On sign changes of Fourier coefficients of Hermitian cusp forms of degree two

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.

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

$$\begin{aligned} D_K={\left\{ \begin{array}{ll} -4d &{} \text{ if }~ -d\equiv 2, 3 \pmod 4, \\ -d &{} \text{ if }~ -d \equiv 1 \pmod 4 \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} U_2({\mathcal {O}}_K)=\{M\in M_{4}({\mathcal {O}}_K)\mid {\overline{M}}^{t}J_2M=J_2\}, \end{aligned}$$

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

$$\begin{aligned} SU_2({\mathcal {O}}_K)=U_2({\mathcal {O}}_K)\cap SL_{4}({\mathcal {O}}_K) \end{aligned}$$

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:

$$\begin{aligned} F(Z)=\sum _{T\in \Delta _2^+}A_F(T)e(tr(TZ))=\sum _{\begin{array}{c} n, m\in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#}\\ n, m, nm-N(r)> 0 \end{array}}A_F(n, r, m)q^n\zeta _1^r\zeta _2^{{\overline{r}}}(q')^m, \end{aligned}$$

(1)

where

$$\begin{aligned} \Delta _2^+=\left\{ T=\begin{pmatrix}n &{} r\\ {\overline{r}} &{} m\end{pmatrix} \bigg |~ n,m \in {\mathbb {Z}}, ~r\in {\mathcal {O}}_K^{\#}, ~T> 0 \right\} , Z\in \begin{pmatrix} \tau &{} z_1 \\ z_2 &{} \tau ' \end{pmatrix} \in {\mathcal {H}}_2, \end{aligned}$$

$q=e(\tau ), \zeta _1=e(z_1), \zeta _2=e(z_2), q'= e(\tau '), e(z)=e^{2\pi i z}$. 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

$$\begin{aligned} \mu (T)=\max \{ l \in {\mathbb {N}}\mid l^{-1}T\in \Delta _2^+ \}. \end{aligned}$$

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

$$\begin{aligned} F(\tau , z_1, z_2, \tau ')=\sum _{\begin{array}{c} n, m\in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#}\\ n, m, nm-N(r)\ge 0 \end{array}}A_F(n, r, m)q^n\zeta _1^r\zeta _2^{{\overline{r}}}(q')^m \in M_k(SU_2({\mathcal {O}}_K)). \end{aligned}$$

If $A_F(n, r, m) =0$ for all $0\le n \le \beta$ and $0 \le m \le \beta$, where

$$\begin{aligned} \beta ={\left\{ \begin{array}{ll} \bigg [\frac{k}{2\left( 5-d\right) }\bigg ] &{} \text{ if } ~~ d=1, 2,\\ \\ \bigg [\frac{2k}{(19-d)}\bigg ] &{} \text{ if } ~~d=3, 7, 11, 15, \end{array}\right. } \end{aligned}$$

then

$$\begin{aligned} F=0. \end{aligned}$$

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

$$\begin{aligned} tr(T_1),~tr(T_2) \ll (4c_dk)^{2+\epsilon }, \end{aligned}$$

for any real $\epsilon >0$, where

$$\begin{aligned} c_d={\left\{ \begin{array}{ll} \frac{7+d}{5-d} &{} \text{ if }~~d=1, 2,\\ \frac{29+d}{19-d} &{} \text{ if } ~~d=3, 7, 11, 15, \end{array}\right. } \end{aligned}$$

such that

$$\begin{aligned} A_F(T_1)A_F(T_2)<0. \end{aligned}$$

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

$$\begin{aligned} {\mathcal {H}}_2=\left\{ Z=\begin{pmatrix}\tau &{} z_1\\ z_2 &{} \tau '\end{pmatrix} \in M_2({\mathbb {C}}) \mid \frac{1}{2i}(Z-{\overline{Z}}^t)>0\right\} . \end{aligned}$$

The Hermitian modular group $U_2({\mathcal {O}}_K)$ acts on ${\mathcal {H}}_2$ by

$$\begin{aligned} M\cdot Z=(AZ+B)(CZ+D)^{-1} ~~~ \text{ where }~Z\in {\mathcal {H}}_2, ~\begin{pmatrix} A &{} B \\ C &{} D \end{pmatrix}\in U_2({\mathcal {O}}_K). \end{aligned}$$

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

$$\begin{aligned} (F\mid _k M) (Z)=(\mathrm{{det}}(CZ+D))^{-k}F(M\cdot Z). \end{aligned}$$

For a positive integer N, we define the congruence subgroup $\Gamma ^{(2)}_0(N)$ of $SU_2({\mathcal {O}}_K)$ by

$$\begin{aligned} \Gamma _0^{(2)}(N)=\bigg \{\begin{pmatrix}A &{} B \\ C&{} D\end{pmatrix}\in SU_2({\mathcal {O}}_K)\mid C \equiv \varvec{0}_2 \pmod {N{\mathcal {O}}_K}\bigg \}. \end{aligned}$$

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

$$\begin{aligned} F\mid _k M=F \end{aligned}$$

(2)

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:

$$\begin{aligned}{} & {} F(Z)=F(\tau , z_1, z_2, \tau ')=\sum _{T\in \Delta _2}A_F(T)e(tr(TZ))\nonumber \\{} & {} =\sum _{\begin{array}{c} n, m\in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#}\\ n, m, nm-N(r)\ge 0 \end{array}}A_F(n, r, m)q^n\zeta _1^r\zeta _2^{{\overline{r}}}(q')^m, \end{aligned}$$

(3)

where $q=e(\tau ), \zeta _1=e(z_1), \zeta _2=e(z_2), q'=e(\tau ')$ and

$$\begin{aligned} \Delta _2=\left\{ T=\begin{pmatrix}n &{} r\\ {\overline{r}} &{} m\end{pmatrix} \bigg |~ n,m \in {\mathbb {Z}}, ~r\in {\mathcal {O}}_K^{\#}, ~T\ge 0 \right\} . \end{aligned}$$

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

$$\begin{aligned} {\overline{g}}^t T g = \begin{pmatrix} * &{} * \\ * &{} p=N(\beta )n+\delta r{\overline{\beta }}+\beta {\overline{r}}{\overline{\delta }}+N(\delta )m\end{pmatrix}. \end{aligned}$$

We know that $\textrm{det}(g)=\epsilon$, where $\epsilon \in {\mathcal {O}}_K^{\times }$. Therefore, we can take

$$\begin{aligned} g_1=\begin{pmatrix} \alpha /\epsilon &{} \beta \\ \gamma /\epsilon &{} \delta \end{pmatrix} \in SL_2({\mathcal {O}}_K) \end{aligned}$$

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

$$\begin{aligned} A_F(gT{\overline{g}}^t)=A_F(T), ~~~~\text{ for } \text{ all }~ g\in SL_2({\mathcal {O}}_K), T\in \Delta _2^+. \end{aligned}$$

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:

$$\begin{aligned} \left( g, (\lambda , \mu ) \right) \cdot (\tau , z_1, z_2)=\left( \frac{a\tau +b}{c \tau +d}, \frac{z_1+\lambda \tau +\mu }{c\tau +d}, \frac{z_2+{\overline{\lambda }}\tau +{\overline{\mu }}}{c\tau +d} \right) , \end{aligned}$$

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

$$\begin{aligned} \Gamma ^{(1)}_0(N)=\bigg \{\begin{pmatrix}a &{} b \\ c &{} d\end{pmatrix}\in SL_2({\mathbb {Z}})\mid c \equiv 0 \pmod {N{\mathbb {Z}}}\bigg \}. \end{aligned}$$

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

$$\begin{aligned}{} & {} \phi \left( \frac{a\tau +b}{c\tau +d}, \frac{ z_1}{c\tau +d}, \frac{ z_2}{c \tau +d}\right) =(c\tau +d)^{k}e^{\frac{2\pi imcz_1z_2}{c\tau +d}}\phi (\tau , z_1, z_2), \end{aligned}$$

(4)

$$\begin{aligned}{} & {} \phi \left( \tau , z_1+\lambda \tau +\mu , z_2+{\overline{\lambda }}\tau +{\overline{\mu }} \right) =e^{-2\pi im (N(\lambda )\tau +{\overline{\lambda }}z_1+\lambda z_2)}\phi (\tau , z_1, z_2) \end{aligned}$$

(5)

and $\phi$ has a Fourier series expansion of the form

$$\begin{aligned} \phi =\sum _{\begin{array}{c} n \in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#} \\ nm-N(r)\ge 0 \end{array}} c(n, r)q^n\zeta _1^r\zeta _2^{{\overline{r}}}, \end{aligned}$$

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

$$\begin{aligned} c_s(L)={\left\{ \begin{array}{ll} c\left( n, r\right) &{} \text{ if } ~ r\equiv s\pmod {m{\mathcal {O}}_K} ~ \text{ and }~L=|D_K|(nm-N(r)),\\ 0 &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

The theta decomposition of $\phi \in J_{k, m}(\Gamma ^{(1)}_0(N))$ is given by

$$\begin{aligned} \phi _m(\tau , z_1, z_2)=\sum _{s\in {\mathcal {O}}_K^{\#}/m {\mathcal {O}}_K} h_s \theta _{m, s}, \end{aligned}$$

where

$$\begin{aligned}{} & {} h_{s}(\tau )=\sum _{\begin{array}{c} L=0\\ L \equiv -N(s)|D_K|\pmod {m|D_K|{\mathbb {Z}}} \end{array}}^{\infty } c_s(L)e\left( \frac{L}{|D_K|m}\tau \right) ,\\ {}{} & {} \quad \theta _{m, s}=\sum _{r\equiv s \pmod {m{\mathcal {O}}_K}}e\left( \frac{N(r)}{m}\tau +rz_1+{\overline{r}}z_2\right) . \end{aligned}$$

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

$$\begin{aligned}{} & {} F(\tau , z_1, z_2, \tau ')=\sum _{m=1}^{\infty }\phi _{m} e(m \tau '),\nonumber \\{} & {} \text{ where }~~\phi _m=\sum _{\begin{array}{c} n\in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#}\\ nm-N(r)\ge 0 \end{array}}A_F(n, r, m)e(n \tau +rz_1+{\overline{r}}z_2). \end{aligned}$$

(6)

For $\begin{pmatrix}a &{}b \\ c &{}d\end{pmatrix}\in \Gamma ^{(1)}_0(N)$ and $(\lambda ,\mu )\in {\mathcal {O}}_K^2$, the matrices

$$\begin{aligned} \begin{pmatrix} a &{} 0 &{}b &{} 0\\ 0 &{} 1&{} 0&{} 0\\ c&{} 0 &{}d &{}0\\ 0 &{} 0 &{}0 &{}1 \end{pmatrix} ~~~~\text{ and }~~~~ \begin{pmatrix} 1&{} 0 &{} 0&{} \mu \\ {\overline{\lambda }} &{}1 &{}{\overline{\mu }} &{}0\\ 0 &{}0 &{}1 &{}-\lambda \\ 0 &{} 0&{}0 &{}1 \end{pmatrix} \end{aligned}$$

are in $\Gamma ^{(2)}_0(N)$. These matrices act on ${\mathcal {H}}_2$ by

$$\begin{aligned}{} & {} (\tau , z_1, z_2, \tau ') \mapsto \left( \frac{a\tau +b}{c\tau +d}, \frac{z_1}{c\tau +d}, \frac{z_2}{c\tau +d}, \tau '-\frac{cz_1z_2}{c\tau +d}\right) ,\\{} & {} (\tau , z_1, z_2, \tau ')\mapsto (\tau , z_1+\lambda \tau +\mu , z_2+\lambda \tau +\mu , \tau '+\lambda z_1+\lambda z_2+\lambda ^2\tau +\lambda \mu ) \end{aligned}$$

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:

$$\begin{aligned} \left( g, (\lambda , \mu )\right) \cdot (\tau , z_1, \cdots , z_{\ell })=\left( \frac{a\tau +b}{c\tau +d}, \frac{z_1+\lambda _1 \tau +\mu _1}{c\tau +d}, \cdots ,\frac{z_{\ell }+\lambda _{\ell } \tau +\mu _{\ell }}{c\tau +d}\right) , \end{aligned}$$

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

$$\begin{aligned}{} & {} \psi \left( \frac{a\tau +b}{c\tau +d}, \frac{z_1}{c\tau +d},\cdots , \frac{z_{\ell }}{c\tau +d} \right) =(c\tau +d)^{k}e^{2\pi i\frac{cM[z^t]}{c\tau +d}}\psi (\tau , z_1, \cdots , z_{\ell }), \end{aligned}$$

(7)

$$\begin{aligned}{} & {} \psi (\tau , z_1+\lambda _1\tau +\mu _1, \cdots , z_{\ell }+\lambda _{\ell }\tau +\mu _{\ell })=e^{-2\pi i(\tau M[\lambda ^t]+2\lambda Mz^t)}\psi (\tau , z_1, \cdots , z_{\ell })\nonumber \\ \end{aligned}$$

(8)

and $\psi$ has a Fourier series expansion of the form

$$\begin{aligned} \psi (\tau , z_1,\cdots , z_{\ell })=\sum _{\begin{array}{c} n\in {\mathbb {Z}}, r\in {\mathbb {Z}}^{\ell } \\ 4\mathrm{{det}}(M)n-M^{\#}[r^t]\ge 0 \end{array}}c(n, r)q^n \zeta ^r, \end{aligned}$$

(9)

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

$$\begin{aligned} h_{s_0}(\tau )=\sum _{\begin{array}{c} n=1\\ n \equiv -N(s_0)|D_K|\pmod {m_{0}|D_K|{\mathbb {Z}}} \end{array}}^{\infty } a(n)e\left( \frac{n}{|D_K|m_{0}}\tau \right) , \end{aligned}$$

where

$$\begin{aligned} a(n)= {\left\{ \begin{array}{ll} A_{F}(T) &{} \text{ if }~~T= \begin{pmatrix}\frac{n+N(s_{0})|D_{K}|}{|D_{K}|m_{0}} &{} s_{0} \\ \overline{ s_{0}} &{} m_{0} \end{pmatrix} \in \Delta _2^+, \\ 0 &{} ~\text{ otherwise }. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} S_{k-1}(\Gamma _1^{(1)}(|D_{K}|^{2}m_{0}^{2})) = \bigoplus \limits _{\psi } S_{k-1}(\Gamma _0^{(1)}(|D_{K}|^{2}m_{0}^{2}), \psi ), \end{aligned}$$

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

$$\begin{aligned} h_{s_{0}}(|D_{K}|m_{0}\tau )=\sum \limits _{\psi } f_{\psi }(\tau ). \end{aligned}$$

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

$$\begin{aligned} \sum _{n\geqslant 1} a(n) e(n\tau )=\sum _{\psi } \sum \limits _{n\geqslant 1} a_{\psi }(n)e(n\tau ). \end{aligned}$$

(10)

Let $\lambda =k-1$ and

$$\begin{aligned} {\hat{a}}(n) =\frac{a(n)}{n^{(\lambda -1)/2}}~~\text{ and }~~ {\hat{a}}_{\psi }(n)=\frac{a_{\psi }(n)}{n^{(\lambda -1)/2}}. \end{aligned}$$

Putting these values in (10), we get

$$\begin{aligned} \sum _{n\geqslant 1 } {\hat{a}}(n) n^{(\lambda -1)/2} e(n\tau )=\sum _{\psi } \sum \limits _{n\geqslant 1} {\hat{a}}_{\psi }(n)n^{(\lambda -1)/2}e(n\tau ). \end{aligned}$$

From the above we also have

$$\begin{aligned} {\hat{a}}(n)=\sum _{\psi } {\hat{a}}_{\psi } (n). \end{aligned}$$

(11)

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 ${\hat{a}}(n)$

$$\begin{aligned}{} & {} {\hat{a}}(n) \ll n^{\epsilon },\\{} & {} \sum \limits _{n\leqslant X} {\hat{a}}(n)\ll X^{1/3 +\epsilon }. \end{aligned}$$

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

$$\begin{aligned} \sum _{n\leqslant X} {\hat{a}}^{2}(n)= c X + O(X^{3/5+\epsilon }), \end{aligned}$$

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

$$\begin{aligned} \omega = {\left\{ \begin{array}{ll} \sqrt{-d} &{} \text{ if } ~~-d \equiv 2, 3 \pmod 4, \\ \frac{1+\sqrt{-d}}{2}&{} \text{ if }~~ -d\equiv 1\pmod 4. \end{array}\right. } \end{aligned}$$

We define the following set

$$\begin{aligned} {\mathcal {J}}=\bigg \{ \begin{pmatrix} x &{} s \\ {\overline{s}} &{} y \end{pmatrix}\mid x, y \in {\mathbb {Z}}, s=\alpha + \omega \beta \in {\mathcal {O}}_K, ~ 0\le \alpha , \beta , x, y < p \bigg \}. \end{aligned}$$

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

$$\begin{aligned} G_p=\sum _{\begin{array}{c} T\in \Delta _2^+\\ p^{-1}T\in \Delta _2^+ \end{array}}A_F(T)e(tr(TZ)). \end{aligned}$$

Proof

Let

$$\begin{aligned} G:=\frac{1}{p^4} \sum _{Y\in {\mathcal {J}}} F \mid _k \begin{pmatrix} \varvec{I}_2 &{} p^{-1}Y \\ \varvec{0}_2 &{} \varvec{I}_2 \end{pmatrix}. \end{aligned}$$

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

$$\begin{aligned} \begin{pmatrix} \varvec{I}_2 &{} p^{-1}Y \\ \varvec{0}_2 &{} \varvec{I}_2 \end{pmatrix} \begin{pmatrix} A &{} B \\ C &{} D \end{pmatrix} \begin{pmatrix} \varvec{I}_2 &{} p^{-1}Y \\ \varvec{0}_2 &{} \varvec{I}_2 \end{pmatrix}^{-1} \in \Gamma _0^{(2)} ( p^2). \end{aligned}$$

This implies that $G'\mid _k M=G'$, which asserts our claim. Now the Fourier series expansion of G is given by

$$\begin{aligned} G(Z)= & {} \frac{1}{p^4} \sum _{Y\in {\mathcal {H}}} F(Z+p^{-1} Y)\\= & {} \frac{1}{p^4} \sum _{Y\in {\mathcal {J}}}\sum _{T\in \Delta _{2}^{+}} A_{F}(T) e(tr(TZ+Tp^{-1}Y))\\= & {} \frac{1}{p^4} \sum _{T\in \Delta _{2}^{+}} A_{F}(T) e(tr(TZ)) \sum _{Y\in {\mathcal {J}}} e(tr(p^{-1}TY)). \end{aligned}$$

Now for any $T \in \Delta _{2}^{+}$, we have

$$\begin{aligned} \sum \limits _{Y\in {\mathcal {J}}} e(tr(p^{-1}TY))= {\left\{ \begin{array}{ll} p^{4} &{} \text{ if }~~ p^{-1}T\in \Delta _2^{+},\\ 0 &{} \text{ Otherwise }. \end{array}\right. } \end{aligned}$$

Therefore, the Fourier series expansion of G is given by

$$\begin{aligned} G=\sum _{\begin{array}{c} T\in \Delta _2^+\\ p^{-1}T\in \Delta _2^+ \end{array}}A_F(T)e(tr(TZ)). \end{aligned}$$

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

$$\begin{aligned} G_p=\sum _{\begin{array}{c} T\in \Delta _2^+\\ p^{-1}T\in \Delta _2^+ \end{array}}A_F(T)e(tr(TZ)). \end{aligned}$$

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

$$\begin{aligned} H=\sum _{\begin{array}{c} T\in \Delta _2^+\\ p^{-1}T \not \in \Delta _2^+ \end{array}}A_F(T) e(tr(TZ)). \end{aligned}$$

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

$$\begin{aligned} \phi _{p}(\tau , z_1, z_2)=\sum _{\begin{array}{c} T=\begin{pmatrix}n &{} r \\ {\overline{r}} &{} p \end{pmatrix}\in \Delta _2^+\\ p^{-1}T\not \in \Delta _2^+ \end{array}} A_F(T)e(n\tau +rz_1+ {\overline{r}}z_2). \end{aligned}$$

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

$$\begin{aligned} h_{s_{0}}(\tau )= \sum _{\begin{array}{c} n\geqslant 1 \\ n\equiv -N(s_{0})|D_{K}| \pmod {p |D_{K}|{\mathbb {Z}}} \end{array}} a(n) e\left( \frac{n\tau }{|D_{K}|p}\right) , \end{aligned}$$

where

$$\begin{aligned} a(n)= {\left\{ \begin{array}{ll} A_{F}(T) &{} ~\text{ if } ~~~ T= \begin{pmatrix} \frac{n+N(s_{0})|D_{K}|}{|D_{K}|p} &{} s_{0} \\ \overline{ s_{0}} &{} p \end{pmatrix} \in \Delta _2^+ \text{ and }~ p^{-1}T\not \in \Delta _{2}^{+}, \\ 0 &{} ~\text{ Otherwise }. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} \phi =\sum _{\begin{array}{c} n \in {\mathbb {Z}}, r \in {\mathbb {Z}}^l \\ 4\mathrm{{det}}(M)n - M^{\#}[r^t]\ge 0 \end{array}} a(n, r)q^n\zeta ^{r} \in J_{k, M}(\Gamma _0^{(1)}(N)). \end{aligned}$$

Let

$$\begin{aligned} M=\begin{pmatrix} \alpha _{11} &{} \alpha _{12}/2 &{} \cdots &{} \alpha _{1\ell }/2 \\ \alpha _{12}/2 &{} \alpha _{22} &{}\cdots &{} \alpha _{2\ell }/2 \\ \cdots &{} \cdots &{} \cdots &{}\cdots \\ \alpha _{1\ell }/2 &{} \cdots &{} \cdots &{} \alpha _{\ell \ell } \end{pmatrix}, \end{aligned}$$

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 $X_{v_1, \cdots , v_{\ell }}(\tau )$,

$$\begin{aligned} \phi =\sum _{v_1,\cdots , v_{\ell }\ge 0} X_{v_1, \cdots , v_{\ell }}(\tau ) z_1^{v_1}\cdots z_{\ell }^{v_{\ell }}. \end{aligned}$$

(12)

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

$$\begin{aligned}{} & {} \sum _{v_1,\cdots , v_{\ell } \ge 0} X_{v_1,\cdots , v_{\ell }}\left( \frac{a\tau +b}{c\tau +d}\right) z_1^{v_1}\cdots z_{\ell }^{v_{\ell }}\\{} & {} =(c\tau +d)^{k+v_1+\cdots +v_{\ell }} ~e\left( \frac{\sum _{1\le i\le j \le \ell }c\alpha _{ij} z_iz_j}{c\tau +d}\right) \sum _{v_1,\cdots v_{\ell } \ge 0}X_{v_1, \cdots , v_{\ell }}(\tau )z_1^{v_1}\cdots z_{\ell }^{v_{\ell }}\\{} & {} =(c\tau +d)^{k+v_1+\cdots +v_{\ell }} \left( \sum _{1\le i \le j\le {\ell }} ~\sum _{t_{ij}\ge 0}\frac{1}{t_{ij}!} \left( \frac{2\pi i \alpha _{ij}c}{c\tau +d}\right) ^{t_{ij}}(z_iz_j)^{t_{ij}} \right) \\{} & {} \times \sum _{v_1,\cdots , v_{\ell } \ge 0}X_{v_1,\cdots , v_{\ell }}(\tau )z_1^{v_1}\cdots z_{\ell }^{v_{\ell }}. \end{aligned}$$

This implies that

$$\begin{aligned}{} & {} X_{v_1,\cdots , v_{\ell }} \left( \frac{a\tau +b}{c\tau +d}\right) =(c\tau +d)^{k+v_1+\cdots +v_{\ell }}\\{} & {} \times \sum _{ \begin{array}{c} 1\le i \le j\le \ell , \\ t_{ij} \ge 0,\\ v_j-2t_{jj}-\sum _{s=1}^{j-1}t_{sj}-\sum _{s=j+1}^{\ell }t_{js}\ge 0 \end{array}}\left( \frac{2\pi ic}{c\tau +d}\right) ^{\sum _{1\le i \le j \le \ell }t_{ij}}~~\frac{\prod \limits _{1\le i \le j\le \ell }\alpha _{ij}^{t_{ij}}}{\prod \limits _{1\le i\le j\le \ell }t_{ij}!}~\\{} & {} \quad \times ~~X_{v_1-2t_{11}-\sum _{s=1}^{\ell } t_{1s}, \cdots , v_{\ell }-\sum _{s=1}^{\ell -1}t_{s\ell }-t_{\ell \ell }}(\tau ). \end{aligned}$$

Following Eichler and Zagier [6, p. 31], we define

$$\begin{aligned}{} & {} \zeta _{v_1,\cdots , v_{\ell }}(\tau )= \sum _{ \begin{array}{c} 1\le i \le j\le \ell , ~t_{ij} \ge 0 \\ v_j-2t_{jj}-\sum _{s=1}^{j-1}t_{sj}-\sum _{s=j+1}^{\ell }t_{js}\ge 0 \end{array}} (-2\pi i)^{(\sum _{1\le i \le j \le \ell }t_{ij})}\\ {}{} & {} \times ~~\frac{(k+\sum _{i=1}^{\ell } v_i-\sum _{1\le i \le j\le \ell }t_{ij}-2)!}{(k+\sum _{i=1}^{\ell } v_i-2)!}\frac{\prod \limits _{1\le i \le j\le \ell }\alpha _{ij}^{t_{ij}}}{\prod \limits _{1\le i\le j\le \ell }t_{ij}!}~\\{} & {} \times ~~X^{(\sum _{1\le i \le j \le \ell }t_{ij})}_{(v_1-2t_{11}-\sum _{j=2}^{\ell } t_{1j},\cdots , v_{\ell }-\sum _{i=1}^{\ell -1}t_{i\ell }-2t_{\ell \ell })} (\tau ), \end{aligned}$$

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

$$\begin{aligned} \zeta _{v_1, \cdots , v_{\ell }}(\tau )= (2\pi i)^{v_1+\cdots +v_{\ell }}~~\sum _{n\ge 0}\left( \sum _{\begin{array}{c} r=(r_1, \cdots , r_{\ell }) \in {\mathbb {Z}}^{\ell }\\ 4\mathrm{{det}}(M)n - M^{\#}[r^t]\ge 0 \end{array}} \kappa ~a(n, r)\right) q^n, \end{aligned}$$

where

$$\begin{aligned}{} & {} \kappa = \sum _{\begin{array}{c} 1\le i \le j\le \ell , \\ t_{ij} \ge 0,\\ v_j-2t_{jj}-\sum _{s=1}^{j-1}t_{sj}-\sum _{s=j+1}^{\ell }t_{js}\ge 0 \end{array}} ~\frac{(k+\sum _{i=1}^{\ell } v_i-\sum _{1\le i \le j\le \ell }t_{ij}-2)! }{(k+\sum _{i=1}^{\ell } v_i-2)!}\\{} & {} \times \frac{\prod _{1 \le i \le j \le \ell } (-n\alpha )^{t_{ij}}\prod _{w=1}^{\ell } r_w^{v_w-\sum _{s=1}^{w-1}t_{sw}-\sum _{s=w+1}^{\ell } t_{ws}-2t_{ww}}}{\prod _{1\le i \le j \le \ell }t_{ij}!~~\prod _{w=1}^{\ell } \left( v_w-\sum _{s=1}^{w-1}t_{sw}-\sum _{s=w+1}^{\ell } t_{ws}-2t_{ww}\right) !}. \end{aligned}$$

We further define

$$\begin{aligned} D_{v_1,\cdots , v_{\ell }}(\phi )=(2\pi i )^{-(v_1+\cdots + v_{\ell })}~~\frac{(k+\sum _{i=1}^n v_i-2)!(\sum _{i=1}^{\ell } v_i)!}{\left( k+\beta -2\right) !} ~\zeta _{v_1,\cdots , v_{\ell }}(\tau ), \end{aligned}$$

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

$$\begin{aligned} D:J_{k, M}(\Gamma _0^{(1)}(N))\rightarrow \bigoplus _{\begin{array}{c} (v_1, \cdots , v_{\ell })\\ 0\le v_i \le 2 \alpha _{ii} \end{array}}M_{k+v_1+\cdots +v_{\ell }}~(\Gamma _0^{(1)}(N)), \end{aligned}$$

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

$$\begin{aligned} \phi \mapsto D_{v_1, \cdots , v_{\ell }}(\phi ). \end{aligned}$$

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 $\tau$

$$\begin{aligned}{} & {} X_{v_1, \cdots , v_{\ell }}(\tau )=0 \quad (0\le \forall v_{\ell }< a_{\ell }; \forall v_{{\ell }-1}, \cdots , v_1 \ge 0),\\{} & {} X_{v_1, \cdots , v_{{\ell }-1}, a_{\ell }}(\tau )=0 \quad (0\le \forall v_{{\ell }-1}< a_{\ell -1}; \forall v_{\ell -2}, \cdots , v_1 \ge 0), \cdots ,\\{} & {} X_{v_1, a_2 \cdots , a_\ell }(\tau )=0 \quad (0\le \forall v_{1} < a_{1}). \end{aligned}$$

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

$$\begin{aligned} f_1(\tau , z_1)=\sum _{v_1\ge 0} X_{v_1, a_2, \cdots , a_{\ell }}(\tau )z_1^{v_1} \end{aligned}$$

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 $\tau$

$$\begin{aligned}{} & {} X_{v_1, \cdots , v_{\ell }}(\tau )=0 \quad (0\le \forall v_{\ell }< a_{\ell }; \forall v_{{\ell }-1}, \cdots , v_1, v_{i} \ge 0),\cdots \\{} & {} X_{v_1, b_2, \cdots , b_{i-1}, v_i, b_{i+1}, \cdots , b_{\ell }}(\tau )=0 \quad (0\le \forall v_{1}< b_{1}; \forall v_i \ge 0),\\{} & {} X_{b_1,\cdots ,b_{i-1}, v_i, b_{i+1} \cdots , b_{\ell }}(\tau )=0 \quad (0\le \forall v_{i} < b_{i}). \end{aligned}$$

Again using minimality condition of $b_{\ell }, \cdots , b_{i-1}, b_{i+1}, \cdots , b_1, b_i$, we see that

$$\begin{aligned} f_i(\tau , z_i)=\sum _{v_i\ge 0}X_{b_1, \cdots ,v_i, \cdots , b_{\ell }}(\tau )z_i^{v_i} \end{aligned}$$

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

$$\begin{aligned} a_{\ell }=b_{\ell }, \cdots , a_{i-1}=b_{i-1}, ~~\text{ and },~a_{i}\le b_i. \end{aligned}$$

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

$$\begin{aligned} \phi =\sum _{\begin{array}{c} n \in {\mathbb {Z}}, r \in {\mathbb {Z}}^l \\ 4\mathrm{{det}}(M)n - M^{\#}[r^t]\ge 0 \end{array}} a(n, r)q^n\zeta ^{r} \in J_{k, M}(\Gamma _0^{(1)}(N)). \end{aligned}$$

If $a(n, r)=0$, for all

$$\begin{aligned} n\le \frac{1}{12}(k+2 tr(M))\gamma , \end{aligned}$$

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

$$\begin{aligned} D_{v_1, \cdots , v_{\ell }}(\phi ) =0 \end{aligned}$$

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

$$\begin{aligned} A={\left\{ \begin{array}{ll} \begin{pmatrix} m &{} 0 \\ 0 &{} md &{} \end{pmatrix} &{} \text{ if } ~-d \equiv 2, 3 \pmod 4, \\ \\ \begin{pmatrix} m &{} \frac{m}{2} \\ \frac{m}{2} &{} m\left( \frac{1+d}{4}\right) \end{pmatrix}&\text{ if } ~-d \equiv 1 \pmod 4. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} \eta _1: J_{k. m}(\Gamma _0^{(1)}(N)) \rightarrow J_{k, A}(\Gamma _0^{(1)}(N)) \end{aligned}$$

by

$$\begin{aligned} \phi (\tau , z_1, z_2)\mapsto \phi \left( \tau , z_1+i\sqrt{d}z_2, z_1-i\sqrt{d}z_2\right) . \end{aligned}$$

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

$$\begin{aligned} \phi (\tau , z_1, z_2)=\sum _{\begin{array}{c} n \in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#} \\ nm-N(r)\ge 0 \end{array}} c_{\phi }(n, r)q^n\zeta _1^r\zeta _2^{{\overline{r}}}, \end{aligned}$$

then

$$\begin{aligned} {\hat{\phi }}(\tau , z_1, z_2)=\sum _{\begin{array}{c} n \in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#} \\ nm-N(r)\ge 0 \end{array}}c_{\phi }(n, r)e\left( n\tau +r\left( z_1+i\sqrt{d}z_2\right) +{\overline{r}}\left( z_1-i\sqrt{d}z_2\right) \right) . \end{aligned}$$

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

$$\begin{aligned}{} & {} {\hat{\phi }}(\tau , z_1, z_2)=\sum _{\begin{array}{c} n \in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#} \\ nm-N(r)\ge 0 \end{array}}c_{\phi }(n, r)e(n\tau -\beta z_1-\alpha z_2)\\{} & {} =\sum _{\begin{array}{c} n \in {\mathbb {Z}}, ~\rho \in {\mathbb {Z}}^2 \\ 4\mathrm{{det}}(A)n-A^{\#}[\rho ]\ge 0 \end{array}}c_{{\hat{\phi }}}( n, \rho )e(n\tau + (-\beta )z_1+(-\alpha ) z_2), \end{aligned}$$

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

$$\begin{aligned} \eta _2: J_{k, A}(\Gamma _0^{(1)}(N))\rightarrow J_{k, m}(\Gamma _0^{(1)}(N)) \end{aligned}$$

defined by

$$\begin{aligned} \psi (\tau , z_1, z_2)\mapsto \psi \left( \tau , \frac{z_1+z_2}{2}, \frac{z_1-z_2}{2i\sqrt{d}}\right) \end{aligned}$$

is also a well-defined linear map. Now consider the composition map $\eta _2 \circ \eta _1,$

$$\begin{aligned} (\eta _2 \circ \eta _1)(\phi (\tau , z_1, z_2))= & {} \eta _2\left( \phi \left( \tau , z_1+i\sqrt{d}z_2, z_1-i\sqrt{d}z_2\right) \right) \\= & {} \phi \left( \tau ,\dfrac{z_1+i\sqrt{d}z_2+z_1-i\sqrt{d}z_2}{2},\dfrac{z_1+i\sqrt{d}z_2-z_1+i\sqrt{d}z_2}{2i\sqrt{d}}\right) \\= & {} \phi (\tau ,z_1,z_2). \end{aligned}$$

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

$$\begin{aligned} \eta _1: J_{k. m}(\Gamma _0^{(1)}(N)) \rightarrow J_{k, A}(\Gamma _0^{(1)}(N)) \end{aligned}$$

by

$$\begin{aligned} \phi (\tau , z_1, z_2)\mapsto \phi \left( \tau , \frac{2z_1+z_2+i\sqrt{d}z_2}{2}, \frac{2z_1+z_2-i\sqrt{d}z_2}{2}\right) \end{aligned}$$

and

$$\begin{aligned} \eta _2: J_{k, A}(\Gamma _0^{(1)}(N))\rightarrow J_{k, m}(\Gamma _0^{(1)}(N)) \end{aligned}$$

by

$$\begin{aligned} \psi (\tau , z_1, z_2)\mapsto \psi \left( \tau , \frac{z_1+z_2}{2}-\frac{z_1-z_2}{2i\sqrt{d}}, \frac{z_1-z_2}{i\sqrt{d}}\right) . \end{aligned}$$

Approaching as above we can easily verify that $\eta _1$ and $\eta _2$ are well defined linear maps and also they satisfy

$$\begin{aligned} \eta _2 \circ \eta _1=I_1, ~~\eta _1 \circ \eta _2=I_2. \end{aligned}$$

$\square$

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

$$\begin{aligned} \phi =\sum _{\begin{array}{c} n \ge 0, r \in {\mathcal {O}}_K \\ 4nm - N(r) \ge 0 \end{array}} a_{\phi }(n, r) q^n\zeta _1^r \zeta _2^{{\overline{r}}} \in J_{k, m}(\Gamma _0^{(1)}(N)). \end{aligned}$$

If $a_{\phi }(n, r)=0$ for all $n\le \beta$, where

$$\begin{aligned} \beta = {\left\{ \begin{array}{ll} \frac{1}{12}(k+2m(1+d))\gamma &{} \text{ if }~ ~ -d \equiv 2, 3 \pmod {4}, \\ \frac{1}{12}\left( k+\frac{m(5+d)}{2}\right) \gamma &{} \text{ if }~ ~ -d \equiv 1 \pmod {4}, \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned}{} & {} {\hat{\phi }}(\tau , z_1, z_2)=\sum _{\begin{array}{c} n \in {\mathbb {Z}}, r\in {\mathcal {O}}_K^{\#} \\ nm-N(r)\ge 0 \end{array}}a_{\phi }(n, r)e(n\tau -\beta z_1-\alpha z_2)\\{} & {} =\sum _{\begin{array}{c} n \in {\mathbb {Z}}, ~\rho \in {\mathbb {Z}}^2 \\ 4\mathrm{{det}}(A)n-A^{\#}[\rho ^t]\ge 0 \end{array}}a_{{\hat{\phi }}}( n, \rho )e(n\tau + (-\beta )z_1+(-\alpha ) z_2), \end{aligned}$$

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

$$\begin{aligned} \phi =\sum _{\begin{array}{c} n \ge 0, r \in {\mathcal {O}}_K \\ 4nm - N(r) \ge 0 \end{array}} a(n, r) q^n\zeta _1^r \zeta _2^{{\overline{r}}} \in J_{k, m}(SL_2({\mathbb {Z}})). \end{aligned}$$

If $a(n, r)=0$ for all $n\le \beta$, where

$$\begin{aligned} \beta = {\left\{ \begin{array}{ll} \frac{1}{12}(k+2m(1+d)) &{} \text{ if } ~~ -d \equiv 2, 3 \pmod {4}, \\ \frac{1}{12}\left( k+\frac{m(5+d)}{2}\right) &{} \text{ if }~ ~ -d \equiv 1 \pmod {4}, \end{array}\right. } \end{aligned}$$

then $\phi =0$.

We are now ready to prove Theorem 1.3. We first define some necessary terms. For

$$\begin{aligned} \phi =\sum _{\begin{array}{c} n \ge 0, r \in {\mathcal {O}}_K \\ 4nm - N(r) \ge 0 \end{array}} a(n, r) q^n\zeta _1^r \zeta _2^{{\overline{r}}} \in J_{k, m}(SL_2({\mathbb {Z}})), \end{aligned}$$

we define

$$\begin{aligned} \textrm{ord}(\phi )=\textrm{min}\{n \mid a(n, r) \not = 0 \}. \end{aligned}$$

From Corollary 5.5, Sturm bound for Hermtian Jacobi form when $-d \equiv 2, 3 \pmod 4$ is

$$\begin{aligned} \beta =\frac{1}{12}(k+2m(1+d)). \end{aligned}$$

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

$$\begin{aligned} F=\sum _{m\ge 0}\phi _m(\tau , z_1, z_2)e(m\tau '). \end{aligned}$$

We will show that $\phi _m =0$ for all $m \ge 0$. We first consider that $m\le \frac{k}{2(5-d)}$. Then

$$\begin{aligned} \mathrm{{ord}}(\phi _m)>\frac{k}{2(5-d)} = \frac{k}{2(5-d)}\left( \frac{1+d}{6}+\frac{5-d}{6}\right) =\frac{k}{12}+\frac{m(1+d)}{6}. \end{aligned}$$

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,

$$\begin{aligned} \mathrm{{ord}}(\phi _m)\ge m=m\left( \frac{1+d}{6}+\frac{5-d}{6}\right) >\frac{m(1+d)}{6}+\frac{k}{12}. \end{aligned}$$

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

$$\begin{aligned} E_k^{(K)}= \sum _{M=\begin{pmatrix} A &{} B \\ C &{} D \end{pmatrix}\in ~ \begin{pmatrix}* &{} *\\ 0 &{} * \end{pmatrix} {\setminus } U_2({\mathcal {O}}_K)} \textrm{det}(M)^{-k/2}\mathrm{{det}}(CZ+D)^{-k}, ~~Z \in {\mathcal {H}}_2. \end{aligned}$$

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],

$$\begin{aligned}{} & {} F_{10}^{(K)}=E_{10}^{(K)}-E_4^{(K)}E_{6}^{(K)}\in S_{10}(SU_2({\mathcal {O}}_K)),\\{} & {} F^{(K)}_{12}=E_{12}^{(K)}-\frac{441}{691}\left( E_{4}^{(K)}\right) ^3-\frac{250}{691}\left( E_{6}^{(K)}\right) ^2 \in S_{12}(SU_2({\mathcal {O}}_K)). \end{aligned}$$

If $K={\mathbb {Q}}(i)$ then

$$\begin{aligned} \chi _8=-\frac{61}{230400}\left( E_8^{(K)}-\left( E_4^{(K)}\right) ^2\right) \in S_8(SU_2({\mathcal {O}}_K)). \end{aligned}$$

Let $\beta (k)=[k/8]$ be the Sturm bound for $K={\mathbb {Q}}(i)$ in Theorem 1.3. We define

$$\begin{aligned} H_k(Z)={\left\{ \begin{array}{ll} \left( E_4^{(K)}\right) ^i \left( E_6^{(K)}\right) ^j \chi _8^{\beta (k)} ~(i+j+\beta (k)=k,~ i,j=0,1) &{} \text{ if } ~~k \not \equiv 2 \pmod 8,\\ \chi _8^{\beta (k)-1}F_{10}^{(K)} &{} \text{ if }~~ k \equiv 2 \pmod 8. \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} A_{\chi _8}\left( \begin{pmatrix}1 &{} (1+i)/2 \\ (1-i)/2 &{} 1 \end{pmatrix}\right) =A_{F_{10}^{(K)}}\left( \begin{pmatrix}1 &{} (1+i)/2 \\ (1-i)/2 &{} 1 \end{pmatrix}\right) =1. \end{aligned}$$

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

$$\begin{aligned} tr(T_0)\le \frac{k}{\left( 5-d \right) } \end{aligned}$$

(13)

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

$$\begin{aligned}{} & {} \phi _{m_0}(\tau , z_1, z_2)=\sum _{\begin{array}{c} n\in {\mathbb {Z}}, r \in {\mathcal {O}}_K^{\#}\\ nm_0 -N(r)\ge 0 \end{array}} c(n, r)e(n\tau +rz_1+{\overline{r}}z_2),\\{} & {} ~~~\text{ where }~ c(n, r)=A_F\left( \begin{pmatrix}n &{} r \\ {\overline{r}} &{} m_0 \end{pmatrix}\right) . \end{aligned}$$

We define

$$\begin{aligned} {\hat{\phi }}(\tau , z_1, z_2)=\phi _{m_0}(\tau , z_1+i\sqrt{d}z_2, z_1-i\sqrt{d}z_2). \end{aligned}$$

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

$$\begin{aligned} \varphi (\tau , z_1, z_2)=\prod _{\epsilon _1, \epsilon _2}{\hat{\phi }}(\tau , \epsilon _1z_1, \epsilon _2z_2), \end{aligned}$$

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

$$\begin{aligned} {\hat{\phi }}(\tau , \epsilon _1 z_1, \epsilon _2 z_2)=\sum _{\begin{array}{c} n\in {\mathbb {Z}}, s=(a, b)\in {\mathbb {Z}}^2\\ 4 \mathrm{{det}}(M_0)n- M_0^{\#}[s^t]\ge 0 \end{array}} c_{\epsilon _1,\epsilon _2}(n, s) e(n\tau +az_1+bz_2), \end{aligned}$$

where

$$\begin{aligned} c_{\epsilon _1, \epsilon _2}(n, s)=A_F\left( \begin{pmatrix}n &{} r \\ {\overline{r}} &{} m_0 \end{pmatrix}\right) , ~~~~r=\frac{i}{2\sqrt{d}}(-\epsilon _2b-i\sqrt{d}\epsilon _2a)\in {\mathcal {O}}_K^{\#}. \end{aligned}$$

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 $z_1=z_2=0$

$$\begin{aligned} \varphi (\tau , z_1, z_2)=\sum _{\alpha \ge 0, \beta \ge 0}X_{v_1, v_2}(\tau )z_1^{v_1}z_2^{v_2}. \end{aligned}$$

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

$$\begin{aligned}{} & {} X_{v_1, v_{2}}(\tau )=0 \quad (0\le \forall v_{2}< a_{2}; \forall v_1 \ge 0),\\{} & {} X_{v_1, a_2}(\tau )=0 \quad (0\le \forall v_{1} < a_{1}). \end{aligned}$$

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

$$\begin{aligned} k_1 \le 4(k+2m_0(1+d)). \end{aligned}$$

(14)

Suppose the Fourier series expansion of $\varphi$ is given by

$$\begin{aligned} \varphi (\tau , z_1, z_2)=\sum _{\begin{array}{c} n\in {\mathbb {Z}}, s=(\alpha , \beta )\in {\mathbb {Z}}^2\\ 4 \mathrm{{det}}(4M_0)n- 4M_0^{\#}[s^t]\ge 0 \end{array}} d(n, s) e(n\tau +\alpha z_1+\beta z_2). \end{aligned}$$

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

$$\begin{aligned} f=\frac{1}{(2\pi i)^{a_1+a_2}}\left( \partial _{z_1}^{a_1}\partial _{z_2}^{a_2}\varphi (\tau , z_1, z_2)\right) _{z_1=0, z_2=0}. \end{aligned}$$

If the Fourier series expansion of f is given by $\sum \limits _{n\ge 1} d(n) e(n\tau )$ then we have

$$\begin{aligned} d(n)=\sum _{\begin{array}{c} s=(\alpha , \beta )\in {\mathbb {Z}}^2\\ 4 \mathrm{{det}}(4M_0)n-4M_0^{\#}[s^t]\ge 0 \end{array}}d(n, s) \alpha ^{a_1}\beta ^{a_2}. \end{aligned}$$

Now by [11, Theorem 2] there exists $n_1\ge 1$ such that

$$\begin{aligned} n_1 \ll k_1^{2+\epsilon }, ~~d(n_1)<0. \end{aligned}$$

Therefore, by (14), we have

$$\begin{aligned} n_1 \ll (4(k+2m_0(1+d)))^{2+\epsilon }. \end{aligned}$$

(15)

We have

$$\begin{aligned} d(n_1)=\sum _{\begin{array}{c} s=(\alpha , \beta )\in {\mathbb {Z}}^2\\ 4 \mathrm{{det}}(4M_0)n_0-M_0^{\#}[s^t]\ge 0 \end{array}}d(n_1, s) \alpha ^{a_1}\beta ^{a_2} <0. \end{aligned}$$

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

$$\begin{aligned} A_F\left( T_1\right) <0,~~~~~~~ T_1=\begin{pmatrix}n_{\epsilon _1, \epsilon _2} &{} * \\ * &{} m_0 \end{pmatrix}. \end{aligned}$$

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

$$\begin{aligned}{} & {} tr(T_1)\ll (4(k+2(1+d)tr(T_0)))^{2+\epsilon }+tr(T_0)\\{} & {} \ll \left( 4\left( \frac{7+d}{5-d}\right) k\right) ^{2+\epsilon }. \end{aligned}$$

Now replacing F by $-F$ and proceeding as above, we get a matrix $T_2$ such that

$$\begin{aligned} tr(T_2)\ll \left( 4\left( \frac{7+d}{5-d}\right) k\right) ^{2+\epsilon } \end{aligned}$$

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

$$\begin{aligned} \chi (M)=\textrm{det}(D),~~~~~ M=\begin{pmatrix} A &{} B \\ C &{} D \end{pmatrix}\in \Gamma ^{(2)}_0(N). \end{aligned}$$

Moreover, if the conductor of $\chi$ is N then Theorem 1.2 also holds.