Abstract
We prove a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form of degree 2. In addition, we prove a quantitative result for the number of sign changes of the primitive Fourier coefficients. We give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2 over certain imaginary quadratic extensions.
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1 Introduction and statement of the main results
The distribution of signs of the Fourier coefficients of a non-zero elliptic cusp form has been a subject of study for several mathematicians over the past years. One aspect of this problem is the study of number of sign changes of the Fourier coefficients. Knopp, Kohnen, and Pribitkin in [14] proved that the Fourier coefficients of a non-zero elliptic cusp form f on a congruence subgroup of the full modular group \(SL_2({\mathbb {Z}})\) have infinitely many sign changes. They use the Landau’s theorem on Dirichlet series with non-negative coefficients and the finiteness of the Hecke L-function attached to the elliptic cusp form f to prove their result. In addition, one can see that [16] and [18] are devoted to the study of sign changes of the Fourier coefficients of an elliptic Hecke eigenform. A more subtle problem is to give an explicit upper bound for the first sign change. This has been studied for elliptic cusp forms of square-free level by Choie and Kohnen [2]. Later, their result has been improved by He and Zhao [11]. For elliptic Hecke eigenforms of level N the problem has been dealt in [13, 16].
The theory of elliptic modular forms has been generalized to several variables. Hermitian modular forms over an imaginary quadratic field K are one of those generalizations. In this article, we give a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form F of degree 2. Moreover, we also give a quantitative result for the number of sign changes of the primitive Fourier coefficients. Note that Yamana [22] has established that F is determined by its primitive Fourier coefficients. Also, we provide an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form over certain imaginary quadratic fields. To the best of our knowledge, this is the first attempt to study the distribution of signs of the Fourier coefficients of a Hermitian cusp form. Now, we introduce the necessary notations to state our results.
Let \(d>0\) be a square free integer. Throughout the article, let \(K={\mathbb {Q}}(\sqrt{-d})\) be a fixed imaginary quadratic field. Let
be the discriminant of K. Let \({\mathcal {O}}_K\) be the ring of integers of K and \({\mathcal {O}}_K^{\#}=\frac{i}{\sqrt{|D_K|}}{\mathcal {O}}_K\) be the inverse different of K over \({\mathbb {Q}}\). The Hermitian modular group of degree 2 over K is given by
where \(J_2=\begin{pmatrix}\varvec{0}_2 &{} -\varvec{I}_2 \\ \varvec{I}_2 &{} \varvec{0}_2\end{pmatrix}\), \(\varvec{I}_2\) and \(\varvec{0}_2\) are the \(2 \times 2\) identity matrix and zero matrix respectively. The subgroup
coincides with the full modular group \(U_2({\mathcal {O}}_K)\) if \(D_K \not = -3, -4\). We denote by \(S_k(SU_2({\mathcal {O}}_K))\) the space of Hermitian cusp forms of degree 2 on \(SU_2({\mathcal {O}}_K)\) (defined in Sect. 2.1). Any \(F\in S_k(SU_2({\mathcal {O}}_K))\) has a Fourier series expansion of the form:
where
\(q=e(\tau ), \zeta _1=e(z_1), \zeta _2=e(z_2), q'= e(\tau '), e(z)=e^{2\pi i z}\). The first result of this article gives a quantitative result for the sign changes of the Fourier coefficients of F.
Theorem 1.1
Let \(F\in S_k(SU_2({\mathcal {O}}_K))\) be a non-zero Hermitian cusp form with real Fourier coefficients \(A_F(T)\). Then \(A_F(T)\) changes sign at least once for \( |D_{K}| \mathrm{{det}} (T) \in ( X, X + X^{3/5}]\) for \(X \gg 1\).
For any \(T \in \Delta _2^+\), we define
We say that T is primitive if \(\mu (T)=1\). The Fourier coefficient of F at a primitive T is known as primitive Fourier coefficient. The second result of this article gives the following quantitative result on the number of sign changes of the primitive Fourier coefficients.
Theorem 1.2
Let \(F\in S_k(SU_2({\mathcal {O}}_K))\) be non-zero with real Fourier coefficients \(A_F(T)\). Then the primitive Fourier coefficients \(A_{F}(T) \) changes sign at least once for \(|D_K|\mathrm{{det}}(T) \in ( X, X + X^{3/5} ]\) for \(X \gg 1.\)
Theorem 1.1 implies that there are infinitely many sign changes of the Fourier coefficients of \(F\in S_k(SU_2({\mathcal {O}}_K))\). Next, we focus our attention on establishing an explicit upper bound for the first sign of any \(F\in S_k(SU_2({\mathcal {O}}_K))\). To accomplish this, we first establish a Sturm bound for Hermitian modular forms of degree 2.
Theorem 1.3
Let \(K={\mathbb {Q}}(\sqrt{-d})\) where \(d\in \{1, 2, 3, 7, 11, 15\}\). Also, let
If \(A_F(n, r, m) =0\) for all \(0\le n \le \beta \) and \(0 \le m \le \beta \), where
then
Finally, using Theorem 1.3, we give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2.
Theorem 1.4
Let \(K={\mathbb {Q}}(\sqrt{-d})\) where \(d\in \{1, 2, 3, 7, 11, 15\}\). Suppose \(F \in S_k(SU_2({\mathcal {O}}_K))\) is non-zero with real Fourier coefficients \(A_F(T)\). Then there exist \(T_1, T_2\in \Delta _2^+\) with
for any real \(\epsilon >0\), where
such that
Remark 1.5
For \(F\in S_k(SU_2({\mathcal {O}}_K))\) with complex Fourier coefficients \(A_F(T)\), the Fourier series with \(Re(A_F(T))\) (respectively \(Im(A_F(T))\)) are again in \(S_k(SU_2({\mathcal {O}}_K))\). Therefore, there is an obvious reformulation of Theorems 1.1, 1.2 and 1.4 for arbitrary \(F\in S_k(SU_2({\mathcal {O}}_K))\) with \(A_F(T)\) replaced by \(Re(A_F(T))\) and \(Im(A_F(T))\).
The article is organized as follows: In the next section we recall the definition of three concepts used in this paper; Hermitian modular forms of degree 2, Hermitian Jacobi forms and Jacobi forms with matrix index. We show that Hermitian Jacobi forms occur as the coefficients in the Fourier–Jacobi expansion of a Hermitian modular form of degree 2. In Sects. 3 and 4, we give the proof of Theorems 1.1 and 1.2 respectively. Section 5 is the largest, and contains the proof of Theorem 1.3. We prove Proposition 5.1, Theorems 5.2, and 5.3 in this section, which may be of interest on their own. Finally, in Sect. 6, we prove Theorem 1.4.
Notation For any ring \(R\subset {\mathbb {C}}\), we write by \(R^{n}=\{(\alpha _1, \cdots , \alpha _{n})\mid \alpha _i \in R\}\) the set of row matrices of size \(1\times n\) with entries in R. We denote by \(M_{n}(R)\) the set of all \(n\times n\) matrices with entries in R. Let \(GL_n(R)\) be the group of matrices in \(M_n(R)\) with non-zero determinant and let \(SL_n(R)\) be the group of matrices with determinant 1. For any \(M \in M_n(R)\), we write by \({\overline{M}}\) the complex conjugate of M and by \(M^t\) the transpose of matrix M. We denote by \(\mathrm{{det}}(M)\) and tr(M) the determinant and trace of the matrix M respectively. Also let A[B] denote the matrix \({\overline{B}}^tA B\) for two complex matrices A and B of appropriate sizes. For \(\alpha \in {\mathbb {C}}\), we write \(e(\alpha ):=e^{2\pi i \alpha }\) and \(N(\alpha ):=\alpha {\overline{\alpha }}\). We denote by \({\mathcal {O}}_K^{\times }\), the group of units in \({\mathcal {O}}_K\).
2 Preliminaries
2.1 Hermitian modular forms of degree two
The Hermitian upper-half space of degree 2 is defined by
The Hermitian modular group \(U_2({\mathcal {O}}_K)\) acts on \({\mathcal {H}}_2\) by
For any non-negative integer k, we define the action of \(U_2({\mathcal {O}}_K)\) on the set of functions from \({\mathcal {H}}_2\) to \({\mathbb {C}}\) by
For a positive integer N, we define the congruence subgroup \(\Gamma ^{(2)}_0(N)\) of \(SU_2({\mathcal {O}}_K)\) by
Note that if \(N=1\) then \(\Gamma ^{(2)}_0(1)=SU_2({\mathcal {O}}_K)\).
Definition 2.1
A holomorphic function \(F:{\mathcal {H}}_2 \rightarrow {\mathbb {C}}\) is called a Hermitian modular form of weight k on \(\Gamma ^{(2)}_0(N)\) if it satisfies
for all \(M\in \Gamma ^{(2)}_0(N)\).
We denote by \(M_k(\Gamma ^{(2)}_0(N))\) the space of Hermitian modular forms of degree 2 on the group \(\Gamma ^{(2)}_0(N)\). Any \(F\in M_k(\Gamma ^{(2)}_0(N))\) possesses a Fourier series expansion of the form:
where \( q=e(\tau ), \zeta _1=e(z_1), \zeta _2=e(z_2), q'=e(\tau ')\) and
Moreover, F is called a cusp form if \(A_F(T)=0\) whenever \(\mathrm{{det}}(T)=0\). We denote by \(S_k(\Gamma ^{(2)}_0(N))\) the space of cusp form in \(M_k(\Gamma ^{(2)}_0(N))\). We note down the following result by Yamana [22] which characterizes a Hermitian cusp form by its primitive Fourier coefficients.
Theorem 2.2
Suppose \(F\in S_k(SU_2({\mathcal {O}}_K))\) is non-zero with Fourier coefficients \(A_F(T)\). Then there exists a primitive matrix \(T_0\in \Delta _2^+\) such that \(A_F(T_0)\ne 0\).
The group \(GL_2({\mathcal {O}}_K)\) acts on \(\Delta _2^+\) by \(T \mapsto {\overline{g}}^tT g\), where \(g\in GL_2({\mathcal {O}}_K)\). We have the following lemma.
Lemma 2.3
Let \(T\in \Delta _2^+\) be a primitive matrix. Then there exists \(g\in SL_2({\mathcal {O}}_K)\) such that \({\overline{g}}^t T g=\begin{pmatrix} * &{} * \\ * &{} p \end{pmatrix}\) for some odd prime p.
Proof
By [1, Lemma 3.1] there exists a \(g\in GL_2({\mathcal {O}}_K)\) such that \({\overline{g}}^t T g=\begin{pmatrix} * &{} * \\ * &{} p \end{pmatrix}\). Let \(g=\begin{pmatrix} \alpha &{} \beta \\ \gamma &{} \delta \end{pmatrix}\in GL_2({\mathcal {O}}_K)\) and \(T=\begin{pmatrix} n &{} r\\ {\overline{r}} &{} m \end{pmatrix}\). Then we have
We know that \(\textrm{det}(g)=\epsilon \), where \(\epsilon \in {\mathcal {O}}_K^{\times }\). Therefore, we can take
such that \({\overline{g}}^t_1 Tg_1=\begin{pmatrix} * &{} * \\ * &{} p \end{pmatrix}\). \(\square \)
For any \(g\in SL_2({\mathcal {O}}_K)\), we have \(\begin{pmatrix} ({\overline{g}}^t)^{-1}&{} \varvec{0}_2\\ \varvec{0}_2 &{} g\end{pmatrix} \in SU_2({\mathcal {O}}_K)\). Applying the transformation (2) on \(F\in S_k(SU_2({\mathcal {O}}_K))\), we get the following relation on the Fourier coefficients of F
Now using Theorem 2.2 and Lemma 2.3 we get the following.
Lemma 2.4
Suppose \(F\in S_k(SU_2({\mathcal {O}}_K))\) is non-zero with Fourier coefficients \(A_F(T)\). Then, for some odd prime, p there exists a primitive \(T_0=\begin{pmatrix} * &{} * \\ * &{} p\end{pmatrix}\in \Delta _2^+\) such that \(A_F(T_0)\ne 0\).
2.2 Hermitian Jacobi forms
Let \(G=SL_2({\mathbb {Z}})\ltimes {\mathcal {O}}_K^2\) be the Hermitian Jacobi group over \({\mathcal {O}}_K\). The Jacobi group G acts on \({\mathcal {H}} \times {\mathbb {C}}^2\) as follows:
where \(g=\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in SL_2({\mathbb {Z}})\), \(\tau \in {\mathcal {H}}\), \(\lambda , \mu \in {\mathcal {O}}_K\), \(z_1, z_2 \in {\mathbb {C}}\). For any positive integer N, let
Definition 2.5
A holomorphic function \(\phi :{\mathcal {H}}\times {\mathbb {C}}^2\longrightarrow {\mathbb {C}}\) is a Hermitian Jacobi form of weight k and index m on \(\Gamma ^{(1)}_0(N)\) if for each \(g= \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in \Gamma ^{(1)}_0(N) \), and \(\lambda , \mu \in {\mathcal {O}}_K\), we have
and \(\phi \) has a Fourier series expansion of the form
where \(q=e(\tau )\), \(\zeta _1=e(z_1)\), \(\zeta _2=e(z_2)\).
We denote by \(J_{k, m}(\Gamma ^{(1)}_0(N))\) the vector space of all Hermitian Jacobi forms of weight k and index m on \(\Gamma ^{(1)}_0(N)\).
2.2.1 Theta decomposition
The invariance of \(\phi \) under the action of \((\lambda , 0)\) in (5) yields that the Fourier coefficient c(n, r) is completely determined by \(r\pmod {m{\mathcal {O}}_K}\) and \(nm-N(r)\). We define
The theta decomposition of \(\phi \in J_{k, m}(\Gamma ^{(1)}_0(N))\) is given by
where
The theta components \(h_{s}\) of \(\phi \) are elliptic modular forms on the prinicipal congruence subgroup \(\Gamma ^{(1)}(|D_K|Nm)\) (see [9, 10]).
2.3 Fourier–Jacobi expansion
Let \(F\in S_k(\Gamma ^{(2)}_0(N))\) has Fourier series expansion of the form (3). We write the Fourier series expansion of F as
For \(\begin{pmatrix}a &{}b \\ c &{}d\end{pmatrix}\in \Gamma ^{(1)}_0(N)\) and \((\lambda ,\mu )\in {\mathcal {O}}_K^2\), the matrices
are in \(\Gamma ^{(2)}_0(N)\). These matrices act on \({\mathcal {H}}_2\) by
respectively. Because F satisfies the transformation law (2), we can deduce the two transformation laws of Hermitian Jacobi forms for \(\phi _m\), and therefore, \(\phi _m\in J_{k, m}(\Gamma ^{(1)}_0(N))\). We call (6) the Fourier–Jacobi expansion of F and \(\phi _m\)’s the Fourier–Jacobi coefficients of F.
2.4 Jacobi form with matrix index
The Jacobi group \(\Gamma ^{\ell }=SL_2({\mathbb {Z}})\ltimes ({\mathbb {Z}}^{\ell } \times {\mathbb {Z}}^{\ell })\) acts on \({\mathcal {H}}\times {\mathbb {C}}^{\ell }\) as follows:
where \(g= \begin{pmatrix} a &{} b \\ c &{} d \\ \end{pmatrix} \in SL_2({\mathbb {Z}}) \), \(\tau \in {\mathcal {H}}\), \(\lambda = (\lambda _1, \cdots , \lambda _{\ell }), ~\mu =(\mu _1,\cdots , \mu _{\ell }) \in {\mathbb {Z}}^{\ell }\) and \(z=(z_1,\cdots , z_{\ell }) \in {\mathbb {C}}^{\ell }\).
Definition 2.6
Let M be a symmetric, positive definite, half-integral \(\ell \times \ell \) matrix with integral diagonal entries. A holomorphic function \(\psi : {\mathcal {H}} \times {\mathbb {C}}^{\ell } \longrightarrow {\mathbb {C}}\) is a Jacobi form of weight k and index M on \(\Gamma _0^{(1)}(N)\) if for each \(g \in \Gamma _0^{(1)}(N)\) and \(\lambda , \mu \in {\mathbb {Z}}^{\ell }\), we have
and \(\psi \) has a Fourier series expansion of the form
where \(\tau \in {\mathcal {H}}\), \(z=(z_1, \cdots , z_{\ell })\in {\mathbb {C}}^{\ell }\), \(q=e^{2\pi i \tau }\), \(\zeta ^r=e^{2\pi i rz^{t}}\) and \(M^{\#}\) is the adjugate of M.
3 Proof of Theorem 1.1
Since \(F\ne 0\), there exists a \(m_0\) such that the Fourier–Jacobi coefficient \(\phi _{m_0}\ne 0\) in the Fourier–Jacobi expansion of F. Therefore, there exists \(s_{0}\in {\mathcal {O}}_K^{\#}/m_{0} {\mathcal {O}}_K\) such that the theta component \(h_{s_{0}}\ne 0\) in the theta decomposition of \(\phi _{m_0}\). The Fourier series expansion of \(h_{s_0}\) is given by
where
We have \(h_{s_{0}} \in S_{k-1}(\Gamma ^{(1)}(|D_{K}|m_{0}))\) and hence \(h_{s_{0}}(|D_{K}|m_{0}\tau ) \in S_{k-1}(\Gamma _1^{(1)}(|D_{K}|^{2}m_{0}^{2}))\). We know that
where the direct sum is over all Dirichlet characters modulo \(|D_{K}|^{2}m_{0}^{2}\). For each Dirichlet character \(\psi \) mod \(|D_K|^2m_{0}^{2}\), let \(f_{\psi }\in S_{k-1}(\Gamma _0^{(1)}(|D_{K}|^{2}m_{0}^{2}), \psi )\) be such that
Suppose the Fourier series expansion of \(f_{\psi }\) is given by \(f_{\psi } = \sum \limits _{n\geqslant 1} a_{\psi }(n)e(n\tau )\). Then from the above equation we have
Let \(\lambda =k-1\) and
Putting these values in (10), we get
From the above we also have
Now using the bounds for \({\hat{a}}_{\psi }(n)\) from [12, Theorem 3.4, Corollary 3.5] and applying (11) we achieve the following two estimates for \({\hat{a}}(n)\)
Also, applying Rankin–Selberg method [19, p. 357, Theorem 1], [20, Eq. 1.14] to \(h_{s_0}(|D_K|m_{0} \tau )\) and following similar steps as we have in [12, Corollary 3.2], we get that
where c is a constant depending on \(h_{s_{0}}(|D_K|m_{0}\tau )\) and \(\epsilon \) is any real number greater than 0. Now applying [12, Theorem 2.1] we get that \({\hat{a}}(n)\) changes sign at least once for \(n\in (X, X + X^{3/5}]\) for \(X \gg 1\). This implies that a(n) and hence \(A_F(T)\) where \(T= \begin{pmatrix} \frac{n+N(s_{0})|D_{K}|}{|D_{K}|m_{0}} &{} s_{0} \\ \overline{ s_{0}} &{} m_{0} \end{pmatrix}\), change sign atleast once for \( |D_{K}| \det (T) \in (X, X+X^{3/5}]\) for \(X\gg 1\).
4 Proof of Theorem 1.2
The ring of integers \({\mathcal {O}}_K\) of K is \({\mathbb {Z}}+\omega {\mathbb {Z}}\), where
We define the following set
We first prove the following proposition which will be required to prove Theorem 1.2.
Proposition 4.1
Let \(F=\sum _{T\in \Delta _2^{+}}A_F(T)e(tr(TZ)) \in S_k(SU_2({\mathcal {O}}_K))\). For any prime p there exists a \(G_p\in S_k(\Gamma ^{(2)}_0(p^2))\) such that the Fourier coefficients of \(G_p\) is given by
Proof
Let
We claim that \(G\in S_k(\Gamma ^{(2)}_0(p^2))\). It is enough to show that for any \(Y\in {\mathcal {J}}\), we have \(G'=F \mid _k \begin{pmatrix} \varvec{I}_2 &{} p^{-1}Y \\ \varvec{0}_2 &{} \varvec{I}_2 \end{pmatrix}\in S_k(\Gamma ^{(2)}_0(p^2))\). Let \(M=\begin{pmatrix} A &{} B \\ C &{} D \end{pmatrix} \in \Gamma _0^{(2)} (p^2)\). It is easy to check that
This implies that \(G'\mid _k M=G'\), which asserts our claim. Now the Fourier series expansion of G is given by
Now for any \(T \in \Delta _{2}^{+}\), we have
Therefore, the Fourier series expansion of G is given by
Thus, we get the required \(G_p\). \(\square \)
4.1 Proof of Theorem 1.2
Since \(F \ne 0\), by Lemma 2.4 there exists a primitive \(T_0=\begin{pmatrix} n_0 &{} r_0 \\ {\overline{r}}_0 &{} p \end{pmatrix}\in \Delta _2^+\) for some odd prime p such that \(A_F(T_0)\ne 0\). Applying Proposition 4.1, we construct \(G_p\) from F such that \(G_p\in S_k(\Gamma ^{(2)}_0(p^2))\) and the Fourier series expansion of \(G_p\) is given by
Let \(H=F-G_p\). We observe that \(H\in S_k(\Gamma ^{(2)}_0(p^2))\) and the Fourier series expansion of H is given by
Since \(T_0\) is primitive \(H\ne 0\). We consider the Fourier–Jacobi coefficient \(\phi _{p}\) in the Fourier–Jacobi expansion of H whose Fourier series expansion is given by
We have \(\phi _p \in J_{k, p}(\Gamma ^{(1)}_0(p^2))\). Let \(s_0\in {\mathcal {O}}_K^{\#}/p{\mathcal {O}}_K\) be such that \(s_0\equiv r_0 \pmod {p {\mathcal {O}}_K}\). We consider the theta component \(h_{s_0}\ne 0\) in the theta decomposition of \(\phi _p\). The Fourier series expansion of \(h_{s_0}\) is given by
where
We have \(h_{s_{0}} \in S_{k-1}(\Gamma ^{(1)}(|D_{K}|p^3))\). Now doing the similar calculation as we have done in the proof of Theorem 1.1, we get that \(A_F(T)\) where \(T= \begin{pmatrix} \frac{n+N(s_{0})|D_{K}|}{|D_{K}|p} &{} s_{0} \\ \overline{ s_{0}} &{} p \end{pmatrix}\), \(p^{-1}T\not \in \Delta _{2}^{+}\), changes sign atleast once for \(|D_K|\mathrm{{det}}(T)\in (X, X+X^{3/5}]\) for \(X\gg 1\).
5 Sturm bound
Sturm [21] proved that an elliptic modular form is determined by its first few Fourier series coefficients. The number of these first few Fourier coefficients is known as Sturm bound. Sturm’s result has had a significant impact on the study of elliptic modular forms. In this section we first develop a Sturm bound for Jacobi form with matrix index. Following this we establish a relation between Jacobi form with matrix index and Hermitian Jacobi forms. We use this relation to derive a Sturm bound for Hermitian Jacobi forms. Finally we prove Theorem 1.3 using the Fourier–Jacobi expansion of Hermitian modular form and Sturm bound of Hermitian Jacobi forms..
5.1 Sturm bound for Jacobi form with matrix index
Let
Let
where \(\alpha _{ij} \in {\mathbb {Z}}\) for \(i<j\) and \(\alpha _{ii} \ge 0\). We consider the Taylor series expansion of \(\phi \) at \(z_1=z_2=\cdots =z_{\ell }=0\), with Taylor coefficients \(X_{v_1, \cdots , v_{\ell }}(\tau )\),
For each \(\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in \Gamma _0^{(1)}(N)\), using the transformation property (7) of \(\phi \) and above equation we get
This implies that
Following Eichler and Zagier [6, p. 31], we define
where \(g^{(\nu )}(\tau )=\left( \frac{\partial }{\partial \tau }\right) ^{\nu } g(\tau )\). It can be readily checked that \(\zeta _{v_1,\cdots , v_{\ell }}(\tau )\in M_{k+v_1+\cdots +v_{\ell }}(\Gamma _0^{(1)}(N))\). The Fourier expansion of \(\zeta _{v_1, \cdots , v_{\ell }}(\tau )\) is given by
where
We further define
where we take \(\beta = \frac{\sum _{i=1}^{\ell }v_i}{2}\), if \(\sum _{i=1}^{\ell }v_i\) is even and \(\beta =\frac{1+\sum _{i=1}^{\ell }v_i}{2}\), if \(\sum _{i=1}^{\ell }v_i\) is odd.
Proposition 5.1
We define
where the map from \(J_{k, M}(\Gamma _0^{(1)}(N))\) to \(M_{k+v_1, \cdots , v_{\ell }}(\Gamma _0^{(1)}(N))\) is given by
Then the linear map D is injective.
Proof
We will show that if \(\phi \not = 0\) then \(D(\phi )\not =0\). Let us choose \(a_{\ell }, a_{\ell -1}, \cdots , a_1\) in a minimal way such that Taylor coefficient \(X_{a_1, \cdots , a_{\ell }}(\tau )\) of \(\phi \) in (12) is non-zero and for all \(\tau \)
We claim that \(a_i \le 2\alpha _{ii}\) for all \(1\le i \le \ell \). By [3, Lemma 3.1], we know that the function
is a non-zero classical Jacobi cusp form of weight \(k+a_2\cdots +a_{\ell }\) and index \(\alpha _{11}\). Therefore, by Eichler and Zagier result [6, Theorem 1.2, p. 10] we have \(a_1 \le 2\alpha _{11}\). Now suppose \(2 \le i \le \ell \). We choose \(b_{\ell }, \cdots , b_{i-1}, b_{i+1}, \cdots , b_1, b_i\) in a minimal way such that \(X_{b_1, b_2, \cdots , b_{\ell }}(\tau ) \ne 0\) and for all \(\tau \)
Again using minimality condition of \(b_{\ell }, \cdots , b_{i-1}, b_{i+1}, \cdots , b_1, b_i\), we see that
is a non-zero classical Jacobi form of weight \(k+b_1+\cdots + b_{i-1}+b_{i+1}+\cdots +b_{\ell }\) and index \(\alpha _{ii}\). Therefore, \(b_i\le 2\alpha _{ii}\). Since both \(a_{\ell }, a_{\ell -1}, \cdots , a_1\) and \(b_{\ell }, \cdots , b_{i-1}, b_{i+1}, \cdots , b_1, b_i\) are minimal we must have
This implies that \(a_i\le 2\alpha _{ii}\) for all \(1\le i \le n\). So we have proved that if \(\phi \ne 0\) then there exists a Taylor coefficient \(X_{a_1, \cdots , a_{\ell }}(\tau )\ne 0\) such that \(a_i \le 2\alpha _{ii}\) for all \(1\le i \le \ell \). Again using the minimality of \(a_{\ell }, a_{\ell -1}, \cdots , a_1\), we see that \(D_{a_1, \cdots , a_{\ell }}(\phi )=\alpha X_{a_1, \cdots , a_{\ell }}(\tau )\) for some non-zero \(\alpha \in {\mathbb {C}}\). Thus, \(D(\phi )\not =0\). \(\square \)
In the following theorem we establish a Sturm bound for Jacobi form with matrix index.
Theorem 5.2
Let \(\gamma =[SL_2({\mathbb {Z}}): \Gamma ^{(1)}_0(N)]\). Let
If \(a(n, r)=0\), for all
then \(\phi =0 \).
Proof
If \(a(n, r)=0\) for all \(n\le \frac{1}{12}(k+2 tr(M))\gamma \) then using the Fourier expansion of \(D_{v_1, \cdots , v_{\ell }}(\phi )\) and Sturm result for elliptic modular forms we get that
for all \((v_1, \cdots , v_{\ell })\) satisfying \(0 \le v_i \le 2\alpha _{ii}\) for all \(1\le i \le \ell \). This implies that \(D(\phi )=0\) and hence \(\phi =0\) as D is injective. \(\square \)
5.2 Relation between Hermitian Jacobi and Jacobi form with matrix index
In [17, Theorem 2.3], Meher and the second author proved a relation between Hermitian Jacobi form over \({\mathbb {Q}}(i)\) and Jacobi form with matrix index. In the following theorem, we generalize their result for an arbitrary imaginary quadratic field.
Theorem 5.3
Let \(K={\mathbb {Q}}(\sqrt{-d})\) be an imaginary quadratic field. Suppose
Then the space \(J_{k, m}(\Gamma _0^{(1)}(N))\) is isomorphic to \(J_{k, A}(\Gamma _0^{(1)}(N))\) as a vector space over \({\mathbb {C}}\).
Proof
First let us consider the case \(-d \equiv 2, 3 \pmod 4\). We define a map
by
Let \({\hat{\phi }}(\tau , z_1, z_2)=\phi \left( \tau , z_1+i\sqrt{d}z_2, z_1-i\sqrt{d}z_2\right) \). Using the transformation property of \(\phi \) mentioned in (4) and (5) we can verify that \({\hat{\phi }}\) satisfies (7) and (8). Suppose \(\phi \) has Fourier series expansion
then
Any \(r\in {\mathcal {O}}_K^{\#}\) can be written as \(r=\frac{i}{2\sqrt{d}}\left( \alpha +i\sqrt{d}\beta \right) \), where \(\alpha , \beta \in {\mathbb {Z}}\). We now consider an element \(\rho \in {\mathbb {Z}}^2,\) where \(\rho =(-\beta , -\alpha ).\) Then the correspondence \(r\mapsto \rho \) is clearly a bijection from \({\mathcal {O}}_K^{\#}\) to \({\mathbb {Z}}^2.\) Now the above Fourier series expansion of \({\hat{\phi }}\) can be expressed as
which is of the form given in (9). This implies that \(\eta _1\) is a well defined linear map. In a similar manner, one can show that the map
defined by
is also a well-defined linear map. Now consider the composition map \(\eta _2 \circ \eta _1,\)
Similarly we can also check that \((\eta _1\circ \eta _2)(\psi (\tau ,z_1,z_2))=\psi (\tau ,z_1,z_2)\) and hence \(\eta _2 \circ \eta _1=I_1\), \(\eta _1 \circ \eta _2=I_2\), where \(I_1\) and \(I_2\) are identity maps on the vector spaces \(J_{k, m}(\Gamma ({\mathcal {O}}_K))\) and \(J_{k, A}(\Gamma ^2)\) respectively.
Now consider the case \(-d\equiv 1\pmod 4\). Here we define the maps
by
and
by
Approaching as above we can easily verify that \(\eta _1\) and \(\eta _2\) are well defined linear maps and also they satisfy
\(\square \)
5.3 Sturm bound for Hermitian Jacobi forms
Using Theorems 5.2 and 5.3, we establish a Sturm bound for Hermitian Jacobi forms.
Theorem 5.4
Let \(\gamma =[SL_2({\mathbb {Z}}):\Gamma ^{(1)}_0(N)]\). Suppose \(K={\mathbb {Q}}(\sqrt{-d})\) and
If \(a_{\phi }(n, r)=0\) for all \(n\le \beta \), where
then \(\phi =0\).
Proof
We begin with the case \(-d \equiv 2, 3 \pmod 4\). The Fourier series expansion of \({\hat{\phi }}(\tau , z_1, z_2)\) in Theorem 5.3 is given by
where \(r=\frac{i}{2\sqrt{d}}\left( \alpha +i\sqrt{d}\beta \right) \), \(\rho =(-\beta , -\alpha )\), \(\alpha , \beta \in {\mathbb {Z}}\). Now, if \(a_{\phi }(n, r)=0\) for all \(n\le \beta \), then by Theorem 5.2, we see that \({\hat{\phi }}=0\). Since \(\eta \) is an isomorphism, we get \(\phi = 0\). This completes the proof when \(-d\equiv 2, 3 \pmod 4\). The case of \(-d \equiv 1 \pmod 4\) follows similarly. \(\square \)
If we put \(N=1\) in the above we get the following.
Corollary 5.5
Let \(K={\mathbb {Q}}(\sqrt{-d})\), where \(d>0\) be square free. Let
If \(a(n, r)=0\) for all \(n\le \beta \), where
then \(\phi =0\).
We are now ready to prove Theorem 1.3. We first define some necessary terms. For
we define
From Corollary 5.5, Sturm bound for Hermtian Jacobi form when \(-d \equiv 2, 3 \pmod 4\) is
We put \(\beta =m=t\) in the above and see that it is possible to get a positive value of \(t=\frac{k}{2(5-d)}\) if \(d=1, 2\). Similarly, when \(-d\equiv 1 \pmod 4\), we will get a positive value of \(t=\frac{2k}{19-d}\), if \(d \in \{3, 7, 11, 15\}\). We will use the Fourier–Jacobi expansion and a transformation of Hermitian modular form F to show that if the Fourier coefficients \(A_F(n, r, m)=0\) for all \(n \le t\) and \(m \le t\) then \(F=0\).
5.4 Proof of Theorem 1.3
We will prove the result when \(d\in \{1, 2\}\). The case \(d \in \{3, 7, 11, 15\}\) will follow similarly. We consider the Fourier–Jacobi expansion of F
We will show that \(\phi _m =0\) for all \(m \ge 0\). We first consider that \(m\le \frac{k}{2(5-d)}\). Then
Therefore, by Corollary 5.5, we have \(\phi _m =0\). Assume that \(m> \frac{k}{2(5-d)}\). We use induction on m to show that \(\phi _m =0\). Suppose that \(\phi _{m'}=0\) for all \(m'<m\). Now we consider \(\phi _m\). For \(g=\begin{pmatrix}0 &{} 1 \\ 1 &{} 0 \end{pmatrix}\), the matrix \(M=\begin{pmatrix} ({\overline{g}}^t)^{-1} &{} \varvec{0}_2 \\ \varvec{0}_2 &{} g\end{pmatrix}\in SU_2({\mathcal {O}}_K)\). Using the transformation property (2) of F for M, we get \(F(\tau , z_1, z_2, \tau ')=(-1)^k F(\tau ', z_2, z_1, \tau )\), which shows that \(A_F(n, r, m)=(-1)^k A_F(m, {\overline{r}}, n)=0\) for all \(n <m\). Therefore,
Again by Corollary 5.5, we have \(\phi _m =0\). This completes the proof.
Remark 5.6
-
(1)
The space of cusp form \(S_k(SU_2({\mathcal {O}}_K))=\{0\}\) if \(k < 2(5-d)\) when \(d=1, 2\) and if \(k<\frac{19-d}{2}\) when \(d=3, 7, 11, 15\).
-
(2)
The bound in Theorem 1.3 is sharp for \(K={\mathbb {Q}}(i), {\mathbb {Q}}(\sqrt{2}i)\). We have explained this in the next Example 5.7.
Example 5.7
The Hermitian Eisenstien series of even weight \(k>4\) over a field \(K={\mathbb {Q}}(\sqrt{-d})\) is given by
Moreover, Krieg [15] constructed weight 4 Eisenstein series by Maass lift. The Eisenstein series \(E_k^{(K)}\) has rational Fourier coefficients for \(k\ge 4\) [7, 8, 15]. There are Hermitian cusp forms over any imaginary quadratic field [4, Corollary 2], [8],
If \(K={\mathbb {Q}}(i)\) then
Let \(\beta (k)=[k/8]\) be the Sturm bound for \(K={\mathbb {Q}}(i)\) in Theorem 1.3. We define
Since \(\chi _8\) and \(F_{10}^{(K)}\) are cusp forms, we have \(A_{\chi _8}(n, r, m)=A_{F_{10}^{(K)}}(n, r, m)=0\) whenever \(n=0\) or \(m=0\). We have
Therefore, we check that \(A_{H_k}(n, r, m)=0\) whenever \(n \le \beta (k)-1\) and \(m \le \beta (k)-1\) but \(H_k \not = 0\). Hence the bound is sharp for \(K={\mathbb {Q}}(i)\). Similarly, one can check that the bound in Theorem 1.3 is sharp for \(K={\mathbb {Q}}(\sqrt{2}i)\) using \(E_4^{(K)}\) and Hermitian cusp form \(\phi _6\) and \(\phi _8\) of weight 6 and 8 respectively constructed by Dern and Krieg [5].
6 Proof of Theorem 1.4
We will prove the result when \(d\in \{1, 2\}\). The case \(d \in \{3, 7, 11, 15\}\) will follow similarly. We have assumed that \(F\ne 0\), therefore, by Theorem 1.3 there exists \(T_0=\begin{pmatrix}n_0 &{} r_0 \\ {\overline{r}}_0 &{} m_0\end{pmatrix}\) such that
and \(A_F(T_0)\ne 0\). We consider the Fourier–Jacobi coefficient \(\phi _{m_0} \ne 0\) in the Fourier–Jacobi expansion of F. The Fourier series expansion of \(\phi _{m_0}\) is given by
We define
Using Theorem 5.3, we get that \({\hat{\phi }}\in J_{k, M_0}(SL_2({\mathbb {Z}}))\), where \(M_0=\begin{pmatrix}m_0 &{} 0 \\ 0 &{} m_0d \end{pmatrix}\). We consider
where \(\epsilon _1, \epsilon _2\in \{1, -1\}\). We can also check that \({\hat{\phi }}(\tau , \epsilon _1 z_1, \epsilon _2 z_2)\in J_{k, M_0}(SL_2({\mathbb {Z}}))\) for every \(\epsilon _1, \epsilon _2\in \{1, -1\}\). The Fourier series expansion of \({\hat{\phi }}(\tau , \epsilon _1 z_1, \epsilon _2 z_2)\) is given by
where
Now \(\varphi (\tau , z_1, z_2)\in J_{4k, 4M_0}(SL_2({\mathbb {Z}}))\). Also, by construction, \(\varphi (\tau , z_1, z_2)\) is an even function in the variable \(z_1\) and \(z_2\). We consider the Taylor series expansion of \(\varphi (\tau , z_1, z_2)\) around \(z_1=z_2=0\)
Since \(\varphi \ne 0\) there exists a non-zero Taylor coefficient in the above equation. We choose \(a_2, a_1\) in a minimal way such that \(X_{a_1, a_2}\not =0\) and
Then from the proof of Proposition 5.1 we get that \(a_1\le 8m_0\) and \(a_2\le 8m_0d\). Also \(D_{a_1, a_2}(\varphi )=\alpha X_{a_1, a_2}(\tau )\) in Proposition 5.1, for some non-zero \(\alpha \in {\mathbb {C}}\). This implies that \(X_{a_1, a_2}(\tau )\) is a non-zero elliptic modular form of weight \(k_1=4k+a_1+a_2\). Therefore, we have
Suppose the Fourier series expansion of \(\varphi \) is given by
Let \(f=\frac{a_1!~ a_2!}{(2\pi i)^{a_1+a_2}} X_{a_1, a_2}(\tau )\). It can be easily checked that
If the Fourier series expansion of f is given by \(\sum \limits _{n\ge 1} d(n) e(n\tau )\) then we have
Now by [11, Theorem 2] there exists \(n_1\ge 1\) such that
Therefore, by (14), we have
We have
Since \(\varphi (\tau , z_1, z_2)\) is an even function in each variable \(z_1\), \(z_2\), the integers \(a_1\), \(a_2\) are even. Therefore, there exists \(s_0=(\alpha _0, \beta _0)\) such that \(d(n_1, s_0)<0\). Also \(d(n_1, s_0)\) is a finite sum of product of Fourier coefficients of the form \( A_F\left( \begin{pmatrix}n_{\epsilon _1, \epsilon _2} &{} * \\ * &{} m_0 \end{pmatrix}\right) \) with \(\sum n_{\epsilon _1, \epsilon _2}=n_1\). Therefore, atleast one of the Fourier coefficient
We have \(tr\left( T_1\right) =n_{\epsilon _1, \epsilon _2} +m_0< n_1+ tr(T_0)\). Using (15) and (13) we have
Now replacing F by \(-F\) and proceeding as above, we get a matrix \(T_2\) such that
and \(-A_F(T_2)<0\). This proves the result.
Remark 6.1
We note that Theorem 1.1 is true for Hermitian modular forms on the congruence subgroup \(\Gamma ^{(2)}_0(N)\) of \(SU_2({\mathcal {O}}_K)\) with Dirichlet character \(\chi \pmod N\), where \(\chi \) acts on \(\Gamma ^{(2)}_0(N)\) by
Moreover, if the conductor of \(\chi \) is N then Theorem 1.2 also holds.
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Acknowledgements
The authors would like to thank Prof. Aloys Krieg for his helpful remarks and for providing his preprint [8]. The authors would also like to thank Dr. Karam Deo Shankhadhar for his insightful comments and ideas.
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Nandi, R., Singh, S.K. & Tiwari, P. On sign changes of Fourier coefficients of Hermitian cusp forms of degree two. Ramanujan J 61, 1037–1062 (2023). https://doi.org/10.1007/s11139-023-00716-2
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DOI: https://doi.org/10.1007/s11139-023-00716-2