × η_{v})

There is also a definition of epsilon factors ε_{2}(s, π_{v} × π′_{v} × η_{v}). The main use of this approach for us comes from knowing that L_{2}(s, π × π′ × τ) is non-zero on the line s = 1.

We take T to be a (finite) set containing all the places v where π_{v} or π′_{v} is in the discrete series, and solve (P2) for the relevant types of (π, π′). In such a case, one first notes, by an extension of the local results of [Ik1], [Ik2] by global arguments, that the local gamma factors (called ε′-factors by some) of first and the third candidates agree at all the places, and that their L-factors also agree at all the places v where at least one of the representations π_{v}, π′_{v} is not supercuspidal It is not clear that the L-factors agree at all the places v. But luckily, it turns out not to matter by a base changing trick. We refer to sections 3.3 and 3.4 for details.

Next comes the establishment of (BV). The idea here is to make use of Arthur’s truncation Λ^{T} E(f_{s}) of the Eisenstein series E(f_{s}) on GSp(6)/F asociated to the character det^{s+2} of the (GL(2) occurring in the Levi of the) Siegel parabolic P. (Here T is a sufficiently regular parameter.) The integral representation for L_{1}(s, π_{1}× π_{2} × π_{3}) is given by the integral, over GL(2)^{3}, of E(f_{s}) against a function φ in the space of π_{1} ⊗ π_{2} ⊗ π_{3}. By the standard arguments using th functional equation and boundedness in ℜ(s) >> 0, it suffices to show that this integral is of bounded order in vertical strips Ξ. Since φ decreases rapidly at infinity, it suffices to check that for s in Ξ, E(f_{s})(g) is bounded in absolute value by a function of s of bounded order times ||g||^{N} for a uniform N as N goes to infinity. Now, using the fact that E(f_{s}) is an eigenfunction, though not square-integrable globally, of the Laplacian Δ, we use standard Sobolev estimates and bound |E(f_{s})(g)| by its L^{2}-norm in a small neighborhood V of controlled size. The L^{2}-norm over V of E(f_{s}) − ∧^{T} E(f_{s}) can be estimated by use of the constant terms; these are essentially given by abelian L-functions, which are known by Hecke to be of bounded order. It then suffices to estimate the L^{2} norm of ∧^{T} E(f_{s}) over the whole locally symmetric manifold, which can be understood by the works of Langlands and Arthur.

Finally, to be able to honestly apply the converse theorem, we must also check (3.1.5). The details are all in sections 3.4 and 3.5.

3.2. Weak to strong lifting, and the cuspidality criterion. In this section we prove the following

Proposition 3.2.1 Let π, π′ be in A(2, F). Suppose we have constructed a weak lifting, i.e., an isobaric automorphic representation π ⊠ π′ of GL(4, 𝔸_{F}) satisfying the identity (L_{v}) (of theorem M) at almost all v. Then we have (L_{v}) and (ε_{v}) at all the finite places v, and (L_{∞}) as well. In addition, the cuspidality criterion (of Theorem M) holds.

Proof. Let S be the (finite) set of places outside which (L_{v}) holds. We may assume (see (I) of section 3.1) that π and π′ are both cuspidal and non-dihedral. Note also that the central character of π ⊠ π′ is simply (ωω′)^{2}, where as before, ω, ω′ denote the respective central character of π, π′. This is so because the two idele class characters agree almost everywhere. (In fact, by a theorem of Hecke, it suffices to know that they agree at a set of primes of density > 1/2.)

First some notations. If f(s), g(s) are two meromorphic functions of s such that their quotient is invertible, we will write f(s) ≡ g(s). At any place v, given a character ν of F_{v}^{∗}, we can write it as ν_{0}|.|^{z}, for a unitary character ν_{0} and a complex number z. The real part of z is uniquely defined; we will call it the exponent of ν, and denote it e(ν).

Choose a finite order character μ of C_{F} such that μ_{v} is sufficiently ramified at every finite place u in S so as to make the Euler u-factors of π ⊠ π′, π × π′, and their contragredients, equal 1. This is possible by using the results of [JPSS]. Then, comparing the global functional equations of both L-functions, we get (3.2.2)

∏_{w|∞} L(s, (π ⊠ π′)_{w})L(1 − s, π_{w}^{∨} × π′_{w}^{∨}) ≡ ∏_{w|∞} L(1 − s, (π ⊠ π′)_{w}^{∨})L(s, π_{w} × π′_{w}).

For any place v, archimedean or otherwise, for any n ≥ 1, and for any cuspidal automorphic representation Π of GL(n, 𝔸_{F}) (such as π ⊠ π′), it is known that L(s, Π_{v}) is holomorphic in ℜ(s) > \frac{1}{2} − t, for some t = t(Π, v) > 0 (see [BaR], Prop. 2.1, part B). Consequently, L(s, (π ⊠ π′)_{v}) has no pole in common with L(1 − s, (π ⊠ π′)_{v}^{∨}).

Next we claim that, since π, π′ are (cuspidal) automorphic representations of GL(2, 𝔸_{F}), L(s, π_{v} × π′_{v}) is also holomorphic in ℜ(s) > \frac{1}{2} − t, for some t > 0. If π_{v} and π′_{v} are both in the discrete series, then one has holomorphy even in ℜ(s) > 0 (see [BaR], Lemma 2.3). If π_{v} is a principal series representation defined by (quasi)characters ν, ν′, then by [GJ], we have e(ν) < 1/4 and e(ν′) < 1/4. (Strictly speaking, when v is archimedean, one finds in [GJ] only the assertion that these exponents are ≤ 1/4, but one can eliminate the possibility of exponent 1/4 by using a simple version of the argument of [LRS1].) At the finite places, one can even replace 1/4 by 1/5 by using [Sh2]; see also [LRS2]. Over ℚ, one can do still better and reduce the bound to 5/28 ([BuDHI]). But we will not need these finer results. In any case, the claim follows easily.

Consequently, L(s, π_{v} × π′_{v}) has no pole in common with L(1 − s, π_{v}^{∨} × π′_{v}^{∨}) either. Applying this, along with the earlier remark on the relative primality of L(s, (π ⊠ π′)_{v}) and L(1 − s, (π ⊠ π′)_{v}^{∨}), we get (L_{∞}) from (3.2.2).

Next pick any finite u_{0} ∈ S and choose a μ which is 1 at u_{0}, but is highly ramified at all other finite u in S. Again, by comparing the functional equations, we get an identity such as (3.2.2), with the factors at infinity replaces by the corresponding ones at u_{0}. Using the relative primality results above, now with v = u_{0}, we get (L_{u_{0}}). Since u_{0} was arbotrary, we get (L_{v}) now at every place v.

For the identity of epsilon factors, we fix a u_{0} in S and note that by [JPSS2], we can choose a global character μ which is 1 at u_{0} and sufficiently ramified at any other place u in S such that the epsilon factor of π_{u} × π′_{u} ⊗ μ_{u} depends only on μ_{u} and the square of the product of the central characters ω_{u}, ω′_{u}, and the dependence is simple. Similarly, the epsilon factor of (π ⊠ π′)_{u} has the same dependence on μ_{u} and the central character of (π ⊠ π′)_{u}, which we noticed above to be (ω_{u}ω′_{u})^{2}. The analogous statements hold for the contragredients, and this results in the identity (∀u ∈ S − {u_{0}})

\frac{ε(s, (π ⊠ π′)_{u})L(s, (π ⊠ π′)_{u})}{L(1 − s, (π ⊠ π′)_{u}^{∨})} = \frac{ε(s, π_{u} × π′_{u})L(s, π_{u} × π′_{u})}{L(1 − s, π_{u}^{∨} × π′_{u}^{∨})}.

Comparing the global functional equations again, we get (ε_{u_{0}}) as we already know that the L-factors agree. This finishes the proof of the first part of the Proposition.

It is left to prove the cuspidality criterion. First suppose π ⊠ π′ is cuspidal. Recall that by (I) of section 3.1, π ⊠ π′ is not cuspidal if π or π′ is not. So π and π′ must both be cuspidal. Suppose π′ ≃ π ⊗ χ, for an idele class character χ of F. Then, by [JS2] and (L_{v}) for almost all v, L^{S}(s, π ⊠ π′ ⊗ (χω)^{−1}) must have a pole at s = 1, with S denoting the set of ramified and archimedean places, and ω the central character of π. (We are using the fact that π ⊗ ω^{−1} is the contragredient π^{∨} of π.) Then π ⊠ π′ cannot be cuspidal, leading to a contradiction.

Conversely, suppose that π, π′ are cuspidal and (C) holds, but that π ⊠ π′ is not cuspidal. We will show that it leads to a contradiction.

Since π ⊠ π′ is isobaric, we can decompose it uniquely as

π ⊠ π′ = ⊞_{j=1}^{r}η_{j},

where each η_{j} is a cuspidal automorphic representation of GL(n_{j}, 𝔸_{F}), with ∑_{j=1}^{r} n_{j} being 4. Suppose some n_{j}, say n_{1}, is 1. Then η_{1} is an idele class character, and we have, for any large enough finite set S of places containing the archimedean ones,

L^{S}(s, π × π′ ⊗ η_{1}^{−1}) = ζ_{F}^{S}(s) ∏_{j≠1} L^{S}(s, η_{j} ⊗ η_{1}^{−1}).

One knows ([JS2]) that, since each L^{S}(s, η_{j} ⊗ η_{1}^{−1}) is cuspidal, it cannot vanish at s = 1. Thus the pole (at s = 1) of ζ_{F}^{S}(s) induces one of the function on the left, which is not allowed by (C). Thus n_{j} must be > 1 for each j. Then we are forced to have

n_{1} = n_{2} = 2.

Now the key idea is to compute the exterior square L-function of π ⊠ π′ in two different ways. On the one hand, since it is of the form η_{1} ⊞ η_{2}, with each η_{j} a cuspidal of GL(2)/F, we get

L^{S}(s, π ⊠ π′, Λ^{2}) = L^{S}(s, η_{1} × η_{2})L^{S}(s, ω_{1})L^{S}(s, ω_{2}),

where ω_{j} is the central character of η_{j}. This can be seen easily at the unramified places v. Indeed, if τ_{j,v} is the 2-dimensional representation of W′_{F_{v}} associated to η_{j,v}, then the above identity is induced by the following:

Λ^{2}(τ_{1,v} ⊗ τ_{2,v}) ≃ (τ_{1} ⊗ τ_{2}) ⊕ det(τ_{1,v}) ⊕ det(τ_{2,v}),

which is easy to verify.

Consequently, L^{S}(s, π ⊠ π′, Λ^{2}) is divisible by (two) abelian L-functions, namely L^{S}(s, ω_{1}) and L^{S}(s, ω_{2}).

On the other hand, denoting by σ_{v} (resp. σ′_{v}) the 2-dimensional representation of W′_{F_{v}} associated to π_{v} (resp. π′_{v}), we also have the identity

Λ^{2}(σ_{v} ⊗ σ′_{v}) ≃ Λ^{2}(σ_{v}) ⊗ sym^{2}(σ′_{v}) ⊕ sym^{2}(σ_{v}) ⊗ Λ^{2}(σ′_{v}).

This implies the following equality of L-functions:

L^{S}(s, π ⊠ π′, Λ^{2}) = L^{S}(s, sym^{2}(π) ⊗ ω)L^{S}(s, sym^{2}(π′) ⊗ ω′),

where sym^{2}(π) (resp. sym^{2}(π′)) is the automorphic representation of GL(3, 𝔸_{F}) associated to π (resp. π′) by Gelbart-Jacquet ([GJ]), and ω (resp. ω′) is the central character of π (resp. π′).

Suppose π is not automorphically induced by an idele class character of a quadratic extension. Then the main theorem of [GJ] says that sym^{2}(π) is cuspidal; so L^{S}(s, sym^{2}(π) ⊗ ω′) cannot be divisible by an abelian L-function. Consequently, if neither π nor π′ is automorphically induced, we get a contradiction, and so π ⊠ π′ must be cuspidal.

It remains to consider the case when π′ is of the form I_{K}^{F}(μ), for μ an idele class character of a quadratic extension K. Then by (II) of section 3.1, we know that π ⊠ π′ must be isomorphic to I_{K}^{F}(β), where β = π_{K} ⊗ μ. By hypothesis, π_{K} is cuspidal and not isomorphic to π_{K} ⊗ (μ ∘ θ)μ^{−1}, where θ is the non-trivial automorphism of K/F. Then β is a cuspidal, not isomorphic to β ∘ θ. So by the properties of automorphic induction (Prop. 2.3.3), π ⊠ π′ is cuspidal.

3.3. Triple product L-functions: local factors and holomorphy. Let π, π′, π′′ be unitary, irreducible, cuspidal automorphic representations of GL(2, 𝔸_{F}) of central characters ω, ω′, ω′′ respectively. Put ω = ωω′ω′′. At each place v, let σ_{v}, σ′_{v}, σ′′_{v} be the 2-dimensional representations of W′_{F_{v}} associated to π_{v}, π′_{v}, π′′_{v} respectively, by the local Langlands correspondence for GL(2) ([Ku]).

Recall that L(s, π ×π′ ×π′′) is the (correct) triple product L-function defined by the Euler product ∏_{v} L(s, σ_{v} ⊗σ′_{v} ⊗σ′′_{v}), while L_{1}(s, π ×π′ ×π′′) is the one given as the gcd of the integral representation of Garrett and Piatetski-Shapiro and Rallis, and L_{2}(s, π × π′ × π′′) is the one defined by the Langlands-Shahidi method.

The following basic fact, due to Ikeda in the archimedean case ([Ik3], Theorem 1.1.0), and Rallis and Piatetski-Shapiro in the non-archimedean case ([PS-R2]), will be very important to us.

Proposition 3.3.1 Let v be any place of F where π_{v}, π′_{v}, π′′_{v} are all unramified. Then we have

L(s, π_{v} × π′_{v} × π′′_{v}) = L_{1}(s, π_{v} × π′_{v} × π′′_{v}).

By a slight refinement of known results we will also establish the following

Proposition 3.3.2 Let v be any place of F, where at least one of the local representations π_{v}, π′_{v}, π′′_{v} is in the principal series (possibly complementary). Then

(a)  L(s, π_{v} × π′_{v} × π′′_{v}) = L_{1}(s, π_{v} × π′_{v} × π′′_{v}),

and

(b)  ε(s, π_{v} × π′_{v} × π′′_{v}) = ε_{1}(s, π_{v} × π′_{v} × π′′_{v}).

Remark. When v is finite, if two of the local representations are non-supercuspidal, then such an identity already follows from the works of Ikeda ([Ik2,3]). In the special, but important, case when all of π_{v}, π′_{v}, π′′_{v} are Steinberg, this was established earlier by Gross and Kudla ([GK]). When v is archimedean, this is treated in [Ik3]. In this case, the L_{1}-factor is defined only up to an invertible holomorphic function, and so the content of this Proposition is that we can normalize it appropriately so as to have the stated identities. Note also that one knows by the work of Shahidi ([Sh5]) the identity L(s, π_{v} × π′_{v} × π′′_{v}) = L_{2}(s, π_{v} × π′_{v} × π′′_{v}) for any archimedean v.

We have stated here only what we need, and a stronger assertion will be proved later in section 4.4 after establishing Theorem M.

Proof. By hypothesis, there exist quasi-characters μ, ν of F_{v}^{∗} such that one of the local representations, say π_{v}, is the principal series defined by (μ, ν). The starting point for us is the following

Lemma 3.3.3 The set of poles of L_{1}(s, π_{v} × π′_{v} × π′′_{v}) is contained in the set of poles of L(s, π_{v} × π′_{v} × π′′_{v}).

Proof of Lemma 3.3.3. We first recall the following

Proposition 3.3.4 For any quasi-character λ of F_{v}^{∗} one has

L(s, λ ⊗ π′_{v} ⊗ π′′_{v}) = L(s, λ ⊗ σ′_{v} ⊗ σ′′_{v})

and

ε(s, λ ⊗ π′_{v} ⊗ π′′_{v}) = ε(s, λ ⊗ σ′_{v} ⊗ σ′′_{v}).

For a proof, see the author’s paper with D. Prasad ([PR], Prop. 4.2), where the identity is established via a global argument.

On the other hand, it is immediate that

(3.3.5)  L(s, σ_{v} ⊗ σ′_{v} ⊗ σ′′_{v}) = L(s, μ ⊗ σ′_{v} ⊗ σ′′_{v})L(s, ν ⊗ σ′_{v} ⊗ σ′′_{v}),

and

ε(s, σ_{v} ⊗ σ′_{v} ⊗ σ′′_{v}) = ε(s, μ ⊗ σ′_{v} ⊗ σ′′_{v})ε(s, ν ⊗ σ′_{v} ⊗ σ′′_{v}).

One also has the following result of Ikeda:

Proposition 3.3.6([Ik3], Thm. 1.8) The quotient

\frac{L_{1}(s, π_{v} × π′_{v} × π′′_{v})}{L(s, μ ⊗ π′_{v} ⊗ π′′_{v})L(s, ν ⊗ π′_{v} ⊗ π′′_{v})}

is entire.

Set

γ(s, π_{v} × π′_{v} × π′′_{v}) = ε(s, π_{v} × π′_{v} × π′′_{v})\frac{L(1 − s, π_{v}^{∨} × (π′_{v})^{∨} × (π′′_{v})^{∨})}{L(s, π_{v} × π′_{v} × π′′_{v})}.

Define γ_{j}(s, π_{v} × π′_{v} × π′′_{v}) analogously, for j = 1, 2, using the L_{j} and ε_{j}-factors instead. (The gamma factor is called the ε′-factor by some, eg., in [Ik1,2].)

In view of Propositions 3.3.4, 3.3.6 and identity (3.3.5), Lemma 3.3.3 will follow once we establish

Proposition 3.3.7 We have:

γ(s, π_{v} × π′_{v} × π′′_{v}) = γ_{1}(s, π_{v} × π′_{v} × π′′_{v}).

Using [Sh4], one can show such an identity with γ_{1} replaced by γ_{2}.

Proof of Proposition 3.3.7. By Theorem 3 of [Ik2], we know that

(3.3.8)  γ_{1}(s, π_{v} × π′_{v} × π′′_{v}) = γ_{1}(s, μ ⊗ π′_{v} × π′′_{v})γ_{1}(s, ν ⊗ π′_{v} × π′′_{v}).

By applying the identities (3.3.4), we then see that

(3.3.9)  γ_{1}(s, π_{v} × π′_{v} × π′′_{v}) = γ_{1}(s, μ ⊗ σ′_{v} ⊗ σ′′_{v})γ_{1}(s, ν ⊗ σ′_{v} ⊗ σ′′_{v}).

In view of (3.3.5), the right hand side of (3.3.9) equals γ(s, π_{v} ×π′_{v} ×π′′_{v}) as claimed.

As noted above, this also proves Lemma 3.3.3.

Proof of Proposition 3.3.2: Define the non-temperednes index λ(τ) of a unitary irreducible τ of GL(2, F_{v}) to be t, resp. 0, if τ is a complelementary series attached to (μ|.|^{t}, μ|.|^{−t}) with t > 0 and μ unitary, resp. τ is tempered, and set (as in [Ik1]) λ(π_{v}, π′_{v}, π′′_{v}) = λ(π_{v}) + λ(π′_{v}) + λ(π′′_{v}). Then it is easy to see that

(3.3.10).  L(s, σ_{v} ⊗ σ′_{v} ⊗ σ′′_{v}) is holomorphic in ℜ(s) > λ(π_{v}, π′_{v}, π′′_{v}).

(This is the Galois analog of Lemma 2.1 of [Ik1].) We know by Shahidi ([Sh2]) that λ(τ) is always < \frac{1}{5} for τ ∈ \{π_{v}, π′_{v}, π′′_{v}\}. Consequently, λ(π_{v}, π′_{v}, π′′_{v}) is always < 1/2 unless all the three representations are non-tempered. In the former case, i.e., if λ(π_{v}, π′_{v}, π′′_{v}) < 1/2, then L(s, σ_{v} ⊗ σ′_{v} ⊗ σ′′_{v}) and L(1 − s, σ_{v}^{∨} ⊗ (σ′_{v})^{∨} ⊗ (σ′′_{v})^{∨}) have no common poles. Ditto for the L_{1}-functions by Lemma 3.3.3, and the assertion follows in this case.

It remains to treat the case when λ(π_{v}, π′_{v}, π′′_{v}) ≥ 1/2 (which should not happen!). Suppose π_{v} is in the principal series associated to the characters μ, ν. The unitarity implies that the set \{\overline{μ}, \overline{ν}\} equals \{μ^{−1}, ν^{−1}\}, and so one of the following happens:

(i) \overline{μ} = μ^{−1}; (ii) \overline{μ} = ν^{−1}. In case (i), μ and ν are unitary, so we are in case (ii). So π_{v} is the complementary series representation associated to the pair of characters (μ, \overline{μ}^{−1}). We may write μ = χ|.|^{t}, for some unitary (possibly ramified) character χ and real number t. Then ν = (\overline{χ}^{−1})|.|^{−t} = χ|.|^{−t}, and so π_{v} is π_{1,v} ⊗ χ with π_{1,v} unramified. Similarly, π′_{v} (resp. π′′_{v}) is π′_{1,v} ⊗ χ′ (resp. π′′_{1,v} ⊗ χ′′), with π′_{1,v}, π′′_{1,v} unramified and χ′, χ′′ unitary. Then the integral representation for triple product L-functions gives the identity (with ξ = χχ′χ′′)

L_{1}(s, π_{v} × π′_{v} × π′′_{v}) = L_{1}(s, π_{1,v} × π′_{1,v} × (π′′_{1,v} ⊗ ξ)).

Consequently, if all three local representations are non-tempered, we may assume, by replacing (π_{v}, π′_{v}, π′′_{v}) by (π_{1,v}, π′_{1,v}, π′′_{1,v}⊗ξ), that at most one of them is ramified. If ξ is ramified, then both the L- and the L_{1}-functions of (π_{v}, π′_{v}, π′′_{v}) at s do not share any pole with the corresponding contragredient functions at 1 − s, and the assertion of Proposition 3.3.2 follows. When ξ is unramified, we can appeal to Proposition 3.3.1.

Theorem 3.3.11 Let π, π′, π′′ be cuspidal automorphic representations of GL(2, 𝔸_{F}) satisfying the following:

• At least one of {π, π′, π′′} is non-dihedral; and

• At every finite place v, at least one of {π_{v}, π′_{v}, π′′_{v}} is in the principal series. Then L(s, π × π′ × π′′) is entire.

In view of Proposition 3.3.2, it suffices to show that L_{1}(s, π × π′ × π′′) is entire, and this follows from the work of Ikeda. But a remark is in order, however. In [Ik1], one finds an assertion (see Theorem 2.7 there) that L_{1}(s, π × π′ × π′′) is entire as long as one of the three representations is not dihedral, without any local conditions. This is as it should be, but a small correction needs to be made in his proof. To be precise, in the case when ω^{2} = 1, ω ≠ 1 with associated quadratic extension K/F (see pp. 231-234 of [Ik1]), he first shows that L_{1}(s, π_{K} × π′_{K} × π′′_{K}) is entire and then proceeds to assert that any pole of L_{1}(s, π × π′ × π′′) on ℜ(s) = 1 is a pole of L_{1}(s, π_{K} × π′_{K} × π′′_{K}). We do not see how to verify this claim; the problem is at the bad factors. The way he tries to derive this is to assert the following two identities:

(Ik1)  L_{1}(s, π × π′ × π′′) = L_{2}(s, π × π′ × π′′),

and

(Ik2)  L_{2}(s, π_{K} × π′_{K} × π′′_{K}) = L_{2}(s, π × π′ × π′′)L_{2}(s, π × π′ × π′′ ⊗ ω),

and then appeal to Shahidi’s result ([Sh6]):

(3.3.12)  L_{2}(s, π × π′ × π′′ ⊗ ω) ≠ 0 if ℜ(s) = 1.

We are unable to verify either of the identities (Ik1), (Ik2), though they both hold locally almost everywhere.

This can be easily fixed as follows. First note that at any place v, (3.3.8) holds with L replaced by L_{2}. Since 3/5 is the maximum possible value of λ(π, π′, π′′) (by [Sh1]), the local factors (of either of three L-functions) is invertible on {ℜ(s) = 1}. So (3.3.12) holds with L_{2} replaced by the incomplete L-function L_{2}^{S}, taken over the places outside a finite set S. Choose S to be such that [Ik1], [Ik2] both hold with L_{1}, L_{2} replaced by L_{1}^{S}, L_{2}^{S} repectively. (Over K, we remove the factors at places above S.) It follows that L_{1}(s, π × π′ × π′′) has a pole iff L^{S}(s, π × π′ × π′′) does, which happens iff L^{S}(s, π_{K} × π′_{K} × π′′_{K}) and (hence) L(s, π_{K} × π′_{K} × π′′_{K}) do. The rest of Ikeda’s argument goes through verbatim.

3.4. Boundedness in vertical strips. A very important prerequisite to the application of converse theorems is the knowledge that the relevant L-functions are, in addition to being entire, bounded in vertical strips. This is what we prove here for a class of triple product L-functions. After the proof of the main theorem in Section 3.8, the same will hold for all such L-functions.

Theorem 3.4.1 Let F be a totally complex number field, and let π_{1}, π_{2}, π_{3} be unitary, cuspidal automorphic representations of GL_{2}(𝔸_{F}). Assume that at any finite place v, at least one π_{j,v} is in the principal series. Then the entire function L(s, π_{1} × π_{2} × π_{3}) is bounded in vertical strips of finite width.

Proof. In view of Propositions 3.3.1 and 3.3.2, we only need to prove the assertion of the theorem for L_{1}(s, π_{1}×π_{2}×π_{3}), which has an integral representation involving a Siegel Eisenstein series E(s) on GSp_{6}/F. Our key idea is to appeal to the fact that E(s) is an eigenfunction of the Laplacian and then to make use of Arthur’s truncation of E(s).

We now need to set up some notations and preliminaries. Let

G = GSp(6)/F = {g ∈ GL_{6}/F | ^{t}gJg = λJ, for some λ ∈ 𝔾_{m}},

where J = ( \begin{matrix}0 & −I_{3} \\ I_{3} & 0 \end{matrix} ), with I_{n} denoting, for any n ≥ 1, the identity n × n-matrix. The root system relative to the diagonal torus A in G is of rank 3 and is described by

Φ = {±e_{i} ± e_{j} | 1 ≤ i, j ≤ 3},

where e_{1}, e_{2}, e_{3} are stadard basis vectors of Lie T_{ℂ}^{∗}. Pick the following set of simple roots:

Δ = {α = e_{1} − e_{2}, β = e_{2} − e_{3}, γ = 2e_{3}}.

Denote by P_{0} = AU_{0} the corresponding minimal parabolic subgroup of G, whose elements are matrices of the form ( \begin{matrix}A & B \\ 0 & D \end{matrix}) , A, B, D ∈ M_{3}, with A (resp. D) upper (resp. lower) triangular. For each subset θ of Δ, let A_{θ} denote (∩_{δ∈θ} ker(δ))^{0} ⊂ A, M_{θ} the centralizer of A_{θ} in G, and P_{θ} = M_{θ}U_{θ} the corresponding standard parabolic subgroup of G, split over F, containing P_{0}. The Weyl group W is a semidirect product (ℤ/2)^{3} ⋊ S_{3}, whose elements are made up of permutations of {e_{1}, e_{2}, e_{3}} (possibly) composed with flips (sign changes) e_{i} ↦ −e_{i}, for i ∈ {1, 2, 3}.

Let P = P_{{α,β}} = MU be the (maximal) Siegel parabolic subgroup with modular function δp. We have the matrix representation

P = \{p = ( \begin{matrix}λA & 0 \\ 0 & ^{t}A^{−1} \end{matrix} ) ( \begin{matrix}I & X \\ 0 & I \end{matrix} ) | \begin{matrix}λ ∈ 𝔾_{m}, A ∈ GL_{3} \\ X = ^{t}X \end{matrix}\} ,

with δ_{p}(p) = | det A|^{6}. Let K = ∏_{v} K_{v} be the standard maximal compact mod center subgroup of G(𝔸_{F}).

Let ω_{i} be the central character of π_{i}, for each i ≤ 3. Put ω = ω_{1}ω_{2}ω_{3} and consider the space J(ω, s) of right K-finite functions f_{s} on G(𝔸_{F}) satisfying forall p ∈ P (𝔸_{F}) and g ∈ G(𝔸_{f})):

(3.4.2)  f_{s}(pg) = ω(λ)|λ|^{3s+ \frac{3}{2}} ω(det A)| det A|^{2s+1}f_{s}(g).

We will henceforth take f_{s} to be a decomposable, entire and K-finite section, which is good in the sense of [Ik1], [PS-R]. The corresponding local space at v will be denoted J_{v}(ω, s). The relevant Eisenstein series is then given by the formula (∀ g ∈ G(𝔸_{F}))

(3.4.3)  E(f_{s})(g) = ∑_{γ∈P (F)\G(F)} f_{s}(γg).

(Note that J(ω, s) corresponds to I(ω, 2s − 1) in Proposition 1.6 of [Ik1].)

Let φ = φ_{1} ⊗ φ_{2} ⊗ φ_{3} be a cusp form in the space of π_{1} ⊗ π_{2} ⊗ π_{3} on H(𝔸_{F}), where

H = {(g_{1}, g_{2}, g_{3}) ∈ GL_{2}^{3} | det g_{1} = det g_{2} = det g_{3})}.

Denote the corresponding Whittaker function by W = W_{2} ⊗ W_{2} ⊗ W_{3}. Then one knows (see [Ik1], [PS-R2]) that in Re(s) ≫ 0 one has the identity (with f_{s} = ⊗_{v}f_{v,s}):

(3.4.4) ⟨E(f_{s}), φ⟩_{H} := ∫_{C(𝔸_{F})H(F)\H(𝔸_{F})} E(h, f_{s})φ(h)dh) = ∏_{v} Ψ(f_{v,s}; W_{v}),

where each Ψ(f_{v,s}; W_{v}) is a local integral having L_{1}(s, π_{1,v} × π_{2,v} × π_{3,v}) as its g.c.d. By [PS-R2] one knows that at every unramified v, Ψ(f_{v,s}; W_{v}) coincides with L_{1}(s, π_{1,v} × π_{2,v} × π_{3,v}) for a suitable choice of f_{v,s}. Hence we get

L_{1}(s, π_{1} × π_{2} × π_{3}) ∏_{v∈S} \frac{Ψ(f_{v,s}; W_{v})}{L_{1}(s, π_{1,v} × π_{2,v} × π_{3,v})} = ⟨E(f_{s}), φ⟩_{H},

where S is a finite set of places containing the ramified and archimedean places.

Recall that L_{1}(s, π_{1} ×π_{2} ×π_{3}) has an Euler product in ℜ(s) > 1 and a functional equation. It is clear that it is bounded for ℜ(s) >> 0. So by the Phragman-Lindelöf theorem, Theorem 3.4.1 will be proved if we establish that this L-function is of bounded order in vertical strips of finite width.

Lemma 3.4.5. For each v, the function \frac{Ψ(f_{v,s};W_{v})}{L_{1}(s,π_{1,v}×π_{2,v}×π_{3,v})} is entire and of bounded order, for a suitable choice of f_{v,s}.

Proof of Lemma. Put g(s, π_{1,v} × π_{2,v} × π_{3,v}) = \frac{Ψ(f_{v,s};W_{v})}{L_{1}(s,π_{1,v}×π_{2,v}×π_{3,v})}. The entireness of this function follows from the fact that L_{1}(s, π_{1,v} × π_{2,v} is the gcd of the local integrals Ψ(f_{v,s}; W_{v}) as (f_{v,s}, W_{v}) vary. The fact that g(s, π_{1,v} × π_{2,v} × π_{3,v}) is of bounded order is obvious at any v-adic place as the local integral is a rational function of Nv^{−s}. So let v be archimedean. Since we have taken F to be totally complex, F_{v} is ℂ. There is a local functional equation

g(s, π_{1,v} × π_{2,v} × π_{3,v}) = ε_{1}(s, π_{1,v} × π_{2,v} × π_{3,v})g(1 − s, \overline{π}_{1,v} × \overline{π}_{2,v} × \overline{π}_{3,v}^{∨}),

where the intervening ε-factor is, thanks to Proposition 3.3.2 (b), an exponential function. So it suffices to prove that, for a suitable choice of f_{v,s}, the local integral Ψ(f_{v,s}; W_{v}) is of bounded order in ℜ(s) ≥ 1/2.

Let λ denote λ(π_{1,v}) + λ(π_{2,v}) + λ(π_{3,v}), where, for each j, λ(π_{j,v}) denotes, as in the proof of Proposition 3.3.2 on page 17, the non-temperedness index of π_{j,v}. It is known ([Ik3]) that the integral Ψ(f_{v,s}; W_{v}) is absolutely convergent in ℜ(s) > λ, defining a holomorphic function there. Noting that |f_{v,s}| transforms under the left action of P (F_{v}) by a character depending only on ℜ(s), we see that Ψ(f_{v,s}; W_{v}) is bounded in any vertical strip {a ≤ ℜ(s) ≤ b} if a > λ.

Now suppose that at least one π_{j,v} is tempered. By Gelbart-Jacquet ([GJ]), one has λ(π_{i,v}) < 1/4 for any i. (For F = ℚ, one even has the archimedean estimate: λ(π_{i,v}) < 5/28 by [LRS1].) So λ < 1/2 in this case, and by the remarks above, Ψ(f_{v,s}; W_{v}) is bounded in any vertical strip {1/2 ≤ ℜ(s) ≤ b} of finite width.

It remains to consider when every π_{j} is non-tempered. Arguing as in the proof of Proposition 3.3.2, we may assume that π_{1,v} and π_{2,v} are unramified and that π_{3,v} is the twist of an unramified representation by a unitary character ξ which is ramified if non-trivial. If ξ = 1, i.e., when each π_{j,v} is unramified, it is proved in [Ik3] that there exists a function f_{v,s} such that Ψ(f_{v,s}; W_{v}) equals L_{1}(s, π_{1,v} × π_{2,v} × π_{3,v}). So we are done in this case by the standard properties of the Gamma function. Now suppose ξ is non-trivial. We can then use the ξ-twisted form of the unramified f_{v,s} emplyed by Ikeda, in which case Ψ(f_{v,s}; W_{v}) will be essentially given by the integral expression preceding Lemma 2.3 in [Ik3], the change being that the integrand will get multiplied by ξ(\sqrt{xyz}R^{−1}z)ξ̃(k), where ξ̃ is the character of the maximal compact K_{v} whose restriction to K_{v} ∩ P (F_{v}) is ξ. (Here the notations are as in [Ik3].) Now we appeal to the fact that the Whittaker function W_{1,v} (resp. W_{2,v}, resp. W_{3,v}) satisfies a second order differential equation on diag(x, 1) (resp. diag(y, 1), resp. diag(z, 1)). Using integration by parts, we can then express an elementary function times Ψ(f_{v,s}; W_{v}) in terms of a finite sum of integrals, each of which converges absolutely in ℜ(s) > λ − 1/2 and is bounded in {a ≤ ℜ(s) ≤ b} if a > λ − 1/2. Since λ is < 3/4 by [GJ], we are done.

Since an entire function which is the ratio of two entire functions of bounded order is itself of bounded order, it suffices then to establish that the entire function ⟨E(f_{s}), φ⟩_{H} is of bounded order. (Such a reduction was employed in [RS] for an analogous situation.)

Since φ vanishes rapidly at infinity, it will suffice to establish the following

Proposition 3.4.6 Fix a vertical strip Ξ in ℂ of finite width, and a Siegel set S in G(𝔸_{F}). Then there exist N ∈ ℤ and a function Φ(s) of of bounded order, depending only on the width of Ξ and S, such that

|E(f_{s})(g)| ≤ |Φ(s)| ||g||^{N},

for all g in S.

By a function of bounded order we mean a quotient ξ_{1}(s)/ξ_{2}(s) such that, for some t_{0} ≥ 0, C > 0, and M ∈ ℤ_{+}, we have for all s = x + iy ∈ Ξ with y ≥ t_{0}, the bound |ξ_{j}(s)| ≤ Ce^{|y|^{M}}, for j = 1, 2.

Proof of Proposition. Let us choose a sufficiently small compact, open subgroup K of G(𝔸_{F,f}) such that E(f_{s}) is right invariant under K, and such that Y := G(F)Z_{∞}^{+}\G(𝔸_{F})/K is a smooth manifold. Fix an ϵ > 0, and also a chart on Y defining the differentiable structure. We will work with the Siegel set modulo K, though we will suppress this in our notation.

For every g in S, choose a neighborhood V = V_{g} of g with non-empty interior, which is small enough to lie inside the unit ball of radius ϵ in local coordinates.

Recall that that E(f_{s}) is an eigenfunction of the Laplacian Δ on all of Y, as it is defined by the Casimir operator, and it is moreover locally in L^{p}, for any p ≥ 1. we will exploit this.

Lemma 3.4.7 Let g, V be as above. Then there is a polynomial P (s) independent of g such that

||E(f_{s})(g)|| ≤ |P (s)|.||E(f, s)||_{2,V} .

Proof of Lemma Let V′ = V′_{g} be any neighborhood of g with non-empty interior such that its closure is contained in the interior of V. It suffices to prove the asserted bound for ||E(f, s)||_{∞,V′}. (We write ||.||_{p,X} for the L^{p}-norm over X.) As remarked above, E(f_{s}) is an eigenfunction, say with eigenvalue λ_{s}, of the second order elliptic differential operator Δ. It can be seen that E(f_{s}) is in the localized Sobolev space H_{n}^{loc}(V) for any n. And since V̄′ ⊂ V, E(f_{s}) equals a function in H_{n}(V) on V′.

Lemma 3.4.8 There exists a polynomial Q(X) in ℂ[X], depending only on ϵ and λ_{s} such that

||E(f, s)||_{∞,V′} ≤ |Q(λ_{s})|.||E(f, s)||_{2,V} .

Since λ_{s} is a polynomial, in fact quadratic, in s, Lemma 3.4.7 follows from this. When λ_{s} is real, which happens in our case iff s lies on the unitary cross, the results of section 4 of [So], in particular the identity (4.2) and Lemma 4.1 , easily imply Lemma 3.4.8. The methods of [So] extend to the case of complex eigenvalues as well, but T. Wolff has pointed out to us how this Lemma can be deduced more directly from the local regularity properties of elliptic differential operators; see Lemma 3.4.9 below. (The difficult part of [So] involves precisely bounding the degree of Q(X), but it is not important for our purposes.) For any r, denote by ||.||_{(2,r);X} norm associated to the rth L^{2}-derivative on X so that ||u||_{(2,r);X} equals ∑_{|ν|≤r} ||∂^{ν}u||_{2,X}, with the ∂^{ν} denoting distribution derivatives.

Lemma 3.4.9 Let Ω be a subset of ℝ^{N} contained in the unit ball of radius ϵ, and Ω′ a subset of Ω with non-empty interior such that Ω̄′ ⊂ Ω. Then we have the following:

• (1) For any integer r > N/2, there exists a constant C_{1} > 0 depending only on ϵ and Δ such that, for all u in H_{r}(Ω),

||u||_{∞,Ω} ≤ C_{1} ||u||_{(2,r);Ω}.

• (2) For any integer i ≥ 2, there is a constant C_{i} > 0 depending only on ϵ and Δ, such that for any u ∈ H_{i}(Ω):

||u||_{(2,i);Ω′} ≤ C_{i} (||u||_{2,Ω′} + ||Δu||_{(2,i−2);Ω}) .

Proof of Lemma 3.4.9. If we write H^{0}(Ω) for the subspace of H(Ω) obtained by completing C_{c}^{∞}(Ω), then the first (resp. second) assertion of this lemma is proved on pages 148-151 (resp. 262-267) of [GiT] (resp. [Fo]). Moreover, given any u in H_{r}(Ω), for any r, and a subset Ω′ of non-empty interior with its closure contained in the interior of Ω, we can find a smooth cut-off function ψ such that v = ψu is in H_{r}^{0}(Ω) and v = u on Ω′ (see, for example, pages 275-276 of [Fo]), and we can do this uniformly for all r. Since we can bound ||ψu||_{(2,r),Ω}, by a universal constant times ||u||_{(2,r),Ω}, we get the first part of the Lemma, and for the second part we only need to bound ||Δ(ψu)||_{(2,i−2),Ω} by a constant times ||Δ(u)||_{(2,i−2),Ω} +||u||_{(2,i−1),Ω}. We will suppress Ω henceforth. Recall that ||Δ(ψu)||_{(2,i−2),Ω} is (by definition) ∑_{|ν|≤i−2} ||D^{ν}Δ(ψu)||, which is ∑_{|ν|≤i−2} ||D^{ν}((Δψ)u+(∇ψ)(∇u)+ψ(Δu))||_{2}; this can be bounded by

∑_{|ν|≤i−2} ∑_{β+γ=ν}(||D^{β}(Δψ)D^{γ}(u)||_{2} + …).

It is easy to estimate derivatives of ψ, and moreover, all but one of the terms involves at most i − 1 derivatives of u and is hence bounded by a constant times ||u||_{(}2, i − 1). The remaining term involves i − 2 derivatives of Δu, and this is bounded by a constant times ||Δu||_{(}2, i−2). The constants involved are controllable in terms of ϵ.

Proof of Lemma 3.4.8. Fix a g_{0} and identify an open neighborhood of g_{0} containing V with an open neighborhood of ℝ^{N}, N =dim(Y), contained in the unit ball of radius ϵ. By shrinking V, we may asume that E(f, s) is in H_{i}(V) for all i. By the first part of Lemma 3.4.9, we are left to estimate ||E(f_{s})||_{(2,r),V} for an r > N/2, which we will take to be of the form 2m. Choose neighborhoods V_{j}, 0 ≤ j ≤ m − 3, of g such that we have the nested inclusions

V′ = V_{m−3} ⊂ … V_{1} ⊂ V_{0} = V,

such that the closure of each V_{i} is in the interior of V_{i+1}. Apply the second part of Lemma 3.4.9 succesively with u = E(f_{s}) and (i, Ω′, Ω) = (2(m−j), V_{m−j}, V_{m−j+1}), with j ranging over {0, …, m − 3}, repeatedly making use of the fact that ΔE(f_{s}) is λ_{s}E(f_{s}). The assertion of Lemma 3.4.8 follows at g_{0}. Now if g is any other point, we can translate the neighborhood V to g and see that the G_{∞}-invariance of Δ gives the corresponding result with the same constant.

Let T ∈ Lie A_{ℝ} be a sufficiently regular parameter, i.e., with ⟨ν, T ⟩ being sufficiently large for each ν in Δ. Denote by ∧^{T} E(f_{s}) the Arthur truncation of E(f_{s}) (cf. [A1] or [MW1], p. 35). One knows that ∧^{T} E(f_{s}) decreases rapidly at infinity, and is hence is in L^{p}(Y) for all p. Clearly, ||ψ||_{2,C_{g}} is bounded above by ||ψ||_{2,G} for any square-integrable function ψ on Y. (By abuse of notation, we are writing ||.||_{p,G} instead of ||.||_{p,Y}.) So, in view of Lemma 3.4.8, Proposition 3.4.6 (and the Theorem) will be proved once we establish the following

Proposition 3.4.10 Fix Ξ, S as above. Then there exist N ∈ ℤ and functions Φ_{1}(s), Φ_{2}(s) of bounded order such that, for all g in S,

(i)  || ∧^{T} E(f_{s})||_{2,G} ≤ |Φ_{1}(s)|

and, up to shrinking V = V_{g} to a smaller neighborhood of g with non-empty interior,

(ii)  ||E(f, s) − ∧^{T} E(f, s)||_{∞,V} ≤ |Φ_{2}(s)| ||g||^{N}.

Our first object will be to prove the bound (i) of this Proposition. For the sake of an inductive argument later, we will in fact need to analyze the L^{2}-norm of the truncation of a class of relative Eisenstein series E^{P′}(h_{s}^{w}) (see (3.4.15) below) attached to any standard parabolic subgroup P′ = M′U′ and a Weyl element w representing a double coset in W_{M′}\W/W_{M}. We set

h_{s}^{w} = M_{w}(f_{s}),

where M_{w} is the intertwining operator defined by

M_{w}(f_{s})(g) = ∫_{(U_{0}∩wŪ_{0}w^{−1})(𝔸_{F})} f_{s}(w^{−1}ug) du.

Here Ū_{0} denotes the opposite unipotent subgroup of the maximal unipotent subgroup U_{0}.

Remark 3.4.11. Since f_{s} is good, it factorizes as ∏_{v} f_{v,s} and at any v where ω is unramified, f_{v,s} is the standard function ϕ_{v,s} and M_{w}(ϕ_{v,s}) = c_{v}^{w}(s)ϕ_{v,s}^{w}, where c^{w}(s) = ∏_{v} c_{v}^{w}(s) is a ratio of abelian L-functions – see [Ik1], [PS-R1]. Moreover, the possible poles are contained (cf. [Ik1]) in the set \{\frac{3}{2}, 1, 0, −\frac{1}{2}\}.

Recall that every element w in the Weyl group W is determined by what it does to {e_{1}, e_{2}, e_{3}}, and that each e_{i} gets sent to ±e_{j}, for some j. We will adopt the following notation: If w sends e_{1} to e_{3}, e_{2} to −e_{2} and e_{3} to −e_{1}, for example, we will denote w by [3, ¯2, ¯1]. Put

(3.4.12)  Σ_{M} = {w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6}, w_{7}, w_{8}}

with w_{1} = id. = [1, 2, 3], w_{2} = [¯2, ¯3, 1], w_{3} = [¯1, ¯2, ¯3], w_{4} = [1, 2, ¯3], w_{5} = [1, 3, ¯2], w_{6} = [1, ¯3, ¯2], w_{7} = [2, ¯1, 3] and w_{8} = [2, ¯3, 1].

For each (standard) parabolic P′ = M′U′ ≠ G, define a subset Σ_{M,M′} of Σ_{M} by the following table: (3.4.13)

\begin{matrix}Δ_{M′} & Σ_{M,M′} \\ \{α, β\} & \{w_{1}, w_{2}, w_{4}, w_{6}\} \\ \{α, γ\} & \{w_{1}, w_{2}, w_{5}\} \\ \{βγ\} & \{w_{1}, w_{7}\} \\ \{α\} & \{w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6}\} \\ \{β\} & \{w_{1}, w_{3}, w_{4}, w_{6}, w_{7}, w_{8}\} \\ \{γ\} & \{w_{1}, w_{2}, w_{5}, w_{7}\} \\ ϕ & Σ_{M} \end{matrix}

Lemma 3.4.14 Let P′ = U′M′ be a standard parabolic. Then Σ_{M,M′} is a set of representatives for W_{M′}\W/W_{M}, where W_{M′} (resp. W_{M}) denotes the Weyl group of the roots relative to M′ (resp. M). Moreover, for any w ∈ Σ_{M,M′}, Q′_{w} := wP w^{−1}∩M′ is a parabolic subgroup of M′, P′_{w} := Q′_{w} ·U′ (but not wP w^{−1}∩P′) is a (standard) parabolic subgroup of G contained in P′, and if we put U′_{w} = wP w^{−1}∩U′, then U′ = U′_{w} · V′_{w}, where V′_{w} = wŪ_{0}w^{−1} ∩ U_{0}.

Proof of Lemma. Put

W_{M,M′}^{∗} = \{w ∈ W | \begin{matrix}w(ν) > 0, ∀ ν ∈ Δ_{M}; \\ w^{−1}(ν) > 0, ∀ ν ∈ Δ_{M′} \end{matrix}\} .

Then it is well known that W_{M,M′}^{∗} is a set of representatives in W for W_{M′}\W/W_{M}. Explicitly, one has, when Δ_{M′} is empty,

W_{M,M′}^{∗} = \begin{cases}w_{1}, w̄_{2} := [3, ¯2, ¯1], w̄_{3} := [¯3, ¯2, ¯1], w_{4}, w_{5}, \\ w_{6}, w̄_{7} := [2, ¯3, ¯1], w_{8} \end{cases}\} .

The only difference between this set and the Σ_{M} defined in the previous page is that w_{2}, w_{3}, w_{7} are replaced by w̄_{2}, w̄_{3}, w̄_{7}. But it is easy to see that for j = 2, 3, 7, w̄_{j} = w_{j}w′_{j}, with w′_{j} ∈ W_{M}. Thus, when Δ_{M′} is empty, Σ_{M,M′} is also a set of representatives for W_{M′}\W/W_{M} in W. The other cases of M′ can be checked by comparing with the table on page 21 for Σ_{M,M′}. The remaining assertions of the Lemma follow by a straightforward computation.

The relative Eisenstein series of interest to us attached to (P′, w) is given by

(3.4.15)  E^{P′}(h_{s}^{w})(g) = ∑_{γ∈Q′_{w}(F)\M′(F)} h_{s}^{w}(γg).

Denote by ∧^{T,P′}E^{P′}(h_{s}^{w}) the truncation of E^{P′}(h_{s}^{w}) as defined on page 97 of [A1]. Let ⟨∧^{T,P′}E^{P′}(h_{s}^{w}), ∧^{T,P′}E^{P′}(h_{s}^{w})⟩_{P′,T} denote the scalar product defined on page 45 of [A2], which specializes to usual scalar product when P′ = G.

Proposition 3.4.16 ⟨∧^{T,P′}E^{P′}(h_{s}^{w}), ∧^{T,P′}E^{P′}(h_{s}^{w})⟩_{P′,T} is of bounded order.

Proof of Proposition 3.4.16. To mesh better with the notations of Langlands and Arthur we will shift the parameter s by 1/2 so that the unitary axis is the line ℜ(s) = 0 (instead of ℜ(s) = 1/2), but by abuse of notation we will continue to denote the Eisenstein series by E^{P′}(h_{s}^{w}).

For s, s′ in ℂ, set

(3.4.16)  h(s, s′) = ⟨∧^{T,P′}E^{P′}(h_{s}^{w}), ∧^{T,P′}E^{P′}(h_{−\overline{s′}}^{w})⟩_{P′,T} ,

which is a meromorphic function in (s, s′) whose poles are in a union of vertical lines s = c_{i} and horizontal lines s′ = c′_{j}, depending on the polar set of E^{P′}(h_{s}^{w}), which is in turn governed (cf. [La4]) by the constant term along P_{0}. The intersection of this polar set of h(s, s′) with \{\overline{s} + s′ = 0\} has only a finite number of points in Ξ. So, for large enough t_{0}, the function s → h(s, −\overline{s}) is well defined and finite on {s = x + iy ∈ Ξ | |y| ≥ t_{0}}.

By Langlands we can write E^{P′}(h_{s}^{w}) in terms of residues of cuspidal Eisenstein series. In the notation of [A2], page 43, it is a finite sum of terms of the form

(3.4.17)  Res_{Λ→Λ_{S}} (F_{B}(Λ), Λ + βs),

where S is a suitable flag of affine subspaces 𝔱 = Λ_{𝔱}+Lie(A_{P′_{w}})^{∗} in Lie(A_{B})^{∗}, β in Lie(A_{P′_{w}})^{∗}, B = M_{B}U_{B} a relevant parabolic subgroup of P′_{w}, and F_{B} a meromorphic function of Λ on Lie(A_{B})^{∗} with singularities lying in hyperplanes.

Note that h_{s}^{w} is asociated to the representation induced from P′_{w} by a onedimensional representation, which is residual on M′_{w} associated to a representation induced by a character ξ_{Λ}^{w} of the minimal parabloic P_{0}, which is exponential in Λ and of bounded order in each coordinate direction. The construction of Langlands ([La4]) is natural relative to induction in stages, and we see that E^{P′}(h_{s}^{w}) must be the residue of the cuspidal Eisenstein series defined by ξ_{Λ}^{w}. Hence the only relevant B occurring in (3.4.17) is P_{0}. In this case the function F_{B}(Λ) is right K-finite in the space of the representation induced by ξ_{Λ}^{w}. By the Iwasawa decomposition we see then that F_{B}(Λ) is of exponential type.

By Lemma 3.1 of [A2], we see that h(s, s′) is a finite sum of double residues

(3.4.18)  Res_{Λ→Λ_{S}}Res_{Λ′→Λ_{S′}} ω^{T,P′}(Λ + βs, −Λ′ + βs′, F_{B}(Λ), F_{B}(Λ′)),

where (cf. page 46 of [A2])

ω^{T,P′}(λ, λ′, ϕ, ϕ′) = ∑_{P_{1}⊂P′} ∑_{t,t′∈W^{P′} (B,P_{1})} \frac{⟨M_{t,λ}(ϕ), M_{t′,−\overline{λ′}}(ϕ′)⟩e^{(tλ−t′λ)(T)}}{a(P_{1}, P′) ∏_{ν∈Δ_{P_{1}}^{P′}} (tλ − t′λ′)(ν^{∨})} .

Here a(P_{1}, P′) is a non-zero constant, and W^{P′}(B, P_{1}) is the set of isomorphisms, possibly empty, of LieA_{B} onto LieA_{P_{1}} leaving LieA_{P′} pointwise fixed which arise from the restricted Weyl group W (G, A_{0}). (In our particular case, P_{1} has to be B = P_{0}.) We refer to [A2] for a definition of Δ_{P_{1}}^{P′}.

The c-functions associated to the intertwining operators M_{t,λ} (of (3.4.18)) acting on functions ϕ(λ) coming from induced representations from the Borel are, just like the c_{s}^{w} associated to M_{w} discussed in Remark 3.4.11, ratios of abelian L-functions ([La7]), hence bounded of order 1. Combining this with that fact that F_{B}(Λ) is of pure exponential type, we see that the expression ω^{T,P′}(Λ + βs, −Λ′ + βs, F_{B}(Λ), F_{B}(Λ′)) is of bounded order in Λ and Λ′. Its singularities are in hyperplanes, and evaluating the residues for Λ → Λ_{S} and Λ′ → Λ_{S′}, we see that the double residue of (3.4.18) must be of bounded order in (s, s′) along any direction. This then implies the result we want about h(s, −\overline{s}) in \{s = x+iy ∈ Ξ | |y| ≥ t_{0}\}.

Proof of Proposition 3.4.10. Part (i) of this Proposition clearly follows by applying Proposition 3.4.16 in the special case P′ = G. It remains to prove (ii). First we need to derive a manageable expression for E(f_{s}) − ∧^{T} E(f_{s}). This takes some work and we start with some preliminaries.

For each standard parabolic P′ = M′U′, M′ = Cent_{G}(A′), put

M′_{0} = \{m ∈ M′(𝔸_{F}) | \begin{matrix}|χ(m)| = 1, \\ ∀ χ ∈ Hom(M′, 𝔾_{m}) \end{matrix}\} .

Then there exists a map

log_{M′} : M′(𝔸_{F}) ⟶ Lie(A′)_{ℂ}

defined so as to satisfy ⟨χ, log_{M′}(m)⟩ = log |χ(m)|, for all m ∈ M′(𝔸_{F}) and χ ∈ Hom(M′, 𝔾_{m}). Then M′_{0}(𝔸_{F}) is precisely ker(log_{M′}). On the other hand, the Iwasawa decomposition G(𝔸_{F}) = U′(𝔸_{F})M′(𝔸_{F})K leads to a map

m_{P′} : G(𝔸_{F}) ⟶ M′_{0}(𝔸_{F})\M′(𝔸_{F}),

given by sending g to the coset M′_{0}(𝔸_{F})m, if g = umk, with u ∈ U′(𝔸_{F}), m ∈ M′(𝔸_{F}) and k ∈ K. Thus log_{M′} ∘ m_{P′} is a map from G(𝔸_{F}) into Lie(A′)_{ℂ}.

Note that A_{Δ} is the center Z of G. We have a decomposition Lie A′_{ℝ} = Lie Z_{ℝ} ⊕ σ_{M′}^{G}, where σ_{M′}^{G} is the subspace generated by the set Δ_{M′} of simple roots relative to M′. Denote by τ̂_{P′} the characteristic function of the sum of Lie Z_{ℝ} and the interior of the cone of σ_{M′}^{G} generated by Δ_{M′}. Then one has (see p. 35 of [MW1]) (∀ G ∈ G(𝔸_{F})):

(E(f_{s}) − ∧^{T} E(f_{s}))(g)

= ∑_{P_{0}⊆P′⊂_{+}G}(−1)^{(dim A′−dim A_{0})} ∑_{γ∈P′(F)\G(F)} τ̂_{P′}(log_{M′}(m_{P′}(γg) − T′)E(f_{s})_{P′}(γg),

(3.4.17)

where T′ is the projection of T in Lie A′_{ℝ}, and E(f_{s})_{P′} is the constant term of E(f_{s}) along P′, namely:

(3.4.20)  E(f_{s})_{P′}(g) = ∫_{U′(F)\U′(𝔸_{F})} E(f_{s})(ug) du.

Proposition 3.4.21 (a) For each P′, the constant term of E(f_{s}) along P′ is given by

E(f_{s})_{P′}(g) = ∑_{w∈Σ_{M,M′}} E^{P′}(h_{s}^{w})(g)

for all g in G(𝔸_{F}).

(b) For T large enough (and regular), we have

E(f_{s}) − ∧^{T} E(f_{s})

= ∑_{P_{0}⊆P′⊂_{+}G}(−1)^{(dim A′−dim A_{0})} ∑_{w∈Σ_{M,M′}} τ̂_{P′}(log_{M′}(m_{P′}(g) − T′)E^{P′}(M_{w}(f_{s}))(g).

Remark: For P′ = P_{0}, the expression given in part (a) is well known by the work of Rallis and Piatetski-Shapiro ([PS-R1]) – see also [Ik1]. For cuspidal Eisenstein series (which is not what we have here), there is a general formula for any reductive group G – see Section II.1.7 of [MW1], for example.

Proof of Proposition 3.4.21 (a) Combining (3.4.3) and Lemma 3.4.14, and remembering that U′_{w} denotes wP w^{−1} ∩ P′, we get

E(f_{s})_{P′}(g) = ∑_{w∈Σ_{M,M′}} ∑_{γ∈Q′_{w}(F)\M′(F)} ∫_{U′_{w}(F)\U′(𝔸_{F})} f_{s}(w^{−1}uγg) du.

Now we use the decomposition U′ = U′_{w} · V′_{w} of Lemma 3.4.14, and recall the P (F)-invariance of f_{s} together with the fact that the intertwining operator M_{w} is defined (see (3.4.12) by integration over V′_{w}(𝔸_{F}). We see that, up to normalizing the measures so that volume of the compact group (wUw^{−1}∩U′)(F)\(wUw^{−1}∩U′)(𝔸_{F}) is 1,

E(f_{s})_{P′}(g) = ∑_{w∈Σ_{M,M′}} ∑_{γ∈Q′_{w}(F)\M′(F)} M_{w}(f_{s})(γg) du.

This proves (a).

(b) By definition, Q′_{w}\M′ = P′_{w}\P′. Since m_{P′} is left invariant under elements of P′(F), the assertion is a consequence, in view of the identity of (a), of the following

Lemma 3.4.22 For g lying in a fixed Siegel domain, we have for T sufficiently large (and regular):

τ̂_{P′}(log_{M′}(γg)) − T′)E_{w}^{P′}(f_{s}(γg)) = 0 if γ ∈ G(F) − P′(F).

Proof of Lemma. We may restrict ourselves to a Siegel domain of the form 𝔖_{c} = ΩA′_{c}K, where Ω is a compact subset of P′(𝔸_{F}), c a positive real number, and A′_{c} the set of elements t in A′(𝔸_{F}) such that |ν(t)| > c for each ν in Δ_{M′}. As usual, one chooses c to be small enough and Ω large enough so that G(𝔸_{F}) = P′(F)𝔖_{c}.

Let γ ∈ G(F) − P′(F). Then by the Bruhat decomposition, γ = ūw, for some Weyl element w, γ ∈ P′(F) and ū ∈ Ū′_{w}(F) := wU′(F)w^{−1}.

By the Iwasawa decomposition relative to P′, we can write any g in G(𝔸_{F}) as a product uxtk, with u ∈ U′(𝔸_{F}), x ∈ M′_{0}(𝔸_{F}), t ∈ A′(𝔸_{F}) and k ∈ K. For any root ν (in Φ_{M′}) let us write H_{ν}(g) for |ν(t)|, which is well defined. By the Iwasawa decomposition relative to P_{0}, we can also write g = nt_{0}k_{1} with n ∈ U_{0}(𝔸_{F}), t_{0} ∈ A_{0}(𝔸) and k ∈ K. Then H_{ν}(g) identifies with ν evaluated at the projection of t_{0} in A′(𝔸_{F}). For w ∈ W, we will write ν^{w}(t) = ν(t^{w}) := ν(wtw^{−1}).

Claim 3.4.23. Let ū ∈ Ū′_{w}(F). Then ∃ ν ∈ Φ_{M′}^{+} such that ν^{w} = ν^{−1} on A′ and H_{ν}(ū) ≤ 1.

Suppose the claim holds. Put γ = yūw ∈ G(F). Then for any g = uzk ∈ G(𝔸_{F}), with z ∈ U′(𝔸_{F})M′(𝔸_{F}), u ∈ U′(𝔸_{F}), k ∈ K, we have

H_{ν}(γg) = H_{ν}(ūwuzk) = Hν(ūu^{w}z^{w}) = H_{ν}(z^{w})H_{ν}((z^{w})^{−1}(ūu^{w})z^{w}).

Note that H_{ν}(z^{w}) = H_{ν^{w}}(z) = H_{ν}(g)^{−1} < c^{−1}. On the other hand, since the conjugate of ūu^{w} by z^{w} lies in Ū′_{w}(F), we have by the claim, H_{ν}((z^{w})^{−1}(ūu^{w})z^{w}) ≤ 1. So H_{ν}(γg) < c^{−1}. Since ν is a positive sum of roots in Δ_{M′}, we can take T to be large enough so that ν(T′) > c^{−1}. Then we must have τ̂_{P′}(log_{M′}(γg)) − T′) = 0.

It remains to prove the claim. We can define local “heights” H_{ν,v} in the obvious way to get H_{ν}(g) = ∏_{v} H_{ν,v}(g_{v}). So it suffices to find a ν with ν^{w} = ν^{−1} such that H_{ν,v}(ū_{v}) ≤ 1 everywhere. It is instructive to begin by looking at the situation for the basic case G′ = SL_{2}, and the argument for G is simply a jazzed up version. Recall that at each v, the standard maximal compact subgroup if G′(F_{v}) is G′(O_{v}) of F is p-adic, SO(2) if F_{v} = ℝ, and SU(2) if F_{v} = ℂ. Let ū = ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ), with x ∈ F^{∗}. By the Iwasawa decomposition, we can find a ∈ F_{v}^{∗} and y ∈ F_{v} such that

k := ( \begin{matrix}a^{−1} & 0 \\ 0 & a \end{matrix} ) ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} ) = ( \begin{matrix}a^{−1}(1+xy) & a^{−1}y \\ ax & a \end{matrix} ) lies in K_{v}. In the non-archimedean

case, if |x| ≤ 1, we take a = 1 (and y = 0) and if |x| > 1, we are forced to have |a| ≤ |x|^{−1}. Suppose F_{v} = ℝ. Then k^{t}k must be the identity matrix, and this forces |a| = \frac{1}{\sqrt{x^{2}+1}} (and y = −x). The complex case is similar and we get |a| = \frac{1}{\sqrt{|x|^{2}+1}}.

Now consider the global case and write a for the idele (a_{v}) with a_{v} being (for each v) the element chosen above (and denoted just by a for brevity of notation). Then we get

|a| ≤ ∏_{v|∞} \frac{1}{\sqrt{1 + |x|_{v}^{2}}} ∏_{v finite} |x|_{v}^{−1} ≤ ∏_{v|∞} \frac{|x|_{v}}{\sqrt{1 + |x|_{v}^{2}}} ≤ 1,

by Artin’s product formula, which gives ∏_{all v} |x|_{v}^{−1} = 1 (since x ∈ F^{∗}). Take ν to be the root ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) ↦ a/b. Clearly, Ū = w ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) w^{−1} and ν^{w} = ν^{−1}, with w = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). Moreover, since |a| ≤ 1, we have H_{ν}(ū) ≤ 1.

In our case (G = GSp(6)), we will now give the argument for the most basic case P′ = P_{0} and Ū_{w} = Ū_{0}. The other cases can be easily extracted from this. We write

ū = ( \begin{matrix}\begin{matrix}1 & 0 & 0 \\ t_{1} & 1 & 0 \\ t_{3} & t_{2} & 1 \end{matrix} & 0 \\ \begin{matrix}u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix} & \begin{matrix}1 & −t_{1} & t_{1}t_{2} − t_{3} \\ 0 & 1 & −t_{2} \\ 0 & 0 & 1 \end{matrix} \end{matrix} )

with u_{i}, v_{i}, w_{i}, t_{i} in F. Let v be a place. By the Iwasawa decomposition (over F_{v}), we can find t = diag(a, b, c, a^{−1}, b^{−1}, c^{−1}) ∈ A_{0}(F_{v}) and n ∈ U_{0}(F_{v}) such that k := t^{−1}n^{−1}u lies in K′_{v}, the standard maximal compact subgroup of Sp(6, F_{v}). If we write

n^{−1} = ( \begin{matrix}\begin{matrix}1 & s_{1} & s_{3} \\ 0 & 1 & s_{2} \\ 0 & 0 & 1 \end{matrix} & \begin{matrix}z_{1} & z_{2} & z_{3} \\ y_{1} & y_{2} & y_{3} \\ x_{1} & x_{2} & x_{3} \end{matrix} \\ 0 & \begin{matrix}1 & 0 & 0 \\ −s_{1} & 1 & 0 \\ −s′_{1} & −s_{2} & 1 \end{matrix} \end{matrix} ) , with s′_{3} = s_{3} − s_{1}s_{2},

then we have:

(3.4.24)  k = ( \begin{matrix}* & * \\ X & Y \end{matrix} )

where

X = (\begin{matrix}au_{1} & au_{2} & au_{3} \\ b(−s_{1}u_{1} + v_{1}) & b(−s_{1}u_{2} + v_{2}) & b(−s_{1}u_{3} + v_{3}) \\ c(−s′_{3}u_{1} − s_{2}v_{1} + w_{1}) & c(−s′_{3}u_{2} − s_{2}v_{2} + w_{2}) & c(−s′_{3}u_{3} − s_{2}v_{3} + w_{3}) \end{matrix})

and

Y = (\begin{matrix}a & −at_{1} & −at′_{3} \\ −bs_{1} & b(s_{1}t_{1} + 1) & b(s_{1}t′_{3} − t_{2}) \\ −s′_{3}c & c(s′_{3}t_{1} − s_{2}) & c(s′_{3}t′_{3} + s_{2}t_{2} + 1) \end{matrix}) .

In the non-archimedean case, we are then forced to have: |a|, |b|, |c| ≤ 1. Moreover, suppose some element, call it d, of the set {u_{1}, u_{2}, u_{3}, t_{1}, t′_{3}} is non-zero. Then we can take a = 1 if d is integral at v and we are forced to have |a| ≤ |d|^{−1} if d is non-integral at v. In the archimedean case, just as in the SL(2) case, we have a = _{\frac{1}{\sqrt{1+|x|_{v}^{2}}}} . Again the global a has absolute value bounded above by 1. In this

case, we are done by taking ν = 2e_{1} and w to be the longest root. Suppose every element of this set is zero. then t_{3} is also zero, and we get

ū = ( \begin{matrix}\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & t_{2} & 1 \end{matrix} & 0 \\ \begin{matrix}0 & 0 & 0 \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix} & \begin{matrix}1 & 0 & 0 \\ 0 & 1 & −t_{2} \\ 0 & 0 & 1 \end{matrix} \end{matrix} ) k_{1},

with k_{1} in K′_{v}. So we may replace ū by ūk_{1}^{−1}. Then we get a condition on b analogous to the one on a above if some element of the set {v_{1}, v_{2}, v_{3}, s_{1}, t_{2}} is non-zero. In this case we take ν to be 2e_{2}. If every element of this set is also zero, then since ū is non-zero, some w_{j} must be non-zero, which leads to a condition on |c|. We are done by taking ν to be 2e_{3} in this case. This finishes the proof of the Claim, Lemma 3.4.22 and Proposition 3.4.21.

Proof of Proposition 3.4.10 (contd.) In view of part (b) of Proposition 3.4.21, it suffices to show that, for all P′ = M′U′ and w in W_{M,M′}^{∗}, we have the bound

(3.4.25)  ||E^{P′}(h_{s}^{w})||_{2,V′_{g}} ≤ |Φ(s)| ||g||^{N},

for some function Φ(s) of bounded order and integer N.

Since Y has finite volume, say A, which is bigger than or equal to vol(U_{g}), we get the bound

(3.4.26)  ||E^{P′}(h_{s}^{w})||_{2,V′_{g}} ≤ A ||E^{P′}(h_{s}^{w})||_{∞,V′_{g}}.

Let us call (P′, w) convenient if P′_{w} = P′. This happens in two cases: (i) when P′ = P_{0} and w arbitrary, and (ii) when P′ = P and w = w_{1} (identity). Suppose we are in either of these cases. Then E^{P′}(h_{s}^{w}) = M_{w}(f_{s}). Recall that the effect of the intertwining operator M_{w} on f_{s} is given by a ratio of abelian L-functions (see Remark 3.4.11), which are functions of bounded order by Hecke. Moreover, the definition of f_{s} shows that its value at g is bounded by a constant (independent of s) times ||g||^{r}, for some integer r depending on the width of Ξ. Making use of (3.4.26), and possibly shrinking V_{g} slightly, we can find an N for which (3.4.25) holds in such a convenient case.

Now suppose (P′, w) is inconvenient. Then E^{P′}(M_{w}(f_{s})) is left-invariant under U′(𝔸_{F})M′(F). Let A′ denote the center of M′ and put K′ = K ∩ M(𝔸_{F,f}). Since P′ commutes with A′, we get the requisite behavior of E^{P′}(M_{w}(f_{s})) along the root directions along A′. It remains to study the growth along the root directions which are trivial on A′. By the decomposition G(𝔸_{F}) = P′(𝔸_{F})K_{∞}K, the problem reduces, in view of (3.4.26), to bounding the growth, for each k ∈ K_{∞}/(K_{∞} ∩M′_{∞}), of the function on Y′ := M′(F)(Z′_{∞})^{+}\M′(𝔸_{F})/K′ given by g → E^{P′}(h_{s}^{w})(gk). Let W_{g} denote the corresponding neighborhood (of the image of g), which we may shrink to be in the open ball of radius ϵ in the local coordinates, and let W′_{g} be the subset defined by U′_{g} such that \overline{W′_{g}} ⊂ W_{g}. Note that the relative Eisenstein series is an eigenfunction of the Laplacian Δ′ defined by the Casimir operator on M′(F_{∞}) with eigenvalue λ′_{s}, which is again a quadratic function of s. Applying the earlier argument (see Lemmas 3.4.8, 3.4.9), and making use of the K_{∞}-finitenes of f_{s}, we get for a polynomial P′(s) independent of g and k:

(3.4.27)  ||E^{P′}(h_{s}^{w})||_{∞,W′_{g}} ≤ |P′(s)|.||E^{P′}(h_{s}^{w})||_{2,W′_{g}}.

We can truncate E^{P′}(h_{s}^{w}), and in view of Proposition 3.4.16, it is enough to estimate ||E^{P′}(h_{s}^{w}) − ∧^{T,P′}E^{P′}(h_{s}^{w})||_{2,W′_{g}} . In view of the obvious analog of Proposition 3.4.21, it will suffice to bound the function ||E^{P′′}(H_{s}^{u})||_{∞,W′_{g}} for any parabolic P′′ contained in P′, Weyl element u in W_{M′_{w}}\W_{M′}/W_{M′′} and function H_{s}^{u} = M_{u}(h_{s}^{w}). In the convenient case the proof goes as above. In the inconvenient cases, the parabolic taking the place of P′_{w} is necessarily the minimal parabolic P_{0}. So if we truncate one more time, all the resulting cases will be convenient. Done.

This finishes the proof of Propositions 3.4.10, 3.4.6, and hence Theorem 3.4.1.

Remark 3.4.28. There is clearly an inductive argument buried here which should help understand the nature of integral representations of L-functions for larger groups.

3.5. Modularity in the good case. Let F be a totally complex number field. We will call a pair (π, π′) of cuspidal automorphic representations of GL(2, 𝔸_{F}) good if at each finite place v, either π_{v} or π′_{v} is not supercuspidal.

Theorem 3.5.1 For every good pair (π, π′), there exists an isobaric automorphic representation π ⊠π′ of GL(4, 𝔸_{F}) such that the identities (L_{v}), (ε_{v}), (L_{∞}) asserted in Theorem M hold, where v is an arbitrary finite place.

Proof. In view of Lemma 3.1.1 we may, and we will, assume that neither π nor π′ is dihedral, i.e., automorphically induced from a grossencharacter of a quadratic extension, and that π′ is a not a character twist of π.

Our first object is to define, for each place v, an irreducible admissible representation Π_{v} of GL(4, F_{v}) canonically associated to the pair (π_{v}, π′_{v}). As before, let σ_{v} (resp. σ′_{v}) be the 2-dimensional representation of W′_{F_{v}} attached to π_{v} (resp. π′_{v}) by [Ku]. First let v be archimedean. Then the local Langlands correspondence exists, and we may (and will) take Π_{v} to be the representation associated to σ_{v} ⊗ σ′_{v} by the method of [La 1].

Next consider the non-archimedean case. Here we will, for every n ≥ 1 with d a divisor of n and a supercuspidal representation β of GL(d, F_{v}), denote the associated (generalizd) special representation of GL(n, F_{v}) by St_{n}(β). By hypothesis, one of the local representations, say π_{v}, is not supercuspidal. If π_{v} is a principal series representation attached to quasi-characters μ, ν of F_{v}^{∗}, then let us set

Π_{v} := (μ ⊗ π′_{v}) ⊞ (ν ⊗ π′_{v}).

It remains to consider when π_{v} is the special representation St_{2}(ν) defined by a quasi-character ν. If π′_{v} is supercuspidal, then let us set

Π_{v} := St_{4}(π′_{v} ⊗ ν).

If π′_{v} is a principal series representation defined by quasi-characters μ′, ν′, then we set

Π_{v} := (St_{2}(ν) ⊗ μ′) ⊞ (St_{2}(ν) ⊗ ν′).

Finally, if π′_{v} is also a special representation St(ν′), for a quasi-character ν′, then we set

Π_{v} := St_{3}(νν′) ⊞ νν′.

If we put Π := ⊗_{v}Π_{v}, then Π is an irreducible, admisible, generic representation of GL(4, 𝔸_{F}). We want to apply the converse theorem for GL(4) (cf. [CoPS], which is recalled in Theorem 3.1.3). Let T be a finite set of finite places containing those v where both π_{v} and π′_{v} are in the discrete series, and let η be an arbitrary cuspidal automorphic representation of GL(m, 𝔸_{F}) with m = 1 or 2, such that η_{v} is unramified at every v in T. We are interested in the formally defined Rankin-Selberg L-function L(s, Π×η), and more precisely in the local integrals which show up in the proof of the converse theorem in [CoPS]. By [JPSS] we know that, at each finite place v, the L-factor L(s, Π_{v} × η_{v}) is the gcd of the corresponding local integrals Ψ(s, W_{v}, W′_{v}, Φ_{v}); when v is archimedean, we know (cf. [JS4]) that the poles of Ψ(s, W_{v}, W′_{v}, Φ_{v}) are contained in the poles of the L-factor. Moreover, we have a local functional equation involving the correct L- and ε-factors. See also [MW2]. (When v is archimedean, it is not known if the L-factor is realized in terms of the local integrals, unless one extends the function spaces [JS4], but this will not play any role for us.) Using this in conjunction with Theorems 3.3.11 and 3.4.1, we see that we can apply Theorem 3.1.3 and conclude the near-automorphy of Π once we note the following

Lemma 3.5.2 At every place v of F, we have the identities

L(s, Π_{v} × η_{v}) = L(s, π_{v} × π′_{v} × η_{v})

and

ε(s, Π_{v} × η_{v}) = ε(s, π_{v} × π′_{v} × η_{v}).

Proof. We have For m = 1, this is simply a rewording of Proposition 3.3.4. So assume that m = 2, and denote, for each v, the 2-dimensional representation of W′_{F_{v}} associated to η_{v} by τ_{v}. Suppose v is non-archimedean. If π_{v} is a principal series representation defined by quasi-characters μ, ν, then σ_{v} is simply μ ⊕ ν, and L(π_{v} ×π′_{v} ×η_{v}) (resp. ε(π_{v} ×π′_{v} ×η_{v})) coincides with L(s, μ⊗σ′_{v} ⊗τ_{v})L(s, ν⊗σ′_{v} ⊗τ_{v}) (resp. ε(s, μ ⊗ σ′_{v} ⊗ τ_{v})ε(s, ν ⊗ σ′_{v} ⊗ τ_{v})). Again the assertion follows from Prop. 3.3.4, thanks to the definition of Π_{v}. It works similarly when π′_{v} is in the principal series. So we may assume that both π_{v} and π′_{v} are in the discrete series. Then by the choice of (T, η), η_{v} is an unramified principal series. If π_{v} and π′_{v} are both supercuspidal, the assertion follows once again by Prop. 3.3.4. It remains to consider when (i) π_{v} is special and π′_{v} is supercuspidal, and (ii) π and π′ are both special. The Lemma followsm in both cases, due to our choice of Π_{v}, from prop. 81 and Theorem 8.2 of [JPSS]. Finally let v be archimedean. But here, one knows by [La1] that L(s, Π_{v} × η_{v}) (resp. ε(s, Π_{v} × η_{v})) equals L(s, σ_{v} ⊗ σ′_{v} ⊗ τ_{v}) (resp. ε(s, σ_{v} ⊗ σ′_{v} ⊗ τ_{v})). Done.

So now we know that Π is nearly automorphic, i.e., that there exists an automorphic representation Π_{1} of GL(4, 𝔸_{F}) such that Π_{v} is isomorphic to Π_{1,v} for almost all v.

Lemma 3.5.3. There is an isobaric automorphic representation Π_{2} of GL(4, 𝔸_{F}) such that Π_{v} ≃ Π_{2,v} for almost all v.

Proof. Indeed, this is obvious if Π_{1} is cuspidal. So assume not. Then Π_{1} is a Jordan-Hölder component, thanks to Prop.2 of [La6], of a representation induced from a cuspidal automorphic representation β of (the Levi component) of a standard parabolic subgroup P of GL(4). There is by definition an isobaric constituent Π_{2} of this same induced representation, whose local components are at the unramified places the same as those of Π_{1} (and those of Π). Moreover, by the reverse implication of prop.2 of [La6], Π_{2} is also automorphic.

Now Theorem 3.5.1 follows by applying Proposition 3.2.1 by taking π ⊠ π′ to be Π_{2}.

Remark 3.5.4. It is natural to wonder, since π is generic, if it is exactly the same as Π_{2}. We will show later (in section 4.3) that this is indeed the case. We will then have a uniqueness statement for π ⊠ π′.

3.6. A descent criterion. The object of this section is to prove the following simple extension of the Proposition in section 4.2 of [BR]. (There is a mistake in [BR], but this does not affect sections 1-4. In any case, the proof given below is completely self-contained.)

Proposition 3.6.1 Fix n, p ∈ ℕ with p prime. Let F a number field, {K_{j} | j ∈ ℕ} a family of cyclic extensions of F with [K_{j} : F] = p, and for each j ∈ ℕ, π_{j} a cuspidal automorphic representation of GL(n, 𝔸_{K_{j}}). Suppose that, for all j, r ∈ ℕ, the base changes of π_{j}, π_{r} to the compositum K_{j}K_{r} satisfy

(DC)  (π_{j})_{K_{j}K_{r}} ≃ (π_{r})_{K_{j}K_{r}}.

Then there exists a unique cuspidal automorphic representation π of GL(n, 𝔸_{F}) such that

(π)_{K_{j}} ≃ π_{j},

for all but a finite number of j.

Proof. First we need the following

Lemma 3.6.2 Let π be a cuspidal automorphic representation of GL(n, 𝔸_{F}). Then there exist atmost a finite number of idele class characters χ such that

π ≃ π ⊗ χ.

Proof of Lemma. Suppose π is isomorphic to its self-twist by an idele class character χ. First we note that, if ω is the central character of π, then the central chacter of π ⊗ χ is simply ωχ^{n}. Thus χ must have order dividing n.

Next we claim that χ must be unramified at any finite place v where π is un-

ramified. Indeed, for any such v, L(s, π_{v}) is by definition, of the form ∏_{j=1}^{n} L(s, μ_{j}),

for some unramified characters μ_{j} of F_{v}^{∗}; this L-factor is not 1. On the other hand,

if χ_{v} is ramified, L(s, π_{v} ⊗ χ_{v}) is none other than ∏_{j=1}^{n} L(s, μ_{j}χ), which equals 1.

Hence the claim.

Consequently, if S is the (finite) support of the conductor of π, then the number of possible self-twists is bounded by the number of idele class characters χ of F such that the (finite) support of the conductor of χ is in S. Every such character cuts out, by class field theory, a cyclic extension F (χ)/F of degree m ≤ n, which is ramified only at the places in S. But the number of such extensions is, for each m ≤ n, known to be finite by a well known variant of a classical theorem of Hermite. The assertion now follows.

Now we continue with the proof of the Proposition. For each j, let θ_{j} be a generator of Gal(K_{j}/F), and δ_{j} a character of F cutting out K_{j} (by class field theory). Note that, for each i ≥ 1, the pull back to K_{i} of δ_{j} by the norm map N_{i} from K_{i} to F cuts out the compositum K_{i}K_{j}.

We claim that

(3.6.3)  π_{j} ∘ θ_{j} ≃ π_{j} (∀j).

For all j, r ≥, let θ_{j,r} denote the automorphism of K_{j}K_{r} such that (i) θ_{j,r}|_{K_{j}} = θ_{j}, and (ii) θ_{j,r}|_{K_{r}} = 1 (where 1 denotes the identity automorphism). It is easy to see that the base change of π_{j} ∘ θ_{j} to K_{j}K_{r} is simply (π_{j})_{K_{j}K_{r}} ∘ θ_{j,r}. Applying (DC), we then have

(π_{j} ∘ θ_{j})_{K_{j}K_{r}} ≃ (π_{r})_{K_{j}K_{r}} ∘ θ_{j,r} ≃ (π_{r})_{K_{j}K_{r}} ≃ (π_{j})_{K_{j}K_{r}},

since θ_{j,r} is trivial on K_{r}. Since K_{j}K_{r} is a cyclic extension of K_{j} of prime degree, we have by Arthur-Clozel (Prop. 2.3.1)

π_{j} ∘ θ_{j} ≃ π_{j} ⊗ (δ_{r} ∘ N_{j})^{m_{r}},

for some m_{r} ∈ {0, 1, …, p − 1}. For every fixed r ≥ 1, and for all k ≠ r, we then have the self-twist identity

π_{j} ≃ π_{j} ⊗ (δ_{r} ∘ N_{j})^{m_{r}}(δ_{k} ∘ N_{j})^{−m_{k}}.

Note that δ_{r} ∘ N_{j} and δ_{k} ∘ N_{j} must be distinct unless their ratio is a power of δ_{j}. So the Lemma above forces m_{r} to be 0 for all but a finite number of r. Then, since (3.6.3) is independent of r, the claim follows.

As a result, by applying Prop. 2.3.1 again, we see that, there exists, for each j ≥ 1, a cuspidal automorphic representation π(j) of GL(n, 𝔸_{F}) such that

π_{j} ≃ (π(j))_{K_{j}}.

Such a π(j) is of course unique only up to twisting by a power of δ_{j}.

It is important to note that, for any r ≠ j, we have the following compatibility for base change in (cyclic) stages:

(3.6.4)  ((π(j))_{K_{j}})_{K_{j}K_{r}} ≃ ((π(j))_{K_{r}})_{K_{j}K_{r}}.

We see this as follows. Let v be a finite place of K_{j}K_{r} which is unramified for the data. Denote by u (resp. w, resp. w′) the place of F (resp. K_{j}, resp. K_{r}) below v. If σ_{u} denotes the representation of W′_{F_{u}} associated to π(j)_{u}, then

res_{(K_{j}K_{r})_{v}}^{(K_{j})_{w}}(res_{(K_{j})_{w}}^{F_{u}}(σ_{u})) ≃ res_{(K_{j}K_{r})_{v}}^{(K_{r})_{w′}}(res_{(K_{r})′_{w}}^{F_{u}}(σ_{u})).

Then (2.3.0) implies the local identity (for all such v)

((π(j)_{u})_{(K_{j})_{w}})_{(K_{j}K_{r})_{v}} ≃ ((π(j)_{u})_{(K_{r})′_{w}})_{(K_{j}K_{r})_{v}}.

The global isomorphism (3.6.4) follows by the strong multiplicity one theorem. We can then rewrite (DC) as saying, for all j, r ≥ 1,

((π(j))_{K_{j}})_{K_{j}K_{r}} ≃ ((π(r))_{K_{j}})_{K_{j}K_{r}}.

Applying Prop. 2.3.1, we get

(π(j))_{K_{j}} ≃ (π(r))_{K_{j}} ⊗ (δ_{r} ∘ N_{j})^{m(r,j)},

for some integer m(r, j). We can replace π(r) by π(r) ⊗ δ_{r}^{−m(r,j)} and get

(π(j))_{K_{j}} ≃ (π(r))_{K_{j}}.

Then, by replacing π(j) by a power of δ_{j}, we can arrange for π(j) and π(r) to be isomorphic. In sum, we have produced, for every pair (j, r), a common descent, say π(j, r), of {π_{j}, π_{r}}.

Fix a, b ∈ {1, …, p − 1}, and consider the possible isomorphism

(3.6.5)  π(j, r) ≃ π(j, r) ⊗ δ_{j}^{a}δ_{r}^{−b}.

We claim that this cannot happen outside a finite set S_{a,b} of pairs (j, r). To see this fix a pair (i, ℓ) and consider the relationship of π(i, ℓ) to π(j, r). Since π(i, ℓ) and π(j, ℓ) have the same base change to K_{ℓ}, they must differ by twisting by a power of δ_{ℓ}. Similarly, π(j, ℓ) and π(j, r) differ by a character twist as they have the same base change to K_{r}. Put together, this shows that π(i, ℓ) and π(j, r) are twists of each other. Then (3.6.5) would imply that

π(i, ℓ) ≃ π(i, ℓ) ⊗ δ_{j}^{a}δ_{r}^{−b}.

The claim now follows since (by Lemma above) π(i, ℓ) admits only a finite number of self-twists and since the characters δ_{j}^{a}δ_{r}^{−b} are all distinct for distinct pairs (j, r) (as a, b are fixed).

Now choose a pair (j, r) not belonging to S_{a,b} for any (a, b) ∈ (ℤ/p)^{∗} × (ℤ/p)^{∗}, and set

π = π(j, r).

We assert that, for all but a finite number of indices m,

π_{K_{m}} ≃ π_{m}.

It suffices to show that, for any large enough m, π = π(j, r) is isomorphic to either π(j, m) or π(m, r). Suppose neither is satisfied. Then there exist a, b ∈ {1, …, p − 1} such that

π(j, m) ≃ π(j, r) ⊗ δ_{j}^{a} and π(m, r) ≃ π(j, r) ⊗ δ_{r}^{b}.

We also have π(j, m) ≃ π(m, r) ⊗ δ_{m}^{c}, for some c ∈ {0, 1, …, p − 1}. Putting these together, we get the self-twisting identity

π(j, r) ≃ π(j, r) ⊗ δ_{j}^{a}δ_{r}^{−b}δ_{m}^{−c}.

By our choice of (j, r), c cannot be 0. For each non-zero c, the set of indices m for which such an identity can hold is finite (again by the Lemma). Hence we get a contradiction for large enough m, which implies that a or b should be 0. The Proposition is now proved.

3.7. Modularity in the general case. In this section we will complete the proof of Theorem M without any hypothesis on (π, π′). we begin by noting the following (well known)

Lemma 3.7.1 Let v be a finite place where π_{v} is supercuspidal. Then there is a finite solvable, normal extension E of F_{v} such that the base change π_{v,E} is in the principal series.

Proof. Let σ_{v} denote as before the two-dimensional representation of W′_{F_{v}} associated to π_{v}. By [Ku], σ_{v} is an irreducible of W_{F_{v}}. The structure of the Weil group implies that σ_{v} must in fact be of the form τ_{v}⊗ν, where τ_{v} is an irreducible ℂrepresentation of W_{F_{v}} factoring through a finite quotient, and ν a quasi-character. So there is a finite (necessarily solvable) normal extension E/F_{v} such that the restriction of τ_{v} to the Weil group of E(v) is trivial. (Take E to be the fixed field of \overline{F}_{v} under the kernel of τ_{v}.) Then by the functoriality of base change, (π_{v})_{E(v)} must be in the principal series, in fact of the form (1 ⊞ 1) ⊗ (ν ∘ N_{E(v)/F}).

It should be noted that by using Krasner’s lemma, we can find a solvable Galois extension k′/k of number fields with local extension E/F_{v} such that Gal(k′/k) is isomorphic to Gal(E/F_{v}) (see [PR], sec.4, Lemma 3, for example). But we cannot hope to be able to take k to be F ! But we can find (see below) a finite chain of cyclic extensions of F with good properties, but the field on top need not be normal over F.

Proof of Theorem M We may, by Proposition 3.1.1, assume that we are not in one of the special situations (I), (II), (III) treated there.

Given any finite solvable group G, define its length ℓ(G) to be the length ℓ of any chain {1} = G_{0} ⊂ G_{1} ⊂ … ⊂ G_{ℓ} = G, with each G_{i} normal in G_{i+1} of prime index. Let S = S(π) denote the (possibly empty) finite set of finite places where either π_{v} is not supercuspidal. At each v ∈ S, let ℓ(π_{v}) denote the length of a minimal Galois extension L(v)/F_{v} such that the base change (π_{v})_{L(v)} is in the principal series. Let ℓ(π) denote the maximum of {ℓ(π_{v}) | v ∈ S}, and let S′ denote the subset of S where this maximum is attained. Further, for each v in S′, let p(v) denote the maximum over all L(v), of the degree, required to be a prime or 1, of the largest cyclic extension K(v) of F_{v} contained in L(v). Let p = p(π) be the maximum of p(v) over all v in S′, and let S′′ denote the subset of S′ where p(v) = p (and ℓ(π_{v}) = ℓ(π)). Note that p is a prime unless π is good over F, i.e., has no supercuspidal components, in which case p = 1.

Let us set

(3.7.2)  r(π) = (ℓ(π), p(π)).

We will order these pairs as follows: (ℓ, p) < (ℓ′, p) if either ℓ < ℓ′ or if ℓ = ℓ′ and p < p′.

Suppose r = r(π) is (0, 1). Then, for any quadratic extension K′ which is totally complex, the assertion holds by Theorem 3.5.1. Also, given any finite place u, we can find a totally complex, quadratic extension K′(u) such that u splits in K′(u). Applying the descent criterion (Proposition 3.6.1), we can find a common descent having the requisite properties.

Now let r > (0, 1), and assume by induction that the Theorem is proved (over all number fields K) for pairs (π_{1}, π_{2}) of cuspidal automorphic representations of GL(2, 𝔸_{K}) with r(π_{1}, π_{2}) < r.

Fix, at every place v in S′′, a character χ_{v} of F_{v}^{∗} cutting out a K(v) (as above) of degree p.

Enumerate the set of finite places where π is unramified, and list them as {v_{1}, v_{2}, …, v_{j}, … }.

Fix an index j ≥ 1 (for the moment), and let S(j) be the union of v_{j} with S′′. Let χ_{v_{0}} denote the trivial character of F_{v_{0}}^{∗}.

Now by the Grunewald-Wang theorem (see [AT], chap. 10, Theorem 5), we can find a global character χ(j) of C_{F} of order ℓ whose local restrictions are given by χ_{v} at every v in S(j). (Note that the set S_{0} which occurs in loc. cit. is empty in our case as each local degree is p or 1 and cannot be divisible by 4, thus allowing us to find χ(j) of order p, not just 2p.) Let K_{j} be the p-extension of F cut out by χ(j).

By construction we have, for every j ≥ 1,

(3.7.3)  r(π) < r.

Thus, by induction, the Theorem holds for π_{K_{j}} for each j. Note that if the automorphic representation π_{K_{j}} is not cuspidal for some j, it must be dihedral (with p = 2), and by the remark earlier, we may assume that we are not in this case. Put

(3.7.4)  Π_{j} = π_{K_{j}} ⊠ π′_{K_{j}}.

The non-dihedrality of π also gives the following

Lemma 3.7.5 There exist at most a finite number of indices j such that Π_{j} is not cuspidal.

Proof. It suffices to prove that, outside a finite set of indices, we have

1. π_{K_{j}} is not dihedral;

2. π_{K_{j}} is not the twist of π′_{K_{j}} by an idele class character of K_{j}.

Suppose π_{K_{j}} is dihedral. Then, by Gelbart-Jacquet ([GJ]), the symmetric square of π_{K_{j}} is Eisensteinian, while by asumption, sym^{2}(π) is not. This forces p to be 3 and sym^{2}(π) to be automorphically induced by a character μ of K_{j}. In other words, the symmetric squre of π admits a self-twist relative to the character of F corresponding to K_{j}. By Lemma 3.6.2, this can only happen for a finite number of j.

We may then asume that π_{K_{j}} is not dihedral and consider the second assertion. Suppose π_{K_{j}} is the twist of π′_{K_{j}} by an idele class character. Then, for a character μ of C_{K_{j}}, L^{T′}(s, Π_{j} ⊗ μ) must have a pole at s = 1, where T′ is any finite set of places of K_{j}. By the inductivity of L-functions, the same holds for the (L-function of the) automorphic induction I_{K_{j}}^{F}(Π_{j}). Let us choose T′ to be the inverse image of the set T of places of F where π or π′ or K_{j} is ramified. Then it is easy to see that we have the identity:

L^{T} (s, I_{K_{j}}^{F}(Π_{j} ⊗ μ)) = L^{T} (s, π × (π′ ⊠ I_{K_{j}}^{F}(μ))).

Since T contains all the ramified places, the right hand side L-function is an incomplete form of any of the triple product L-functions attached to (π, π′, I_{K_{j}}^{F}(μ)). By [Ik1], it cannot have a pole unless π is dihedral. Done.

Consequently, after shrinking the index set by removing an appropriate finite subset and renumbering it, we may assume that

Π_{j} is cuspidal for every j.

Next we fix a pair (j, r) of indices and consider the descent criterion (DC) of Proposition 3.6.1. Let w be a finite place where ((Π_{j})_{K_{j}K_{r}})_{w}, ((Π_{r})_{K_{j}K_{r}})_{w} and K_{i}K_{j} are all unramified. Then, by construction, both these local representations correspond to the restriction (to the Weil group of (K_{j}K_{r})_{w}) of σ_{v} ⊗ σ′_{v}, where v signifies the place of F below w. (Recall that σ_{v}, σ′_{v} are associated to π_{v}, π′_{v} respectively.) This leads to the identity

L^{Y} (s, (Π_{j})_{K_{j}K_{r}}) = L^{Y} (s, (Π_{r})_{K_{j}K_{r}}),

for a large enough finite set X of places, which gives (DC) by the strong multiplicity one theorem.

Applying Proposition 3.6.1, we then get a unique cuspidal descent Π on GL(4)/F such that, for all but a finite number of indices, we have

Π_{K_{j}} ≃ Π_{j}.

Finally, by construction, each (unramified) finite place v_{j} splits completely in K_{j}; let w_{j} be a divisor of v_{j} in K_{j}. This implies (by the definition of base change) that, for almost all j, we have

L(s, Π_{v_{j}}) = L(s, (Π_{j})_{w_{j}}) = L(s, (π_{K_{j}})_{w} × (π′_{K_{j}})_{w}) = L(s, π_{v} × π′_{v}).

Similarly for the ε-factors. Thus Π is a weak (tensor product) lifting of (π, π′). By Proposition 3.2.1, it is also the strong lifting.

4. Applications

4.1. A multiplicity one theorem for SL(2). Denote by L_{0}^{2}(SL(2, F)\SL(2, 𝔸_{F})) the subspace of cusp forms in L^{2}(SL(2, F)\SL(2, 𝔸_{F})). It is acted upon by SL(2, 𝔸_{F}) by right translation, and this action is unitary relative to the natural scalar product on L^{2}(SL(2, F)\SL(2, 𝔸_{F})). The main result of this section is

Theorem 4.1.1 The representation of SL(2, 𝔸_{F}) on L_{0}^{2}(SL(2, F)\SL(2, 𝔸_{F}) has multiplicity one.

This was conjectured by Labesse and Langlands (see [LL], [Lab1]). The Theorem says that the commutant of this representation is abelian, and equivalently, since the space of cusp forms is completely reducible, every irreducible unitary representation of SL(2, 𝔸_{F}) oocurring in (the decomposition of) this space appears with multiplicity one.

The stable trace formula for SL(2) was analyzed deeply, and in detail, in [LL]. We refer to this paper for all the background material. The results there show in particular that multiplicity one for SL(2) is a consequence of the following

Theorem 4.1.2 Let π, π′ be unitary, cuspidal automorphic representations of GL(2, 𝔸_{F}). Suppose we have, for almost all v,

(LL(v))  Ad(π_{v}) ≃ Ad(π′_{v}).

Then there exists an idele class character χ, which is unique if π is not automorphically induced by a character of a quadratic extension, such that

π′ ≃ π ⊗ χ.

If π and π′ have the same central character, then χ is quadratic.

Here Ad(π) denotes the automorphic representation sym^{2}(π)⊗ω^{−1} of GL(3, 𝔸_{F}), where ω is the central character of π.

As noted in the Introduction, in the special case when π and π′ are defined by holomorphic eigenforms, with F necessarily totally real, various people have known the proof of this result by making use of the associated ℓ-adic repreentations. But this does not work in general, and the main point here is to give a unified proof using the modularity of π ⊠ π′.

Before beginning the proof of this Theorem, we would like to indicate the following concrete

Corollary 4.1.3 Let π, π′ be unitary, cuspidal automorphic representations of GL(2, 𝔸_{F}) of trivial character. Suppose we have at every unramified finite place v,

(Sq(v))  a_{v}(π)^{2} = a_{v}(π′)^{2},

where a_{v}(π) =tr(A_{v}(π)) is the v-th Hecke eigenvalue of π. Then there is a unique quadratic idele class character χ such that

a_{v}(π′) = χ_{v}(ϖ_{v})a_{v}(π′),

for almost all v, where ϖ_{v} denotes the uniformizer at v. If in addition, the conductors N, N′ of π, π′ are square-free, then χ = 1.

Theorem 4.1.2 implies Corollary 4.1.3:

Let π, π′ be as in the Corollary, and let v be a finite place where both π and π′ are unramified. Since by hypothesis, the central characters are trivial, we may write A_{v}(π) = diag(α_{v}, α_{v}^{−1}) and A_{v}(π′) = diag(α′_{v}, α′_{v}^{−1}). Then the Langlands class of Ad(π_{v}) (resp. Ad(π′_{v})) is given by {α_{v}^{2}, 1, α_{v}^{−2}} (resp. {α′_{v}^{2}, 1, α′_{v}^{−2}}). Consequently, we have (in ℜ(s) > 1)

log(L(s, Ad(π_{v}))) = 1 + ∑_{m≥1} \frac{α_{v}^{2m} + 1 + α_{v}^{−2m}}{m(Nv)^{ms}}

and

log(L(s, Ad(π′_{v}))) = 1 + ∑_{m≥1} \frac{α′_{v}^{2m} + 1 + α′_{v}^{−2m}}{m(Nv)^{ms}} .

We claim that, for every m ≥ 1,

α_{v}^{2m} + α_{v}^{−2m} = α′_{v}^{2m} + α′_{v}^{−2m}.

By induction on m, it suffices (by the binomial formula) to verify

(α_{v}^{2} + α_{v}^{−2})^{m} = (α′_{v}^{2} + α′_{v}^{−2})^{m},

which follows immediately from the hypothesis (Sq(v)). Hence the claim.

Since (LL(v)) then holds for almost all v, the Corollary follows by applying Theorem 4.1.2.

Proof of Theorem 4.1.2.

First note that, since (LL(v)) holds for almost all v, the automorphic representations Ad(π) and Ad(π′) are isomorphic by the strong multiplicity one theorem.

Suppose π is dihedral, i.e., of the form I_{K}^{F}(μ), for a character μ of C_{K}, for some quadratic extension K of F. Its central character then identifies with μ_{0}δ, where μ_{0} is the restriction of μ to C_{F} and δ the quadratic character of C_{F} associated to K/F by class field theory. Denote by θ the non-trivial automorphism of K/F. We claim that

(ad)  Ad(π) = δ ⊞ I_{K}^{F}(μ/(μ ∘ θ)).

Since (μ_{0})_{K} = μ(μ ∘ θ), this is equivalent to the identification of sym^{2}(π) with μ_{0} ⊞ I_{K}^{F}(μ^{2}), which is easy to verify at the unramified places. So the claim follows by the strong multiplicity one theorem ([JS2]).

A particular consequence of the claim is that L(s, Ad(π)⊗δ) has a pole at s = 1. This forces π′ also to be dihedral, for otherwise, L(s, Ad(π′) ⊗ δ) will be entire ([GJ]). Then δ must in fact occur in the isobaric decomposition of Ad(π′_{K}), which in turn is a summand of π′ ⊠ π′^{∨}. Then L(s, π′ ⊠ π′^{∨} ⊗ δ) also has a pole at s = 1. This implies that

π′ ≃ π′ ⊗ δ.

In other words, π′ is of the form I_{K}^{F}(μ′), for a character μ′ of C_{K}.

Now, base changing Ad(π) =Ad(π′) to K, and using (ad), we get the identity

μ/(μ ∘ θ) ⊞ (μ ∘ θ)/μ ≃ μ′/(μ′ ∘ θ) ⊞ (μ′ ∘ θ)/μ′.

Replacing μ′ by μ′ ∘ θ if necessary, we can then deduce that

μ/μ′ = (μ/μ′) ∘ θ.

In other words, there exists a character χ of C_{F} such that μ′ = μχ_{K}. Then we have

π′ = I_{K}^{F}(μ′) ≃ I_{K}^{F}(μ) ⊗ χ ≃ π ⊗ χ,

as desired. To verify the middle isomorphism, one first checks it locally almost everywhere and then apply the strong multiplicity one theorem.

So we may assume henceforth in the main case of interest, namely when neither π nor π′ is dihedral. Put

Π = π ⊠ π′.

We know that it is an isobaric automorphic representation of GL(4, 𝔸_{F}). The main point here is to prove the following Lemma 4.1.4. Π is not cuspidal.

Proof of Lemma. Let S be a finite set of places containing the ramified places (for π and π′) and the archimedean ones.

Suppose Π is cuspidal. Then, by [JS2], we know that the Rankin-Selberg Lfunction L^{S}(s, Π × Π^{∨}) must have a simple pole at s = 1.

On the other hand, at any finite place v, if we denote (as usual) by σ_{v} (resp. σ′_{v}) the representation of W′_{F_{v}}, we get, as σ_{v} ⊗ σ_{v}^{∨} ≃ Ad(σ_{v}) ⊕ 1, the following identity:

(σ_{v} ⊗ σ′_{v}) ⊗ (σ_{v} ⊗ σ′_{v})^{∨} ≃ 1 ⊕ Ad(σ_{v}) ⊕ Ad(σ′_{v}) ⊕ Ad(σ_{v}) ⊗ Ad(σ′_{v}).

At the unramified places v, we know that σ_{v} ⊗ σ′_{v} corresponds to Π_{v}. So this translates to the L-function identity

(Id) L^{S}(s, Π × Π^{∨}) = ζ_{F}^{S}(s)L^{S}(s, Ad(π))L^{S}(s, Ad(π′))L^{S}(s, Ad(π) × Ad(π′)).

But we know that Ad(π) is isomorphic to Ad(π′). Moreover, since Ad(π) = sym^{2}(π) ⊗ ω^{−1}, we have

Ad(π)^{∨} ≃ sym^{2}(π^{∨}) ⊗ ω ≃ Ad(π),

as sym^{2}(π^{∨}) is sym^{2}(π) ⊗ ω^{−2}.

Consequently, L^{S}(s, Ad(π) × Ad(π′)) has a pole at s = 1. Also, since π and π′ are non-dihedral, L^{S}(s, Ad(π) and L^{S}(s, Ad(π′) are entire (cf. [GJ]). Then the identity (Id) implies that L^{S}(s, Π × Π^{∨}) has at least a double pole at s = 1, leading to the desired contradiction.

Thus π, π′ are non-dihedral cusp forms on GL(2)/F with π ⊠ π′ not cuspidal. Then by the cuspidality criterion of Theorem M (see page 9), which was proved in section 3.2 (on pp. 14-15), we see that π′ must be isomorphic to π ⊗ χ for an idele class character χ. Done.

4.2. Some new functional equations. Let π_{1}, π_{2}, π_{3}, π_{4} be cuspidal automorphic representations of GL(2, 𝔸_{F}), and let S be a finite set of places containing the archimedean and ramified places. Let us set

L^{S}(s, π_{1}×π_{2}×π_{3}×π_{4}) = ∏_{v∉S} det(I−(Nv)^{−s}A_{v}(π_{1})⊗A_{v}(π_{2})⊗A_{v}(π_{3})⊗A_{v}(π_{4}))^{−1},

L^{S}(s, sym^{2}(π_{1})×π_{2}×π_{3}) = ∏_{v∉S} det(I−(Nv)^{−s}sym^{2}(A_{v}(π_{1}))⊗A_{v}(π_{2})⊗A_{v}(π_{3}))^{−1},

and

L^{S}(s, sym^{3}(π_{1}) × π_{2}) = ∏_{v∉S} det(I − (Nv)^{−s}sym^{3}(A_{v}(π_{1})) ⊗ A_{v}(π_{2}))^{−1}.

These functions of s converge absolutely in some right half plane.

Theorem 4.2.1. Each of the three (incomplete) Euler products above admit meromorphic continuations to the whole s-plane. In addition, each of them can be completed to a convergent Euler product over all places in ℜ(s) > 1 , admitting meromorphic continuation and a functional equation of the usual form (as conjectured by Langlands). They are non-zero on the line s = 1 with no pole except possibly at the point s = 1. The first two L-functions are in fact analytic everywhere except for possible poles at s = 0, 1.

Remark: The meromorphic continuation and functional equation of the third of the L-functions above is already known by the work of Shahidi using the theory of Eisenstein series ([Sh2]). But his approach does not seem to be able to give invertibility in ℜ(s) ≥ 1 except for a possible pole at s = 1.

Proof. Since by Theorem M, there exist π_{1} ⊠ π_{2} and π_{3} ⊠ π_{4} in A(4, F), we may set

L(s, π_{1} × π_{2} × π_{3} × π_{4}) = L(s, (π_{1} ⊠ π_{2}) × (π_{3} ⊠ π_{4})).

By the definition of ⊠, the unramified local factors of the L-function on the right agree with those of L^{S}(s, π_{1} × π_{2} × π_{3} × π_{4}). This way we get a complete Euler product in ℜ(s) > 1, and the remaining assertions follow immediately for the first Lfunction from the corresponding ones, due to Jacquet, Piatetski-Shapiro and Shalika ([JPSS]), Moeglin-Waldspurger ([MW2]) Shahidi [Sh6], for the Rankin-Selberg Lfunctions on GL(4)×GL(4).

We complete the second and third L-functions by respectively setting

L(s, sym^{2}(π_{1}) × π_{2} × π_{3}) = L(s, sym^{2}(π_{1}) × (π_{2} ⊠ π_{3}))

and

L(s, sym^{3}(π_{1}) × π_{2}) = \frac{L(s, sym^{2}(π_{1}) × (π_{1} ⊠ π_{2}))}{L(s, (π_{1} ⊠ π_{2}) ⊗ ω_{1}},

where ω_{1} is the central character of π_{1}.

The verification of the fact that the unramified factors of these L-functions are the right ones is an easy exercise, which is left to the reader. Since the Rankin-Selberg L-functions on GL(n)×GL(m) are non-zero in {ℜ(s) ≥ 1} and are analytic everywhere except for possible poles at s = 0, 1, all the assertions of Theorem 4.2.1 follow except for knowing that the third L-function is non-zero at s = 1. So we will be done if we prove

Lemma 4.2.2.

−ord_{s=1}L(s, π_{1} ⊠ π_{2} ⊗ ω_{1}) ≤ −ord_{s=1}L(s, sym^{2}(π_{1}) × (π_{1} ⊠ π_{2})).

Suppose L(s, π_{1} ⊠ π_{2} ⊗ ω_{1}) has a pole at s = 1. Since π_{1}, π_{2} are cuspidal, the pole has to be a simple pole, and besides, this can happen iff π_{2} is isomorphic to the contragredient of π_{1} ⊗ ω_{1}. It then follows, since π_{1} ⊠ π_{1}^{∨} = Ad(π_{1}) ⊞ 1, that

(F ac)  L(s, sym^{2}(π_{1}) × (π_{1} ⊠ π_{2})) = L(s, Ad(π_{1}) × Ad(π_{1}))L(s, Ad(π_{1})).

We have seen earlier that Ad(π_{1}) is self-dual. Hence L(s, Ad(π_{1}) × Ad(π_{1})) has a pole at s = 1. We claim that L(s, Ad(π_{1})) is holomorphic at s = 1, which is clear if π is non-dihedral, as Ad(π_{1}) is then cuspidal by [GJ]. So suppose π is of the form I_{K}^{F}(μ) for a character of (the idele classes of) a quadratic extension K of F, with non-trivial automorphism θ over F. Then, by the identity (ad) appearing in the proof of Theorem 4.1.2, L(s, Ad(π_{1})) can have a pole at s = 1 iff μ/(μ ∘ θ) is the pull back by norm to K of either the trivial character of C_{F} or the quadratic character δ (attached to K). In either case, we will have μ = μ ∘ θ, contradicting the cuspidality of π_{1}.

Consequently, the right (and hence the left) hand side of (Fac) has a pole at s = 1. This proves the Lemma and completes the proof of the theorem.

Remark 4.2.3. It is now clear how we can also define, in an analogous way, the epsilon factors, both locally and globally, for the L-functions under consideration.

4.3. Root numbers and representations of orthogonal type. Let k be a non-archimedean local field of characteristic zero. Our first aim is to prove the following, which is not clear even if one knew the local Langlands conjecture for GL(4).

Proposition 4.3.1. Let n ≤ 4, {η_{j}|j ≤ n} supercuspidals of GL(2, k), with corresponding 2-dimensional irreducibles {τ_{j}|j ≤ n}. Then we have

L(s, η_{1} × … × η_{n}) = L(s, τ_{1} ⊗ … ⊗ τ_{n})

and

ε(s, η_{1} × … × η_{n}) = ε(s, τ_{1} ⊗ … ⊗ τ_{n}).

Proof of Proposition. For n = 1, this is contained in [Ku]. It is also well known for n = 2; see Prop. 4.2 of [PR], for example. So assume n ≥ 3. If either of the η_{j} is the local automorphic induction of a (quasi)character μ of E^{∗}, for some quadratic extension E/k, then the Proposition is the same as Prop. 4.2 of loc. cit. Even otherwise, we can modify the argument there and appeal to the global Rankin-Selberg product to get what we want. We give the details for completeness. We start by appealing to Lemma 3, sec. 4 of [PR], and find a number field F with k = F_{u} for some place u, and irreducible 2-dimensional representations σ_{j} of Gal(\overline{ℚ}/F) with solvable image such that their respective restrictions to the decomposition group at u identify with τ_{j}. Solvability allows us to apply the theorem of Langlands ([La5]) and Tunnell ([Tu]), and find corresponding cuspidal automorphic representations π_{j} with π_{j,u} ≃ η_{j}. Then we choose a finite order character μ of C_{F} which is highly ramified at all the ramified places except u and which is 1 at u. Comparing the functional equations of L(s, π_{1} × … × π_{n}) and L(s, σ_{1} ⊗ … ⊗ σ_{n}), we proceed as in Proposition 3.2.1 and deduce the assertions above.

Remark 4.3.2. In chapter 3 we asociated, to each pair (π, π′) of cupidal automorphic representations of GL(2, 𝔸_{F}) an irreducible admissible representation Π of GL(4, 𝔸_{F}) and an isobaric automorphic representation π ⊠ π′ such that Π_{v} ≃ (π ⊠ π′)_{v} for almost all v. Using Proposition 4.3.3 we can conclude that such an identity holds at every v.

To clarify this remark, we will digress a bit about the local Langlands conjecture for GL(4), which predicts a canonical bijection, preserving local factors, between the set A^{0}(4, k) of irreducible, supercuspidal representations η of GL(4, k) and the set R^{0}(4, k) of irreducible, continuous, 4-dimensional ℂ-representations τ of W_{k}. There is a numerical bijection between the sets ([He4], compatible with twisting by unramified characters. A theorem of M. Harris ([Ha]) gives a (geometrically defined) bijection τ → η, which gives moreover an equality of epsilon factors of pairs

ε(s, τ ⊗ τ′) = ε(s, η × η′),

for all irreducibles τ′, with corresponding η′, of dimension m, for all m ≤ 4, provided the residual characteristic is prime to 2m. On the other hand, a recent result of Jeff Chen ([Ch]) asserts that any η in A^{0}(4, k) is uniquely determined by the collection of epsilon factors ε(s, η × η′), as η′ ranges over A^{0}(m, k) for m = 1 and m = 2. Consequently, as noted in [PR], sec. 4, the local Langlands conjecture follows for GL(4) in the odd residual characteristic case by combining these results of [Ha] and [Ch]. In the case of even residual characteristic, one known by [He2], one knows the bijection for representations of “cyclic type”, i.e., those which admit a quartic self-twist. Moreover, Prop. 4.1 of [PR] shows how to extend this and treat also the case of representations admitting just a quadratic self-twist.

Now let GO(4, ℂ) denote the subgroup of GL(4, ℂ) consisting of orthogonal similitudes with similitude factor μ. Let GSO(4, ℂ) denote the kernel of the homomorphism GO(4, ℂ) → {±1}, g → μ(g)^{−2}det(g). In this section we will be concerned with the subset R^{1}(4, k) of τ in R^{0}(4, k) whose image lie in GSO(4, ℂ). It is well known that there is a short exact sequence

1 → ℂ^{∗} → GL(2, ℂ) × GL(2, ℂ) → GSO(4, ℂ) → 1,

inducing a surjection

Hom(W_{k}, GL(2, ℂ) × GL(2, ℂ)) → Hom(W_{k}, GSO(4, ℂ)).

See, for example, the proof of Lemma 4 in sec. 5 of [PR]. Consequently, every τ in R^{1}(4, k) is a tensor product τ_{1} ⊗ τ_{2}, with each τ_{j} an irreducible of dimension 2. The pair (τ_{1}, τ_{2}) is not unique, as it can be replaced by (τ_{1} ⊗ ν, τ_{2} ⊗ ν^{−1}), but this is the only ambiguity.

Motivated by this, we define A^{1}(4, k) to be the irreducible supercuspidal representations η of GL(4, k) which are of the form η_{1} ⊠ η_{2} (see sec. 3.7 for a definition). In what follows, we will make use of the local Langlands correspondence for GL(2) and GL(3), established respectively by Kutzko ([Ku]) and Henniart ([He1]).

Proposition 4.3.3. There is a unique bijection τ → η between R^{1}(4, k) and A^{1}(4, k), compatible with twisting by characters and taking duals, such that the following identity of epsilon factors holds for all τ′ ∈ R^{0}(m, k), with corresponding η′ ∈ A^{0}(m, k), for m = 1 and m = 2:

ε(s, τ ⊗ τ′) = ε(s, η × η′).

Proof. Let τ ∈ R^{1}(4, k). Then we know by the remark above that it is of the form τ_{1} ⊗ τ_{2}, with τ_{1}, τ_{2} irreducible 2-dimensionals of W_{k}. Let η_{1}, η_{2} be the respective supercuspidals of GL(2, k) given by the local Langlands correspondence. Put

η = η_{1} ⊠ η_{2}.

Suppose η = η_{1} ⊠ η_{2} is not supercuspidal. Then in its isobaric sum decomposition, there must occur a supercuspidal, say β, of GL(m, k), for some m < 4. (Since η is a representation of W_{k}, there will be no summand of Steinberg type.) Then L(s, β × β^{∨}), and hence L(s, η × β^{∨}) will have a pole at s = 0 (see [JPSS], [JS2]).

Now let λ denote the irreducible m-dimensional representation of W_{k} attached to β by the local Langlands correspondence, which we can do by [Ku] and [He1] as m ≤ 3. Then, by the Proposition above, L(s, τ ⊗ λ^{∨}) equals L(s, η × β^{∨}), and hence must have a pole at s = 0 as well. This cannot hapen as τ is an irreducible of dimension 4, and λ is one of dimension < 4. This proves that η is supercuspidal, and we get a “reciprocity map”

r : R^{1}(4, k) → A^{1}(4, k).

The ε-factor identity of Prop. 4.3.1, when used in conjunction with Jeff Chen’s theorem ([Ch]), shows the injectivity of r. It is also easy to see that r is compatible with taking duals and twisting by characters.

The surjectivity of r is shown by reversing the construction of τ → η. The ireducibility of τ is checked by using Prop. 4.3.2.

Let π_{1}, π_{2}, π_{3}, π_{4} be cuspidal automorphic representations of GL(2, 𝔸_{F}). Define its root number to be

W (π_{1} × π_{2} × π_{3} × π_{4}) = ∏_{v} W (π_{1,v} × π_{2,v} × π_{3,v} × π_{4,v}),

where the local root numbers are defined by

W (π_{1,v} × π_{2,v} × π_{3,v} × π_{4,v}) = ε(\frac{1}{2}, π_{1,v} × π_{2,v} × π_{3,v} × π_{4,v}).

Note that the functional equation implies that the root number is ±1 in the self-dual situation.

Theorem 4.3.4. Let π_{j}, j ≤ 4, be as above. Assume that the central character of each π_{j} is trivial. Then we have

W (π_{1} × π_{2} × π_{3} × π_{4}) = 1.

When all the π_{j}’s are isomorphic, say to π, the root number is W (Sym^{2}(π) × Sym^{2}(π))W (Sym^{2}(π))^{2}W (1), and this theorem follows in that case from Proposition 6.1 of [PR].

Proof. This is an immediate consequence of Proposition 4.3.1 in conjunction with the method of [PR] (see also [Ro]). We give the argument for completeness.

It suffices to show that the local root number at any place v is 1. Clearly, for each j ≤ 4, the associated representation σ_{j,v} of W′_{F_{v}} is self-dual with determinant 1 as π_{j} has trivial central character. Put

β_{v} = σ_{1,v} ⊗ σ_{2,v} ⊗ σ_{3,v} ⊗ σ_{4,v}.

By Prop. 4.3.1, we are reduced to showing that W (β_{v}) (with the obvious definition) is 1. Being the tensor product of an even number of symplectic representations, β_{v} is orthogonal, and by Deligne ([De]) the root number of β̃_{v} := β_{v} ⊖ 4[1] is given by its second Stieffel-Whitney class w_{2}(β̃_{v}). But since Sp(2, ℂ) is simply connected, and since the image of β_{v} in O(4, ℂ) factors through a 4-fold product of Sp(4, ℂ), this image is simply connected. It follows that β̃_{v} lifts to the Spin group, which is a 2-fold cover of SO(4, ℂ). Consequently, w_{2}(β̃_{v}) is trivial, from which it follows that W (β_{v}) = W (β̃_{v})W (1)^{4} is 1.

4.4. Triple product L-functions revisited. Let π_{1}, π_{2}, π_{3} be cuspidal automorphic representations of GL(2, 𝔸_{F}) of respective central characters ω_{1}, ω_{2}, ω_{3}. Recall from section 3.1 the definitions of the three candidates for the triple product Lfunctions and ε-factors associated to (π_{1}, π_{2}, π_{3}). The object of this section is to extract the following precise statement from Theorem M.

Theorem 4.4.1. Let π_{j}, j ≤ 3, be as above. Then we have, at every place v,

L_{1}(s, π_{1,v} × π_{2,v}, π_{3,v}) =^{(a)} L(s, π_{1,v} × π_{2,v}, π_{3,v}) =^{(b)} L_{2}(s, π_{1,v} × π_{2,v}, π_{3,v}),

and

ε_{1}(s, π_{1,v} × π_{2,v}, π_{3,v}) =^{(a)} ε(s, π_{1,v} × π_{2,v}, π_{3,v}) =^{(b)} ε_{2}(s, π_{1,v} × π_{2,v}, π_{3,v})

Proof. Since we know this (see section 3.3) at archimedean places, we may restrict our attention to finite places; fix such a place u. Let p be the residual characteristic of v. Recall from the discussion in section 3.3 that (a) (resp. (b)) is known (i) when all the representations are unramified, and (ii) when one of the representations is not supercuspidal (resp. when all the representations are tempered).

One has the following (at any place v) by the works of Shahidi:

(4.4.2)  γ(s, π_{1,v} × π_{2,v} × π_{3,v}) = γ_{2}(s, π_{1,v} × π_{2,v} × π_{3,v}).

In view of the remark above, the only case to check is when one of the representations, say π_{1,v}, is a subquotient of a principal series representation defined by quasi-characters μ_{1}, μ_{2} of F_{v}^{∗}. Then one has by the multiplicativity of local factors ([Sh4]), the identity

γ_{2}(s, π_{1,v} × π_{2,v} × π_{3,v}) = γ_{2}(s, μ_{1} ⊗ π_{2,v} × π_{3,v})γ_{2}(s, μ_{⊗}π_{2,v} × π_{3,v}).

The analog for γ(s, π_{1,v} ×π_{2,v} ×π_{3,v}) holds by the decomposition of σ_{1,v} ⊗σ_{2,v} ⊗σ_{3,v}. Done.

Our first object is to establish a weak analog of (4.4.2) for the γ_{1}-factor in the place of the γ_{2}-factor.

Lemma 4.4.3. Let S(p) be the set of places of F above p. Then we have

∏_{v∈S(p)} γ(s, π_{1,v} × π_{2,v} × π_{3,v}) = ∏_{v∈S(p)} γ_{1}(s, π_{1,v} × π_{2,v} × π_{3,v}).

Remark 4.4.4. In [Ik1], page 229, corollary to Lemma 2.2, one finds an argument to prove Thm. 4.4.1 (a) in the special case when each π_{j,u} is dihedral, being associated to a character χ_{j,u} of a quadratic extension K_{j,u} of F_{u}. But in the course of the argument, there is an assertion that there is a quadratic extension K_{j} of F with local extension K_{j,u}/F_{u}, together with an idele class character χ of K, such that (1) χ agrees with χ_{j,u} on K_{j,u} (which is fine) and that (2) the cuspidal automorphic representation η_{j} of GL(2, 𝔸_{F}) defined by χ_{j} has principal series components at all the places v outside u. We are unable to make sure that (2) can be chieved. (One cannot in general control the behavior of χ_{j} at an infinite number of places!) However, the argument can be made to work with a slight refinement as follows. We can find K_{j}, χ_{j} satisfying (1) and also (2’): χ_{j} is trivial at every place in S(p) − {u}; this is possible by the Grunewald-Wang theorem ([AT]). By applying Lemma 4.4.3 above, and using Lemma 2.1 of [Ik1], the desired equality then follows.

Proof of Lemma 4.4.3. Let T be the set of finite places where at least one of the representations {π_{1}, π_{2}, π_{3}} is ramified. Then, comparing the functional equations of L(s, π_{1} × π_{2} × π_{3}) and L_{1}(s, π_{1} × π_{2} × π_{3}), we get the equality

∏_{v∈T} γ(s, π_{1,v} × π_{2,v} × π_{3,v}) = ∏_{v∈T} γ_{1}(s, π_{1,v} × π_{2,v} × π_{3,v}).

When v is not in S(p), its norm is a power of a prime ℓ distinct from p, and the corresponding local factors are inverses of polynomials in ℓ^{−s}. Since the identity above holds for all s, it is easy to separate the contribution from the primes in S(p) and those outside.

Proof of Theorem 4.4.1 (contd.). Now we need the following

Proposition 4.4.5. Let v be a finite place, and let L_{∗} denote L, L_{1} or L_{2}. Then, for any quasi-character ν of which is sufficiently ramified, we have

L_{∗}(s, π_{1,v} × π_{2,v} × π_{3,v} ⊗ ν) = 1.

Proof of Proposition 4.4.5. First we consider when L_{∗} = L. Since by definition L(s, π_{1,v} × π_{2,v} × π_{3,v} ⊗ ν) is the same as L(s, σ_{1,v} ⊗ σ_{2,v} ⊗ σ_{3,v} ⊗ ν), it suffices, by the additivity of the L-factor, to show that, for any irreducible summand τ of σ_{1,v} ⊗ σ_{2,v} ⊗ σ_{3,v}, viewed as a representation of W_{F_{v}} × SL(2, ℂ), we have L(s, τ ⊗ ν) is 1 for ν of large enough conductor. Any irreducible τ is of the form β ⊗ S^{k}, where β is an irreducible of W_{F_{v}} and S^{j} denotes the symmetric j-th power representation of SL(2, ℂ). When β is one-dimensional, L(s, τ ⊗ ν) equals L(s, βν|.|^{j/2}), which is trivial (by Tate’s thesis) if ν is sufficiently ramified. If β is higher dimensional, then L(s, τ ⊗ ν) is a product, for suitable half-integers t, of factors of the form L(s, β ⊗ ν|.|^{t}), which are 1 for any ν (because β is irreducible of dimension > 1).

Next consider the case L_{∗} = L_{i}, with i = 1, 2. We may asume that at least one of the π_{j,v}, say π_{3,v}, is tempered. Suppose π_{1,v} is not supercuspidal. Then by using Proposition 3.3.6 for L_{1}, and (4.4.2) for L_{2} together with the temperedness of π_{3,v}, we deduce that (for any ν)

\frac{L_{1}(s, π_{1,v} × π_{2,v} × π_{3,v} ⊗ ν)}{L(s, π_{1,v} × π_{2,v} × π_{3,v} ⊗ ν)} is entire.

The assertions (a) and (b) of the Theorem about L-factors follows in this case by the triviality of the denominator for sufficiently ramified ν. The identities for the ε-factors also follows because the respective γ-factors are the same.

So we may take all the π_{j,v} to be supercuspidal. We will now use a slight variant of the inductive argument utilized in the proof of section 3.7. Let ℓ(π_{1,v}), ℓ(π_{1}), p(v), p = p(π_{1}), r = r(π_{1}) = (ℓ(π_{1}), p(π_{1})) be as in section 3.7, together with the (lexicographic) ordering < introduced there. We will use induction on r. If r = (0, 1), we are done, and so take that (0, 1) < r and assume by induction that the Proposition holds for L_{1} at all places for any triple (π, π′, π′′) of cuspidal automorphic representations o GL(2) over a number field if r(π) < r. Fix at every place v in S′′, a character χ_{v} cutting out a K(v) of degree p such that the base change of π_{1,v} to K(v) has ℓ equal to ℓ(π_{1})/p. Choose, using the Grunewald-Wang theorem, a global character χ of order p, cutting out a cyclic extension K/F with local extensions K(v)/F_{v}. Then r(π_{1,K} is less than r and we may use induction in conjunction with the functional equations of L_{i}(s, π_{1} × π_{2} × π_{3} ⊗ ν) and L_{i}(s, π_{1,K} × π_{2,K} × π_{3,K} ⊗ ν) to conclude that, for any idele class character ν of F which is sufficiently ramified at each of the places in S′′, we have (for i = 1, 2)

∏_{v∈S′′} γ_{i}(s, π_{1,v} × π_{2,v} × π_{3,v} ⊗ ν) = 1.

As remarked earlier, we may assume that, for every j ≤ 3 and every v ∈ S′′, the local representation π_{j,v} is supercuspidal. We may then apply Lemma 2.1 of [Ik1] to conclude that L_{i}(s, π_{1,v} × π_{2,v} × π_{3,v} ⊗ ν) has no pole in common with L_{i}(1 − s, π_{1,v}^{∨} × π_{2,v}^{∨} × π_{3,v}^{∨} ⊗ ν^{−1}). The proposition follows for L_{i}.

Remark 4.4.6. One can also prove Proposition 4.4.5 for L_{2} by arguing as in the proof of Lemma 5 of [PR]. Moreover, Shahidi has informed us that he can now prove a very general statement of this sort for L_{2}.

Proof of Theorem 4.4.1 (contd.). Fix any finite place u of residual characteristic p, and let S(p) be as in Lemma 4.4.3. Choose an idele class character ν of F such that (i) ν_{v} is trivial, and (ii) ν_{v} is highly ramified at each v in S(p). Then, comparing the functional equations of L_{1}(s, π_{1} × π_{2} × π_{3} ⊗ ν) and L(s, π_{1} × π_{2} × π_{3} ⊗ ν) and using (4.4.2), Lemma 4.4.3 and Proposition 4.4, we get

(4.4.7)  γ_{1}(s, π_{1,u} × π_{2,u} × π_{3,u} ⊗ ν) = γ(s, π_{1,u} × π_{2,u} × π_{3,u} ⊗ ν).

Again, we can reduce to the case when each π_{j,u} is supercuspidal and so, applying Lemma 2.1 of [Ik1] and the obvious analog for L_{2}, we deduce the assertions (a) and (b) of the theorem for the L-factors. The identities for the ε-factors also follow by (4.4.7).

4.5. The Tate conjecture for 4-fold products of modular curves. Let Y denotes the moduli scheme over ℚ parametrizing elliptic curves E with level infinity structure, i.e., equipped with an isomorphism of the Tate module T_{f}(E) =lim_{n}E[n] with ℤ̂^{2}. It comes with a natural right action of GL(2, 𝔸^{f}), and a smooth compactification X ([KM]). For every compact, open subgroup K of GL(2, 𝔸^{f}), let Y_{K} (resp. X_{K} = Y_{K} ∪Y_{K}^{∞}) denote Y/K (resp. X/K). Then X_{K} is a smooth projective curve over ℚ, and one has the identifications

Y_{K}(ℂ) = GL(2, ℚ)\H^{±}/K,

where H^{±} denotes ℂ − ℝ, and

Y_{K}^{∞} = ℙ^{1}(ℤ̂)/K.

A cofinal system of compact open subgroups is provided by {K_{1}(N)|N ≥ 1}, where

K_{1}(N) = {k = (k_{ij}) ∈ GL(2, ℤ̂) | k_{21}, k_{22} − 1 ∈ Nℤ̂}.

For every N ≥ 1, we will also be interested in

K_{0}(N) = {k = (k_{ij}) ∈ GL(2, ℤ̂) | k_{21} ∈ Nℤ̂}.

We will write X_{1}(N), X_{0}(N) instead of X_{K_{1}(N)}, X_{K_{0}(N)} respectively.

Let V be a smooth projective variety over a number field k. For every j ≥ 0, denote by C^{j}(V/k) the ℚ-vector space spanned by codimension j algebraic cycles on V over k modulo homological equivalence. Then, for every prime ℓ, one has an ℓ-adic cycle class map

C^{j}(V/k) ⟶ H_{et}^{2j}(V\overline{k}, ℚ_{ℓ})(j),

which is injective and lands in the group of codimension j Tate cycles over k, namely

Ta_{ℓ}^{j}(V/k) = H_{et}^{2j}(V\overline{k}, ℚ_{ℓ})(j)^{Gal(\overline{k}/k)}.

Let S be a finite set of places containing the ramified places for the Gal(\overline{ℚ}/k)- module H_{et}^{2j}(V\overline{k}, ℚ_{ℓ}), the archimedean places, and also the places above ℓ. The associated L-function is

L^{(2j)}(s, V/k) = ∏_{v: finite} L_{v}^{(2j)}(s, V/k),

where

L_{v}^{(2j)}(s, V/k) = det(I − Fr_{v}T |H_{et}^{2j}(V\overline{k}, ℚ_{ℓ})^{I_{v}})^{−1}|_{T =(Nv)^{−s}}.

Here Fr_{v} denotes the geometric Frobenius at v and I_{v} the inertia group at v. In computing L_{v}^{(2j)}(s, V/k), one takes ℓ to be prime to v. For each ℓ, let S = S(ℓ) be the finite set of places consisting of the ramified places for the Gal(\overline{ℚ}/k)-module H_{et}^{2j}(V\overline{k}, ℚ_{ℓ}), together with the places above ℓ. One knows by Deligne’s proof of the Weil conjectures that the incomplete L-function L^{(2j),S}(s, V/k), obtained from L^{(2j)}(s, V/k) by deleting the factors over S, converges absolutely in ℜ(s) > j + 1. One also has a definition of L_{v}^{(2j)}(s, V/k) for any archimedean place v (see [Ra3], sec.5.5, for example). Write L_{∞}^{(2j)}(s, V/k) for the product ∏_{v|∞} L_{v}^{(2j)}(s, V/k), and put

L̃^{(2j)}(s, V/k) : = L_{∞}^{(2j)}(s, V/k)L^{(2j)}(s, V/k).

One also has epsilon factors ε_{v}^{(2j)}(s, V/k) for each v, and the global one

ε^{(2j)}(s, V/k) = ∏_{all v} ε_{v}^{(2j)}(s, V/k),

which is an invertible holomorphic function.

Theorem 4.5.1. Fix positive integers N_{j}, 1 ≤ j ≤ 4, and put

V = X_{1}(N_{1}) × X_{2}(N_{2}) × X_{3}(N_{3}) × X_{4}(N_{4}).

Let k be a finite solvable, normal extension of ℚ, and let ℓ be a prime. Then the following hold:

(a) L^{(4)}(s, V/k) admits a meromorphic continuation to the whole s-plane, has a convergent Euler product in {Re(s) > 3}, and satisfies the exact functional equation:

L̃^{(4)}(s, V/k) = ε^{(4)}(s, V/k)L̃^{(4)}(5 − s, V/k).

(b) L^{(4)}(s, V/k) has no pole anywhere except possibly at the “edge” point s = 3.

(c) If we let −ord_{s=3} stand for the order of pole at s = 3,

−ord_{s=3}L^{(4)}(s, V/k) = dim_{ℚ_{ℓ}}Ta_{ℓ}^{2}(V/k).

(d) Suppose some N_{j} is square-free. Then

−ord_{s=3}L^{(4),S}(s, V/k) = dim_{ℚ}C^{2}(V/k).

In this case, for any number field k, dim_{ℚ_{ℓ}}Ta_{ℓ}^{2}(V/k) and dim_{ℚ}C^{2}(V/k) are equal.

Parts (a) and (b) are as predicted by the Hasse-Weil hypothesis, while parts (c) and (d) verify certain conjectures of Tate (see [Ta2]; [Ra3], chap.5).

Proof. First we recall some well known basic facts about the cohomology of the modular curves X_{1}(N). For j ∈ {0, 1, 2}, put (for any prime ℓ)

W_{ℓ}^{j}(N) : = H_{et}^{j}(X_{1}(N)\overline{ℚ}, ℚ_{ℓ}).

For any field extension E of ℚ, denote by H_{E}(N) the E-algebra of Hecke correspondences at level N. This algebra is semisimple and acts on W_{ℓ}^{j}(N)⊗_{ℚ} E, commuting with the Galois action. If π = π_{∞} ⊗π_{f} is an irreducible automorphic representation of GL(2, 𝔸_{ℚ}), with π_{f} rational over \overline{ℚ}, then the K_{1}(N)-invariants in the space of π_{f} over \overline{ℚ} is naturally an H\overline{ℚ}(N)-module. Conversely, any irreducible H\overline{Q}(N)-module

occurring in W_{ℓ}^{j}(N) is a summand of some π_{f}^{K_{1}(N)}. Denote by Coh^{j}(N) the set of irreducible automorphic representations π contributing to the cohomology in degree j with π_{f}^{K_{1}(N)} non-zero. Then one knows that π is one-dimensional for j = 0, 2, and that for j = 1 it is cuspidal with π_{∞} in the lowest discrete series, corresponding, in the classical language, to a holomorphic newform of weight 2 and level N. Decomposing according to the Hecke algebra, one gets

(4.5.2)  W_{ℓ}^{j}(N) ⊗_{ℚ_{ℓ}} \overline{Q}_{ℓ} ≃ ⊕_{π∈Coh^{j}(N)} W_{ℓ}^{j}(π)^{m(π_{f} ,N)},

where W_{ℓ}^{j}(π) is an irreducible \overline{ℚ}_{ℓ}-representation of Gal(\overline{ℚ}/ℚ) of dimension 2 (resp.

1) when j = 1 (resp. j = 0, 2), and m(π_{f}, N) the dimension of the K_{1}(N)-invariants in the space of π_{f}. When a π occurs in degree 1, then any of its Galois conjugate π^{τ} := π_{∞} ⊗ π_{f}^{τ} also occurs. One knows the following at any place v by the Eichler-Shimura theory and its refinement due to Igusa, Deligne, Langlands and Carayol([Ca]):

(4.5.3)  L_{v}(s, W_{ℓ}^{0}(π)) = L(s, π_{v}) = L_{v}(s + 1, W_{ℓ}^{2}(π)),

and

L_{v}(s, W_{ℓ}^{1}(π)) = L(s − 1/2, π_{v}).

In fact this also holds for one-dimensional twists. The analogous identities hold for the ε-factors as well, implying in particular the identification of the conductors.

Now let us turn our attention to the variety V at hand. In the following i will denote a any 4-tuple (i_{1}, i_{2}, i_{3}, i_{4}) of integers in {0, 1, 2} with ∑_{j=1}^{4} i_{j} = 4, and for each such i, π will denote (π_{1}, π_{2}, π_{3}, π_{4}) with π_{j} ∈Coh^{i_{j}}(N_{j}) for each j. Let m(i, π) signify ∏_{j=1}^{4} m(π_{j,f}, N_{j}).Applying the Künneth formula in conjunction with the decomposition (4.5.2), we get

(4.5.4)  H^{4}(V\overline{ℚ}, \overline{ℚ}_{ℓ}) ≃ ⊕_{i} ⊕_{π} H_{ℓ}^{i}(π)^{m(i,π)},

where

H_{ℓ}^{i}(π) ≃ W_{ℓ}^{i_{1}}(π_{1}) ⊗ W_{ℓ}^{i_{2}}(π_{2}) ⊗ W_{ℓ}^{i_{3}}(π_{3}) ⊗ W_{ℓ}^{i_{4}}(π_{4}).

Applying Proposition 4.3.1 in conjunction with (4.5.3), we get moreover,

(4.5.5)  L(s, H_{ℓ}^{i}(π)) = L(s − 2, π_{1} × π_{2} × π_{3} × π_{4}).

Similarly for the ε-factors.

Since the cycle class maps are functorial for the action of correspondences, we can split the problem and prove the assertions of the theorem for the image of the codimension 2 cycles in H_{ℓ}^{i}(π), for every i and π. Let Ta_{ℓ}^{i}(π)_{k} denote the space of Tate classes over k in H_{ℓ}^{i}(π).

Suppose two of the indices, say i_{3}, i_{4}, are zero. Then H_{ℓ}^{i}(π) is W_{ℓ}^{2}(π_{1})⊗W_{ℓ}^{2}(π_{2})⊗ W_{ℓ}^{0}(π_{3}) ⊗ W_{ℓ}^{0}(π_{4}), which is one-dimensional, in fact of the form μ ⊗ \overline{ℚ}_{ℓ}(−2) with μ of finite order. Then for any number field k, we have L(s, H_{ℓ}^{i}(π)) = L(s−2, μ). The assertions (a), (b) follow by Hecke. The order of pole at s = 3 is 1 or 0 depending n whether or not μ is trivial when restricted to Gal(\overline{Q}/k). When μ is trivial, the corresponding algebraic cycle is given by the intersection of the the π-components of the threefolds {P }×X_{2}(N)_{2})×X_{3}(N_{3})×X_{4}(N_{4}) and X_{1}(N_{1})×{Q}×X_{3}(N_{3})× X_{4}(N_{4}), for points P, Q on X_{1}(N_{1}), X_{2}(N_{2}) respectively. So (d) also follows in this case.

So we may assume from now on that i = (1, 1, 1, 1) and consequently that each π_{j} is cuspidal. In view of (4.5.5), parts (a) and (b) hold by Theorem 4.2.1. Note that the determinant of each W_{ℓ}^{1}(π_{j}) is necessarily of the form ω_{j} ⊗ \overline{ℚ}_{ℓ}(−1), where ω_{j} is of finite order corresponding to the central character of π_{j} by class field theory.

Suppose all the π_{j} are dihedral, defined by a Hecke character χ_{j} of a (necessarily imaginary) quadratic field K_{j}. Then H_{ℓ}^{i}(π) is the \overline{ℚ}_{ℓ}-realization of a CM motive (in the category of pure motives for absolute Hodge cycles), and the assertion (c) is well known (see [DMOS]). The conductor M_{j}, say, of each π_{j} is the conductor of an abelian variety factor A(π_{j}) over ℚ, up to isogeny, of the Jacobian of the corresponding modular curve. One knows by a theorem of Fontaine ([Fon]) that there is no abelian variety over ℚ of good reduction everywhere. On the other hand, since π_{j} is automorphically induced by χ_{j}, it has no local component which is special. Then by the description of th conductor of π_{j} in [Ge], for example, we see that M_{j}, and hence N_{j} which it divides, cannot be square-free. So the hypothesis of part (d) precludes the case when all the π_{j} are dihedral.

Thus we may, and we will, assume that some π_{j}, say π_{1}, is non-dihedral.

Lemma 4.5.6. Let i = (1, 1, 1, 1) and π = (π_{1}, π_{2}, π_{3}, π_{4}), with each π_{j} cuspidal automorphic of weight 2 at infinity and π_{1} non-dihedral. Fix any number field k. Then the dimension of Ta_{ℓ}^{i}(π)_{k} is at most 2. It is non-zero iff we can find, after possibly renumbering \{π_{2}, π_{3}, π_{4}\}, a character μ of Gal(\overline{ℚ}/k) such that

(i) as Gal(\overline{ℚ}/k)-modules,

W_{ℓ}^{1}(π_{2}) ≃ W_{ℓ}^{1}(π_{1}) ⊗ μ

and (ii) (μω_{1})^{−1} occurs in W_{ℓ}^{1}(π_{3}) ⊗ W_{ℓ}^{1}(π_{4}) over k. Moreover, when this happens, the dimension of Ta_{ℓ}^{i}(π) is 1 unless we are in one

of the following situations:

(a) Over k, W_{ℓ}^{1}(π_{3})) is irreducible and non-dihedral, with its symmetric square being isomorphic to S^{2}(W_{ℓ}^{1}(π_{1})) ⊗ ω_{3}ω_{1}^{−1};

(b) W_{ℓ}^{1}(π_{3}) and W_{ℓ}^{1}(π_{4}) are both reducible over k, and dim Hom(μω_{1}^{−1}, W_{ℓ}^{1}(π_{3})⊗ W_{ℓ}^{1}(π_{4}) is 2.

Proof. Since π_{1} is non-dihedral, we know by a result of Ribet ([Ri1]) that W_{ℓ}^{1}(π_{1}) is irreducible upon restriction to any open subgroup of Gal(\overline{ℚ}/ℚ). It follows easily that there are no Tate classes over k unless (i) holds after a possible renumbering. There will be a Tate cycle iff (ii) also holds. Now suppose there is a Tate class.

If π_{3} is irreducible, (ii) will hold iff π_{4} is also irreducible and

(ii′)  W_{ℓ}^{1}(π_{4}) ≃ W_{ℓ}^{1}(π_{3}) ⊗ (μω_{1}ω_{3})^{−1}.

We claim that, since π_{1} is non-dihedral, S^{2}(W_{ℓ}^{1}(π_{1})) must irreducible under restriction to any open subgroup. Indeed, if the claim is false, then over a number field the symmetric square would admit a one-dimensional and would force W_{ℓ}^{1}(π_{1}) would have to be dihedral (over the extension field), contradicting the openness of the image of Gal(\overline{ℚ}/ℚ) in GL(W_{ℓ}^{1}(π_{1})). Hence the only way there could be more than 1 Tate class over k is to have (a), and this is not possible if π_{3} is dihedral. Suppose π_{3} is non-dihedral. Then each π_{j} is non-dihedral and S^{2}(W_{ℓ}^{1}(π_{j})) is irreducible (by the above argument) for any j. Then the number of (independent) such classes can evidently not be more than 2.

Finally, consider the case when (i), (ii), hold, and π_{3}, π_{4} are both reducible. Looking at the Hodge-Tate types we see that the dimension of Ta_{ℓ}^{i}(π) can never exceed 2. It is then also clear that to have dimension 2, we need (b).

Remark 4.5.7. The reason for requiring in Theorem 4.5.1 that some N_{j} is squarefree is just to rule out the case when all the π_{j} are dihedral, in which case the number of Tate classes can at the extreme be 6, and only four of those can be shown to be algebraic by the argument below. When one of them is non-dihedral, the numbers of tate and algebraic cycles in H_{ℓ}^{i}(π) coincide over every number field, not necessarily solvable. The reason for restricting to solvable extensions is to be able to appeal to base change in conjunction with Theorem 4.2.1 and deduce the standard properties of the L-function.

Proof of Theorem 4.5.1 (contd.). Let k be a number field. We need to match the number of Tate classes with the order of pole. By using the standard results on Hecke L-functions, we may assume that one of the π_{j}, say π_{1}, is non-dihedral. In view of (4.5.5), part (c) of the Theorem will follow once we establish the following analog of Lemma 4.5.6:

Lemma 4.5.8 Let i = (1, 1, 1, 1) and π = (π_{1}, π_{2}, π_{3}, π_{4}), with each π_{j} cuspidal automorphic of weight 2 at infinity and π_{1} non-dihedral. Fix any number field k, and denote (∀j) by π_{j,k} the base change of π to k Then the order of pole, m(π_{k}) say, of L(s, π_{k}) := L(s, π_{1} × π_{2} × π_{3} × π_{4}) at s = 1 is at most 2. It is non-zero iff we can find, after possibly renumbering {π_{2}, π_{3}, π_{4}}, an idele class character μ of k such that

(i) π_{2,k} ≃ π_{1,k} ⊗ μ, and

(ii) (μ_{k}ω_{1,k})^{−1} occurs in the isobaric sum decomposition of π_{3,k}) ⊠ π_{4,k}.

Moreover, when this happens, m(π_{k}) is 1 unless we are in one of the following situations:

(a) π_{3,k} is cuspidal and non-dihedral, with its symmetric square being isomorphic to S^{2}((π_{1})) ⊗ ω_{3,k}ω_{1,k}^{−1};

(b) π_{3,k} and π_{4,k} are both Eisensteinian over k, and the multiplicity of (μ_{k}ω_{1,k}^{−1} in π_{3,k} ⊠ π_{4,k} is 2.

Proof of Lemma. The conditions (i) and (ii) are easy to verify if every π_{j} is twist equivalent to π_{1}. So suppose not. Then, after a renumbering, we may assume that π_{1,k} ⊠ π_{3,k} is cuspidal. Using Theorem M we can write L(s, π_{k}) as L(s, (π_{1} ⊠ π_{3}) × (π_{2} ⊠ π_{4})). So by [JS2], it has a pole at s = 1 iff we have

π_{2,k} ⊠ π_{4,k} ≃ (π_{1,k} ⊠ π_{3,k}) ≃ π_{1,k} ⊠ π_{3,k} ⊗ (ω_{1,k}ω_{3,k})^{−1}.

Up to a renumbering, this can happen iff (i) and (ii) hold.

If π_{3,k} is cuspidal, then (ii) will hold iff π_{4,k} is also cuspidal and

(ii′)  π_{4,k} ≃ π_{3,k} ⊗ (μ_{k}ω_{1,k}ω_{3,k})^{−1}.

We claim that S^{2}(π_{1,k}) is cuspidal. Suppose not. Then by [GJ], π_{1,k} will be dihedral. This would imply, by (4.5.3) and the functoriality of base change, that W_{ℓ}^{1}(π_{1}) is dihedral when restricted to Gal(\overline{ℚ}/k), contradicting the openness of the image of Galois ([Ri1]). Hence the claim. Applying [JS2], we then see that m(π_{k}) could be more than 1 iff we have (a), and this is evidently not possible is π_{3} is dihedral. Then each π_{j} is non-dihedral and m(π_{k}) cannot possibly be more than 2. when π_{3}, π_{4} are both reducible, we see by looking at the infinity type that m(π_{k}) can never exceed 2, and that it can be 2 iff (b) holds.

It remains to show (d). In view of (c), it suffices to show, over any number field k, that Ta_{ℓ}^{i}(π) consists of algebraic classes. Suppose there is a non-zero Tate class. Then (i) holds and this is represented by a twisting correspondence R_{μ} on X_{1}(N_{1})_{k}×X_{1}(N_{2})_{k} (see [Ri2], [Mu]). If π_{3,k} is also cuspidal, then (ii’) holds and this is represented by another twisting correspondence R_{ν} on X_{1}(N_{3}) × X_{1}(N_{4}), with ν = (μω_{1}ω_{3})^{−1}. We get a codimension 2 algebraic cycle on V_{k} whose π_{k}-component has cycle class in Ta_{ℓ}^{i}(π) by taking the intersection γ of R_{μ} × X_{1}(N_{3}) × X_{1}(N_{4}) and X_{1}(N_{1}) × X_{1}(N_{2}) × R_{ν}. Now suppose there are two independent Tate classes. If π_{3,k} is cuspidal, then we would have the symmetric square twisting equivalence of (a), and this will give us a twisting correspondence R_{λ} directly on the 4-fold V, with λ = ω_{3}ω^{−1}. This cycle is evidently not multiple of γ. So we are done in this case. If we are in the situation of (b), the abelian varieties A(π_{3}) and A(π_{4}) corresponding to π_{3} and π_{4} will necessarily be of CM type, and we will have two independent Tate classes in Hom(A(π_{3}), A(π_{4})) over k which are algebraic by Faltings; they correspond to divisors δ_{1}, δ_{2} on X_{1}(N_{3}) × X_{1}(N_{4}) rational over k. If we take ξ_{j}, for j = 1, 2 to be R_{μ} × δ_{j}, they define the needed cycles on V.

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Dinakar Ramakrishnan Department of Mathematics California Institute of Technology, Pasadena, CA 91125. dinakar@cco.caltech.edu