REMARK ON THE PAPER ON PRODUCTS OF FOURIER COEFFICIENTS OF CUSP FORMS 1. INTRODUCTION

Transcription

1 REMARK ON THE PAPER ON PRODUCTS OF FOURIER COEFFICIENTS OF CUSP FORMS YUK-KAM LAU, YINGNAN WANG, DEYU ZHANG ABSTRACT. Le a(n) be he Fourier coefficien of a holomorphic cusp form on some discree subgroup of SL (R). This noe is o refine a recen resul of Hofmann and Kohnen on he non-posiive (and non-negaive resp.) produc of a(n)a(n + r) for a fixed posiive ineger r. 1. INTRODUCTION Le Γ be a subgroup of SL (R). Assume as in [] ha (i) Γ is a finiely generaed Fuchsian group of he firs kind, (ii) I Γ where ( I is he ideniy, 1 b (iii) Γ conains exacly if b is an ineger. 1) Condiions (ii) and (iii) ( may be formulaed as: Γ has a cusp a i and is sabilizer 1 1 Γ i is generaed by ±. 1) Le k > be any even ineger. Wrie S k (Γ) for he space of all ellipic cusp forms of weigh k on Γ (wih rivial muliplier sysem). Throughou, a cusp form is acily assumed o be holomorphic. Suppose all he coefficiens of f in is Fourier expansion a i are real. In [], Hofmann and Kohnen showed he infiniude of non-vanishing erms in he sequence {a f (n)a f (n + r)} n 1. Moreover, when Γ is a congruence subgroup, hey showed ha he sequence has infiniely many non-negaive (resp. nonposiive) erms, i.e. for ɛ = + or respecively, (1.1) C ɛ f,r(x) := #{n [1, x] : ɛ a f (n)a f (n + r) } as x, where #{ } denoes he cardinaliy of he se { }. I is also poined ou ha (1.1) holds for more general, bu sill resriced as in [5], subgroups Γ; in paricular, i requires he discree subgroups Γ (considered here) ha have a cusp a. This noe aims a some refinemen: Firs we show ha (1.1) will hold when Γ saisfies merely Condiions (i)-(iii). Second we give a quaniaive version for he case of congruence subgroups a lower bound for C ± f,r (x). Dae: December 8, 16. Mahemaics Subjec Classificaion. 11F1, 11F3. Key words and phrases. Fourier coefficiens, Inegral weigh modular forms To appear in Archiv der Mahemaik. 1

2 Y,-K. LAU, Y. WANG, D. ZHANG Theorem 1.1. Suppose Γ saisfies Condiions (i)-(iii) and f is a cusp form of even inegral weigh k > on Γ whose Fourier coefficiens are all real. For any r N, we have (x) as x. C ± f,r Theorem 1.. Le Γ be a congruence subgroup and f a cusp form of even inegral weigh k > on Γ. Suppose all he Fourier coefficiens of f are real. There exis posiive consans c 1 = c 1 (f, r) and x 1 = x 1 (f, r) such ha for all x x 1, a f (n 1 )a f (n 1 + r) and a f (n )a f (n + r) for some inegers n 1, n (x, x + c 1 x 1/ ]. In paricular, we have C ± f,r (x) f,r x 1/ for all x x 1. Remark Compared wih [, Theorem ], he condiion on Γ here is relaxed in Theorem 1.1 while an explici lower bound is given in Theorem 1. (for congruence subgroup Γ).. Boh resuls are derived, similarly o he argumen in [], via couning he sign changes of a f (n) in arihmeic progressions A = A a,r where (1.) A a,r := {n N : n a mod r} for r N and a Z. However we deec he sign-changes in ways differen han he mehod in [5] (used in []). 3. In Theorem 1., f is no necessarily a Hecke eigenform or primiive form. The Fourier coefficiens of a primiive form are muliplicaive. Recenly Maomäki and Radziwill [8] obained very srong resuls on sign-changes via heir heory on muliplicaive funcions. Hence i is plausible o ge a lower bound much beer han x 1/ for primiive forms. 4. Our resuls hold for maass cusp forms (of weigh and rivial muliplier sysem) wih real coefficiens. In view of he proof (in Secion 4), he analogue of Theorem 1. is clear while for Theorem 1.1, one may apply [9, Theorem 5.1], he poinwise bound a(n) n /5+ε in [1, Corollary 1] and [4, (8.3)] insead (cf. Secion ).. PROOF OF THEOREM 1.1 Le f be given as in Theorem 1.1. By [1, Theorem ] and is Corollary, we have a f (n) n (k 1)/+1/3 and a f (n) C f x k as x n x where C f > is a consan depending on f. Thus, here exiss 1 a r such ha (.1) a f (n) f,r x k/+1/6. x/<n x n a mod r

3 REMARK ON THE PAPER ON PRODUCTS OF FOURIER COEFFICIENTS OF CUSP FORMS 3 On he oher hand, from [3, Corollary 5.4], i follows ha for any 1 b r, a f (n) x k/ log x. n x n b mod r Hence for all large x, here are u, v (x, x] A a,r for some a such ha a f (u)a f (v) <, and consequenly a f (n)a f (n+r) for some n (x, x] A a,r, implying he infiniude of non-posiive a f (n)a f (n + r). Suppose a f (n)a f (n + r) < for all n. Then a f (n)a f (n + r) > for all n which conradics o he las asserion wih r in place of r. This complees he proof. Define for M, N N, Γ (M, N) = Γ(M, N) = 3. PRELIMINARIES FOR THEOREM 1. { } SL c d (Z) : M c, N b, { } Γ c d (M, N) : a d 1 (mod [M, N]) where [M, N] denoes he leas common muliple of M and N. Then Γ (M, 1) = Γ (M), Γ(M, 1) = Γ 1 (M) and Γ(M, M) = Γ(M) of he usual noaion. Recall a subgroup Γ of SL (Z) is a congruence subgroup of level N if Γ(N) Γ for some N N. Also we wrie ( 1 x 1 n(x) = and W =. 1) 1 For any γ = GL c d (R) and any funcion g, define g k cz + d γ (z) = g(γz) where γz = az + b de γ cz + d. Lemma 3.1. (i) Suppose f S k (G(M, N)) where G(M, N) = Γ (M, N) or Γ(M, N). We have f W S k (G(N, M)). (ii) If f S k (Γ(M, N)) and r N, hen f n( Nu r ) is a cusp form on Γ(Mr, N) for any u (mod r). (iii) Le 1 A be he characerisic funcion on A = A a,r (cf. (1.)). The wis f 1 A (z) := n 1 a f (n)1 A (n)e(nz/n) is a cusp form on Γ(Mr, N). d c Proof. (i) Le γ =. Then W γw c d 1 =. So W γw b a 1 G(M, N) if γ is belonged o G(N, M). In [7], he produc MN is used insead of he leas common muliple.

4 4 Y,-K. LAU, Y. WANG, D. ZHANG (ii) For any γ = Γ(Mr c d, N), ( 1 Nu/r 1 Nu/r a + N cu = r 1 c d 1 c ) b + Nu d a N cnu r r d cnu r which is in Γ(M, N) as a d 1 mod [Mr, N] and c mod Mr. Thus, f n(nu/r)γ = f n(nu/r). Our asserion follows readily. (iii) I is a direc consequence from (ii) and he fac f 1 A (z) = 1 au e f r r. u mod r n( Nu r ) Any cusp form f on Γ(M, N) has a Fourier expansion a i, f(z) = n 1 a f (n)e(nz/n) where e(x) = e πix. Moreover is associaed L-funcion L(s, f) := n 1 is enire and saisfies a funcional equaion. a f (n) n s n (k 1)/ Lemma 3.. Suppose f S k (Γ(M, N)) and le g = f W. Then s N Γ(s + k 1 k 1 M )L(s, f) = i k 1 s M Γ(1 s + k 1 )L(1 s, g). π N π Proof. For sufficienly large Re s, we have s+(k 1)/ d f(i) = s+(k 1)/ N Γ(s + k 1 )L(s, f). π The cuspidaliy of f ensures he absolue convergence of he inegral for all s C. Wrie g(z) = n 1 a g(n)e(nz/m) for g = f W S k (Γ(N, M)) by Lemma 3.1 (i). Wih a change of variable ino 1/, i is apparen ha f( 1 d ) (s+(k 1)/) i = (i) k f W (i) = i k n 1 a g (n) (s+(k 1)/) d e πn/m 1 s+(k 1)/ d yielding he funcional equaion. = i k ( M π ) 1 s+(k 1)/ Γ(1 s + k 1 )L(1 s, g),

5 REMARK ON THE PAPER ON PRODUCTS OF FOURIER COEFFICIENTS OF CUSP FORMS 5 4. PROOF OF THEOREM 1. Le us assume more generally f S k (Γ(M, N)). Wrie A = A a,r for any given a<r. Lemma 3.1 (iii) implies F := f 1 A S k (Γ(Mr, N)), hus is L-funcion L(s, F ) is enire and saisfies a funcional equaion wih gamma facors. Separaing ino he wo cases of F = or no, he following proposiion follows from [6, Remark (iii)] wih a n = a F (n)/n (k 1)/. Proposiion 4.1. There exis posiive consans c = c (F ) and x = x (F ) such ha a F (u)a F (v) for some u, v (x, x + c x 1/ ] for all x x. Arguing as before, we apply he proposiion wih a =. There exis consans c and x such ha for all x x, a f (n)a f (n + l) for some n (x, x + c x 1/ ] A,l (l = r or r). If a f (n)a f (n + r) < for all n A,r (x, x + c x 1/ ], hen all a f (n) wih n A,r (x, x + c x 1/ ] are nonzero and have he same sign, conradicing o Proposiion 4.1 for f 1 A,r. Now we are done. Acknowledgemens Lau is suppored by GRF of he Research Grans Council of Hong Kong. Wang is suppored by he Naional Naural Science Foundaion of China (Gran No ), Guangdong Province Naural Science Foundaion (Gran No. 15A33141) and Naural Science Foundaion of Shenzhen Universiy (Gran No. 1541). Zhang is suppored by Naural Science Foundaion of Shandong Province (Gran No. ZR15AM1) and he Naional Naural Science Foundaion of China (Gran No ). This paper is sared during he visi of Wang and Zhang a The Universiy of Hong Kong (HKU) in 16. They would like o hank he Deparmen of Mahemaics a HKU for hospialiy and excellen working condiions.

6 6 Y,-K. LAU, Y. WANG, D. ZHANG REFERENCES [1] A. Good, Cusp forms and eigenfuncions of he Laplacian, Mah. Ann. 55 (1981), [] E. Hofmann and W. Kohnen, On producs of Fourier coefficiens of cusp forms, o appear in Forum Mah., available a ArXiv, hp://arxiv.org/pdf/ pdf. [3] H. Iwaniec, Topics in classical auomorphic forms, Graduae Sudies in Mahemaics, 17 American Mahemaical Sociey, Providence, RI, [4] H. Iwaniec, Specral mehods of auomorphic forms, nd ediion, Graduae Sudies in Mahemaics, 53 American Mahemaical Sociey, Providence, RI,. [5] M. Knopp, W. Kohnen and W. Pribikin, On he signs of Fourier coefficiens of cusp forms, Rankin memorial issues, Ramanujan J. 7 (3), [6] Y.-K. Lau, J. Liu and J. Wu, Local behavior of arihmeical funcions wih applicaions o auomorphic L-funcions, manuscrip, acceped for publicaion in IMRN, available a hp://hkumah.hku.hk/ imr/imrpreprinseries/16/imr16-4.pdf. [7] W.C.W. Li, Newforms and funcional equaions, Mah. Ann. 1 (1975), [8] K. Maomäki and M. Radziwill, Muliplicaive funcions in shor inervals, Ann. of Mah., o appear. [9] W.A. Müller, The Rankin-Selberg mehod for non-holomorphic auomorphic forms, J. Number Theory 51 (1995), [1] Y.N. Peridis, On squares of eigenfuncions for he hyperbolic plane and a new bound on cerain L -series, Inerna. Mah. Res. Noices 1995, YUK-KAM LAU, DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF HONG KONG, POKFULAM ROAD, HONG KONG address: yklau@mahs.hku.hk DEYU ZHANG, SCHOOL OF MATHEMATICAL SCIENCES, SHANDONG NORMAL UNIVERSITY, JINAN, SHAN- DONG 514, P.R. CHINA address: zdy 78@homail.com YINGNAN WANG, COLLEGE OF MATHEMATICS AND STATISTICS, SHENZHEN UNIVERSITY, SHENZHEN, GUANGDONG 5186, P.R. CHINA address: ynwang@szu.edu.cn