SELBERG S ORTHOGONALITY CONJECTURE FOR AUTOMORPHIC L-FUNCTIONS

Transcription

1 SELBERG S ORTHOGONALITY CONJECTURE FOR AUTOMORPHIC L-FUNCTIONS JIANYA LIU 1 AND YANGBO YE 2 Abstract. Let π an π be automorphic irreucible unitary cuspial representations of GL m (Q A ) an GL m (Q A ), respectively. Assume that π an π are self contragreient. Uner the Ramanujan conjecture on π an π, we euce a prime number theorem for L(s, π π ), which can be use to asymptotically escribe whether π = π, or π = π et( ) for some non-zero τ 0 R, or π = π et( ) it for any t R. As a consequence, we prove the Selberg orthogonality conjecture, in a more precise form, for automorphic L-functions L(s, π) an L(s, π ), uner the Ramanujan conjecture. When m = m = 2 an π an π are representations corresponing to holomorphic cusp forms, our results are unconitional. 1. Introuction. Let π be an irreucible unitary cuspial representation of GL m (Q A ). Then the global L-function attache to π is given by proucts of local factors for Re s > 1 (Goement an Jacquet [3]): L(s, π) = p L p (s, π p ), Φ(s, π) = L (s, π )L(s, π), where an m ( L p (s, π p ) = 1 απ (p, j)p s) 1 j=1 m L (s, π ) = Γ R (s + µ π (j)). j=1 Here Γ R (s) = π s/2 Γ(s/2), an α π (p, j) an µ π (j), j = 1,..., m, are complex numbers associate with π p an π, respectively, accoring to the Langlans corresponence. De Mathematics Subject Classification (MSC) 11F70, 11M26, 11M41. 1 Partially supporte by China National Natural Science Founation grant # Project sponsore by the USA National Security Agency uner Grant Number MDA The Unite States Government is authorize to reprouce an istribute reprints notwithstaning any copyright notation herein. 1

2 note (1.1) a π (p k ) = Then for Re s > 1, we have (1.2) 1 j m α π (p, j) k. s log L(s, π) = Λ(n)a π (n) n s, n 1 where Λ(n) = log p if n = p k an = 0 otherwise. If π is an automorphic irreucible cuspial representation of GL m (Q A ), we efine L(s, π ), α π (p, i), µ π (i), an a π (p k ) likewise, for i = 1,..., 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 L(s, π) was propose in 1989 (Selberg [19]). See also Ram Murty [15] [16]. Conjecture 1.1. (i) For any automorphic irreucible cuspial representation π of GL m (Q A ) (1.3) a π (p) 2 = log log x + O(1). p p x (ii) For any automorphic irreucible cuspial representations π an π of GL m (Q A ) an GL m (Q A ), respectively, (1.4) if π is not equivalent to π. p x a π (p)ā π (p) p 1, The asymptotic formula in (1.3) was prove by Runick an Sarnak [17] uner a conjecture on the convergence of a series on prime powers (Hypothesis H below), an unconitionally for m 4. Hypothesis H. For k 2, p a π (p k ) 2 log 2 p p k <. This Hypothesis H is trivial for m = 1, an follows from bouns towar the Ramanujan conjecture for m = 2. For m = 3 it was prove by Runick an Sarnak [17], while the case of m = 4 was prove by Kim an Sarnak [9]. For m > 4, Hypothesis H is an easy consequence of the Ramanujan conjecture. In this paper, we will assume the Ramanujan conjecture for primes p: 2

3 Conjecture 1.2. Let π be an irreucible unitary cuspial representation of GL m (Q A ). For any unramifie p, we have (1.5) α π (p, j) = 1. Note that in Conjecture 1.2 we o not inclue the Archimeean Ramanujan conjecture, Re µ π (j) = 0. What we will prove as a consequence of Conjecture 1.2 is the following orthogonality. Denote α(g) = et(g). Theorem 1.3. Let π an π be irreucible unitary cuspial representations of GL m (Q A ) an GL m (Q A ), respectively, such that at least one of π an π is self contragreient: π = π or π = π. Assume the Ramanujan Conjecture 1.2 for both π an π. Then (log n)λ(n)a π (n)ā π (n) = x1+ log x x1+iτ0 1 + (1 + ) 2 + O{x exp( c log x)} n x if π = π α for some τ 0 R; =O{x exp( c log x)} if π = π α it for any t R, where c is a positive constant. Note that for m = m = 2 an π an π being representations corresponing to holomorphic cusp forms, i.e., when their Archimeean local components are iscrete series or limits of iscrete series, the Ramanujan conjecture was prove by Deligne. Therefore in this case, Theorem 1.3 is an unconitional result. Theorem 1.3 is inee a version of prime number theorem for the the Rankin-Selberg L-function L(s, π π ). It is stronger than the following Mertens theorem on orthogonality with a weight (log n)λ(n)/n: Corollary 1.4. Let π an π be given as in Theorem 1.3. Assume either (i) Ramanujan Conjecture 1.2 for both π an π, or (ii) that m = m = 2 an π an π are 3

4 representations corresponing to holomorphic cusp forms. Then (1.6) n x (log n)λ(n)a π (n)ā π (n) n = 1 2 log2 x + c 1 + O{exp( c log x)} if π = π; = x 1 log x + xiτ0 τ0 2 + c 2 + O{exp( c log x)} if π = π α for some τ 0 R ; =c 3 + O{exp( c log x)} if π = π α it for any t R, where c, c 1,..., c 3 are positive constants. A remarkable feature of this corollary is that it escribes the orthogonality of a π (n) an a π (n) in three cases with ifferent main terms. As we are assuming Ramanujan an hence the Hypothesis H, we can control sums over prime powers an easily get an orthogonality over primes. Selberg s orthogonality conjecture 1.1 is then a consequence of Corollary 1.4 by partial summation. Corollary 1.5 (Selberg s orthogonality). Let π an π be given as in Theorem 1.3. Assume either (i) Ramanujan Conjecture 1.2 for both π an π, or (ii) m = m = 2 an that π an π are representations corresponing to holomorphic cusp forms. Then (1.7) p x a π (p)ā π (p) p = log log x + c 4 + O{exp( c log x)} if π = π; =c 5 + Ei( log x) + O{exp( c log x)} if π = π α for some τ 0 R ; =c 6 + O{exp( c log x)} if π = π α it for any t R. Here Ei is the exponential integral, an c, c 4,..., c 6 are positive constants. Recall that Ei( log x) = x ( n k! ) log x ( log x) k + O ( (log x) n 2). k=0 Our Corollary 1.5 is thus in a more precise form than Selberg s Conjecture

5 The error terms in our theorem an corollaries are in a form that reflects very much our presesnt knowlege of zero free regions for our Rankin-Selberg L-functions (see 4). The proofs of Corollaries 1.4 an 1.5 procee along stanar arguments, base on variations of Abel summation. We will thus not give these proofs here, but only point out that in the proof of Corollary 1.5, Hypothesis H is use to control sums over prime powers in the expression on the left sie of (1.6). This way we can obtain a sum taken over primes as in (1.7). We woul like to express our heartfelt thanks to F. Shahii for information on zero-free regions of automorphic L-functions, to the referee for etaile, in epth comments an suggestions, an to P. Sarnak for constructive remarks on an early raft of this paper. 2. Rankin-Selberg L-functions. We will use the Rankin-Selberg L-functions L(s, π π ) as evelope by Jacquet, Piatetski-Shapiro, an Shalika [4], Shahii [20], an Moeglin an Walspurger [12], 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 given by local factors: (2.1) L(s, π π ) = p L p (s, π p π p) where m m L p (s, π p π p) ( = 1 απ (p, j)ᾱ π (p, k)p s) 1. j=1 k=1 The Archimeean local factor L (s, π π ) is efine by L (s, π π ) = m m j=1 k=1 Γ R (s + µ π π (j, k)) where the complex numbers µ π π (j, k) satisfy the trivial boun (2.2) Re µ π π (j, k) > 1. Denote Φ(s, π π ) = L (s, π π )L(s, π π ). We will nee the following properties of the L-functions L(s, π π ) an Φ(s, π π ). RS1. The Euler prouct for L(s, π π ) in (2.1) converges absolutely for Re s > 1 (Jacquet an Shalika [5]). 5

6 RS2. The complete L-function Φ(s, π π ) has an analytic continuation to the entire complex plane an satisfies a functional equation Φ(s, π π ) = ε(s, π π )Φ(1 s, π π ), with ε(s, π π ) = τ(π π )Q s π π where Q π π > 0 an τ(π π ) = ±Q 1/2 π π (Shahii [20], [21], [22], an [23]). RS3. Denote α(g) = et(g). When π = π α it for any t R, Φ(s, π π ) is holomorphic. When m = m an π = π α for some τ 0 R, the only poles of Φ(s, π π ) are simple poles at s = an 1 + coming from L(s, π π ) (Jacquet an Shalika [5], [6], Moeglin an Walspurger [12]). We will only consier the latter case in the proof of Theorem 1.3. RS4. Φ(s, π π ) is meromorphic of orer one away from its poles, an boune in vertical strips (Gelbart an Shahii [2]). RS5. Φ(s, π π ) an L(s, π π ) are non-zero in Re s 1. (Shahii [20]) 3. Estimation of logarithmic erivatives. Let C(m) be the complex plane with the iscs s 2n + µ π π (j, k) < 1, n 0, 1 j, k m, 8m2 exclue. Here we give a remark about the structure of C(m). For j, k = 1,, m, enote by β(j, k) the fractional part of Re(µ π π (j, k)). In aition we let β(0, 0) = 0 an β(m + 1, m + 1) = 1. Then all β(j, k) [0, 1], an hence there exist β(j 1, k 1 ), β(j 2, k 2 ) such that β(j 2, k 2 ) β(j 1, k 1 ) 1/(3m 2 ) an there is no β(j, k) lying between β(j 1, k 1 ) an β(j 2, k 2 ). It follows that the strip S 0 = {s : β(j 1, k 1 ) + 1/(8m 2 ) Re s β(j 2, k 2 ) 1/(8m 2 )} is containe in C(m). Consequently, for all n = 0, 1, 2,, the strips (3.1) S n = {s : n + β(j 1, k 1 ) + 1/(8m 2 ) Re s n + β(j 2, k 2 ) 1/(8m 2 )} are subsets of C(m). This structure of C(m) will be use later. In Liu an Ye [10] an [11], we prove the following lemmas. It is believe that a much sharper result than Lemma 3.1 coul be obtaine using Selberg s explicit formula. This an generalization to a wier class of L-functions will be stuie in a subsequent paper. Lemma 3.1. Assume m = m an π = π α for some nonzero τ 0 R. (a) Let T > 2. The number N(T ) of zeros of L(s, π π ) in the region 0 Re s 1, Im s T satisfies N(T + 1) N(T ) log(q π π T ) 6

7 an N(T ) T log(q π π T ). (b) For any T > 2, we have T Im ρ >1 1 (T Im ρ) 2 log(q π π T ). (c) Let s = σ + it with 2 σ 2, t > 2. If s C(m) is not a zero of L(s, π π ), then s logl(s, π π ) = t Im ρ 1 1 s ρ 1 s 1 1 s + O ( log(q π π t ) ). () For T > 2, there exists τ with T τ T + 1 such that when 2 σ 2 s log L(σ + iτ, π π ) log 2 (Q π π τ ). (e) For T > 2, there exists τ with T τ T + 1 such that when 2 σ 2 2 s 2 log L(σ + iτ, π π ) log 3 (Q π π τ ). Lemma 3.2. Assume m = m an π = π α s is in some strip S n as in (3.1) with n 2, then for some nonzero τ 0 R as before. If 2 s 2 log L(s, π π ) m Zero free regions. We nee a zero free region for the Rankin-Selberg L-function L(s, π π ) which was prove by Moreno [13] an [14]. See also Gelbart, Lapi, an Sarnak [1], an Sarnak [18]. In orer for later usage, we formulate the theorem for automorphic L-functions attache to cuspial representations of GL m over an algebraic number fiel F. Similar formulation can also be mae to Moreno s zero free region near the possible pole. As in [13] an [14], the constant c in (4.1) below can be mae more precise in terms of the infinite types of the representations. Denote by q v the number of elements in the resiue fiel of F v at a non Archimeean place v of F. Let π an π be any automorphic irreucible cuspial representations of GL m (F A ) an GL m (F A ), respectively. Then their Rankin-Selberg L-function is efine by L(s, π π ) = L v (s, π v π v ) v< 7

8 where L v (s, π v π v) = m m ( 1 απ (v, j)ᾱ π (v, k)q s j=1 k=1 for Re s > 1 an by analytic continuation to C. We efine its Archimeean local factors in the stanar way. This Rankin-Selberg L-function satisfies the same properties RS1 through RS5, an the lemmas in 3 also hol. Theorem 4.1. Let π (resp. π ) be any automorphic irreucible cuspial representation of GL m (F A ) (resp. GL m (F A )). Assume that at least one of π an π is self-contragreient: π = π or π = π. Then there is an effectively computable constant c such that there is no zero of L(s, π π ) in the region (4.1) σ 1 c, t 1. log(q t + 1) Here Q is the largest of Q π π, Q π π = Q π π, an Q π π. v ) 1 5. Proof of Theorem 1.3. We now prove Theorem 1.3 when π = π α for some τ 0 R. The proof for case of π an π being not twiste equivalent, in particular, when m m, is the same with all terms relate to τ 0 remove. By RS1, we have for Re s > 1 that an therefore s log L(s, π π ) = K(s) := 2 s 2 log L(s, π π ) = n=1 n=1 Λ(n)a π (n)ā π (n) n s, (log n)λ(n)a π (n)ā π (n) n s. By RS3 an RS5, K(s) is holomorphic in Re s > 1. On Re s = 1, L(s, π π ) is nonzero (RS5) an has only a simple pole at s = 1 +. Thus (5.1) K(s) = 1 (s 1 ) 2 + G(s) has only a ouble pole in Re s 1, where G(s) is analytic for Re s 1. On C, K(s) also has a ouble pole at each of the pole at, trivial zeros, an nontrivial zeros of L(s, π π ). By Conjecture 1.2 an (1.1), we have a π (p k ) m, a π (p k ) m. Therefore, (log n)λ(n)a π (n)ā π (n) mm log 2 n, 8

9 an for σ > 1, n=1 (log n)λ(n)a π (n)ā π (n) n σ 1 (σ 1) 2. Let T = exp( log x) an set b = 1 + 1/ log x. By Perron s summation formula (see e.g. Theorem 5.1 in [7]), we have (5.2) (log n)λ(n)a π (n)ā π (n) n x = 1 2πi = 1 2πi b+it b it b+it b it ( K(s) xs s s + O x b ) ( x log 3 x ) + O T (b 1) 2 T ( K(s) xs x log 3 s s + O x ). T Here we use the Ramanujan Conjecture 1.2 to control the size of error terms in (5.2). Choose a with 2 < a < 1 such that the line Re s = a is containe in the strip S 2 C(m); this is guarantee by the structure of C(m). Let T be the τ such that Lemma 3.1(e) hols, by aing a constant with 0 < < 1 to our T = exp( log x) if necessary. Now we consier the contour C 1 : b σ a, t = T ; C 2 : σ = a, T t T ; C 3 : a σ b, t = T. Note that the two poles, some trivial zeros, an certain nontrivial zeros of L(s, π π ), as well as the pole at s = 0 are passe by the shifting of the contour. The trivial zeros can be etermine by the functional equation in RS2: s = µ π π (j, k) with a < 1 Re(µ π π (j, k)) < 0. Here we use the facts that Re(µ π π (j, k)) > 1 an 2 < a < 1. 9

10 Then we have (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) 1 2πi b+it b it K(s) xs s s = 1 ( 2πi C 1 + C 2 + xs + Res K(s) s=0 s C 3 + Res K(s) xs s=,1+ s + )K(s) xs s s a+1< Re(µ π π (j,k))<1 + Im ρ T xs Res K(s) s=ρ s. Res s= µ π π (j,k) K(s)xs s By Lemma 3.1(e), we get (5.9) C 1 b a log 3 (Q π π T ) xσ T σ x log3 (Q π π T ), T log x an the same upper boun also hols for the integral on C 3. By Lemma 3.2, then (5.10) Obviously, (5.5) is C 2 T T x a log T t t + 1 x. xs (5.11) Res K(s) s=0 s = K(1). Since the poles at s = an s = 1 + are ouble poles, the resiues in (5.6) give lim s s (s ) 2 K(s) xs For the secon term, we have, by (5.1), lim s 1+ = lim s 1+ = lim s 1+ = x1+ s + lim s 1+ s (s 1 ) 2 K(s) xs s. s (s 1 ) 2 K(s) xs s ( 1 + (s 1 iτ0 ) 2 G(s) ) x s s s x s s s (5.12) log x x1+iτ0 1 + (1 + ) 2. 10

11 The first term can be estimate similarly, an we get Consequently (5.6) is lim s s (s ) 2 K(s) xs s = lim s x s s s log x. (5.13) x 1+ log x x1+iτ0 + O(log x). 1 + (1 + ) 2 Near a trivial zero s = µ π π (j, k) of orer l in (5.7), we can express K(s) as l/(s + µ π π (j, k)) 2 plus an analytic function, like in (5.1). The resiues in (5.7) can therefore compute similar to what we i in (5.12). By (2.2), we know that Re(µ π π (j, k)) δ 1 for some δ > 0. Consequently (5.7) is boune (5.14) Res s= µ π π (j,k) a< 1 Re(µ π π (j,k))<0 K(s) xs s x1 δ log x. To compute the resiues corresponing to nontrivial zeros in (5.8), we note that Φ(s, π π ) is of orer 1 (RS4), an Φ(1, π π ) 0 (RS5). Using a stanar argument, we see that Consequently, (5.8) becomes Im ρ T γ T xs Res K(s) s=ρ s = 1 ρ log2 T. Im ρ T Im ρ T Res s=ρ xρ log x. ρ Using Moreno s zero free region in Theorem 4.1, we get ( (5.15) x exp c log x ) (log x) log 2 T log T 1 x s (s ρ) 2 s By taking T = exp( log x) + for some with 0 < < 1, we can boun (5.9), (5.10), (5.15), an the error term in (5.2) by O{x exp( c 2 log x)} for the c in Theorem 4.1. Using the main term from (5.13) an other error terms from (5.10), (5.11), (5.14), an (5.15), we get a proof of Theorem 1.3. References [1] S.S. Gelbart, E.M. Lapi, an P. Sarnak, A new metho for lower bouns of L-functions, C.R. Aca. Sci. Paris, Ser. I 339 (2004),

12 [2] S.S. Gelbart an F. Shahii, Bouneness of automorphic L-functions in vertical strips, J. Amer. Math. Soc., 14 (2001), [3] R. Goement an H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math., 260, Springer-Verlag, Berlin, [4] H. Jacquet, I.I. Piatetski-Shapiro, an J. Shalika, Rankin-Selberg convolutions, Amer. J. Math., 105 (1983), [5] H. Jacquet an J.A. Shalika, On Euler proucts an the classification of automorphic representations I, Amer. J. Math., 103 (1981), [6] H. Jacquet an J.A. Shalika, On Euler proucts an the classification of automorphic representations II, Amer. J. Math., 103 (1981), [7] A.A. Karatsuba, Basic Analytic Number Theory, Springer, Berlin [8] H. Kim, Functoriality for the exterior square of GL 4 an the symmetric fourth of GL 2, J. Amer. Math. Soc., 16 (2003), [9] H. Kim an P. Sarnak, Refine estimates towars the Ramanujan an Selberg conjectures, Appenix to Kim [8]. [10] Jianya Liu an Yangbo Ye, Superposition of zeros of istinct L-functions, Forum Math., 14 (2002), [11] Jianya Liu an Yangbo Ye, Weighte Selberg orthogonality an uniqueness of factorization of automorphic L-functions, preprint. [12] C. Moeglin an J.-L. Walspurger, Le spectre résiuel e GL(n), Ann. Sci. École Norm. Sup., (4) 22 (1989), [13] C.J. Moreno, Explicit formulas in the theory of automorphic forms, Lecture Notes Math. vol. 626, Springer, Berlin, 1977, [14] C.J. Moreno, Analytic proof of the strong multiplicity one theorem, Amer. J. Math., 107 (1985), [15] M. Ram Murty, Selberg s conjectures an Artin L-functions, Bull. Amer. Math. Soc., 31 (1994), [16] M. Ram Murty, Selberg s conjectures an Artin L-functions II, Current trens in mathematics an physics, Narosa, New Delhi, 1995, [17] Z. Runick an P. Sarnak, Zeros of principal L-functions an ranom matrix theory, Duke Math. J., 81 (1996), [18] P. Sarnak, Non-vanishing of L-functions on R(s) = 1, Shalika s 60th Birthay, preprint,

13 [19] A. Selberg, Ol an new conjectures an results about a class of Dirichlet series, Collecte Papers, vol. II, Springer, 1991, [20] F. Shahii, On certain L-functions, Amer. J. Math., 103 (1981), [21] F. Shahii, Fourier transforms of intertwining operators an Plancherel measures for GL(n), Amer. J. Math., 106 (1984), [22] F. Shahii, Local coefficients as Artin factors for real groups, Duke Math. J., 52 (1985), [23] F. Shahii, A proof of Langlans conjecture on Plancherel measures; Complementary series for p-aic groups, Ann. Math., 132 (1990), Jianya Liu: Department of Mathematics, Shanong University Jinan , China. jyliu@su.eu.cn Yangbo Ye: Department of Mathematics, The University of Iowa Iowa City, Iowa , USA. yey@math.uiowa.eu 13