WEIGHTED SELBERG ORTHOGONALITY AND UNIQUENESS OF FACTORIZATION OF AUTOMORPHIC L-FUNCTIONS

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1 WEIGHTED SELBERG ORTHOGONALITY AND UNIQUENESS OF FACTORIZATION OF AUTOMORPHIC L-FUNCTIONS JIANYA LIU AND YANGBO YE 3 Abstract. We prove a weighte version of Selberg s orthogonality conjecture for automorphic L-functions attache to irreucible cuspial representations of GL m over Q. Using this weighte orthogonality, we obtain the uniqueness of factorization of general L-functions. As a consequence, we prove that the L-function attache to an automorphic irreucible cuspial representation of GL m over Q cannot be factore further.. Introuction. It is a far-reaching conjecture of Langlans [] that the most general L-function is inee the L-function Ls, Π attache to an automorphic representation Π of GL n over an algebraic number fiel. It was further conjecture that this Ls, Π is equal to a prouct of L-functions Ls, π j attache to automorphic irreucible cuspial representations π j of GL mj over Q in a unique way:. Ls, Π = Ls, π Ls, π k. These Ls, π j are calle principal or primitive L-functions over Q in the sense that they are suppose to be L-functions that cannot be factore further. They are believe to be the builing blocks of all L-functions. A known special case of the unique factorization. is for Π being an automorphic irreucible cuspial representation of GL n over a cyclic algebraic number fiel F, when Π is invariant uner the action of the GalF/Q. Accoring to Arthur an Clozel [], such a representation Π is the base change of π η, where π is an automorphic irreucible cuspial representation of GL n over Q, an η is any iele class character of Q trivial on N F/Q F A. In terms of L-functions, we have the factorization Ls, Π = η Ls, π η 000 Mathematics Subject Classification MSC F70, M6, M4. Partially supporte by China NNSF Grant Number 050 Partially supporte by a China NNSF Dual Research Base Grant 3 Project sponsore by the USA NSA uner Grant Number MDA The Unite States Government is authorize to reprouce an istribute reprints notwithstaning any copyright notation herein.

2 uniquely. In this paper we will prove the uniqueness of the factorization in.. That is, if any general L-function can be written as a prouct of principal L-functions Ls, π j for GL mj over Q, we will show that this factorization is unique. Theorem.. Let π j an π i, j =,..., k, i =,..., l, be automorphic irreucible cuspial representations of GL mj Q A an GL m i Q A with unitary central characters, respectively. Then. Ls, π Ls, π k = Ls, π Ls, π l cannot hol, if there is a π j which is not equivalent to any π i. By taking k =, Theorem. implies that Ls, π cannot be factore further. Corollary. The L-function Ls, π attache to an automorphic irreucible cuspial representation π of GL m Q A cannot be factore into Ls, π Ls, π l nontrivially, where π i is an automorphic irreucible cuspial representation of GL m Q A with unitary central i character. Unique factorization of L-functions in the Selberg class Selberg [8] has been stuie by Conrey an Ghosh [] an Ram Murty [4], uner the Selberg orthogonality conjecture Conjecture. below. For automorphic L-functions, Ram Murty [5] prove that Ls, π is primitive, i.e., cannot be factore further, when π is an automorphic irreucible cuspial representation of GL Q A, uner the Ramanujan conjecture. Our proof of Theorem. an its Corollary is unconitional. The proof of Theorem. will utilize a weighte version of Selberg s orthogonality for automorphic L-functions. Let π be an automorphic irreucible cuspial representation of GL m Q A. Then the global L-functions attache to π are given by proucts of local factors for Re s > Goement an Jacquet [5]: Ls, π = p L p s, π p, Φs, π = L s, π Ls, π, where m L p s, π p = απ p, jp s j=

3 an m L s, π = Γ R s + µ π j. j= Here Γ R s = π s/ Γs/, an α π p, j an µ π j, j =,..., m, are comple numbers associate with π p an π, respectively, accoring to the Langlans corresponence. Denote a π p k = α π p, j k. Then for Re s >, we have j m.3 s log Ls, π = Λna π n n s, n where Λn = log p if n = p k an = 0 otherwise. If π is an automorphic irreucible cuspial representation of GL m Q A, we efine Ls, π, α π p, i, µ π i, an a π p k likewise, for i =,..., m. If π an π are equivalent, then m = m an {α π p, j} = {α π p, i} for any p. Hence a π n = a π n for any n = p k, when π = π. The Selberg orthogonality conjecture for automorphic L-functions Ls, π was propose in 989 Selberg [8]. See also Ram Murty [4]. Conjecture.. i For any automorphic irreucible cuspial representation π of GL m Q A.4 a π p p p = log log + O. ii For any automorphic irreucible cuspial representations π an π of GL m Q A an GL m Q A, respectively,.5 if π is not equivalent to π. p a π pā π p p, A weaker version of this conjecture is Conjecture.. i For any automorphic irreucible cuspial representation π of GL m Q A.6 log nλn a π n = n log + Olog. n 3

4 ii For any automorphic irreucible cuspial representations π an π of GL m Q A an GL m Q A, respectively,.7 if π is not equivalent to π. n log nλna π nā π n n log, Note that in.6 an.7, the sums are taken over primes n = p an prime powers n = p k, k >, while in Conjecture., the sums are taken over primes only. Runick an Sarnak conjecture that the sums taken over prime powers containe in.6 an.7 converge as : Conjecture.3. [7] For any fie k, p log pa π p k p k <. Conjectures. an.3 imply Conjecture.. Conjecture. Part i in.6 was prove by Runick an Sarnak [7]. They also prove Conjecture.3 for GL an GL 3. Recently, Kim an Sarnak [0] prove Conjecture.3 for GL 4. Therefore, part i of Selberg s original Conjecture. is known up to GL 4. On the other han, part ii of Conjectures. an. is still open. In [7], Runick an Sarnak first prove a weighte version of.6:.8 n log nλn aπ n = n log + Olog n an euce.6 from.8 using the fact that the left sie of.6 is a sum of positive terms. What we want to prove here is a weighte version of.7. Theorem.. For any automorphic irreucible cuspial representations π an π of GL m Q A an GL m Q A, respectively,.9 n log nλnaπ nā π n log, n n if π is not equivalent to π. We note that we cannot remove the weight n/ from the left sie of.9 using stanar methos, because the sum there is not of positive terms. For our purpose of proving Theorem., however, there is no nee to remove this weight. There is also no nee to remove terms on prime powers from the sum on the left sie of.9. That is, we o not assume Conjecture.3. 4

5 When π an π are not twiste equivalent, i.e., when π = π α it for any t R, where αg = etg,.9 was prove by the authors in []. In this paper, we will prove the remaining case:.9 is true when m = m an π = π α iτ 0 for some non-zero τ 0 R. We woul like to epress our sincere thanks to Peter Sarnak an Freyoon Shahii for helpful information an iscussion.. Proof of Theorem.. We first prove Theorem. using.8 an Theorem.. A similar argument was first use by Conrey an Ghosh [] who euce the uniqueness of factorization in the Selberg class by assuming Conjecture. cf. Ram Murty [4]. get Suppose that. hols. Taking logarithmic erivatives of both sies like in.3, we n for Re s >. Consequently Λn n s a π n + + a πk n = Λn n s a π n + + a π n l n a π n + + a πk n = a π n + + a π l n for any n = p k. Consier the sum n Λn log n. a π n + + a πk n ā πj n n n = n n Λn log n n a π n + + a π l n ā πj n. Since π j is not equivalent to any π i, the right sie of. is log, by Theorem.. On the left sie of., we get at least one, an possibly more copies of log + Olog by.8. This contraiction proves Theorem.. 3. Rankin-Selberg L-functions. We will use the Rankin-Selberg L-functions Ls, π π as evelope by Jacquet, Piatetski-Shapiro, an Shalika [6], Shahii [9], an Moeglin an Walspurger [3], where π an π are automorphic irreucible cuspial representations of GL m an GL m, respectively, over Q with unitary central characters. This L-function is also given by local factors: 3. Ls, π π = p L p s, π p π p where m L p s, π p π p = απ p, jα π p, kp s. m j= k= 5

6 The Archimeean local factor L s, π π is efine by L s, π π = m where, when π an π are unramifie, } {µ π π j, k : j m, k m = m j= k= Γ R s + µ π π j, k {µ π j + µ π k : j m, k m }. Denote Φs, π π = L s, π π Ls, π π. We will nee the following properties of the L-functions Ls, π π an Φs, π π. RS. The Euler prouct for Ls, π π in 3. converges absolutely for Re s > Jacquet an Shalika [7]. RS. The complete L-function Φs, π π has an analytic continuation to the entire comple plane an satisfies a functional equation Φs, π π = εs, π π Φ s, π π, with εs, π π = τπ π Q s π π where Q π π > 0 an τπ π = ±Q / π π Shahii [9], [0], [], an []. RS3. When π = π α it for any t R, Φs, π π is holomorphic. When m = m an π = π α iτ 0 for some τ 0 R, the only poles of Φs, π π are simple poles at s = iτ 0 an + iτ 0 coming from Ls, π π Jacquet an Shalika [7], [8], Moeglin an Walspurger [3]. We will only consier the latter case in the proof of Theorem.. RS4. Φs, π π is meromorphic of orer one away from its poles, an boune in vertical strips Gelbart an Shahii [4]. RS5. Φs, π π an Ls, π π are non-zero in Res. Shahii [9] In aition, we will also use the following simple properties of the Γ-function. If then λs = min s n, n 0 3. log Γs + log s +, s λs 6

7 an 3.3 log Γs s λs. For the proof of 3., see e.g. Pan an Pan [6] p.49. To show 3.3, one appeals to Pan an Pan [6] p.48, to get where γ is Euler s constant. Therefore s log Γs = s + γ + n + s n s log Γs = s + n= n= n + s = n=0 n + s λs. Let Cδ be the comple plane with the iscs s n < δ, n = 0,,, eclue. Then by 3. an 3.3, for s Cδ we have s log Γs δ log s +, s log Γs δ. Lemma 3.. Assume m = m an π = π α iτ 0 for some nonzero τ 0 R. For any T + Im T log Q π π T + where runs over all the non-trivial zeros of Ls, π π. Proof. Since Φs, π π is of orer one RS4, we have see e.g. Davenport [3] Chapter Φs, π π = e A+Bs s + iτ 0 s iτ 0 s e s/, where A = A π π an B = B π π are constants epening on π π. Take logarithmic erivative 3.4 s log Φs, π π = B s + iτ 0 + s iτ 0 s +, where here an throughout we set log = 0. By the efinition of Φs, π π, we have 3.5 s log Φs, π π = s log L s, π π + s log Ls, π π. 7

8 The efinition of L s, π π an 3. give s log L s, π π = j,k j,k Let Cm be the comple plane with the iscs s log π s+µ π π j,k/ s + µπ πj, s log Γ k + j,k s + µπ π j, k λ + log s +. s n + µ π π j, k <, n 0, j, k m, 8m eclue. Then for s Cm an all j, k, s + µπ π j, k λ Thus in Cm 3.6 By 3.4, 3.5, an 3.6 we have 3.7 6m. s log L s, π π m log s +. s logls, π π = B + s + s + iτ 0 s iτ 0 + Olog s +. Here we give a remark about the structure of Cm. For j, k =,, m, enote by βj, k the fractional part of Reµ π π j, k. If π an π are unramifie at R, the Ramanujan conjecture preicts that Reµ π π j, k = 0. We will not assume the Ramanujan conjecture in this paper. In aition we let β0, 0 = 0 an βm +, m + =. Then all βj, k [0, ], an hence there eist βj, k, βj, k such that βj, k βj, k /3m an there is no βj, k lying between βj, k an βj, k. It follows that the strip S 0 = {s : βj, k + /8m Res βj, k /8m } is containe in Cm. Consequently, for all n = 0,,,, the strips 3.8 S n = {s : n + βj, k + /8m Res n + βj, k /8m } are subsets of Cm. This structure of Cm will be use later. We will prove in a moment that 3.9 ReB = Re + Olog Q π π. 8

9 Now taking real part in 3.7, we get by 3.9 that 3.0 Re s log Ls, π π = Re s Re s + iτ 0 Re s iτ 0 + O logq π π s +. Let s = σ + it with σ 3 such that s Cm; this is possible by the structure of Cm. We want to point out that the left sie of 3.0 is O. In fact, by RS, we have for Res > that s log Ls, π π = n= Λna π na π n n s. From a π p k = m j= α πj, p k an the boun towar the Ramanujan conjecture prove by Runick an Sarnak [7] we obtain for n = p k that α π j, p p / /m +, a π n = a π p k mp k/ /m + m n / /m +. A similar estimate hols for a π n too. For n p k, we have Λn = 0. Thus we conclue that n= Λna π nα π n n s m for Res = σ lying between an 3. Therefore from 3.0 we euce that σ Re σ Re + T Im logq π π T +. This gives the assertion of Lemma 3. because 0 Re. It remains to prove 3.9. We start from the efinition of Φs, π π. By a π n = ā π n an the fact that {µ π π j, k} = {µ π π j, k}, we have Ls, π π = L s, π π an L s, π π = L s, π π. It follows that 3. Φs, π π = Φ s, π π, i.e., 3. ep A π π + sb π π s + iτ 0 s + iτ 0 π π = ep Ā π π + s B π π s + iτ 0 s + iτ 0 9 s e s/ π π s e s/.

10 Taking s = 0 in 3., we get 3.3 A π π = Āπ π. By 3., Φ, π π = 0 if an only if Φ, π π = 0, an consequently 3.4 π π s e s/ = s e s/. π π Inserting 3.4 an 3.3 into 3., we obtain 3.5 B π π = B π π. On the other han, by 3.4 an the functional equation in RS, we get 3.6 B π π = s log Φ0, π π + iτ 0 iτ 0 Applying 3.5, we conclue = s log Φ, π π + Olog Q π π = B π π + + Olog Q π π. ReB π π = π π π π Re + Re + Olog Q π π. By the functional equation an 3., we have Φ, π π = 0 if an only if Φ, π π = 0. Therefore in the above formula we can write in place of, an 3.9 follows. 4. Estimation of logarithmic erivatives. Lemma 4.. Assume m = m an π = π α iτ 0 for some nonzero τ 0 R. a Let T >. The number NT of zeros of Ls, π π in the region 0 Res, Ims T satisfies NT + NT logq π π T an NT T logq π π T. b For any T >, we have T Im > T Im logq π π T. 0

11 c Let s = σ +it with σ, t >. If s Cm is not a zero or pole of Ls, π π, then s logls, π π = t Im s + O logq π π t. s iτ 0 s iτ 0 For T >, there eists τ with T τ T + such that when σ s log Lσ + iτ, π π log Q π π τ. e For T >, there eists τ with T τ T + such that when σ s log Lσ + iτ, π π log 3 Q π π τ. Proof. a The first estimate follows from Lemma 3. an the observation that NT + NT = + T Im. T <Im T + The secon inequality can be euce from the first. b The left sie here is less than T Im > T <Im T + + T Im, which in combination with Lemma 3. gives the esire result. c By the structure of Cm, there eists σ 0 3 such that s 0 = σ 0 + it Cm. We euce from 3.7 that 4. s log Ls,π π s log Ls 0, π π = + + s iτ 0 s iτ 0 s 0 iτ 0 s 0 iτ 0 + s + O logq π π t. s 0 Because σ 0, the secon term on the left sie is O. By the same reason, /s 0 iτ 0 + /s 0 iτ 0. By a, then Im t Also it follows from b an σ that s s 0 Im t > s 0 logq π π t. Im t > Inserting these estimates into 4., we get the esire result. t Im logq π π t.

12 By the counting of zeros in a, we can choose c > 0 an τ with T τ T + satisfying the following properties: For any σ + it with σ an t τ c log Q π π T, σ + it Cm, Lσ + it, π π 0, an σ + it is away from poles + iτ 0 an iτ 0 of Ls, π π by at least /3. Then for s = σ + iτ by c an a. s logls, π π c logq π π τ τ Im + OlogQ π π τ s iτ 0 s iτ 0 log Q π π τ e The proof is similar an easier than that for. Now, instea of 3.4, 3.5, an 3.6, we have s log Φs, π π = s + s iτ 0 + s iτ 0, an in Cm, s log Φs, π π = s log L s, π π + s log Ls, π π, s logl s, π π = j,k j,k s log π s+µ π π j,k/ + j,k s + µπ π j, k λ by 3.3. Then from the efinition of Cm we get s log L s, π π m for s Cm. Thus, instea of 3.7, we now have s logls, π π s + µπ πj, s log Γ k = s log Φs, π π s log L s, π π = s + s iτ 0 + s iτ 0 + O.

13 By an argument similar to that leaing to c we get s logls, π π = t Im s + s iτ 0 + s iτ 0 + O logq π π t for s = σ + it as in the statement of c. As in the proof of, we can choose c > 0 an τ with T τ T + satisfying the following properties: For any σ + it with σ an 4. t τ c log Q π π T, σ + it Cm, Lσ + it, π π 0, an σ + it is away from poles + iτ 0 an iτ 0 of Ls, π π by at least /3. Then the esire result of e follows from s loglσ + iτ, π π c log Q π π τ log 3 Q π π τ τ Im by 4.. Lemma 4.. Assume m = m an π = π α iτ 0 s is in some strip S n as in 3.8 with n, then s log Ls, π π m. Proof. By the functional equation in RS, we have that 4.3 s logl s, π π + s log Ls, π π for some nonzero τ 0 R as before. If = s log εs, π π + s log L s, π π + s log L s, π π. Using the efinition of L s, π π an 3.3 we get s logl s, π π = j,k j,k s log π s+µ π π j,k/ + j,k s + µπ π j, k. λ 3 s + µπ πj, s log Γ k

14 As in the proof of Lemma 3., for all s Cm an all j, k, we have s + µπ π j, k λ 6m. Thus in Cm 4.4 s log Ls, π π m. Since Φs, π π = L s, π π Ls, π π is analytic for Re s > RS an has only a simple pole + iτ 0 on Re s = coming from Ls, π π RS3, the Archimeean local factor L s, π π is analytic for Re s. Consequently Reµ π π j, k > for all j, k. Now we take s = σ + it S n with n in 4.3; trivially σ <. Thus 4.5 Using σ < again, we get 4.6 s logl s, π π s + µ π π j, k λ j,k σ + Reµ π π j, k λ j,k m. s log L s, π π. The esire result now follows from Proof of Theorem.. We will prove Theorem. when m = m an π = π α iτ 0 for some nonzero τ 0 R. The case of π an π being not twiste equivalent was prove by Liu an Ye []. By RS, we have for Res > that s log Ls, π Λna π nā π n π =, n s an therefore Ks := s log Ls, π π = n= n= log nλna π nā π n n s. By RS3 an RS5, Ks is holomorphic in Res >. On Res =, Ls, π π is nonzero RS5 an has only a simple pole at s = + iτ 0. Thus 5. Ks = s iτ 0 + Gs has only a ouble pole in Re s, where Gs is analytic for Re s. On C, Ks has a ouble pole at each of the pole at iτ 0, trivial zeros, an nontrivial zeros of Ls, π π. 4

15 Note that πi b y s { /y if y, ss + s = 0 if 0 < y <, where b means the line Res = b > 0. Then we have n n log nλnaπ nā π n n = s Ks + πi ss + s = +it + πi it it +i i +it. The last two integrals are clearly boune by T /t t /T. Thus 5. n n log nλnaπ nā π n n = πi +it it s Ks + ss + s + O T. Choose σ 0 with < σ 0 < such that the line Res = σ 0 is containe in the strip S Cm; this is guarantee by the structure of Cm. Let T be the τ such that Lemma 4.e hols. Now we consier the contour C : σ σ 0, t = T ; C : σ = σ 0, T t T ; C 3 : σ 0 σ, t = T. Note that the two poles, some trivial zeros, an certain nontrivial zeros of Ls +, π π, as well as s = 0, are passe by the shifting of the contour. The trivial zeros can be etermine by the functional equation in RS: s + = µ π π j, k with σ 0 < Reµ π π j, k < 0. Here we use the facts that Reµ π π j, k > an < σ 0 < 5

16 . Then we have πi +it it s Ks + ss + s = πi C + C + C 3 s + Res Ks + s=0, ss + s + Res Ks + s=iτ 0, +iτ 0 ss + + Res s= µ π π j,k σ 0 < Reµ π π j,k<0 + Im T Res s= s Ks + ss +. s Ks + ss + By Lemma 4.e, we get C σ log 3 Q π πt T σ log3 Q π πt T, σ 0 an the same upper boun also hols for the integral on C 3. By Lemma 4., then C T T σ 0 t + t. By taking T, the three integrals in 5.4 are 5.9 log3 Q π π. Obviously, 5.5 is 5.0 Res Ks + s=0, ss + = K + K0. s Since the poles at s = iτ 0 an s = + iτ 0 are ouble poles, The resiues in 5.6 are 5. lim s iτ 0 s s iτ 0 Ks + ss + + lim s +iτ 0 s s + iτ 0 s Ks + ss +. 6 s

17 Using 5. we get 5. lim s iτ 0 s s iτ 0 Ks + ss + = lim + s iτ0 Gs + s s iτ 0 s ss + s = lim log. s iτ 0 s ss + s Similarly the secon term in 5. is log /, an hence 5.6 is boune 5.3 Res Ks + s=iτ 0, +iτ 0 ss + s log. Near a trivial zero s = µ π π j, k of orer l in 5.7, we can epress Ks + as l/s + + µ π π j, k plus an analytic function, like in 5.. The resiues in 5.7 can therefore compute similar to what we i in 5.. Because Reµ π π j, k > an Reµ π π j, k < 0, 5.7 is boune 5.4 Res s= µ π π j,k σ 0 < Reµ π π j,k<0 s Ks + log. ss + To compute the resiues corresponing to nontrivial zeros in 5.8, we note that Φs, π π is of orer RS4, an Φ, π π 0 RS5. Using an argument like 3.6, we see that Consequently, 5.8 becomes 5.5 Im T Res s= = <. s Ks + ss + Im T Im T log. Res s= log s s + ss + Using 5.9, 5.0, an 5.3 through 5.5, we get a boun for 5.3 πi +it it s Ks + s log. ss + The esire estimate.9 now follows from this, 5., an the fact that T. 7

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19 [8] A. Selberg, Ol an new conjectures an results about a class of Dirichlet series, Collecte Papers, vol. II, Springer, 99, [9] F. Shahii, On certain L-functions, Amer. J. Math., 03 98, [0] F. Shahii, Fourier transforms of intertwining operators an Plancherel measures for GLn, Amer. J. Math., , 67-. [] F. Shahii, Local coefficients as Artin factors for real groups, Duke Math. J., 5 985, [] F. Shahii, A proof of Langlans conjecture on Plancherel measures; Complementary series for p-aic groups, Ann. Math., 3 990, Jianya Liu: Department of Mathematics, Shanong University Jinan 5000, China. jyliu@su.eu.cn Yangbo Ye: Department of Mathematics, The University of Iowa Iowa City, Iowa 54-49, USA. yey@math.uiowa.eu 9