Functoriality for the exterior square of and the symmetric fourth of

By Henry H. Kim, with an appendix by Dinakar Ramakrishnan and with an appendix co-authored by Peter Sarnak

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 -functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of and the absolute convergence of the exterior square -functions of . Second, we prove that the fourth symmetric power -functions of cuspidal representations of 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 -adic places and of Langlands Reference 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)

Theorem A.

Let be the set of places where and is a supercuspidal representation. Then there exists an automorphic representation of such that if . Moreover, is of the form , where the тАЩs are all cuspidal representations of .

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 th power (the -dimensional irreducible representation of 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)

Theorem B.

is an automorphic representation of . If is cuspidal, is either cuspidal or induced from cuspidal representations of and .

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 -functions of the symmetric fourth (cf. Reference 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 -functions of cuspidal representations of . 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 -functions which we need for the converse theorem, namely, , where is a cuspidal representation of , , and is a cuspidal representation of . The automorphic -functions appear in the constant term of the Eisenstein series coming from the split spin group (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 -function defined by either integral representations Reference J-PS-S or the Langlands-Shahidi method. They are the same, and they are an Artin -function due to the local Langlands correspondence. However, the -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 -functions which are defined by completely different methods is not obvious at all. The same is true for -factors. Indeed, a priori we do not know the equality when 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 -factors as in Reference 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 тАЩs are (unitary) cuspidal representations of 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 -function and obtain our assertion that the local lift we constructed is equivalent to the one given by the local Langlands correspondence. (If , 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 -function and obtain our assertion.

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 -factors for supercuspidal representations directly (see Remark 7.2 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.

Theorem 2.1 (Reference Co-PS1).

Suppose is an irreducible admissible representation of such that is a gr├╢ssencharacter of . Let be a finite set of finite places, and let be a set of cuspidal representations of that are unramified at all places . Suppose is nice (i.e., entire, bounded in vertical strips and satisfies a functional equation) for all cuspidal representations , . Then there exists an automorphic representation of such that for all .

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 -factors imply the equality of -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 -functions in the next section; (2) 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 -functions on the left and on the right are defined in completely different manners. The right-hand side is the Rankin-Selberg -function Reference J-PS-S defined by either integral representations or the Langlands-Shahidi method, which in turn is an Artin -function due to the local Langlands correspondence. We note that if 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 -functions and -factors on the left are those of Artin factors. It is clearly true if is unramified. Shahidi has shown that (1) is true when Reference Sh7.

Remark 2.1.

Eventually we are going to prove in Section 5 that on the right side of (1) is generic in our case. However, is not generic in general. For example, if is given by the principal series