1 TWISTED SYMMETRIC-SQUARE L-FUNCTIONS AND THE NONEXISTENCE OF SIEGEL ZEROS ON GL3) William D. Banks 1. Introduction. In a lecture gien at the Workshop on Automorphic Forms at the MSRI in October 1994, D. Goldfeld emphasized the importance of establishing the conjecture that the standard L-function of a cuspidal automorphic representation π on GLn) does not, when n > 1, admit a Siegel zero, i.e., a real zero of Ls, π) that lies close to s = 1. This is also conjectured for n = 1, although the problem is much deeper in that case. For n = 2, the Siegel zero conjecture was proed unconditionally by Hoffstein and Ramakrishnan [5]. They were also able to establish the result when n = 3 under the assumption of a certain hypothesis Hypothesis 6.1 of [5] ). The main result of this paper is the proof of that hypothesis, which thus establishes the Siegel zero conjecture unconditionally for n = 3. Let F be an algebraic number field, A its ring of adeles, and let π be a unitary cuspidal automorphic representation of GL3, A). In this paper, we will proe the following: Theorem 1. Ls, π) does not admit a Siegel zero. Theorem 2. Let ω be the central character of π, and let χ be a Hecke character. Then the partial Langlands L-function L S s, π, 2 χ) defined below) extends to a meromorphic function of s C, with possible simple poles occurring at s = 0 and s = 1. There is no pole unless χ 3 ω 2 = 1. Theorem 2 is a restatement of Theorem 7, which is proed below in 3. An immediate corollary to Theorem 2 is the proof of Hypothesis 6.1 of [5], which implies Theorem 1. Although the theorems aboe are new, the techniques of proof are entirely contained in the papers listed as references, most notably [1], [2], [5], [6], and [9]. Mathematics Subject Classification Numbers: 11R42, 11F70, 11F27.
2 The author wishes to thank J. Hoffstein for suggesting the problem, D. Goldfeld for a useful discussion, and D. Bump for inspiration. It is also a pleasure to thank the Mathematical Sciences Research Institute MSRI) in Berkeley and the Centre Interuniersitaire en Calcul Mathématique Algébrique CICMA) in Montréal for proiding the author with support during the preparation of this paper. 2. The Nonexistence of Siegel Zeros on GL3). Let F be an algebraic number field, with A = A F its ring of adeles. For eery integer n 1, let A nf) denote the set of unitary cuspidal automorphic representations of GLn, A) up to equialence. Let π = π A n F), and let fπ) be the conductor of π. If is a finite place of F not diiding fπ), the Satake parameterization associates to the local representation π a diagonal matrix ρ π ) in the L-group L GLn) := GLn, C), which is uniquely determined up to a permutation of the diagonal entries. For eery finite-dimensional representation r of L GLn) and eery finite set S of places containing all of the archimedean places and all of the finite places diiding fπ), the partial Langlands L-function is defined as an Euler product absolutely conergent for all s in some right half-plane): L S s, π, r) := S L s, π, r), where: L s, π, r) := Det I r ρ π ) ) q s ) 1 for all S. Here q denotes the cardinality of the residue field of F. If S and either is archimedean or n 3, one knows how to associate to π an n-dimensional representation σ of the extended Weil group of F, and we define L s, π, r) := L s, r σ ) in these cases cf. [5] ). Following [5], the completed L-function: Ls, π, r) := L s, π, r) is said to be nice if: 1) all of the local factors are defined, and the product is absolutely conergent in { Res) > 1 }, 2) Ls, π, r) continues meromorphically to all s C, bounded
3 in ertical strips, with no poles outside { Res) [0, 1] }, and 3) it satisfies a functional equation of the form: Ls, π, r) = ǫs, π, r) L1 s, π, r), where: ǫs, π, r) := D Dimr) F Nfπ, r) ) 1/2 s Wπ, r). Here D F denotes the absolute alue of the discriminant of F, fπ, r) a certain conductor ideal in F, N the absolute norm, and Wπ, r) the root number. Note that since π is unitary, its contragredient can be identified with the complex conjugate representation π. When r is the standard representation of L GLn) in C n, we write Ls, π) for Ls, π, r). This L-function is always nice with fπ, r) = fπ). Moreoer, Ls, π) is entire unless n = 1 and π is triial, in which case Ls, π) is the completed Dedekind zeta function ζf s) of F. The Rankin-Selberg L-function attached to a pair of representations π = π A n F) and π = π A n F) is defined as an Euler product absolutely conergent in some right half-plane): Ls, π π ) := L s, π π ) where: L s, π π ) := Det I ρ π ) ρ π ) ) q s ) 1 for all finite places not diiding fπ)fπ ). It is known cf. Remark 1.2 of [5] ) that Ls, π π ) is nice, with no pole at s = 1 unless n = n and π = π, in which case the pole at s = 1 is simple. Moreoer, the exponent of D F occurring in the epsilon factor is nn 1/2 s), and Nfπ π ) Nfπ) n Nfπ ) n. Let A F) := A nf), and let π A F) be such that Ls, π, r) is nice for some r. n 1 If c > 0 and R > 1, then Ls, π, r) is said to hae a Siegel zero relatie to c and R if there exists a real number β in the open interal 1 c/logr, 1) such that Lβ, π, r) = 0. Theorem 3. Hoffstein Ramakrishnan) Let R > 1 be fixed, and let A F, R) denote the subset of A F) consisting of cuspidal representations π such that if π A n F), then D n2 F Nfπ π) R. Then there exists an absolute effectie constant c > 0 such that
4 there is at most one cusp form in A F, R) for which Ls, π) has a Siegel zero relatie to c and R. Proof: This is Theorem A of [5]. The constant c of the preceding theorem is independent of R. If π A nf), the L-function Ls, π) is said to hae a Siegel zero if Ls, π) has a Siegel zero relatie to c and R, where c is the constant of Theorem 3, and R := D n2 F Nfπ π). Corollary 4. Hoffstein Ramakrishnan) Suppose that π is a cusp form in A F) that is not self-dual. Then Ls, π) does not admit a Siegel zero. Proof: This is Corollary 3.2 of [5]. In particular, if π is a cusp form in A 3 F), then Ls, π) can hae a Siegel zero only if π is self-dual. Thus, let π A 3 F) be fixed such that π = π. Since π is self-dual, its central character ω = ω π must be either triial or quadratic. Theorem 5. Goldfeld Hoffstein Lieman) Suppose that π A 3 F) is self-dual with triial central character. Then Ls, π) does not admit a Siegel zero. Proof: The proof is identical to the argument gien in [3] for the symmetric-square of a cusp form on GL2). Briefly, let S denote the set of archimedean places, and consider the Dirichlet series: Here 1 Ds) := L S s, 1 π) 1 π) ). π denotes the isobaric sum of 1 and π. Then Ds) is nice, has non-negatie coefficients, and its conductor is bounded aboe by M C for some absolute constant C and M as in Definition 1.4 of [5]. Since Ord s=1 Ds) = 2, Ds) can hae at most two zeros in the interal 1 c/logr, 1). On the other hand, one factorizes Ds), using the fact that π is self-dual with triial central character, as follows: Ds) = ζ F s) L S s, π) 3 L S s, π, 2 ).
5 Here 2 : L GL3) GL6, C) is the symmetric-square representation. Now ζ F s) and L S s, π, 2 ) hae no poles in the real open interal 1 c/logr, 1) the latter result is due to Bump and Ginzburg [1] ). If Ls, π) were to admit a Siegel zero, this would imply that Ds) has three zeros in 1 c/logr, 1), a contradiction. Theorem 6. Suppose that π A 3 F) is self-dual with nontriial, quadratic central character ω. Then Ls, π) does not admit a Siegel zero. Proof: In Theorem D of [5], it is shown that this result follows immediately from a certain hypothesis, namely that the partial Langlands L-function L S s, π, 2 ω) has no pole in { Res) 0, 1) }. This will be proed in 3 Theorem 7). Combining the results of Corollary 4, Theorem 5, and Theorem 6, it follows that if π is any unitary cuspidal automorphic representation of GL3, A), then Ls, π) does not admit a Siegel zero. Hence, Theorem 1 follows from Theorem 7 of the next section. 3. Twisted Symmetric-Square L-functions on GL3). In this section, we will establish the alidity of the hypothesis required in the proof of Theorem 6 of 2. In fact, we will proe a more general result. The basic references for this section are the papers of Bump and Ginzburg [1], Gelbart and Piatetski-Shapiro [2], Kazhdan and Patterson [6], and Patterson and Piatetski-Shapiro [9]. Theorem 7. Let π = π A 3 F), and let χ = χ be a Hecke character. Let S be a finite set of places of F containing all of the archimedean places and all of the finite places diiding 2fπ)fχ). Then the partial Langlands L-function L S s, π, 2 χ) extends to a meromorphic function of s C, with possible simple poles at s = 0 and s = 1. There is no pole unless χ 3 ω 2 = 1. Proof: In the case of a global field F with CharF) 0, 2, this result was proed as Theorem 6.1 of [9]. In characteristic 0, the proof proceeds in a similar way. We will simply reiew the main ideas here, which follow closely the ideas in [1] and [9]. The important
6 difference in the case of an algebraic number field is the necessity of showing that the data can be chosen so that the local zeta factors are nonanishing. The L-function we are considering is defined as an Euler product absolutely conergent in some right half-plane): L S s, π, 2 χ) := S L s, π, 2 χ ), where: L s, π, 2 χ ) := Det I 2 ρ π ) ) χ ) q s ) 1 for all S. Here denotes a fixed prime element in the nonarchimedean) field F. For fixed S, if ρ π ) is the diagonal matrix in L GL3) := GL3, C) with diagonal entries { α i, C i = 1, 2, 3 }, then: 1) L s, π, 2 χ ) = 1 i j 3 ) 1 αi, α j, χ ) q s 1. Let G := GL3). Let G := GL2) be the subgroup of G obtained by embedding a copy of GL2) in the upper left-hand corner, and let N be the subgroup of G consisting of upper triangular, unipotent matrices. Let φ be a cusp form in L 2 G F \G A ) associated with π. By [10] or [11], φ has the Fourier expansion: 2) φg) = Wγg), γ N F \G F where W = W ψ φ is the global Whittaker function defined by: Wg) := φng) ψn) dn. N F \N A Here N is the subgroup of G consisting of upper triangular, unipotent matrices, and ψ is a nontriial, N F -inariant Whittaker character of N A. Let G [ resp. ] G denote the two-fold metaplectic coer of G [ resp. G ], constructed both locally and globally in [6]. The parameter c 0.1 of [6] ) is taken to be zero in all cases.
We construct a theta function Θ on G A as follows. Let ω denote the unitary) central character of π. Following 1 of [1], let θ be an exceptional automorphic representation of G A with determinantal character ω 1. Since θ : G A EndV ) is exceptional, by Theorem II.2.1 of [6], we can represent V = V as a restricted tensor product oer all the places of F, where the V s are the spaces of the corresponding local representations. Since θ is automorphic, there exists a nonzero, G F -inariant linear functional l : V C. Here G F is regarded as a subgroup of G A ia the canonical splitting of the metaplectic group oer the group of rational points. If x V, then Θ l,x g) := l θg)x ) is an automorphic form on G A. With l and x fixed, we will simply write Θ for Θ l,x. Next, we construct a twisted Eisenstein series E χ on G A. Let r χ0, V ) denote the global Weil representation of G A associated to the Hecke character χ 0 := χω 2 as constructed in [2]. Let P denote the standard parabolic of type 2, 1) in G, with Lei decomposition P = MU, and let P [ resp. M ] be the preimage of P [ resp. M ] in G. If Z denotes the center of G, then the preimage Z in G is the center of G, and we hae M = G Z. We extend rχ0 to an irreducible representation r χ 0, V ) of M A such that the central character is ω 2. Since r χ 0 is an automorphic representation, there exists a nonzero, M F -inariant linear functional l : V C. We next extend r χ 0 to all of P A by letting U A act triially. Let F s Ind G A P A r χ 0 δ s P A ) for all s C unnormalized induction). Here δ PA : P A C denotes the modular character, which pulls back ia the natural projection P A P A to a character on P A. Define f s g) := l F s g) ) for all g G A. The Eisenstein series that we consider is gien by: 3) E χ g, s, f s ) := γ P F \G F f s γg), the sum being absolutely conergent for Res) sufficiently large. The function Θg)E χ g, s, f s ) factors through the natural map G A G A, hence we can consider the function φg)θg)e χ g, s, f s ) as a function on G A that is G F -inariant. Let s : G A G A denote the preferred section g g, 1). Then if z = zi Z A, φzg) = ωz)φg), Θ sz)g ) = ω 3 z)θg), and E χ sz)g, s, fs ) = ω 2 z)e χ g, s, f s ), and 7
8 therefore φg)θg)e χ g, s, f s ) is inariant by Z A as well. The Rankin-Selberg integral that we consider is the following: 4) φg) Θg) E χ g, s, f s ) dg, Z A G F \G A which is well-defined and absolutely conergent for all s, except where E χ g, s, f s ) has poles. Here dg denotes a left Haar measure on G A. To unfold the Rankin-Selberg integral when Res) 0, we substitute 3) into 4) and use the fact that φ and Θ are automorphic forms. Thus, 4) equals: Z A G F \G A φγg) Θγg) f s γg) dg = φg) Θg) f s g) dg. γ P F \G F Z A P F \G A Let N i, i = 1, 2, denote the subgroup of N for which the only nonzero entries aboe the diagonal occur in the i + 1)-th column. Thus, N = N 1, and N = N 1 N 2. Then P = ZN 2 G, and the preceding integral equals: φg) Θg) f s g) dg = Wγg) Θγg) f s γg) dg Z A N 2,F G F \G A Z A N 2,F G F \G A γ N F \G F = Wg) Θg) f s g) dg. Z A N 2,F N 1,F \G A Here we hae used the relation 2) and the fact that Θ and f s are G F-inariant. Continuing to unfold the integral as in [1] or [9], we find that 4) equals the zeta integral: 5) ZW, Q, R s ) := Z A N A \G A Wg) Qg) R s g) dg, where: Qg) := Θn 2 n 1 g) ψn 2 ) dn 2 dn 1, N 1 F)\N 1 A) N 2 F)\N 2 A) and: R s g) := f s n 1 g) ψn 1 ) dn 1. N 1 F)\N 1 A)
Arguing as in Theorem 3.3 of [1], using the fact that r χ0 has a unique ψ-whittaker model, the integral 5) decomposes into an Euler product: 9 ZW, Q, R s ) = Z W, Q, R s, ), where: 6) Z W, Q, R s, ) := W g ) Q g ) R s,g ) dg. Z N \G The notation is explained as follows. For any algebraic group H, let H denote the set of F -rational points of H. The character ψ has the form ψ, where ψ is the corresponding local character of N, assumed to be unramified for all S. From [11], it is known that π has a unique ψ-whittaker model W = W, where W is the ψ -Whittaker model for π. The cusp form φ is chosen so that Wg) decomposes into W g ). Here W W, and W W for all. Next, we assume the ector x V employed in the construction of Θ is of the form x, where x V for all, and x is the standard K -fixed ector for all S. Here K is the standard maximal compact subgroup of G for all, which can be regarded as a subgroup of G when S ia the canonical splitting of G oer K. Then Θg) decomposes into Θ g ), where Θ g ) := l θ g )x ). By Proposition 3.2 of [1], Qg) decomposes into Q g ), where {Q } is a collection of genuine functions such that the Q are identically one on K for all S, and the product is independent of the decomposition of g. Finally, we choose F s in the construction of E χ so that F s = F s,, where F s, is an element of Ind G P r χ 0, δs P ) for all, and F s, is the standard K -fixed ector for all S. Here δ P is the modular character of P. Because r χ0 has a unique ψ-whittaker model, R s g) decomposes into R s, g ), where {R s, } is a collection of genuine functions such that the R s, are identically one on K for all S, and the product is independent of the decomposition of g. The function R s, can be interpreted as belonging to the space Ind G W P δp s ), where W is the unique semi-ψ-whittaker model ) for rχ 0, cf. pp. 164-165 of [1].
10 Now suppose that S. Since the integrand in 6) is K -inariant, we can replace the integral by: W b ) Q b ) R s,b ) db, Z N \B where B is the standard Borel subgroup of G, and db is a left Haar measure. Since the integrand is right N -inariant, this equals: 7) W t ) Q st ) ) R s, st ) ) δ B t ) 1 dt, Z \T where T is the subgroup of G consisting of diagonal matrices, and δ B character of B. For eery λ = λ 1, λ 2 ) Z 2, let: t λ := λ1 λ2 1. is the modular As the integrand in 7) is T K -inariant, the integral equals: 8) W t λ ) Q stλ ) ) R s, stλ ) ) δ B t λ ) 1. λ Z 2 Shintani [12] proed that: δ B t λ ) 1/2 s λ α ) if λ 1 λ 2 0, W t λ ) = 0 otherwise, where s λ is the symmetric function defined in Section I.3 of Macdonald [8] here λ is regarded as a triple λ 1, λ 2, 0) Z 3), and s λ α ) denotes the alue of s λ applied to the Satake parameters {α 1,, α 2,, α 3, }. From p. 170 of [1], we hae that: Q stλ ) ) δ B t λ ) 1/4 ω 1 Dettλ ) ) if λ 1 and λ 2 are een, = 0 otherwise. Similarly, since χ is unramified, hence een, and χ 0, = χ ω 2, the representation r χ 0, can be embedded the unnormalized) induced representation Ind G 1/2 χ B 0, ) δ1/4 B, where
11 B denotes the standard Borel subgroup of G, B its metaplectic preimage in G, and δ B the modular character of B. This is Proposition 2.3.3 of [2]. Then r χ 0, is exceptional, and we hae that: R s, stλ ) ) = δ P t λ ) s δ B t λ ) 1/4 χ ω 2 Dettλ ) 1/2) if λ 1 and λ 2 are een. Consequently, 8) equals: λ 1 λ 2 0 λ 1,λ 2 een s λ α ) δ P t λ ) s 1/4 χ Dettλ ) 1/2) = λ 1 λ 2 0 λ 1,λ 2 een s λ α ) q s 1/4)λ 1+λ 2 ) χ ) λ 1+λ 2 )/2. Here we e used the fact that δ B = δ B δ P. Now, the corrected) formula on p. 171 of [1] states that if X is an indeterminate, then: 1 i j 3 1 α i, α j, X) 1 = λ 1 λ 2 0 λ 1 λ 2 0 Mod 2) s λ α ) X λ 1+λ 2 )/2 λ 0 λ 0 Mod 2) X 3λ/2 ω ) λ. Taking X := χ ) q 2s+1/2, it follows from 1) and the aboe calculation that: Z W, Q, R s, ) = L 2s 1 2, π, 2 χ ) L 6s 3 2, χ3 ω2 ) 1. The preceding results, alid for all S and Res) sufficiently large, imply the following relation: L6s 3 2, χ3 ω 2 ) φg) Θg) E χ g, s, f s ) dg G F Z A \G A 9) = L S 2s 1 2, π, 2 χ) Z W, Q, R s, ) L 6s 3 2, χ3 ω). 2 S It is possible to assume that the collection { R s s C } has the following form. For eery s C, let Φ s be the standard spherical ector in Ind G A P A δ s P A ). Thus, Φ s pk) = δ PA p) s for all p P A and k K A, where K A := K is the standard maximal compact subgroup of G A. Then Φ s pulls back ia the natural projection G A G A to a function on G A. Let R 0 be a fixed element of Ind G A P A W ), where W = W is the unique semi-ψ-whittaker model for r χ 0. Then R s := Φ s R 0 is an element of Ind G A P A W δ s P A ) for all s C. We can
12 and will assume that the collection { } { } F s s C was chosen so that Rs s C is of this form. From 2 of [9] or the general theory of Eisenstein series [7], it is known that: E χ g, s, f s) := L6s 3 2, χ3 ω 2 ) E χ g, s, f s ) continues to a holomorphic function of s C for eery g G A, excepting that if χ 3 ω 2 = 1, it has simple poles at s = 1/4 and s = 3/4. Note that the Eisenstein series of [9] were obtained by normalized induction. This can also be established by modifying the proof of Theorem 7.4 of [1]. Relation 9) implies that: L S 2s 1 2, π, 2 χ) S Z W, Q, R s, ) L 6s 3 2, χ3 ω 2 ) defines a meromorphic function of s C, with possible poles occurring only at the poles of Eχ g, s, f s). Since the local L-functions L 6s 3 2, χ3 ω2 ) are meromorphic and nonanishing, it remains only to show that the local zeta factors Z W, Q, R s, ) extend to meromorphic functions of s C, and the data can be chosen at any fixed point s = s 0 so that the local zeta factors are nonanishing for all S. The meromorphic continuation of the local zeta factors is accomplished as in the proof of Proposition 5.2 of [1]. The nonanishing result was proed in the nonarchimedean case as Proposition 5.3 of [9]. The proof for an arbitrary local field archimedean or nonarchimedean) is Theorem 7.2 of [1], which follows through without modification for the representations considered here. References. [1] D. Bump and D. Ginzburg, Symmetric Square L-Functions on GLr), Ann. of Math. 136 1992) pp. 137-205. [2] S. Gelbart and I. Piatetski-Shapiro, Distinguished Representations and Modular Forms of Half-Integral Weight, Inent. Math. 59 1980) pp. 145-188. [3] D. Goldfeld, J. Hoffstein, and D. Lieman, An Effectie Zero Free Region, Appendix to [4]. [4] J. Hoffstein and P. Lockhart, Coefficients of Maass Forms and the Siegel Zero, Ann. of Math. 140 1994) pp. 161-181.
[5] J. Hoffstein and D. Ramakrishnan, Siegel Zeros and Cusp Forms, International Mathematics Research Notices 6 1995), pp. 279-308. [6] D. Kazhdan and S. Patterson, Metaplectic Forms, Publ. Math. IHES 59 1984) pp. 35-142. [7] R. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics 544, Springer-Verlag, 1976. [8] I. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Uniersity Press, Oxford, 1979. [9] S. Patterson and I. Piatetski-Shapiro, The Symmetric-Square L-Function Attached to a Cuspidal Automorphic Representation of GL3), Math. Ann. 283 1989) pp. 551-572. [10] I. Piatetski-Shapiro, Euler Subgroups, in Lie Groups and Their Representations, John Wiley and Sons, New York, 1975. [11] J. Shalika, The Multiplicity One Theorem for GLn), Ann. of Math. 100 1974) pp. 171-193. [12] T. Shintani, On an Explicit Formula for Class-1 Whittaker Functions on GLn) oer P-Adic Fields, Proc. Japan Acad. 52 1976) pp. 180-182. 13 William D. Banks, Concordia Uniersity. banks@abacus.concordia.ca). Reised: June 1, 1996.