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

$$\begin{aligned} \theta (z)= \sum _{r\in \mathbf{Z}}q^{r^2} \end{aligned}$$

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

$$\begin{aligned} L:S_k\rightarrow S_{k+1/2}^+, \quad f(z)\mapsto f(4z)\theta (z). \end{aligned}$$

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

$$\begin{aligned} \alpha _n:=\#\lbrace r\in \mathbf{Z} \,|\, c(4n+r^2)\ne 0\rbrace \end{aligned}$$

and if in addition the c(m) are real

$$\begin{aligned} \alpha ^+_n:=\#\lbrace r\in \mathbf{Z} \,|\, c(4n+r^2)>0\rbrace ,\quad \alpha ^-_n:=\#\lbrace r\in \mathbf{Z} \,|\, c(4n+r^2)< 0\rbrace . \end{aligned}$$

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

$$\begin{aligned} a(L^*g,n)={{(4\pi n)^{k-1}}\over {(k-2)!}}\langle L^*g, P_{k,n}\rangle \end{aligned}$$

by the usual Petersson formula, where \(P_{k,n}\) denotes the n-th Poincaré series in \(S_k\).

By definition

$$\begin{aligned} \langle L^*g, P_{k,n}\rangle = \langle g(z), P_{k,n}(4z)\theta (z)\rangle \\ =\int _{{\mathcal {F}}}G(z)\overline{ P_{k,n}(4z)}y^kdV \end{aligned}$$

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

$$\begin{aligned} a(L^*g,n)=C_k\cdot n^{k-1}\cdot \ell (g,n) \end{aligned}$$
(1)

where \(C_k\) is a real positive constant depending only on k and

$$\begin{aligned} \ell (g,n):=\sum _{r\in \mathbf{Z}}{{c(4n+r^2)}\over {(4n+r^2)^{k-1/2}}}. \end{aligned}$$
(2)

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

$$\begin{aligned} v(n)=\Omega (w(n)) \end{aligned}$$

if

$$\begin{aligned} \limsup _{n\rightarrow \infty }{{|v(n)|}\over {w(n)}}>0, \end{aligned}$$

and if in addition v is real-valued

$$\begin{aligned} v(n)=\Omega _+ (w(n)) \end{aligned}$$

if

$$\begin{aligned} \limsup _{n\rightarrow \infty }{{v(n)}\over {w(n)}}>0, \end{aligned}$$

and

$$\begin{aligned} v(n)=\Omega _- (w(n)) \end{aligned}$$

if

$$\begin{aligned} \liminf _{n\rightarrow \infty }{{v(n)}\over {w(n)}}<0. \end{aligned}$$

Now recall that for \(f\ne 0\) it was proved in [11] that

$$\begin{aligned} a(n)=\Omega \Bigl ( n^{(k-1)/2}\,\exp (c{{\log n}\over {\log \log n}})\Bigr ), \end{aligned}$$
(3)

and if in addition f is a normalized Hecke eigenform

$$\begin{aligned} a(n)=\Omega _\pm \Bigl ( n^{(k-1)/2}\, \exp (c_\pm {{\log n}\over {\log \log n}})\Bigr ), \end{aligned}$$
(4)

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

$$\begin{aligned} A(n_\nu )>0 \end{aligned}$$
(5)

for all \(\nu \) and

$$\begin{aligned} \lim _{\nu \rightarrow \infty }{{A(n_\nu )}\over {n_\nu ^{(k-1)/2}}}\, \exp (-c_+{{\log n_\nu }\over {\log \log n_\nu }})>0. \end{aligned}$$
(6)

We claim that

$$\begin{aligned} \lim _{\nu \rightarrow \infty }{{\alpha ^+_{n_\nu }}\over {n_\nu ^\sigma }}=\infty , \end{aligned}$$

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

$$\begin{aligned} {{\alpha ^+_{n_{\nu _\mu }}}\over {n_{\nu _\mu }^\sigma }}\le K, \end{aligned}$$
(7)

for all \(\mu \ge 1\).

It follows from (1) and (2) that

$$\begin{aligned}A(n_{\nu _\mu })=C_k\cdot n_{\nu _\mu }^{k-1}\cdot \Bigl (\sum \nolimits _r^+{{c(4n_{\nu _\mu }+r^2)}\over {(4n_{\nu _\mu }+r^2)^{k-1/2}}} + \sum \nolimits _r^- {{c(4n_{\nu _\mu }+r^2)}\over {(4n_{\nu _\mu }+r^2)^{k-1/2}}}\Bigr )\nonumber \\\quad\le C_k\cdot n_{\nu _\mu }^{k-1}\cdot\sum \nolimits _r^+ {{c(4n_{\nu _\mu }+r^2)}\over {(4n_{\nu _\mu }+r^2)^{k-1/2}}}, \end{aligned}$$
(8)

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

$$\begin{aligned} c(m)\ll _{g,\epsilon }m^{k/2-\delta +\epsilon }\quad (\epsilon >0) \end{aligned}$$
(9)

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

$$\begin{aligned} A(n_{\nu _\mu })\ll _{g,\epsilon } n_{\nu _\mu }^{k-1}\cdot \sum \nolimits _r^+ {1\over {(4n_{\nu _\mu }+r^2)^{k/2-1/2+\delta -\epsilon }}}\\ \ll _{g,\epsilon } n_{\nu _\mu }^{k-1}\cdot {{\alpha ^+_{n_{\nu _\mu }}}\over {(4n_{\nu _\mu })^{k/2-1/2+\delta -\epsilon }}}\\ \ll _{g,\epsilon ,K}n_{\nu _\mu }^{k/2-1/2-\delta +\epsilon +\sigma } \end{aligned}$$

where in the last line we have used (7). Choosing \(\epsilon =\delta -\sigma ={1\over {16}}-\sigma \) we therefore find that

$$\begin{aligned} A(n_{\nu _\mu })\ll _{g,\epsilon ,K} n_{\nu _\mu }^{(k-1)/2}. \end{aligned}$$

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 \(L^*g\ne 0\).