On Products of Fourier Coefficients of Cusp Forms
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 nonvanishing and nonnegativity.^{1}^{1}1 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 nonvanishing and nonnegativity. For a precise statement of our results see section 2 below.
2 Statement of results
Let denote the upper halfplane 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]:

is a finitely generated Fuchsian group of the first kind.

The negative identity matrix is contained in .

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 nonzero.
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 nonvanishing 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 nonzero.
Proof.
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 [6] and later studied by Good [1], see above.
Further, assuming is a congruence subgroup of level , we use a result of Knopp, Kohnen and Pribitkin [4], 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 nonnegative terms.
Remark.
In the same way, one can prove that the sequence has infinitely many nonpositive terms.
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
To the sequence , we attach a Dirichlet series [see 1, 6] by setting
(1) 
The theorem we shall prove is the following:
Theorem 1.
Assume that for almost all and that the sum
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 nonzero. Then, the Dirichlet series from (1) becomes a Dirichlet polynomial
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 complexvalued invariant functions on that are square integrable on with respect to the invariant measure . If and are in we denote the inner product by
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 [1]. 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
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 nonholomorphic Eisenstein series of weight zero attached to the cusp (cf. [1, pp. 104f] or e.g. [5, Chapter II]) is defined as
This sum converges absolutely in the halfplane . The Eisenstein series has meromorphic continuation to the entire plane. Its Fourier expansion is given by
with coefficient functions
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]:
(2)  
where is given by [see 1, p. 119],
(3)  
Here, is the hypergeometric function
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]
(4) 
Since by hypothesis, converges for all , we obtain from (2), (3) and (4) by continuation, that the identity
(5)  
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 nonzero residues only from the second and third terms on the righthand side of (5); in the third term, the only nonzero contribution comes from the second (nonconstant) term of in (3). Note that for .
More precisely, we have
We note that
and  
Thus, we obtain
Whence, further
Also, we have
Therefore altogether, we obtain
for all . It follows that for all
By the definition of the hypergeometric function, we have
This is a polynomial in of degree , which we denote as . All of its coefficients , are nonzero. Set and write . We find that
Since all the are nonzero, we find immediately that for all ,
(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
Whence for all ,
Here, as grows arbitrarily large, the righthand side is positive and bounded, whereas,on the lefthand side, there exist with (and with ), thus, the lefthand side becomes arbitrarily large. This is a contradiction, hence for all .
Now, from (6) we get the system of linear equations with
the determinant of which is easily seen to be nonzero. 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
defines a cusp form for a congruence subgroup of higher level.
Proof.
With the usual operation, we can write in the form
Here, denotes the totient function, and summation runs over a minimal system of representatives . Clearly, the righthand 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 nonzero 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
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 [4], 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 [4] 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
 [1] Anton Good. On various means involving the Fourier coefficients of cusp forms. Math. Z., 183(1):95–129, 1983.
 [2] Jeff Hofstein, Thomas A. Hulse, and Andre Reznikov. Multiple Dirichlet series and shifted convolutions. ArXiv eprints, 2015, 1110.4868v3.
 [3] Winfried Hohnen. Period polynomials and congruences for Hecke algebras [eigenvalues]. Proc. Edinburgh Math. Soc. (2), 42(2):217–224, 1999.
 [4] Marvin Knopp, Winfried Kohnen, and Wladimir Pribitkin. On the signs of Fourier coefficients of cusp forms. Ramanujan J., 7(13):269–277, 2003. Rankin memorial issues.
 [5] Tomio Kubota. Elementary theory of Eisenstein series. Kodansha Ltd., Tokyo; Halsted Press [John Wiley & Sons], New YorkLondonSydney, 1973.
 [6] 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.
 [7] JeanPierre 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.
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