Holomorphy of the 9th Symmetric Power L-Functions for Re(s) >1. Henry H. Kim and Freydoon Shahidi

Transcription

1 IMRN International Mathematics Research Notices Volume 2006, Article ID 59326, Pages 1 7 Holomorphy of the 9th Symmetric Power L-Functions for Res >1 Henry H. Kim and Freydoon Shahidi We proe the holomorphy of the 9th symmetric power L-functions for arbitrary cusp forms on GL2 oer a number field for Res >1which was not aailable earlier in the case of Maass forms. This complements our earlier results for inertibility and in fact absolute conergence of the twisted symmetric power L-functions of these forms up to degree 8 oer the same interal. Let F be a number field and denote by A its ring of adeles. Let π = π be a cuspidal automorphic representation of GL 2 A. We normalize π so that ω π R + 1, where ω π is the central character of π and A = A 1 R +. One of the consequences of the existence of Sym 3 π and Sym 4 π as automorphic representations of GL 4 A and GL 5 A, respectiely, recently proed in [2, 7], is a proof of certain analytic properties of Ls, π, Sym m ρ 2 for m 9 cf. [6, 8]. While we succeeded in proing that when m 8, each of these symmetric power L-functions are inertible and een absolutely conergent for Res >1cf. [6, 13], no such results were obtained for Ls, π, Sym 9 ρ 2. In fact, all that we knew een though it is not stated explicitly in [6] was that L S s, π, Sym 9 ρ 2 is holomorphic and nonanishing for Res 1, except possibly for finitely many poles and zeros on the real axis 1 s 2. On the other hand, Gelbart and Lapid [1] hae recently established zero free regions for each of the L-functions obtained from Langlands-Shahidi method to the left Receied 13 January 2006 Reised 14 February 2006 Accepted 14 June 2006 Communicated by James W. Cogdell

2 2 H. H. Kim and F. Shahidi of the line Res = 1. This, in particular, shows the existence of such regions for L S s, π, Sym 9 ρ 2. One then expects this L-function to be nonanishing and holomorphic for Res 1, except for a possible pole at s = 1. While we still cannot proe the nonanishing for 1 s 2, in this paper we proe the holomorphy of Ls, π, Sym 9 ρ 2 for 1<s 2. The reader must note that this result is much stronger than the general results obtained from the theory of Eisenstein series cf. [10 12], where one cannot rule out the possibility of a finite number of poles on the real axis. On the other hand, our method fails when it comes to nonanishing for Res >1. In fact, using our method, anonanishing result for Res >1requires ruling out the poles for the corresponding Eisenstein series for Res >0cf. [10, 11] which presently is not possible een by means of unitary duals cf. [5]. One can of course use the fact that the Eisenstein series in this case is holomorphic for Res >1as we establish here, but this only proes the nonanishing for Res >2which can be concluded from the absolute conergence of the L-function for Res >2which was proed in the full generality of our method in [11, 12]. The proof relies upon a criterion proed in [8, Proposition 4.1] by the authors of this paper. We refer to Lemma 2 here for a reference to that. We should point out that the fact that one can find such regions to the left of Res = 1 without the knowledge of similar results to the right of this line as in [1], must be of no surprise. In fact, this phenomenon was already noticed in [10], where the nonanishing for a product of these L-functions at s = 1 was proed while similar results to the right of Res = 1 are still not aailable in general. They are both consequences of the main characteristic of Eisenstein series in determining the continuous spectrum. This paper was written to answer the question posed by Erez Lapid that, whether in iew of the results in [1] discussed earlier, one can show the holomorphy of Ls, π, Sym 9 ρ 2 for Res >1. We like to thank him for that. We will now state and proe our theorem. In it S denotes a set of places of F including the archimedean ones such that π is spherical for eery S. Each π with S is parametrized by a diagonal element diagα,β GL 2 C. For each positie integer m, let L S s, π, Sym m ρ 2 = m S j=0 1 α j β m j q s 1. 1 If m 9, we can define a local factor Ls, π, Sym m ρ 2 for S inductiely cf. [12]. For m 8, we use the Rankin-Selberg L-functions see [6]. For example, we define

3 9th Symmetric Power L-Functions 3 Ls, π, Sym 5 ρ 2 using the relation L s, Sym 2 π Sym 3 π s, π, Sym 5 ρ 2 L s, Sym 3 π ωπ L s, π ω 2 π. 2 Similarly, Ls, π, Sym 8 ρ 2 comes from Ls, Sym 4 π Sym 4 π. Form = 9, we use the relation 6 below. Hence if m 9, we can define the completed L-functions Ls, π, Sym m ρ 2. We refer to [6, 8] for any unexplained notation. If Sym 4 π is not cuspidal, π is of Galois type, and we know that Sym m π is an automorphic representation of GL m+1 A for all m see [3]. Hence Ls, π, Sym m ρ 2 is holomorphic for all s C, except possibly at s = 1. It is also nonanishing for Res 1. So we assume that Sym 4 π is cuspidal. Theorem 1. a Let π be a cuspidal representation of GL 2 A such that Sym 4 π is cuspidal. Then the 9th symmetric power L-function Ls, π, Sym 9 ρ 2 is holomorphic for Res 1, except possibly for a simple pole at s = 1. It is nonanishing for Res 1, except possibly for finitely many simple zeros on the real axis 1 s 2. b Both statements are alid for L S s, π, Sym 9 ρ 2. Proof. Let Sym 3 π and Sym 4 π be the symmetric cube and fourth of π which are automorphic representations of GL 4 A and GL 5 A, respectiely. Let A 3 π = Sym 3 π ω 1 π, where ω π is the central character of π. Let us recall how we obtained L S s, π, Sym 9 ρ 2 in [6]. Consider the case E 8 2 of [11] cf. [4]. LetM be a Lei subgroup of P, the maximal parabolic subgroup obtained by deleting α 5, and denote by A its center which is connected since G is adjoint. Since G = E 8 is also simply connected, the deried group M D of M is simply connected as well, and hence M D = SL 4 SL 5. Thus M = GL 1 SL 4 SL 5 / A MD. 3 As in [2, 4, 7] there is a natural map φ : M GL 4 GL 5.Gieni = 1, 2, let π i be cuspidal representations of GL 4 A and GL 5 A with central characters ω i, i = 1, 2, respectiely. Let Σ be a cuspidal representation of MA, induced by φ and π 1,π 2.Then the central character of Σ is ω Σ = ω 5 1 ω8 2. We then get the L-function LS s, π 1 π 2,ρ 4 2 ρ 5 as our first L-function for the triple G, M,Σ cf. [11, 12]. In fact, there are fie

4 4 H. H. Kim and F. Shahidi L-functions in the constant term of the corresponding Eisenstein series namely L S s, Σ, r 1 S s, π 1 π 2,ρ 4 2 ρ 5 L S s, Σ, r 2 S s, π 1 π 2 ω 2, 2 ρ 4 ρ 5 L S s, Σ, r 3 S s, π 1 π 2 ω 1 ω 2 4 L S s, Σ, r 4 S s, π 2, 2 ρ 5 ω 1 ω 2 2 L S s, Σ, r 5 S s, π 1 ω 1 ω 2 2. More precisely, let Ms, Σ be the main part of the constant term for the Eisenstein series attached to G, M,Σ.Then M S s, Σ = 5 i=1 L S is, Σ, r i L S 1 + is, Σ, r i S A s, Σ, 5 where M S s, Σ is the restriction of Ms, Σ to functions in the induced space Is, Σ of the form f = f S S f 0, where f S S Is, Σ, while each f 0 is the GO -inariant spherical function in Is, Σ whose alue at e equals to f 0 e = x, S. Here the ectors {x HΣ S} are the fixed MO -inariant ectors which were used to decompose Σ to Σ = Σ. The operators As, Σ are the standard local intertwining operators originating from Is, Σ. We apply the aboe to π 1 = A 3 π and π 2 = Sym 4 π. By standard calculations, we hae L S s, π 1 π 2,ρ 4 2 ρ 5 S s, π, Sym 9 ρ 2 L S s, π, Sym 7 ρ 2 ω π L S s, π, Sym 5 ρ 2 ω 2 2 π 6 L S s, Sym 3 π ω 3 2L S π s, π ω 4 π. Here L S s, π, Sym 7 ρ 2 ω π L S s, π, Sym 5 ρ 2 ω 2 π 2 L S s, Sym 3 π ω 3 π 2 L S s, π ω 4 π is inertible for Res > 1 by [6, Propositions 4.2 and 4.5]. Soitisenoughtoproethat L S s, Σ, r 1 is holomorphic for Res >1. To proceed we need the following. Lemma 2. Suppose Σ is spherical and tempered. Then for each s with Res > 1, the induced representation Is, Σ is irreducible, and hence it is not unitary. Proof. By [8, Proposition 4.1], we only need to proe that 5 i=1 L1 is, Σ,r i is holomorphic for Res >1. Here Ls, Σ,r i = j 1 α jq s 1, and since Σ is tempered, α j = 1 for all j. Our result follows.

5 9th Symmetric Power L-Functions 5 Ramakrishnan has proed in [9] that π = π has infinitely many tempered spherical components. In fact, he showed that the lower Dirichlet density of the set of places, where π is tempered, is greater than 9/10. In [6], we improed this to 34/35. Choose a π which is spherical and tempered. Then Σ is spherical and tempered. Since Is, Σ is not unitary for Res >1, M S s, π is holomorphic at least up to Res >1. In fact, if M S s, Σ has a pole at a point s with Res >1, its residue will identify a quotient of Is, Σ as a constituent of the space of automorphic forms. This implies that the corresponding quotient of Is, Σ is unitary for Res >1which is a contradiction. It then follows from 5 that 5 L S is, Σ, r i i=1 L S 1 + is, Σ, r i 7 is holomorphic for Res >1since each local intertwining operator As, Σ is nonanishing. Since L S s, Σ, r i is inertible for Res>2cf. [11], we conclude that L S s, Σ, r 1 is holomorphic for Res >1. For S, let Ls, Σ, r i be the local L-factors defined in [12], i = 1,...,5.We showed in [4, Proposition 4.9] that the local L-factors Ls, Σ, r i are holomorphic for Res 1. Hence the completed L-function Ls, Σ, r 1 is holomorphic for Res >1.This proes that the completed L-function Ls, π, Sym 9 ρ 2 is holomorphic for Res >1. All other assertions are implicit in the proof of [6, Proposition 4.7]. For the sake of completeness, we state it here explicitly. Due to the normalization of π, the poles of the Eisenstein series for Res >0are on the real axis. So the poles of 5 i=1 LS is, Σ, r i / L S 1 + is, Σ, r i for Res >0are all real. Howeer, we showed in the proof of [6, Proposition 4.7] that for each i = 2, 3, 4, 5, L S s, Σ, r i is inertible for Res 1. Hence the poles of L S s, Σ, r 1 /L S 1 + s, Σ, r 1 for Res 1/2 are all on the real axis. Inductiely, this implies that the poles of L S s, Σ, r 1 for Res 1/2 are on the real axis. Since L S s, π, Sym 7 ρ 2 ω π L S s, π, Sym 5 ρ 2 ω 2 π 2 L S s, Sym 3 π ω 3 π 2 L S s, π ω 4 π is inertible for Res 1, 6 implies that the possible poles of L S s, π, Sym 9 ρ 2 for Res 1 are all real. Hence L S s, π, Sym 9 ρ 2 has no poles for Res 1 except possibly at s = 1. We proed in [8, Proposition 7.2] that L S s, π, Sym 9 ρ 2 has at most a simple pole or a simple zero at s = 1. The Eisenstein series is holomorphic for Res = 0, and the possible poles for Res >0are on the real axis, and they are simple. Therefore by considering the nonconstant term of the Eisenstein series, we conclude that the same is true for the possible zeros of 5 i=1 LS 1 + is, Σ, r i for Res >0. Consequently, since for each i = 2, 3, 4, 5, L S s, Σ, r i is inertible for Res 1, the possible zeros of L S s, Σ, r 1 for Res 1 are on

6 6 H. H. Kim and F. Shahidi the real axis, and they are simple. Since L S s, Σ, r 1 is absolutely conergent for Res > 2, it is nonanishing there. Using the fact that L S s, π, Sym 7 ρ 2 ω π L S s, π, Sym 5 ρ 2 ω 2 π 2 L S s, Sym 3 π ω 3 π 2 L S s, π ω 4 π is inertible for Res 1, from 6, we see that L S s, π, Sym 9 ρ 2 is nonanishing for Res 1, except possibly for finitely many simple zeros on the real axis with 1 s 2. This completes the proof of Theorem 1. Remark 3. We refer to [5] for a generalization of our argument and other results for 1/2 < Res <1. For the sake of completeness, we conclude by recording a result from [13] due to the authors of this paper, proing the absolute conergence of twisted symmetric power L-functions for m 8. We refer to [13] for some of its consequences. Let χ = χ be an idele class character of A and assume χ is unramified for all S.Let L S s, π, Sym m ρ 2 χ = m S j=0 1 α j β m j χ ϖ q s 1. 8 Theorem 4 cf. [13]. The partial L-functions L S s, π, Sym m ρ 2 χ are all absolutely conergent for Res > 1 for all m 8 and eery idele class character of A, unramified outside S. Acknowledgments The first author was partially supported by an NSERC grant, and the second author was partially supported by NSF Grant DMS References [1] S. Gelbart and E. Lapid, Lower bounds for L-functions at the edge of the critical strip, American Journal of Mathematics , no. 3, [2] H. H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourth of GL 2, Journal of the American Mathematical Society , no. 1, , with appendix 1 by D. Ramakrishnan and appendix 2 by H. H. Kim and P. Sarnak. [3], An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type, Inentiones Mathematicae , no. 3, [4], On local L-functions and normalized intertwining operators, Canadian Journal of Mathematics ,no. 3,

7 9th Symmetric Power L-Functions 7 [5], Langlands-Shahidi method and poles of automorphic L-functions III exceptional groups, preprint. [6] H. H. KimandF. Shahidi, Cuspidality of symmetric powers with applications, Duke Mathematical Journal , no. 1, [7], Functorial products for GL 2 GL 3 and the symmetric cube for GL 2, Annals of Mathematics. Second Series , no. 3, , with an appendix by C. J. Bushnell and G. Henniart. [8], On simplicity of poles of automorphic L-functions, Journal of the Ramanujan Mathematical Society , no. 4, [9] D. Ramakrishnan, On the coefficients of cusp forms, Mathematical Research Letters , no. 2-3, [10] F. Shahidi, On certain L-functions, American Journal of Mathematics , no. 2, [11], On the Ramanujan conjecture and finiteness of poles for certain L-functions, Annals of Mathematics. Second Series ,no. 3, [12], A proof of Langlands conjecture on Plancherel measures complementary series for p-adic groups, Annals of Mathematics. Second Series , no. 2, [13], Infinite dimensional groups and automorphic L-functions, Pure and Applied Mathematics Quarterly , no. 3, , special issue: in memory of Armand Borel, part 2 of 3. Henry H. Kim: Department of Mathematics, Uniersity of Toronto, Toronto, ON, Canada M5S 2E4 address: henrykim@math.toronto.edu Freydoon Shahidi: Department of Mathematics, Purdue Uniersity, West Lafayette, Indiana, IN 47907, USA address: shahidi@math.purdue.edu