THE FOURTH POWER MOMENT OF AUTOMORPHIC L-FUNCTIONS FOR GL(2) OVER A SHORT INTERVAL YANGBO YE

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1 THE FOURTH POWER MOMENT OF AUTOMORPHIC -FUNCTIONS FOR G2 OVER A SHORT INTERVA YANGBO YE Abstract. In this paper we will prove bounds for the fourth power moment in the t aspect over a short interval of automorphic -functions s, g for G2 on the central critical line Re s = 1/2. Here g is a fixed holomorphic or Maass Hecke eigenform for the modular group S 2 Z, or in certain cases, for the Hecke congruence subgroup Γ 0 N with N > 1. The short interval is from a large to + 103/135+ε. The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg -function for Maass forms by Jianya iu and Yangbo Ye [18] and Yuk- am au, Jianya iu, and Yangbo Ye [17], which in turn is relied on the uznetsov formula [16] and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak [21] and by [17]. 1. Introduction. For the Riemann zeta function and Dirichlet -functions, estimates for their power moments on the critical line Re s = 1/2 played central roles in analytic number theory. Classical results on short intervals + α 1 ζ 2 + it 4 dt α+ε were proved for α = 7/8 by Heath-Brown [9] and for 2/3 by Iwaniec [11], for any ε > 0. In this paper, we want to prove a similar result for automorphic -functions attached to a certain holomorphic or Maass cusp form g for Γ 0 N. To describe our results, we need bounds towards the Ramanujan conjecture for Maass forms. In terms of representation theory, let π be an automorphic cuspidal 2000 Mathematics Subject Classification MSC 11F66. Project sponsored by the National Security Agency under Grant Number MDA The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. 1

2 representation of G 2 Q A with unitary central character and local Hecke eigenvalues α j π p for p < and µ j π for p =, j = 1, 2. Then bounds toward the Ramanujan conjecture are α j π 1.1 p p θ for p at which π is unramified, Re µ j π θ if π is unramified at. These bounds were proved for θ = 1/4 by Selberg and uznetsov [16], for θ = 1/5 by Shahidi [22] and uo, Rudnick, and Sarnak [19], for θ = 1/9 by im and Shahidi [14], and most recently for θ = 7/64 by im and Sarnak [13]. The automorphic -functions we will consider are s, g = λ g n n s n 1 1 λg pp s + p 2s 1. 1 λg pp s 1, = p N p N where g is a holomorphic or Maass cusp Hecke eigenform for Γ 0 N, and its twist by a real primitive character χ modulo Q with N Q: s, g χ = n 1 λ g nχn n s = p Q 1 χpλg pp s + p 2s 1 for Re s > 1. Following the setting in Conrey and Iwaniec [2], we will assume that Q is odd and square-free, and χ is the real, primitive character modulo Q, i.e., the Jacobi symbol, so that the twisted cusp form g χ see 2.1 and 2.2 below is a cups form for Γ 0 N 2. As pointed in [2], p.1176, g χ is primitive even if the Hecke eigenform g itself is not primitive. Our results, nevertheless, are valid in other cases, as long as the twisted -function s, g χ has a standard functional equation as in 2.3 cf. Atkin and i [1]. In particular, our theorem below is valid for s, g when N = 1. We will assume that g is self-contragredient. If g is holomorphic, we denote its weight by l. If g is Maass, we denote its aplace eigenvalue by 1/4 + l 2. Theorem 1.1. et g be a fixed self-contragredient holomorphic or Maass Hecke eigenform for Γ 0 N, and χ a real, primitive character mod Q with N Q. Then it, g χ 4 dt ε,n,g,q +ε 2

3 for = 1 1/4+2θ+ε. Here θ is given by bounds toward the Ramanujan conjecture in 1.1, and we can take θ = 7/64 with 1 1/4 + 2θ = 103/135. A subconvexity bound for s, g in the t aspect was deduced by Good for holomorphic cusp form g in [6], [7], and [8], and by Meurman for Maass g in [20]: it, g g 1 + t 1/3 log 5/6 2 + t. The goal of the present paper is not an improvement to this subconvexity bound for s, g. By a standard argument cf. Ivic [10], p.197 though, our Theorem 1.1 implies it, g g 1 + t 1/2 1/16+8θ+ε = 1 + t 119/270+ε. Certainly our 1.3 is not as good as 1.2. Using 1.2, however, one can only get a fourth power moment bound of 4/3 1+ε, not as good as our Theorem 1.1. Subconvexity bounds in the level N aspect and the l aspect were studied extensively by Duke, Friedlander, and Iwaniec [3] [4] [5], and by owalski, Michel, and Vanderam [15]. The proof of Theorem 1.1 is based on an argument in Jianya iu and Yangbo Ye [18] and Yuk-am au, Jianya iu, and Yangbo Ye [17]; see 3 below. [18] subconvexity bounds for Rankin-Selberg -functions s, f g were proved as the aplace eigenvalue of the Maass cusp form f goes to, where g is a fixed holomorphic or Maass cusp form. While the exponent 3 + 2θ/4 + ɛ as claimed in [18] does not hold because of a gap in 4.14 and 4.15, the paper did prove a subconvexity bound 1.4 1/2 + it, f g N,t,g,ɛ k 15+2θ/16+ɛ In as pointed out in the first sentence in In [17] 1.4 was improved to a better bound Ok 1 1/8+4θ+ε. What was done in [18] and [17] is to express 1/2, f g in terms of spectral decomposition of f and g by an approximate functional equation. Using the uznetsov trace formula [16] the spectral sum of f is rewritten in terms of loosterman sums. Therefore the central value of s, f g is essentially expressed as a spectral sum 3

4 of g with loosterman sums as coefficients. An application of bounds for shifted convolution sums of Fourier coefficients of g Sarnak [21] with improvement given in [17] gives a subconvexity bounds for s, f g. What we will proceed in this paper is to consider the continuous spectrum of the aplacian in place of the Maass form f. This approach is motivated by Conrey and Iwaniec [2]. The author would like to express thanks to Chaohua Jia, Jianya iu, and Wenzhi uo for informative discussions, and to the referee for detailed, helpful comments and suggestions. Thanks are also due to the Obermann Center for Advanced Studies, the University of Iowa, for support services and nice academic environment. 2. The approximate functional equation. We know that the twisted -function s, g χ is entire, where χ is a real, primitive character modulo Q with N Q. Note that s, g χ is indeed the -function attached to a twisted cusp form g χ. It is 2.1 g χ z = n k 1/2 χnλ g nenz, n 1 when g is holomorphic, and 2.2 g χ z = y 1/2 n 0 χnλ g n il 2π n yenx when g is Maass. In any case, denote by the complete -function, where Λs, g χ = s, g χs, g χ s, g χ = 2 Γ R s + µ gχ j j=1 with Γ R s = π s/2 Γs/2. Here µ gχ 1 and µ gχ 2 are complex numbers associated to g χ at. According to Conrey and Iwaniec [2], p.1188, our twisted cusp form g χ satisfies the standard functional equation 2.3 Λs, g χ = εs, g χ Λ1 s, g χ, 4

5 where εs, g χ = τg χ Q s g χ. Here Q gχ > 0 is the conductor of g χ and τg χ C satisfies τg χ τ g χ = Q gχ. Since g is self-contragredient and χ is real, we have τg χ 2 = Q gχ. We actually want a functional equation for a product of two such -functions: s + ir, g χs ir, g χ = γs1 s ir, g χ1 s + ir, g χ, where according to 2.3 γs =τg χ 2 Q 2s g χ π 2 2s 1 = Q gχ 2 j=1 2 j=1 Γ R 1 s + ir + µ gχ jγ R 1 s ir + µ gχ j Γ R s + ir + µ gχ jγ R s ir + µ gχ j Γ1 s + ir + µ gχ j/2γ1 s ir + µ gχ j/2. Γs + ir + µ gχ j/2γs ir + µ gχ j/2 By Stirling s formula, we get see similar computations in [21] and [18] 4π 2 /Q gχ 2s 1 γs = 1 + ηr s µ gχ 1 + r 2 1/2 µ gχ 2 + r 2 1/2 where the error term η r s 1 + s 3 /1 + r. We will consider the case of large r with fixed g; hence γs is asymptotically Q gχ r 2 /4π 2 1 2s. Following [21] and [18] again, we can express the central value of s + ir, g χs ir, g χ as ir, g χ 2 ir, g χ = 1 X s πi 2 + s + ir, g χ 2 + s ir, g χ Gs ds s where Re s=2 + O Re s=2 X s 1 η r 2 s 2 + s + ir, g χ 2 + s ir, g χ Gs ds, s X = Q g χ 4π 2 µgχ 1 + r 2 1/2 µgχ 2 + r 2 1/2. Here Gs is an analytic function in B Re s B for a fixed B > 0 satisfying G0 = 1, Gs = G s, Gs 1 + s A for a fixed large constant A. We note that X is real and positive, because of our assumption on g being self-contragredient. 5

6 We may shift the contour in the integral of the big O term in 2.4 to Re s = 1/2 + ε. This way X s r 1+ε. Recall that η r 1/2 s 1 + s 3 /1 + r. Moreover, 2 + s + ir, g χ 2 + s ir, g χ ε,g 1 as r, for Re s = 1/2 + ε, because its Dirichlet series is absolutely convergent. All these show that the big O term in 2.4 is r ε. To compute the main term in 2.4, we expand 1/2 + s + ir, g χ1/2 + s ir, g χ into its Dirichlet series. For Re s > 1/2, we have s + ir, g χ 2 + s ir, g χ = m,n 1 As g is a Hecke eigenform, we have λ g mλ g n χmn m 1/2+s+ir n. 1/2+s ir λ g mλ g n = d m,n λ g mn d 2. Apply this to the right side of 2.5 and set m = ad, n = bd. Then s + ir, g χ 2 + s ir, g χ = a,b,d 1 χabd 2 λ g ab a 1/2+s+ir b 1/2+s ir d 1+2s =1 + 2s, χ 2 n 1 χnλ g n d n 1/2+s ir n where d s n = a s. b ab= n We remark that in 2.6, the series is actually taken over n which are relatively prime to Q. Consider the Eisenstein series for any, fixed cusp a of Γ = Γ 0 N defined by E a z, s = γ Γ a \Γ 6 Im σ 1 a γz s

7 for Re s > 1 and by analytic continuation for all s C. Here Γ a is the stability group of a, while σ a S2, R is given by σ a = a and σa 1Γ aσ a = Γ. This Eisenstein series is an eigenfunction of the Hecke operators T n E a z, s = η a n, se a z, s, if n, N = 1. As pointed out in Conrey and Iwaniec [2], for any n relatively prime to N, η a n, s = d s 1/2 n. Consequently, from 2.6 we get 2.7 for Re s > 1/2. Denote 2 + s + ir, g χ 2 + s ir, g χ =1 + 2s, χ 2 n 1 χnλ g nη a n, 1/2 + ir n 1/2+s Substituting 2.7 into the integral of the main term in 2.4, we get 2 + ir, g χ 2 ir, g χ = 1 χnm 2 λ g nη a n, 1/2 + ir X s Gs ds πi nm 2 1/2+s s + Orε Re s=2 =2 m,n 1 m,n 1 χnm 2 λ g nη a n, 1/2 + ir m n V y = 1 2πi 1 2πi Re s=2 Re s=2 s ds Gsy s. nm 2 s ds Gs X s + Orε Then as in [21] and [18], lim y 0 V y = 1 and V y B 1 + y B because of our choice of the function Gs. Therefore ir, g χ 2 ir, g χ =2 χ2 m m 1 m X 1/2+ε n 1 + Or ε, χnλ g nη a n, 1/2 + ir V n nm 2 X because the outer series is negligible if taken over m > X 1/2+ε. 7

8 3. Averaging and the uznetsov trace formula. According to 2.8, estimation of the central value of our -function is reduced to estimation of S Y g, r = n n χnλ g nη a n, 1/2 + irh Y for fixed g, where H is a fixed smooth function of compact support contained in 1, 2. To prove our results on short intervals, let be a number which satisfies /4 for large. et ht be an even analytic function in Im t 1/2 satisfying h n t 1 + t N for any N > 0 in this region. Thus h is a Schwartz function on R. We also assume that ht 0 for real t. For example, we may simply take ht = 1/ cosht. Denote ζ N s = p N1 p s 1. We want to estimate 3.1 r + r SY I, = h + h g, r 2 ζ N 1 + 2ir 2 ζ1 + 2ir 2 R = n m χn χmλ g n λ g mh H Y Y m,n R r + r h + h η a n, 1/2 + ir η a m, 1/2 + ir ζ N 1 + 2ir 2 ζ1 + 2ir 2 dr. dr As in iu and Ye [18], we apply the uznetsov trace formula to the integral on the right side of 3.1: 3.2 π n m χn χmλ g n λ g mh H Y Y m,n 3.3 kj + kj h + h λ fj n λ fj m f j + n m χn χmλ g n λ g mh H Y Y m,n 8

9 R δ n,m π r h R + r + h η a n, 1/2 + ir η a m, 1/2 + ir ζ N 1 + 2ir 2 ζ1 + 2ir 2 dr = n m χn χmλ g n λ g mh H Y Y m,n rdr r tanhπr h n Y +2i m,n χn χmλ g n λ g mh Sn, m; c c c 1 R r + r h + h 4π nm J 2ir c + r + h m H Y rdr coshπr. Here in 3.2 f j are Hecke eigenforms, with aplace eigenvalues 1/4+kj 2 and Fourier coefficients λ fj n, which form an orthonormal basis of the space of Maass cusp forms for Γ 0 N, while in 3.5 Sn, m; c is the classical loosterman sum. Recall that χnλ g n is the nth Fourier coefficient of the twisted cusp form g χ as in 2.1 or 2.2. We want to apply the main estimation in iu and Ye [18] and au, iu, and Ye [17], 2.2, to our 3.4 and 3.5 above. Note that these estimations are based on bounds for shifted convolution sums of Fourier coefficients of cusp forms proved by Sarnak [21], Appendix, and by au, iu, and Ye [17]. More precisely, Y 1+ε for = 1 1/4+2θ+ε. Since 3.2 and 3.3 are both positive, this implies that 3.3, i.e., I,, is bounded by OY 1+ε for the same. By ζ1 + 2ir log1 + r, this estimate of I, implies that SY g, r 2 dr Y 1+ε for the above. Now we can go back to the fourth power moment of 1/2 + ir, g χ. Since g 9

10 is self-contragredient and χ is real, we have from 2.8 that ir, g χ 4 dr = χ2 m m 1 m X 1/2+ε n ir, g χ 2 + ir, g χ 2 dr χnλ g nη a n, 1/2 + ir V n nm 2 2 dr. X Here we can take X = 2 and get ir, g χ 4 dr + χnλ g nη a n, 1/2 + ir χ 2 m V nm 2 / 2 2 dr. nm2 / n m 1+ε Now we apply a smooth dyadic subdivision to χ 2 m V nm 2 / 2 nm2 /, 2 1 m 1+ε by dividing the interval [1, 1+ε ] into subintervals of the form [a, 1.8a] and covering, with overlapping, each subinterval by a smooth, nonnegative function of compact support. The total number of subinterval is Olog. This way, we can find a smooth function H of compact support in, 2 so that + log ir, g χ 4 dr + max 1 B 2+ε n 1 n 2 χnλ g nη a n, 1/2 + irh dr. 2 /B The sum inside the absolute value signs is indeed S 2 /Bg, r. By 3.6, the maximum contribution is from B = 1: ir, g χ 4 log dr 2 + S 2g, r 2 dr log ε +ε 10

11 for = 1 1/4+2θ+ε. This completes the proof of Theorem 1.1. References [1] A.O.. Atkin and Wen-Ch ing W. i, Twists of newforms and pseudo-eigenvalues of W -operators, Invent. Math., , [2].B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic -functions, Ann. Math., , [3] W. Duke, J.B. Friedlander, and H. Iwaniec, Bounds for automorphic -functions, Invent. Math., , 1-8. [4] W. Duke, J.B. Friedlander, and H. Iwaniec, Bounds for automorphic -functions II, Invent. Math., , ; erratum, ibid , [5] W. Duke, J.B. Friedlander, and H. Iwaniec, Bounds for automorphic -functions III, Invent. Math., , [6] A. Good, Beiträge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind, J. Number Theory, , [7] A. Good, The square mean of Dirichlet series associated with cusp forms, Mathematika, , [8] A. Good, The convolution method for Dirichlet series, in: D. Hejhal, P. Sarnak, A. Terras Eds., The Selberg trace formula and related topics Brunswick, Maine, 1984, Amer. Math. Soc. Contemporary Mathematics, Vol. 53, Amer. Math. Soc., Provindence, RI, 1986, [9] D.R. Heath-Brown, The fourth power moment of the Riemann zeta-function, Proc. ondon Math. Soc., , [10] A. Ivic, The Riemann zeta-function The theory of the Riemann zeta-function with applications, John Wiley & Sons, New York [11] H. Iwaniec, Fourier coefficients of cusp forms and the Riemann zeta function, Séminaire de Théorie des Nombres de Bordeaux, Année , exposé n 18, [12] H. im, Functoriality for the exterior square of G 4 and the symmetric fourth of G 2, J. Amer. Math. Soc., ,

12 [13] H. im and P. Sarnak, Appendix: Refined estimates towards the Ramanujan and Selberg conjectures, Appendix to im [12]. [14] H. im and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J., , [15] E. owalski, P. Michel, and J. Vanderkam, Mollification of the fourth moment of automorphic -functions and arithmetic applications, Invent. Math., , [16] N.V. uznetsov, Petersson s conjecture for cusp forms of weight zero and innik s conjecture. Sums of loosterman sums, Math. U.S.S.R. Sbornik, , [17] Yuk-am au, Jianya iu, and Yangbo Ye, Improvement to subconvexity bounds for Rankin-Selberg -functions of cusp forms, preprint, [18] Jianya iu and Yangbo Ye, Subconvexity for Rankin-Selberg -functions of Maass forms, Geom. Funct. Anal., , [19] W. uo, Z. Rudnick, and P. Sarnak, On the generalized Ramanujan conjecture for Gn, in: Automorphic forms, automorphic representations, and arithmetic Fort Worth, TX, 1996, , Proc. Sympos. Pure Math., 66, Part 2, Amer. Math. Soc., Providence, RI, [20] T. Meurman, On the order of the Maass -function on the critical line, in:. Györy, G. Halász Eds., Number Theory, Vol. I Budapest, 1987, Colloquium of Mathematical Society Janos Bolyai, Vol. 51, North-Holland, Amsterdam, 1990, [21] P. Sarnak, Estimates for Rankin-Selberg -functions and quantum unique ergodicity, J. Functional Analysis, , [22] F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain -functions, Ann. of Math., , no. 3, Department of Mathematics The University of Iowa, Iowa City, Iowa , USA. yey@math.uiowa.edu 12