On Products of Fourier Coefficients of Cusp Forms

# On Products of Fourier Coefficients of Cusp Forms

Eric Hofmann    Winfried Kohnen
###### Abstract

The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, for a fixed positive integer , concerning both non-vanishing and non-negativity.111 2010 Mathematics Subject Classification: 11F12, 11F30

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## 1 Introduction

Let be an elliptic cusp form of positive integral weight with real Fourier coefficients . Let be a fixed positive integer. Then, the purpose of this paper is to study the products , regarding both non-vanishing and non-negativity. For a precise statement of our results see section 2 below.

We remark that these products have been previously investigated under different aspects, namely first by Selberg  and by Good  who studied growth properties of the sums where , and more recently by Hoffstein and Hulse  in connection with shifted Dirichlet convolutions.

## 2 Statement of results

Let denote the upper half-plane of complex numbers , with . The modular group acts on by fractional linear transformations, as usual.

In the following, we denote by a discrete subgroup of , which satisfies the following conditions [cf. 1, p. 98]:

1. is a finitely generated Fuchsian group of the first kind.

2. The negative identity matrix is contained in .

3. contains exactly if is an integer.

Condition (iii) means that has a cusp at , the stabilizer of which is generated by the two matrices .

We shall prove the following theorem

###### Theorem 1.

Let be a cusp form of integer weight on with real Fourier coefficients . Let and assume that is not identically zero. Then, in fact, infinitely many terms of the sequence are non-zero.

The following corollary is an immediate consequence of the theorem. Recall only that for a normalized Hecke eigenform, the Fourier coefficients are real and .

###### Corollary 1.

Let be a cusp form of integer weight and level that is a normalized Hecke eigenform with Fourier coefficients . Let and suppose that . Then, the sequence has infinitely many non-vanishing terms.

From Corollary 1, we obtain

###### Corollary 2.

Let be a normalized Hecke eigenform of integer weight on with Fourier coefficients . Then, there are infinitely many such that and are both non-zero.

###### Proof.

By results from  and , the second Fourier coefficient is non-zero modulo any prime lying above (in an appropriate finite extension of ), hence is non-zero. The assertion follows. ∎

The proof of Theorem 1 which will begin in section 3 makes use of a Dirichlet series associated to the sequence and of its analytic properties. This series was introduced by Selberg  and later studied by Good , see above.

Further, assuming is a congruence subgroup of level , we use a result of Knopp, Kohnen and Pribitkin , Theorem 1, on sign changes of Fourier coefficients of cusp forms to show the following:

###### Theorem 2.

Let be a cusp form of integer weight on , with real Fourier coefficients . Let . Then the sequence has infinitely many non-negative terms.

###### Remark.

In the same way, one can prove that the sequence has infinitely many non-positive terms.

The proof of this theorem is carried out in section 4. We note that, in fact a somewhat more general statement holds, as the requirements of , which need to be considered in addition to our conditions (i)–(iii), are minimal (see the remark on p. 4 below).

Ideally, one might hope to prove a sign change result for the sequence by combining Theorems 1 and 2 appropriately. However we have not been able to do this.

## 3 Proof of Theorem 1

We shall actually prove a more general statement than the assertion of Theorem 1.

We assume that the Fourier expansion of is given by with . Let be fixed. For , set

 cn\vcentcolon=a(n)a(n+r).

To the sequence , we attach a Dirichlet series [see 1, 6] by setting

 D(s,r)\vcentcolon=∑n≥1cn(n+r2)s(σ\vcentcolon=Re(s)>k). (1)

The theorem we shall prove is the following:

###### Theorem 1′.

Assume that for almost all and that the sum

 D(s,r)\vcentcolon=∑n≥1cn(n+r2)−s(σ>k)

in fact converges for all . Then for all .

We postpone the proof of this Theorem, and give the proof of Theorem 1 first.

###### Proof of Theorem 1.

It suffices to assume that all but a finite number of the coefficients are zero and to derive a contradiction.

Thus, let , with be those which are non-zero. Then, the Dirichlet series from (1) becomes a Dirichlet polynomial

 D(s,r)=p∑i=1cni(ni+r2)s,

and hence converges in the entire -plane. Also, by hypothesis almost all of its coefficients are . Now, by applying Theorem 1, we find that for all . This is a contradiction, and the proof is complete. ∎

The rest of this section is dedicated to the proof of Theorem 1. First, we fix some notation: In the following, denote by a complex variable with real part and imaginary part . By denote a fundamental domain for the action of on , and by the Hilbert space of complex-valued -invariant functions on that are square integrable on with respect to the invariant measure . If and are in we denote the inner product by

 ⟨f,g⟩=∫Ff(z)¯¯¯¯¯¯¯¯¯g(z)dν(z).

Even if and do not both belong to we continue to use the notation for the above integral, as long as it converges absolutely. Finally, set .

###### Proof of Theorem 1′.

We loosely follow the notation used by Good in . Fix a maximal system of -inequivalent cusps, , . (Note that by our assumptions on , this system is finite and contains .)

For , the Poincaré series [see 1, p. 13] is defined as

 P0(z,s,r)=(πr)s−1/2Γ(s+12)∑γ∈Γ∞∖ΓI(γz)se(rR(γz)),

for a complex variable with . Here, stands for the stabilizer of in , consisting of the matrices , .

For each cusp , denote by an element with and for which is the stabilizer of in . Then, the non-holomorphic Eisenstein series of weight zero attached to the cusp (cf. [1, pp. 104f] or e.g. [5, Chapter II]) is defined as

 Ei(z,s)=∑γ∈Γξi∖Γ(I(g−1iγz))s(σ>1).

This sum converges absolutely in the half-plane . The Eisenstein series has meromorphic continuation to the entire -plane. Its Fourier expansion is given by

 Ei(z,s)=δi,1ys+ϕi(s)y1−s+∑m≥1ϕi(m;s)2y1/2Ks−1/2(2π|m|y)e(mx),

with coefficient functions

 ϕi(s)=√π⋅Γ(s+12)Γ(s)L(0)i(s),ϕi(m;s)=πs|m|s−12Γ(s)L(m)i(s),whereL(m)i=∑c>0c−2s∑dmodce(mdc)((∗∗cd)∈Γξi).

Note that, by the functional equation of , one has .

Now, consider for . As a function in , it has holomorphic continuation to a small neighborhood of the line and there, the functional equation holds [cf. 1, Lemma 5, p. 115f]:

 ⟨P0(⋅,s,r),F⟩ =12s−1κ∑i=1ϕi(−r;1−s)⟨Ei(⋅,s),F⟩+⟨P0(⋅,s−1,r),F⟩ (2) +(4π)12−k(n4)s−12Γ(k+s−1)Γ(s+12)∑n≥1cn(n+r2)k+s−1Δr(s,n),

where is given by [see 1, p. 119],

 Δr(s,n)\vcentcolon=1 −F(k+s−12,k+s2,s+12;(r2n+r)2) (3) −Γ(k−s)Γ(s−12)Γ(k+s−1)Γ(12−s)(4n+2rr)2s−1× ×[F(k−s2,k−s+12,32−s;(r2n+r)2)−1].

Here, is the hypergeometric function

 F(a,b,c;z)=∞∑w=0(a)w(b)w(c)ww!zw(|z|<1),(a)w=Γ(a+w)Γ(a).

From the expression in (3) one sees immediately that the series on the right hand side of (2) is convergent at least for .

Also, note that the Dirichlet series satisfies [cf. 1, Lemma 5 (i), p. 116]

 D(s+k−1,r)=(4π)k−12(4r)s−1/2Γ(s+12)Γ(k+s−1)⟨P0(⋅,s,r),F⟩(σ>1). (4)

Since by hypothesis, converges for all , we obtain from (2), (3) and (4) by continuation, that the identity

 (4π)12−k (r4)s−12Γ(k+s−1)Γ(s+12)D(k+s−1,r)= (5) 12s−1κ∑i=1ϕi(−r;1−s)⟨Ei(⋅,s),F⟩ +(4π)12−k(r4)12−sΓ(k−s)Γ(32−s)⋅D(k−s,r) +(4π)12−k(r4)s−12Γ(k+s−1)Γ(s+12)∑n≥1cn(n+r2)k+s−1⋅Δr(s,n)

holds between meromorphic functions for all .

Now, take residues at on both sides of (5). By [5, Theorems 4.3.4, 4.3.5, p. 43], the Eisenstein series are holomorphic in except for finitely many poles in the interval , which are precisely the poles of the constant coefficients . It follows that the are holomorphic at and do not contribute to the residue.

Thus, we pick up non-zero residues only from the second and third terms on the right-hand side of (5); in the third term, the only non-zero contribution comes from the second (non-constant) term of in (3). Note that for .

More precisely, we have

 0=(r4)12−k−2m1Γ(32−k−2m)1(2m)!⋅D(−2m,r)−(r4)k+2m−12Γ(2k+2m−1)Γ(k+2m+12)∑n≥1cn(n+r2)2k+2m−1(2mn)−1⋅Γ(k+2m−12)Γ(2k+2m−1)⋅Γ(12−k−2m)××(4n+2rr)2k+4m−1[F(−m,−m+12,32−k−2m,(r2n+r)2)−1].

We note that

 Γ(32−k−2m) =(12−k−2m)⋅Γ(12−k−2m) and Γ(k+2m+12) =(k+2m−12)⋅Γ(k+2m−12).

Thus, we obtain

 0= (r4)12−k−2m1(12−k−2m)D(−2m,r) −(r4)k+2m−121(k+2m−12)∑n≥1cn(n+r2)2k+2m−1× ×(4n+2rr)2k+4m−1[F(−m,−m+12,32−k−2m;(r2n+r)2)−1].

Whence, further

 0= (r4)12−k−2mD(−2m,r)+(r4)k+2m−12∑n≥1cn(n+r2)2k+2m−1× ×(4n+2rr)2k+2m−1[F(−m,−m+12,32−k−2m;(r2n+r)2)−1].

Also, we have

Therefore altogether, we obtain

 D(−2m,r)=∑n≥1cn(n+r2)2m⋅[−1+F(−m,−m+12,32−k−2m;(r2n+r)2)],

for all . It follows that for all

 0=∑n≥1cn(2n+r)2mF(−m,−m+12,32−k−2m;(r2n+r)2).

By the definition of the hypergeometric function, we have

 F(−m,−m+12,32−k−2m;z)= ∑w≥0(−m)⋯(−m+w−1)(−m+12)⋯(−m+w−1+12)(−k−2m+32)⋯(−k−2m+32+w−1)zww!.

This is a polynomial in of degree , which we denote as . All of its coefficients , are non-zero. Set and write . We find that

 ∑n≥1cnPm(2n+r)=0.

Since all the are non-zero, we find immediately that for all ,

 ∑n≥1cn(2n+r)2ν=0. (6)

Now recall that (by hypothesis) almost all the coefficients are . Trivially, if there are non negative coefficients, since the right hand side is zero, there can be no positive coefficients, either and so for all , in which case we are finished. Thus, we assume that a finite number of coefficients, , with are negative. For all , we can write the equation (6) in the form

 ∑n≥1n≠n1,…,ntcn(2n+r)2ν=−cn1(2n1+r)2ν−…−cnt(2nt+r)2ν

Whence for all ,

 ∑n≥1n≠n1,…,ntcn(2n+r2nt+n)2ν=−cn1(2n1+r2nt+r)2ν−…−cnt.

Here, as grows arbitrarily large, the right-hand side is positive and bounded, whereas,on the left-hand side, there exist with (and with ), thus, the left-hand side becomes arbitrarily large. This is a contradiction, hence for all .

Now, from (6) we get the system of linear equations with

 nt∑n=1cn(2n+r)2ν=0.

the determinant of which is easily seen to be non-zero. It follows that the remaining coefficients vanish, too. Thus, to conclude, for all . ∎

## 4 Proof of Theorem 2

Recall that for this theorem, we require to be a congruence subgroup of level (then, condition (i) from p. 2 is satisfied automatically). For the proof, we need the following lemma.

###### Lemma 1.

Let be cusp form with respect to , with Fourier coefficients . Let . Then, the Fourier expansion

 g(z)\vcentcolon=∑n≥1n≡n0mod2ra(n)e2πinz

defines a cusp form for a congruence subgroup of higher level.

###### Proof.

With the usual -operation, we can write in the form

 g(z) =1ϕ(2r)∑smod2re−πin0srzf∣k(1s2r01) =1ϕ(2r)∑n≥n0a(n)e2πinz∑smod2re2πi(n−n0)s2r

Here, denotes the totient function, and summation runs over a minimal system of representatives . Clearly, the right-hand side is modular with level a multiple of , and is cuspidal if is. ∎

Now, we are ready to prove Theorem 2.

###### Proof.

Set for all . Assume that holds for only finitely many . Then, there is an index , with the property that for all .

Since it follows that and are both non-zero and have opposite sign. By induction, it follows that all coefficients with have the same sign. Now, by Lemma 1, the resulting sequence of coefficients defines a cusp form

 g(z)=∑n≥1n≡n0mod2ra(n)e2πinz

with real Fourier coefficients, of which, by construction only a finite number are less than or equal to zero. However, by the results of Knopp, Kohnen and Pribitkin , Theorem 1, if all Fourier coefficients of a cusp form are real, the sequence of its coefficients has infinitely many terms of either sign. This is a contradiction. ∎

###### Remark.

Actually, for the result from  we use here, the only requirement is that the group mentioned in Lemma 1, besides being a Fuchsian group of the first kind with finite covolume, have and as parabolic fixed points. However, if already satisfies conditions (i)–(iii) (see p. 2), this imposes only a mild further restriction. Thus, Theorem 2 holds somewhat more generally then only for congruence subgroups.

## References

•  Anton Good. On various means involving the Fourier coefficients of cusp forms. Math. Z., 183(1):95–129, 1983.
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•  Winfried Hohnen. Period polynomials and congruences for Hecke algebras [eigenvalues]. Proc. Edinburgh Math. Soc. (2), 42(2):217–224, 1999.
•  Marvin Knopp, Winfried Kohnen, and Wladimir Pribitkin. On the signs of Fourier coefficients of cusp forms. Ramanujan J., 7(1-3):269–277, 2003. Rankin memorial issues.
•  Tomio Kubota. Elementary theory of Eisenstein series. Kodansha Ltd., Tokyo; Halsted Press [John Wiley & Sons], New York-London-Sydney, 1973.
•  Atle Selberg. On the estimation of Fourier coefficients of modular forms. In Proc. Sympos. Pure Math., Vol. VIII, pages 1–15. Amer. Math. Soc., Providence, R.I., 1965.
•  Jean-Pierre Serre. Valeurs propres des opérateurs de Hecke modulo . In Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), pages 109–117. Astérisque, Nos. 24–25. Soc. Math. France, Paris, 1975.

Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205,
D-69120 Heidelberg, Germany

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