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# «To my teacher, Herv´ Jacquet e Introduction Let F be a number ﬁeld, and (ρ, V ) a continuous, n-dimensional representation of the absolute Galois ...»

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GO(4)-TYPE

## DINAKAR RAMAKRISHNAN

To my teacher, Herv´ Jacquet

e

Introduction

Let F be a number ﬁeld, and (ρ, V ) a continuous, n-dimensional representation

of the absolute Galois group Gal(F /F ) on a ﬁnite-dimensional C-vector space V.

Denote by L(s, ρ) the associated L-function, which is known to be meromorphic

with a functional equation. Artin’s conjecture predicts that L(s, ρ) is holomorphic everywhere except possibly at s = 1, where its order of pole is the multiplicity of the trivial representation in V. The modularity conjecture of Langlands for such representations ([La3]), often called the strong Artin conjecture, asserts that there should be an associated (isobaric) automorphic form π = π∞ ⊗πf on GL(n)/F such that L(s, ρ) = L(s, πf ). Since L(s, πf ) possesses the requisite properties ([JS]), the modularity conjecture implies the Artin conjecture.

For any ﬁeld k, let GO(n, k) denote the subgroup of GL(n, k) consisting of orthogonal similitudes, i.e., matrices g such that t gg = λg I, with λg ∈ k ∗.

We will say that a k-representation (ρ, V ) is of GO(n)-type iﬀ dim(V ) = n and it factors as σ ρ = [Gal(F /F ) −→ GO(n, k) ⊂ GL(V )].

In this article we prove Theorem A Let F be a number ﬁeld and let (ρ, V ) be a continuous, 4-dimensional C-representation of Gal(F /F ) whose image is solvable and lies in GO(4, C). Then ρ is modular, i.e., L(s, ρ) = L(s, πf ) for some isobaric automorphic representation π = π∞ ⊗ πf of GL(4, AF ). Moreover, π is cuspidal iﬀ ρ is irreducible.

Among the ﬁnite groups G showing up as the images of such ρ are those ﬁtting into an exact sequence 1 → C → H × H → G → {±1} → 1, where H is any ﬁnite solvable group in GL(2, C) with scalar subgroup C, the embe- ding of C in H × H is given by x → (x, x−1 ), and the action of {±1} on (H × H)/C is induced by the permutation of the two factors of H × H. This is due to the well known fact (see section 1) that GO(4, C) contains a subgroup of index 2 which is a quotient of GL(2, C)× GL(2, C) by C∗. Of particular interest are the examples where H is a central extension of S4 or A4 (cf. section 8).

PARTIALLY SUPPORTED BY THE NSF GRANT DMS-9801328 1 2 Dinakar Ramakrishnan One can ask if this helps furnish new examples of Artin’s conjecture, and the answer is yes.

Corollary B Let F be a number ﬁeld, and let ρ, ρ be continuous C-representations of Gal(F /F ) of solvable GO(4)-type. Then Artin’s conjecture holds for ρ ⊗ ρ.

We will show in section 8 that in fact there is, for each F, a doubly inﬁnite family of such examples where the representations ρ ⊗ ρ are irreducible and primitive (i.e., not induced) of dimension 16. Primitivity is important because Artin Lfunctions are inductive, and one wants to make sure that these examples do not come by induction from known (solvable) cases in low dimensions. We will then show (in secton 9) that, given any ρ as in Theorem A with corresponding extension K/F, the strong Dedekind conjecture holds for certain non-normal extensions N/F contained in K/F, for instance when [N : F ] = 3a, a ≥ 1 (see Theorem 9.3); the assertion is that the ratio ζN (s)/ζF (s) is the standard L-function of an isobaric automorphic form η on GL(3a − 1)/F. It implies that for any cusp form π on GL(n)/F, the formally deﬁned Euler product L(s, πN ) (see the discussion before Corollary 9.4) admits a meromorphic continuation and functional equation, and more importantly, it is divisible by L(s, π). It should be noted that we do not know if πN is automorphic. For a curious consequence of this result see Remark 9.6.

It has been known for a long time, thanks to the results of Artin and Hecke, that monomial representations of Gal(F /F ), i.e., those induced by one-dimensional representations of Gal(F /K) with K/F ﬁnite, satisfy Artin’s conjecture. (In fact this holds for any multiple of a monomial representation.) But the strong Artin conjecture is still open for these except when K/F is normal and solvable ([AC]) and when [K : Q] = 3 ([JPSS1]. The work [AC] of Arthur and Clozel implies that the strong conjecture holds for ρ with nilpotent image, in fact whenever ρ is accessible ([C]), i.e., a positive integral linear combination of representations induced from linear characters of open, subnormal subgroups. It should however be noted that Dade has shown ([Da]) that all accessible representations of solvable groups are monomial.

The odd dimensional orthogonal representations with solvable image are simpler than the even dimensional ones. Indeed we have Proposition C Let ρ be a continuous, irreducible, solvable C-representation of Gal(F /F ) of GO(n)-type with n odd. Then ρ is monomial and hence satisﬁes the Artin conjecture. If ρ is in addition self-dual, it must be induced by a quadratic character.

By the groundbreaking work of Langlands in the seventies ([La1]), and the ensuing theorem of Tunnell in 1980 ([Tu]), one knows that the strong Artin conjecture holds for all two-dimensional representations σ with solvable image. The raison d’etre for this article is the desire to ﬁnd irreducible, solvable, but non-monomial, even primitive, examples in higher dimensions satisfying the Artin cojecture. We were led to this problem a few years ago by a remark of J.-P.Serre. One could look at the symmetric powers of a two-dimensional solvable σ, but they are all in the linear span of monomial representations and 1-dimensional twists of σ. The facts that the symmetric square of σ is modular (by Gelbart-Jacquet [GeJ]) and monomial (for σ solvable) were used crucially in [La1] and [Tu].

Solvable Artin representations of GO(4)-type 3 In the non-solvable direction, which is orthogonal (no pun intended) to the one pursued in this paper, there has been some spectacular progress recently. For odd 2-dimensionals ρ of Gal(Q/Q) with projectivization ρ of A5 -type, a theorem of Buzzard, Dickinson, Shepherd-Barron and Taylor ([BDST]) establishes the modularity conjecture assuming the following: (i) ρ is unramiﬁed at 2 and 5, and (ii) ρ(Frob2 ) has order 3. In a sequel ([T]), this is shown with diﬀerent ramiﬁcation conditions, namely that under ρ (i’) the inertia group at 3 has odd order, and (ii’) the decomposition group at 5 is unramiﬁed at 2. See also [BS] for some explicit examples.

Some positive examples were given earlier in [Bu], and then in [Fr]. Moreover, the very recent theorems of Kim and Shahidi ([KSh]), and Kim ([K]), establishing the automorphy of the symmetric cube, and the symmetric fourth power, on GL(2), establishes the strong Artin conjecture for sym3 (σ), and sym4 (σ), for all the σ proved modular by [BDST].

If σ, σ are Galois representations which are modular, then one can deduce the Artin conjecture for σ⊗σ by applying the Rankin-Selberg theory on GL(m)×GL(n) developed in the works of Jacquet, Piatetski-Shapiro, Shalika ([JPSS2], [JS]) and of Shahidi ([Sh1,2]); see also [MW]. This explains why the strong form of Artin’s conjecture is really a bit stronger than the original conjecture that Artin made, at least given what one knows today.

If σ, σ are 2-dimensional representations which are modular, then the strong Artin conjecture for σ ⊗ σ follows from the main theorem of [Ra1], hence the Artin conjecture holds, by the remark above, for 4-fold tensor products of such representations. Now let K/F be a quadratic extension with non-trivial automorphism θ, and let σ θ denote the θ-twisted representation deﬁned by x → σ(θxθ−1 ), where θ is ˜˜ ˜ any lift to Gal(F /F ). (The equivalence class of σ θ is independent of the choice of θ.) ˜ Given any irreducible, non-monomial 2-dimensional representation σ of Gal(F /K) which is not isomorphic to any one-dimensional twist of σ θ (see section 3), there is an irreducible 4-dimensional representation As(σ) of Gal(F /F ) whose restriction to Gal(F /K) is isomorphic to σ ⊗ σ θ. When σ is of solvable type, one can combine the theorem of Langlands-Tunnell with that of Asai ([HLR]) to deduce the Artin conjecture for As(σ).

One of the main steps in our proof of Theorem A is that even modularity can be established for any Asai representation As(σ) (in the solvable case). To begin, there exists, by Langlands-Tunnell, a cusp form π on GL(2)/K such that L(s, σ) = L(s, π). It follows that L(s, σ ⊗ σ θ ) equals the Rankin-Selberg L-function L(s, π × (π ◦ θ)). By [Ra1], there exists an automorphic form π (π ◦ θ) on GL(4)/K such that L(s, σ ⊗ σ θ ) = L(s, π (π ◦ θ)). Since π (π ◦ θ) is θ-invariant, one can now deduce the existence of a cusp form Π on GL(4)/F whose base change to K is π (π ◦ θ), but even when π (π ◦ θ) is cuspidal, Π is unique only up to twisting by the quadratic character δ, say, of the idele class group of F corresponding to K/F by class ﬁeld theory. But it is not at all clear that Π, or Π ⊗ δ, should correspond precisely to As(σ), with an identity of the corresponding L-functions. (It is easy to see that the local factors agree at half the places.) Put another way, one can construct an irreducible admissible representation As(π) of GL(4, AF ) which has the same local factors as does As(σ). But the problem is that there is no simple reason why As(π) should be automorphic, even when π is dihedral. (Recall that π is dihedral, or of CM type, iﬀ it is associated to a representation σ of the global Weil group WK induced by a character χ of a quadratic extension M of K, which 4 Dinakar Ramakrishnan

–  –  –

Then we show, still in section 3, how to deduce Theorem A modulo Theorem D;

in fact we need Theorem D only in the (crucial) case when (iii) occurs. When (i) or (ii) occurs, the modularity already follows from Theorem M of [Ra1] and base change ([AC]), with the desired Π being in case (i) (resp. (ii)) the Rankin-Selberg F product π(τ ) ⊗ π(τ ) (resp. the automorphic induction IL (π(η))); here τ → π(τ ) the Langlands-Tunnell map on solvable 2-dimensional Galois representations.

The proof of Theorem D is accomplished in sections 4 through 7. The approach

is similar to, but somewhat more subtle than, the proof of the existence of :

A(GL(2)/F )×A(GL(2)/F ) → A(GL(4)/F ) in [Ra1]. (For any n ≥ 1, A(GL(n)/F ) denotes the set of isomorphism classes of isobaric automorphic representations of GL(n, AF ).) In section 4 we show why it suﬃces to have the requisite properties at almost all places. Then in section 5 the distinguished case, i.e., when π ◦ θ is an abelian twist of π, is treated separately. In the general situation, i.e., when π (π ◦ θ) is cuspidal, we crucially use the converse theorem for GL(4) due to Cogdell and Piatetski-Shapiro ([CoPS1], which requires knowledge of the niceness of the twisted L-functions L(s, As(π) × π ) for automorphic forms π of GL(2)/F for a suitable class of π. Many properties of certain closely related functions, to be denoted L1 (s, As(π) × π ), were established by Piatetski-Shapiro and Rallis ([PSR]), and by Ikeda ([Ik1,2]), via an integral representation, which we use. There is another possible approach to studying L(s, As(π) × π ) via the Langlands-Shahidi method [Sh1], which yields another family of closely related L-functions, denoted L2 (s, As(π) × π ), with boundedness properties established recently in [GeSh], but we will not use it and our argument here is hewed to make use of the integral representation.

For almost all ﬁnite places v, the local factors L(s, As(πv )×πv ) and L1 (s, As(πv )× πv ) agree. But a thorny problem arises however, due to our inability to identify the bad and archimedean factors. In fact, when Fv = R, one does not even have a computation of the corresponding L1 -factor when π, π are unramiﬁed. Luckily, things simplify quite a bit under suitable, solvable base changes K/F with K totally complex, and after constructing the base-changed candidates As(π)K for an inﬁnitude of such K, we descend to F as in sections 3.6, 3.7 of [Ra1]. We also have to control the intersection of the ramiﬁcation loci of As(π), π and K/F.

One of the reasons why we have to work with L(s, As(π) × π ) is that we know how its local ε-factors behave under twisting by a highly ramiﬁed character, and this is not apriori the case with Lj (s, As(π) × π ) for j = 1 or 2. Indeed this will present a diﬃculty for the lifting of (generic) automorphic forms from GO(2n) to GL(2n), and for large n one cannot, as of yet, make use of base change and descent as we do here. For the lifting from GO(2n + 1) to GL(2n), see [CoKPSSh].

We would like to thank R.P. Langlands and the Institute for Advanced Study, Princeton, for their hospitality during the year 1999-2000, and the American Institute of Mathematics, Palo Alto, for support during the month of August in

1999. This project was partially funded by the NSF and AMIAS. We also want to acknowledge with thanks some helpful conversations with Michael Aschbacher and David Wales concerning the representations of ﬁnite groups. The main result here was presented at a conference on Automorphic Forms in Luminy, France (May 1999), and then at the IAS, Princeton (April 2000), and ICTP, Trieste, Italy (August 2000), where we received some helpful feedback. We would like to thank a number of people, including Jim Arthur, Don Blasius, Kevin Buzzard, Jim Cogdell, 6 Dinakar Ramakrishnan Bill Duke, Farshid Hajir, Henry Kim, Jeﬀ Lagarias, Nick Katz, Dipendra Prasad, A.Raghuram, Peter Sarnak, Jean-Pierre Serre, Freydoon Shahidi and Joe Shalika who have shown interest and/or made comments.

–  –  –

Here we collect some basic facts, which we will need.

Let k be a ﬁeld of characteristic diﬀerent from 2, with separable algebraic closure k. If V is a ﬁnite dimensional vector space with a non-degenerate, symmetric bilinear form B, the associated orthogonal similitude group is (1.1) GO(V, B) := {g ∈ GL(V ) | B(gv, gw) = λ(g)B(v, w), with λ(g) ∈ k ∗, ∀v, w ∈ V }.

The character λ : GO(V, B) → k ∗, g → λ(g), is the similitude factor. The kernel of λ is the orthogonal group O(V, B), whose elements necessarily have determinant ±1, and the kernel of det is the special orthogonal group SO(V, B).

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