Abstract
We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Starting with the paper [6] many authors have investigated sign change properties of Fourier coefficients of cusp forms, in various directions. In particular, the case of half-integral weight has been the focus of much research. If g is a cusp form of half-integral weight \(k+{1\over 2}\) with real Fourier coefficients \(c(m)\,(m\ge 1)\) and in addition g is a Hecke eigenform, then there are at least two important themes in this area: on the one hand the study of sign changes of \((c(tn^2))_{n\ge 1}\) where t is a fixed positive integer, and on the other hand the corresponding question for the sequence \((c(t))_{t\ge 1 squarefree}\) where t runs over positive squarefree integers only. Of course, similar questions can be studied for forms of weight \(k+{1\over 2}\) in the plus subspace in which case t has to be replaced by |D| where D is a fundamental discriminant with \((-1)^kD>0\). For a good (at least partial) survey the reader may look up the literature given in [4].
Note that sign change results trivially imply corresponding non-vanishing results and in general non-vanishing properties of Fourier coefficients a priori are easier to handle. We recall that non-vanishing of products of Fourier coefficients was studied in [3].
In this short note we will investigate sign change and non-vanishing properties of the double sequence \((c(4n+r^2))_{n\ge 1, r\in \mathbf{Z}}\) where g is a cusp form of weight \(k+{1\over 2}\) with k even and level 4 in the plus subspace \(S_{k+1/2}^+\) (so \(c(m)=0\) unless \(m\equiv 0,1 \pmod 4\), see [7]). These coefficients turn up naturally when one considers the adjoint linear map with respect to the Petersson scalar products of (essentially) the linear map “multiplication with \(\theta \)”, where
is the standard theta function of weight \({1\over 2}\) and level 4. Here as throughout \(q=e^{2\pi iz}\) for \(z\in {{\mathcal {H}}}\), the complex upper half-plane.
Our results will be stated in the next section; the proofs will be given in section 3. They rely on a detailed study of the above mentioned adjoint map, on growth properties of Fourier coefficients of cusp forms of integral weight due to Ram Murty and on a strong bound for the Fourier coefficients of cusp forms of half-integral weight due to Blomer-Harcos. Detailed references will be given below.
2 Statement of results
If \(M\subset \mathbf{Z}\) we denote by \(\#M\) the cardinality of M (thus \(\#M\) is either a non-negative integer or \(\infty \)).
By k we always understand a positive even integer. We let \(S_k\) be the space of cusp forms of weight k on \(\Gamma _1:=SL_2(\mathbf{Z})\). There is a linear map
Note that in general L is not Hecke equivariant.
We denote by \(L^*:S_{k+1/2}^+ \rightarrow S_k\) the linear map adjoint to L with respect to the Petersson scalar products. Note that since L is injective, \(L^*\) is surjective.
Let \(g\in S_{k+1/2}^+\) be fixed, with Fourier coefficients \(c(m)\, (m\ge 1)\). For each \(n\in \mathbf{N}\) we then put
and if in addition the c(m) are real
Theorem 1
Let \(g\in S_{k+1/2}^+\) with real Fourier coefficients \(c(m)\, (m\ge 1)\) and suppose that \(L^*g\) is a normalized Hecke eigenform. Then there are sequences \((n_\nu )_{\nu \ge 1}\) and \((m_\mu )_{\mu \ge 1}\) in \(\mathbf{N}\) such that for any \(\sigma <{1\over {16}}\) one has \(\lim _{\nu \rightarrow \infty }{{\alpha ^+_{n_\nu }}\over {n_{\nu }^\sigma }}=\infty \) and \(\lim _{\mu \rightarrow \infty }{{\alpha ^-_{m_\mu }}\over {m_{\mu }^\sigma }}=\infty \). In particular one has \(\lim _{\nu \rightarrow \infty } \alpha ^+_{n_\nu }=\infty \) and \(\lim _{\mu \rightarrow \infty } \alpha ^-_{m_\mu }=\infty \).
Remark
It is easy to see that for any normalized Hecke eigenform \(F\in S_k\) there exists \(g\in S_{k+1/2}^+\) with real Fourier coefficients such that \(F=L^*g\).
If we drop the assumption that \(L^*g\) is an eigenform, we still can get non-vanishing results for the Fourier coefficients. Let us put \(V:=im L\) and denote by \(V^\bot \) the orthogonal complement of V in \(S_{k+1/2}^+\).
Theorem 2
Let \(g\in S_{k+1/2}^+\) with real Fourier coefficients \(c(m)\, (m\ge 1)\) and suppose that g is not contained in \(V^\bot \). Then there exists a sequence \((n_\nu )_{\nu \ge 1}\) in \(\mathbf{N}\) such that for any \(\sigma <{1\over {16}}\) one has \(\lim _{\nu \rightarrow \infty }{{\alpha _{n_\nu }}\over {n_{\nu }^\sigma }}=\infty \). In particular one has \(\lim _{\nu \rightarrow \infty } \alpha _{n_\nu }=\infty \).
Remark
Applying the above result with g replaced by \(g-g_0\) where \(g_0\in V^\bot \) has Fourier coefficients \(c_0(m)\), we obtain a corresponding statement with “\(c(4n+r^2)\ne 0\)” replaced by “\(c(4n+r^2)\ne c_0(4n+r^2)\)” in the definition of \(\alpha _n\). A corresponding assertion mutatis mutandis (and in the case where the \(c_0(m)\) are real) of course is valid also in the context of Theorem 1.
3 Proof of results
We start with briefly indicating the explicit construction of the map \(L^*\) adjoint to L following [9, sect. 5], and [8], mutatis mutandis.
Let \(g\in S_{k+1/2}^+\). The n-th Fourier coefficient of \(L^*g\) is given by
by the usual Petersson formula, where \(P_{k,n}\) denotes the n-th Poincaré series in \(S_k\).
By definition
where \(z=x+iy, dV={{dxdy}\over {y^2}}\) is the invariant measure, \({{\mathcal {F}}}\) is a fundamental domain for \(\Gamma _0(4)\subset \Gamma _1\) and \(G(z):=\sqrt{y} \, g(z)\overline{\theta (z)}\) behaves like a modular form of weight k under \(\Gamma _0(4)\). Recall that \(\Gamma _0(4)\) consists of those matrices in \(\Gamma _1\) whose left lower component is divisible by 4. The integral in the last line above can be computed by the usual unfolding argument.
Altogether one finds that
where \(C_k\) is a real positive constant depending only on k and
The convergence of the sum is clear by the usual Hecke estimate for the coefficients c(m) (observe that we may assume that \(k\ge 4\), otherwise \(S^+_{k+1/2}=\lbrace 0\rbrace \)). This gives an explicit description of the map \(L^*\).
Since the \(P_{k,n}\,(n\ge 1)\) generate \(S_k\), we also see that \(V^\bot =ker L^*\) consists of those g with the property that \(\ell (g,n)=0\) for all \(n\ge 1\).
For the proof of our results we also need \(\Omega \)-results for the Fourier coefficients \(a(n)\, (n\ge 1)\) of cusp forms \(f\in S_k\). Recall that for arithmetic functions v, w with w(n) ultimately strictly positive, one defines
if
and if in addition v is real-valued
if
and
if
Now recall that for \(f\ne 0\) it was proved in [11] that
and if in addition f is a normalized Hecke eigenform
where \(c_,c_\pm \) are positive constants depending only on f.
We shall now prove the first assertion of Theorem 1. We put \(F:=L^*g\) and denote by \(A(n)\,(n\ge 1)\) the Fourier coefficients of F. According to (4) (applied with \(\Omega _+\)) we can choose a sequence \((n_\nu )_{\nu \ge 1}\) in \(\mathbf{N}\) such that
for all \(\nu \) and
We claim that
for any \(\sigma <{1\over {16}}\).
Suppose that this is not true, for a given \(\sigma \). Then we can find a sequence \(n_{\nu _1}<n_{\nu _2}<\dots \) and \(K>0\) such that
for all \(\mu \ge 1\).
It follows from (1) and (2) that
where r in \(\sum \nolimits _r^+\) runs over those \(r\in \mathbf{Z}\) with \(c(4n_{\nu _\mu }+r^2)>0\) and r in \(\sum \nolimits _r^-\) runs over those r with \(c(4n_{\nu _\mu }+r^2)\le 0\). Note that the sum \(\sum \nolimits _r^+\) is non-empty by (1) and (5) and for each fixed \(\mu \) is finite by (7).
By [1] the Fourier coefficients c(m) of g can be estimated by
where one can take \(\delta ={1\over {16}}\). This estimate is slightly better than the Weil bound with \(\delta =0\). It is important to us that the bound (9) holds for all \(m\ge 1\). Bounds better than the Weil bound for m squarefree were obtained in [2, 5, 10].
Inserting (9) into (8) we obtain
where in the last line we have used (7). Choosing \(\epsilon =\delta -\sigma ={1\over {16}}-\sigma \) we therefore find that
Letting \(\mu \) going to \(\infty \) we obtain a contradiction to (6).
This proves the assertion of Theorem 1 regarding \(\alpha _n^+\). To obtain the assertion with \(\alpha _n^-\) one proceeds in the same way, mutatis mutandis, using (4) with \(\Omega _-\). Finally to prove Theorem 2, one again proceeds in the same way, using (3). Note that the assumption that \(g\not \in V^\bot \) is used to guarantee that \(L^*g\ne 0\).
References
Blomer, V., Harcos, G.: Hybrid bounds for twisted \(L\)-functions. J. Reine Angew. Mathematik 621, 53–79 (2008)
Bykovskii, V.A.: A trace formula for the scalar product of Hecke series and its applications. J. Math. Sci. (New York) 89, 915–932 (1998)
Hofmann, E., Kohnen, W.: On products of Fourier coefficients of cusp forms. Forum Math. 29(1), 245–250 (2017)
Inam, I., Wiese, G.: Fast computation of half integral weight modular forms. Preprint (2020)
Iwaniec, H.: Fourier coefficients of modular forms of half integral weight. Invent. Math. 87, 385–401 (1987)
Knopp, M., Kohnen, W., Pribitkin, W.: On the signs of Fourier coefficients of cusp forms. Ramanujan J. 7(1–3), 269–277 (2003)
Kohnen, W.: Modular forms of half-integral weight on \(\Gamma _0(4)\). Math. Ann. 248(3), 249–266 (1980)
Kohnen, W.: Cusp forms and special values of certain Dirichlet series. Math. Z. 207, 657–660 (1991)
Kohnen, W.: On squares of Hecke eigenforms (to appear in Pure Appl. Math. Quarterly)
Petrov, I., Young, M.P.: A generalized cubic moment and the Petersson formula for newforms (to appear in Math. Ann)
Ram Murty, M.: Oscillations of Fourier coefficients of modular forms. Math. Ann. 262, 431–446 (1983)
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jens Funke.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kohnen, W. A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms. Abh. Math. Semin. Univ. Hambg. 92, 85–90 (2022). https://doi.org/10.1007/s12188-021-00253-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-021-00253-z