# Functoriality for the exterior square of and the symmetric fourth of

## Abstract

In this paper we prove the functoriality of the exterior square of cusp forms on as automorphic forms on and the symmetric fourth of cusp forms on as automorphic forms on We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain . obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of -functions and the absolute convergence of the exterior square of -functions Second, we prove that the fourth symmetric power . of cuspidal representations of -functions are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound for Hecke eigenvalues of Maass forms over any number field.

## 1. Introduction

Let where , be the map given by the exterior square. Then LanglandsтАЩ functoriality predicts that there is a map from cuspidal representations of , to automorphic representations of which satisfies certain canonical properties. To explain, let , be a number field, and let be its ring of adeles. Let be a cuspidal (automorphic) representation of In what follows, a cuspidal representation always means a unitary one. Now by the local Langlands correspondence, . is well defined as an irreducible admissible representation of for all (the work of Harris-Taylor Reference H-T and Henniart Reference He2 on places and of Langlands -adicReference La4 on archimedean places). Let It is an irreducible admissible representation of . Then LanglandsтАЩ functoriality in this case is equivalent to the fact that . is automorphic.

Note that and in fact for a cuspidal representation of , the central character of , Furthermore, . In this case, given a cuspidal representation . of , where , is the contragredient of .

In this paper, we look at the case Let . be a cuspidal representation of What we prove is weaker than the automorphy of . We prove (Theorem .5.3.1)

The reason why we have the exceptional places especially for , is due to the fact that supercuspidal representations of , are very complicated when We use the Langlands-Shahidi method and a converse theorem of Cogdell-Piatetski-Shapiro to prove the above theorem (cf. .Reference Co-PS1, Reference Ki-Sh2). We expect many applications of this result. Among them, we mention two: First, we prove the weak Ramanujan property of cuspidal representations of (Proposition 6.3; see Definition 3.6 for the notation).

Second, we prove the existence of the symmetric fourth lift of a cuspidal representation of as an automorphic representation of More precisely, let . be the symmetric power (the th irreducible representation of -dimensional on symmetric tensors of rank Let ). be a cuspidal representation of with central character By the local Langlands correspondence, . is well defined for all Hence LanglandsтАЩ functoriality predicts that . is an automorphic representation of Gelbart and Jacquet .Reference Ge-J proved that is an automorphic representation of We proved in .Reference Ki-Sh2 that is an automorphic representation of as a consequence of the functorial product corresponding to the tensor product map , .

We prove (Theorem 7.3.2)

Here we stress that there is no restriction on the places as opposed to the case of the exterior square lift.

Theorem B is obtained by applying Theorem A to For simplicity, we write . We prove that .

This implies that is an automorphic representation of and so is , .

An immediate corollary is that we have a new estimate for Ramanujan and SelbergтАЩs conjectures using Reference Lu-R-Sa. Namely, let be a cuspidal representation of Let . be a local (finite or infinite) spherical component, given by Then . If and this condition implies that , where , is the first positive eigenvalue for the Laplace operator on the corresponding hyperbolic space.

In a joint work with Sarnak in Appendix 2 Reference Ki-Sa, by considering the twisted symmetric square of the symmetric fourth (cf. -functionsReference BDHI), we improve the bound further, at least over namely, , As for the first positive eigenvalue for the Laplacian, we have

In Reference Ki-Sh3, we determine exactly when is cuspidal. We show that is not cuspidal and is cuspidal if and only if there exists a non-trivial quadratic character such that or equivalently, there exists a non-trivial gr├╢ssencharacter , of such that where , is the quadratic extension, determined by We refer to that paper for many applications of symmetric cube and symmetric fourth: The analytic continuation and functional equations are proved for the 5th, 6th, 7th, 8th and 9th symmetric power . of cuspidal representations of -functions It has immediate application for Ramanujan and SelbergтАЩs bounds and the Sato-Tate conjecture: Let . be an unramified local component of a cuspidal representation Then it is shown that . where the Hecke conjugacy class of , is given by Furthermore, if . then for every , there are sets , and of positive lower (Dirichlet) density such that for all and for all .

In Reference Ki5, we give an example of automorphic induction for a non-normal quintic extension whose Galois closure is not solvable. In fact, the Galois group is the alternating group on five letters. The key observation, due to Ramakrishnan is that the symmetric fourth of the 2-dimensional icosahedral representation is equivalent to the 5-dimensional monomial representation of , (see Reference Bu). It should be noted that the only complete result for non-normal automorphic induction before this is for non-normal cubic extension due to Reference J-PS-S2 as a consequence of the converse theorem for .

We now explain the content of this paper. In Section 2, we recall a converse theorem of Cogdell and Piatetski-Shapiro and the definition of weak lift and strong lift. In Section 3, we study the analytic properties of the automorphic which we need for the converse theorem, namely, -functions where , is a cuspidal representation of , and , is a cuspidal representation of The automorphic . appear in the constant term of the Eisenstein series coming from the split spin group -functions (the case in Reference Sh3). Hence we can apply the Langlands-Shahidi method Reference Ki1, Reference Ki2, Reference Ki-Sh2, Reference Sh1тАУReference Sh3.

In Section 4, we first obtain a weak exterior square lift by applying the converse theorem to with , being a finite set of finite places, where is unramified for and In this case, the situation is simpler because if . as in the statement of the converse theorem, one of or is in the principal series for Here one has to note the following: In the converse theorem, the . -function is the Rankin-Selberg defined by either integral representations -functionReference J-PS-S or the Langlands-Shahidi method. They are the same, and they are an Artin due to the local Langlands correspondence. However, the -function -function is defined by the Langlands-Shahidi method Reference Sh1 as a normalizing factor of intertwining operators which appear in the constant term of the Eisenstein series. The equality of two which are defined by completely different methods is not obvious at all. The same is true for -functions Indeed, a priori we do not know the equality when -factors. is a supercuspidal representation, even if is a character of Hence we need to proceed in two steps as in .Reference Ra1, namely, first, we do the good case when none of is supercuspidal, and then we do the general case, following RamakrishnanтАЩs idea of descent Reference Ra1. It is based on the observation of Henniart Reference He1 that a supercuspidal representation of becomes a principal series after a solvable base change. Here one needs an extension of Proposition 3.6.1 of Reference Ra1 to isobaric automorphic representations (from cuspidal automorphic representations). Appendix 1 provides the extension. We may avoid using the descent method, hence Appendix 1 altogether, by using the stability of as in -factorsReference CKPSS (see Remark 4.1 for more detail). We hope to pursue this in the future. Indeed, for the special case of the functoriality of hence the symmetric fourth of , we do not need it. (See Remark ,7.2.)

The converse theorem only provides a weak lift which is equivalent to a subquotient of where the , are (unitary) cuspidal representations of тАЩs and If . satisfies the weak Ramanujan property, it immediately implies In general, we show that . by comparing the Hecke conjugacy classes of and .

In Section 5.1, we give a new proof of the existence of the functorial product corresponding to the tensor product map It is originally due to Ramakrishnan .Reference Ra1. However, we give a proof, based entirely on the Langlands-Shahidi method. As a corollary, we obtain the Gelbart-Jacquet lift Reference Ge-J as an automorphic representation of for a cuspidal representation of by showing that .

In Section 5.2, we construct all local lifts in the sense of Definition 2.2 and show that unless and is a supercuspidal representation, is in fact the one given by the local Langlands correspondence ,Reference H-T, Reference He2. Here is how it is done: Note that if any supercuspidal representation of , is induced, i.e., corresponds to where , is an extension of degree 4 (not necessarily Galois) and is a character of (This is the so-called tame case. See, for example, .Reference H, p. 179 for references.) Also thanks to HarrisтАЩ work Reference H, we have automorphic induction for non-Galois extensions. Namely, there exists a cuspidal representation which corresponds to where , , and , is a gr├╢ssencharacter of such that Likewise, if . any supercuspidal representation , of , is induced. We embed , as a local component of a cuspidal representation using automorphic induction. We can compare the functional equations of and the corresponding Artin and obtain our assertion that the local lift we constructed is equivalent to the one given by the local Langlands correspondence. (If -function we need to twist by supercuspidal representations of , where there can be supercuspidal representations which are not induced. The global Langlands correspondence is not available for them.) ,

In Section 5.3, by applying the converse theorem twice to with , where , are any finite places, we prove that is an automorphic representation of .

In Section 7, we prove that if is a cuspidal representation of then , is an automorphic representation of Here we need to be careful because of the exceptional places . in the discussion of the exterior square lift. We first prove that there exists an automorphic representation of such that if Next we show that this is true for . If . any supercuspidal representation of , is monomial, and hence it can be embedded into a monomial cuspidal representation of If . any extraordinary supercuspidal representation of , is of tetrahedral type or octahedral type (see Reference G-L, p. 121). Hence in this case, the global Langlands correspondence is available Reference La3, Reference Tu. We can compare the functional equations of and the corresponding Artin and obtain our assertion. -function

Finally, we emphasize that for the functoriality of we do not need the full functoriality of the exterior square of , first of all, one does not need the comparison of Hecke conjugacy classes in Section ;4.1, since satisfies the weak Ramanujan property. Secondly, one does not need the method of base change and RamakrishnanтАЩs descent argument (hence Appendix 1), because we can prove the equality of for supercuspidal representations directly (see Remark -factors7.2 for the details).

## 2. Converse theorem

Throughout this paper, let be a number field, and let be the ring of adeles. We fix an additive character of Let . be the standard representation of .

First recall a converse theorem from Reference Co-PS1.

Let be a cuspidal representation of In order to apply the converse theorem, we need to do the following: .

- (1)
For all find an irreducible representation , of such that

for all where , , .

- (2)
Prove the analytic continuation and functional equation of the -functions .

- (3)
Prove that is entire for , .

- (4)
Prove that is bounded in vertical strips for , .

Recall the equalities:

Hence the equalities of and imply the equality of -factors -factors.

The -function and the -factor are available from the Langlands-Shahidi method, by considering the split spin group with the maximal Levi subgroup whose derived group is We will study the analytic properties of the . in the next section; ( -functions2) is well known by ShahidiтАЩs work Reference Sh3; (4) is the result of Reference Ge-Sh. We will especially study (3); in general, the -functions may not be entire. Our key idea is to apply the converse theorem to the twisting set where , is highly ramified for Then for . the , -function is entire. Observe that Hence applying the converse theorem with the twisting set . is equivalent to applying the converse theorem for with the twisting set (see Reference Co-PS2).

We will address problem (1) in Section 4. We have a natural candidate for namely, , the one given by the local Langlands correspondence (see Section ,4 for the detail). However, proving the equalities in (1) is not so obvious due to the fact that two on the left and on the right are defined in completely different manners. The right-hand side is the Rankin-Selberg -functions -functionReference J-PS-S defined by either integral representations or the Langlands-Shahidi method, which in turn is an Artin due to the local Langlands correspondence. We note that if -function is not generic, then we write as a Langlands quotient of an induced representation which is generic, and we define the , and - -factors and .

The left-hand side is defined in the Langlands-Shahidi method Reference Sh1 as a normalizing factor of intertwining operators which appear in the constant term of the Eisenstein series. Proving (1) is equivalent to the fact that ShahidiтАЩs and -functions on the left are those of Artin factors. It is clearly true if -factors is unramified. Shahidi has shown that (1) is true when Reference Sh7.