´ SERIES ON AVERAGE NONVANISHING OF POINCARE SOUMYA DAS AND SATADAL GANGULY

Abstract. We show that a positive proportion of the Poincar´ e series do not vanish identically when either the index or the weight varies over an interval of suitable length, the other one being fixed.

1. Introduction A well known open question in the theory of Poincar´e series of integral weight is whether a Poincar´e series can vanish identically if the weight is sufficiently large. We denote by Pn,q,k , the Poincar´e series of index n ≥ 1, weight k > 2, and level q ≥ 1. See sec. 2 for the definition and basic properties of Poincar´e series. This question is related to a famous question of Lehmer [Leh47] which asks if τ (n) = 0 for any n, where τ is the Ramanujan τ -function. It is widely believed that τ (n) is never zero and it is often referred to as “Lehmer’s conjecture” in the literature. There are theoretical as well as numerical grounds for believing in this conjecture. It is immediate from eqn. (2.3) that Lehmer’s conjecture is equivalent to the statement that the unique Poincar´e series in S12 (1) is not identically zero. It is reasonable to expect that no Poincar´e series should vanish identically even for a general level q, provided the weight is not too small (i.e., larger than an effective numerical constant). Although we are far from proving any definite general result towards this, important progress was made by Rankin [Ra80] who showed that there are constants B > 4 log 2 and k0 > 0 such that the Poincar´e series Pn,k,1 does not vanish identically for 
n ≤ k 2 exp −B log k log log k , (1.1)

provided k ≥ k0 . Here the constant k0 is effective but difficult to compute explicitly. Rankin’s result was generalized to arbitrary Fuchsian groups by Lehner [Leh80] and to Hecke congruence groups Γ0 (q) by Mozzochi [Moz89] in both k and q aspect, their methods and the strength of their results being similar to Rankin’s. In a recent work [DG11], it was shown that there is an effective numerical constant k0 such that if k ≥ k0 then Pn,k,q does not vanish identically for 
n ≤ qk 2 exp −B log k log log k . (1.2)

However, on average, we can do a little better. We prove here (see Theorems 4.1 and 4.2 ) that a positive proportion of the Poincar´e series Pn,k,q do not vanish identically when either of the parameters n and k vary over a sufficiently long segment with the other 2000 Mathematics Subject Classification. Primary 11F11; Secondary 11F30. Key words and phrases. Poincar´ e series, Modular form, Kloosterman sum. 1

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SOUMYA DAS AND SATADAL GANGULY

parameters fixed. For proving the first theorem, we invoke a recent result of Kowalski, Robert, and Wu [KRW07] about nonzero Fourier coefficients of non-CM forms in short intervals. For the next theorem, we exploit cancellations in sums of Bessel functions with Lemma 3.1 on norms of Poincar´e series playing a key role. Notation. P d(n) = 1, the number of divisors of a positive integer n. d|n

e(z) is an abbreviation for e2πiz . δi,j = 0 or 1 according as i 6= j or i = j. a ≪j b is an abbreviation for a ≤ C(j)b for some constant C(j) > 0 depending on j. Acknowledgements. The authors thank School of Mathematics, T.I.F.R., Mumbai and Stat-Math Unit, I.S.I., Kolkata, where this work was done, for providing excellent working atmosphere.

2. Basic definitions and results We assume familiarity with the basic theory of holomorphic modular forms for Hecke congruence groups Γ0 (q). For simplicity we consider only forms with trivial nebentypus. We also consider only forms of weight k ≥ 4. We follow the definitions and notation of [Iw97]. We denote the space of cusp forms of weight k and level q by Sk (q). It is a vector space of finite dimension over C. For n = 1, 2, · · · , we define Pn,k,q , the Poincar´e series of index n, weight k and level q by X Pn,k,q (z) = e(nz)|k γ γ∈Γ0 (q)

=

X

(cqz + d)

(c,d)=1

−k



az + b e n cqz + d



,


a b where, for each pair (c, d), exactly one pair of integers (a, b) is chosen such that cq d ∈ Γ0 (q). The totality of all Poincar´e series Pn,k,q ; n ≥ 1 span Sk (q). The Petersson inner product of two Poincar´e series can be written as follows (see [Iw97, Cor. 3.4]). hPm , Pn i

(  √ ) X S(m, n; cq) Γ(k − 1) 4π mn −k √ = , Jk−1 δm,n + 2πi cq cq (4π mn)k−1

(2.1)

c≥1

where S(m, n; c), the Kloosterman sum, is defined by   X⋆ ma + nd S(m, n; c) = e , c a(mod c) ad≡1(mod c)

X⋆

denoting that the sum is over residue classes prime to c. The Weil bound for Kloosterman sums ( see [Weil48]) says 1

1

|S(m, n; c)| ≤ (m, n, c) 2 c 2 d(c).

(2.2)

´ SERIES ON AVERAGE NONVANISHING OF POINCARE

3

The connection between the Fourier coefficients (at the cusp at ∞) af (n) of a cusp form f and Poincar´e series stems from the relation hf, Pn,k,q i =

Γ(k − 1) af (n). (4πn)k−1

(2.3)

If f ∈ Sk (q) is a Hecke eigenform; i.e., an eigenfunction of all the Hecke operators Tn with (n, q) = 1, then the n-th Fourier coefficient af (n) of f for (n, q) = 1 and the eigenvalue λf (n) of Tn are related by the formula af (n) = af (1)λf (n).

(2.4)

Deligne’s bound for Hecke eigenvalues says |λf (n)| ≤ n

k−1 2

d(n)

(2.5)

for all n. 3. A preparatory lemma We now prove a general lemma concerning the Petersson norm of a Poincar´e series. Lemma 3.1. For n and q coprime and k > 2, we have   (2π)k ζ(2) d(q) Γ(k − 1)d(n)2 d(n)2 2 2 , 1+ ||Pn,k,q || ≤ k−1 ||P1,k,q || ≤ n (4πn)k−1 q k−1/2 Γ(k) where ζ(s) is the Riemann zeta function. Consequently, if either q or k is sufficiently large, 2 Γ(k − 1)d(n)2 ||Pn,k,q ||2 ≤ . (4πn)k−1 Proof. Let H be an orthogonal basis of Sk (q) consisting of eigenfunctions of all the Hecke operators Tm with (m, q) = 1. We first write a Poincar´e series Pn,k,q in terms of this basis as X hf, Pn,k,q i Pn,k,q = f. (3.1) ||f ||2 f ∈H

Taking inner product with itself, we get ||Pn,k,q ||2 =

X hf, Pn,k,q i2

f ∈H

||f ||2

.

(3.2)

Therefore, by (2.3), 2 X Γ(k − 1) af (n)2 ||Pn,k,q || = (4πn)k−1 ||f ||2 f    2 d(n)2  Γ(k − 1) X af (1)2  ≤ k−1  (4π)k−1 n ||f ||2  

2

f

2

d(n) ||P1,k,q ||2 , nk−1 using (2.4) and (2.5) to obtain the inequality at the second step and applying (3.2) (with n = 1) and (2.3) at the last. =

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SOUMYA DAS AND SATADAL GANGULY

For the second inequality, we apply (2.1) to get      X S(1, 1; cq) Γ(k − 1) 4π ||P1,k,q ||2 = 1 + 2πi−k . Jk−1 (4π)k−1  cq cq  c≥1

Using the Weil bound (2.2) and the standard bounds for the Bessel functions:   x ν  1 |Jν (x)| ≤ min 1, , Γ(ν + 1) 2

we estimate the sum as follows.  X   k−1 X S(1, 1; qc) d(qc) 4πn 2π ≤ J k−1 1/2 Γ(k) qc qc qc (qc) c≥1 c≥1 ≤

≤

d(q) (2π)k−1 X d(c) q k−1/2 Γ(k) c≥1 ck−1/2 d(q) (2π)k−1 ζ(2) . q k−1/2 Γ(k)



´ series on average 4. Nonvanishing of Poincare 4.1. Varying index. Here we consider Poincar´e series Pn = Pn,k,q of fixed level and weight. If n runs through an interval (X, X + Y ], the question we want to answer is how many of the Poincar´e series are not identically zero. The following theorem says that 7 asymptotically a positive proportion of them are not if Y ≫ X 17 +ε . Theorem 4.1. There is an absolute, effective constant C > 0 such that the following holds whenever max{k, q} > C. For any ε > 0, there is some constant x0 = x0 (k, q, ε) 7 such that whenever x ≥ x0 and y ≥ x 17 +ε , we have #{x < n ≤ x + y : Pn 6= 0} ≫q,k,ε y. Remarks. We claim that if k and q are not both very small; i.e., not smaller than some effective absolute constant, we can always choose a cusp form in Sk (q) which is not a linear combination of CM forms. This can be seen as follows. To each CM form is associated a Hecke character of a quadratic imaginary extensions of Q of discriminant dividing q (see [Ri77, §3 & §4] for the facts concerning CM forms). Now, for a fixed CM form, the Hecke characters associated to it can be obtained from one another by multiplication with characters of the ray class group of some modulus of norm a divisor of q. By the standard bounds on the class number of quadratic fields and the divisor function, it follows that the total number of CM forms of level q of any fixed 1 weight is O(q 2 +ε ) for any ε > 0, where the implied constant depends only on ε. In the above, we make use of the observation that the ratio of the size of a ray class group and the size of the class group is bounded by an absolute constant in the case of a imaginary quadratic field. The claim is now clear. Now we give the proof of the theorem.

´ SERIES ON AVERAGE NONVANISHING OF POINCARE

5

Proof. By the remarks above, the theorem follows straightaway from (2.3) and corollary 1 of Theorem 1 of an article by Kowalski, Robert, and Wu [KRW07]. The corollary says that if a cusp form f of weight two or higher is not in the space spanned by CM forms, then, for any ε > 0, there is some constant x0 (f, ε) such that for any x ≥ x0 (f, ε) and 7 any y ≥ x 17 +ε , #{x < n ≤ x + y : af (n) 6= 0} ≫f,ε y.  4.2. Varying weight. Fixing a level q ≥ 1 and an index n ≥ 1 with (n, q) = 1, our aim is to show that if k varies over even integers in a dyadic segment K ≤ k ≤ 2K, then a positive proportion of the Poincar´e series Pn,k,q do not vanish identically for all K sufficiently large. Precisely, we prove the following. Theorem 4.2. Suppose (n, q) = 1. There exists an absolute constant k0 > 0 such that if K ≥ k0 · (n/q)2/5 d(q), # {K ≤ k ≡ 0(mod 2) ≤ 2K | Pn,k,q 6= 0} ≫ Kd(n)−2 , the implied constant being absolute. Remark. By (1.1) and its generalization (1.2), no Poincar´e series Pn,k,q vanishes identically if k is of size about (n/q)1/2 or larger. Thus the theorem applies to a wider range of k. Proof. We shall derive this by applying the formula (2.1) and then summing the Bessel functions Jk−1 over k using Fourier analysis. We carry this out following §5.5 of [Iw97]. It is easier to sum with a smooth weight. To this end, we fix a positive test function g which satisfies the following properties: supp g ⊂ [K, 2K] and g (j) ≪j K −j

for all j ≥ 0.

We set S(n, q, K) =

X

k≡0(mod 2)

We apply (2.1) to get S(n, q, K) =

g(k − 1)

X

g(k − 1) + 2π

k≡0(mod 2)

=

 

X

k≡0(mod 2)



g(k − 1)

1 + 2πi−k

(4πn)k−1 ||Pn,k,q ||2 . Γ(k − 1)

X S(n, n; qc) c≥1

X S(n, n; qc) c≥1

Following [Iw97, §5.5], we set, for a = ±1, X Ga (x) =

qc

k−1≡a(mod 4)

qc

Jk−1

X

k≡0(mod 2)

  4πn  qc 

i−k g(k − 1)Jk−1

g(k − 1)Jk−1 (x) .

We have the following asymptotic formulae (loc. cit.): 4Ga (x) = g(x) + O xK −3 + (xK −2 )j








4πn qc



.

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SOUMYA DAS AND SATADAL GANGULY

and X

k≡0(mod 2)

g(k − 1) = gˆ(0)/2 + O(K −j ),

for any j ≥ 0. Here gˆ denotes the Fourier transform of g defined by Z ∞ g(y)e(ty) dy. gˆ(t) = −∞

Combining these, we get

S(n, q, K) = gˆ(0)/2 + O(K −j ) + 2π

X S(n, n; qc)  c≥1



qc

G1



4πn qc

n X S(n, n; qc) (4πn)j + q2 K 3 c2 q j+1 K 2j c≥1  3/2    n2 n d(q) +O = gˆ(0)/2 + O , q2 K 4 q 3/2 K 3

= gˆ(0)/2 + O(K −j ) + O 





 4πn qc  X S(n, n; qc)  cj+1 − G−1

c≥1

(4.1)

by taking j = 2 and applyinng the Weil bound for Kloosterman sums. Any function g satisfying the desired conditions must satisfy gˆ(0) ≪ K. We choose a function g for which k1 K ≤ gˆ(0) ≤ k2 K for two positive absolute constants k1 , k2 . For example, we can take the function defined by g(x) := f (2x/K − 3), where   1 if |x| < 1, and 0 otherwise. f (x) = exp x2 − 1 Then it is easy to see that K gˆ(0) = fˆ(0) ≫ K 2 and that g satisfies all the required properties. Therefore, Lemma 3.1 and eqn. (4.1) now yields     X K n2 n3/2 d(q) +O g(k − 1) ≫ +O d(n)2 q 2 K 4 d(n)2 q 3/2 K 3 d(n)2 k≡0(mod 2) Pn,k,q 6≡0

≫ provided K ≫

 2/5 n q

K , d(n)2

(4.2)

d(q).

Since g is absolutely bounded, this means # {K ≤ k ≡ 0(mod 2) ≤ 2K | Pn,k,q 6= 0} ≫ Kd(n)−2 , proving the theorem.

 References DG

[DG11]

S. Das & S. Ganguly, Linear relations among Poincar´ e series, to appear in Bull. Lond. Math. Soc.. Iw

[Iw97]

H. Iwaniec, Topics in classical automorphic forms Graduate Studies in Mathematics, 17, American Mathematical Society, Providence, RI, 1997. xii+259 pp.

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7

KRW [KRW07] E. Kowalski, O. Robert, J. Wu: Small gaps in coefficients of L-functions and B-free numbers in short intervals. Rev. Mat. Iberoam. 23, no. 1, 2007, 281–326. leh [Leh47]

D. H. Lehmer, The vanishing of Ramanujan’s function τ (n). Duke Math. J. 14, 1947, 429–433.

[Leh80]

˙ 23, 1980, J. Lehner, On the nonvanishing of Poincar´ e series, Proc. Edinburgh Math. Soc(2)

lehner

no. 2, 225228. mozzochi [Moz89]

C. J. Mozzochi, On the nonvanishing of Poincar´ e series. Proc. Edinburgh Math. Soc. 32, 1989, 133–137. rankin

[Ra80]

R. A. Rankin: The vanishing of Poincar´ e series. Proc. Edinburgh Math. Soc. 23, 1980, 151– 161.

[Ri77]

K. Ribet, Galois representations attached to eigenforms with nebentypus, Springer Lecture Notes in Mathematics, 601, 17–52, 1977.

[Weil48]

A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. USA 34 (1948), pp. 204–207, Oeuvres I, pp. 386–389.

Ribet

We

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai – 400005, India. E-mail address: [email protected],[email protected]

Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700178, India E-mail address: [email protected]