2.1 Hermitian modular groups

Historically, in the context of Hermitian modular forms, the unitary group $\mathrm {U}(g, g)$ is defined as a classical group: for instance, in [Reference Braun11]. In most parts of Sections 2–5, following all prior literature in this field, we employ this classical notion as opposed to the perspective of algebraic groups but adopt the notation from the latter for coherence of this paper. Let $E/\mathbb {Q}$ be an imaginary quadratic field, specified at the beginning of each section, and let $\mathrm {U}(g, g) (\mathbb {R})$ denote the classical unitary group of signature $(g, g)$ with entries in $\mathbb {C}$ , which in fact corresponds to the real points of a reductive group. Similarly, let $\mathrm {U}(g, g) (\mathbb {Q})$ denote the classical unitary group with entries in E and $\mathrm {U}(g, g) (\mathbb {Z})$ denote the one with entries in the ring of integers ${\mathcal {O}}_E \subseteq \mathbb {C}$ for a fixed embedding . In other words, the classical group $\mathrm {U}(g, g) (\mathbb {R})$ (respectively, $\mathrm {U}(g, g) (\mathbb {Q})$ and $\mathrm {U}(g, g) (\mathbb {Z})$ ) is the subgroup of $\mathrm {GL}_{2 g} (\mathbb {C})$ (respectively, $\mathrm {GL}_{2 g} (E)$ and $\mathrm {GL}_{2 g} ({\mathcal {O}}_E)$ ) whose elements $\gamma $ satisfy the equation

(2.1) $$ \begin{align} \gamma^{\ast} J_{g, g} \gamma = J_{g, g}\,\text{,}\\[-15pt]\nonumber \end{align} $$

for the $2g \times 2g$ matrix

(2.2) $$ \begin{align} J_{g, g} = \left(\begin{smallmatrix} 0_g & I_g \\ - I_g & 0_g \end{smallmatrix}\right)\text{.}\\[-15pt]\nonumber \end{align} $$

Writing $\gamma $ furthermore in blocks of $g \times g$ matrices

$$ \begin{align*} \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\text{,}\\[-15pt] \end{align*} $$

we can characterise these elements $\gamma $ by the condition

(2.3) $$ \begin{align} a^{\ast} c = c^{\ast} a,\, b^{\ast} d = d^{\ast} b ,\, \text{and}\, a^{\ast} d - c^{\ast} b = I_g\text{.}\\[-15pt]\nonumber \end{align} $$

Similar to the Siegel modular group $\mathrm {Sp}_{g} (\mathbb {Z}) = \mathrm {U} (g, g) (\mathbb {Z}) \cap \mathrm {Mat}_{2g}(\mathbb {Z})$ , we call $\mathrm {U} (g, g) (\mathbb {Z})$ the Hermitian modular group of degree g. The notation $\mathrm {Herm}_n (A)$ stands for the set of $n \times n$ Hermitian matrices with entries in the ring $A \subseteq \mathbb {C}$ . For a matrix $x \in \mathrm {Herm}_n (A)$ , the notation $x \geq 0$ (respectively, $x> 0$ ) means x is positive semidefinite (respectively, positive definite), and the set of all such matrices is denoted by $\mathrm {Herm}_n (A)_{\geq 0}$ (respectively, $\mathrm {Herm}_n (A)_{> 0}$ ).

2.2 Hermitian modular forms

Basic notions on Hermitian modular forms can be found in [Reference Braun11, Reference Braun12, Reference Braun13]. For completeness, we recall them briefly in the case of integral weight and arbitrary arithmetic type.

We define the Hermitian upper half-space of degree g, denoted by ${\mathbb {H}}_g$ , to be the set of matrices $\tau \in \mathrm {Mat}_{g} (\mathbb {C})$ such that the Hermitian matrix

(2.4) $$ \begin{align} y := \frac{1}{2i} (\tau - \tau^{\ast})\\[-15pt]\nonumber \end{align} $$

is positive definite. By this definition, the Siegel upper half-space ${\mathcal {H}}_g$ is the set of all the symmetric matrices in ${\mathbb {H}}_g$ . Recall that the symplectic group $\mathrm {Sp}_{g} (\mathbb {R}) = \mathrm {U} (g, g) (\mathbb {R}) \cap \mathrm {Mat}_{2g}(\mathbb {R})$ acts on ${\mathcal {H}}_g$ biholomorphically. Similarly, $\mathrm {U} (g, g) (\mathbb {R})$ acts on ${\mathbb {H}}_g$ biholomorphically via Möbius transformations

(2.5) $$ \begin{align} \tau \longmapsto \gamma \tau := (a \tau + b) (c \tau + d)^{-1}\\[-15pt]\nonumber \end{align} $$

for $\gamma = \left (\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right ) \in \mathrm {U} (g, g) (\mathbb {R})$ . We define the factor of automorphy j via the formula

$$ \begin{align*} j (\gamma, \tau):=\mathrm{det}\ (c \tau + d)\\[-15pt] \end{align*} $$

for $\gamma = \left (\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right ) \in \mathrm {U} (g, g) (\mathbb {R})$ and $\tau \in {\mathbb {H}}_g$ . It is well known that $j (\gamma , \tau )$ is always nonzero and satisfies the $1$ -cocycle condition

$$ \begin{align*} j(\gamma_1 \gamma_2, \tau)=j(\gamma_1, \gamma_2 \tau) j (\gamma_2, \tau)\\[-15pt] \end{align*} $$

for all $\gamma _1, \gamma _2 \in \mathrm {U} (g, g) (\mathbb {R})$ and $\tau \in {\mathbb {H}}_g$ .

Let $g, k$ be integers such that $g \geq 1$ and $\big (\rho , V(\rho )\big )$ a finite-dimensional complex representation of $\mathrm {U} (g, g) (\mathbb {Z})$ that factors through a finite quotient, also called an (arithmetic) type. We define the space of (vector-valued) Hermitian modular forms of degree g, weight k and (arithmetic) type $\rho $ , to be the space of holomorphic (vector-valued) functions $f: {\mathbb {H}}_g \longrightarrow V(\rho )$ satisfying the modularity condition

$$ \begin{align*} f(\gamma \tau) = j(\gamma, \tau)^k \rho(\gamma) f(\tau) \end{align*} $$

for all $\gamma \in \mathrm {U} (g, g) (\mathbb {Z})$ , and if $g = 1$ , being also holomorphic at the cusp of $\mathrm {U} (1, 1) (\mathbb {Z})$ .

If $\rho $ is trivial, $\mathrm {M}_k^{(g)}(\rho )$ is just a direct sum of copies of classical (scalar valued) Hermitian modular forms, and the latter is denoted by $\mathrm {M}_k^{(g)}$ in this paper. For a finite-index subgroup $\Gamma \subseteq \mathrm {U} (g, g) (\mathbb {Z})$ , the space of classical modular forms for $\Gamma $ is denoted by $\mathrm {M}_k(\Gamma )$ . The graded module of Hermitian modular forms of genus g and type $\rho $ is denoted by

$$ \begin{align*} \mathrm{M}_{\bullet}^{(g)}(\rho):= \bigoplus_{k \in \mathbb{Z}} \mathrm{M}_k^{(g)}(\rho) \text{.} \end{align*} $$

Note that if $\rho $ is trivial, $\mathrm {M}_{\bullet }^{(g)}(\rho )$ is a graded algebra and we write $\mathrm {M}_{\bullet }^{(g)}$ for it.

2.3 Hermitian Jacobi groups

Let $g, l$ be positive integers. Consider a discrete Heisenberg group

$$ \begin{align*} H_{{\mathcal{O}}_E}^{(g, l)} := \big\{ [(\lambda, \mu), \kappa] : \lambda, \mu \in \mathrm{Mat}_{l, g} ({\mathcal{O}}_E), \kappa \in \mathrm{Mat}_{l} ({{\mathcal{O}}_E}), \kappa + \mu \lambda^{\ast} \in \mathrm{Herm}_l ({\mathcal{O}}_E) \big\} \text{,} \end{align*} $$

with the following group law:

$$ \begin{align*} [(\lambda, \mu), \kappa] [(\lambda^{\prime}, \mu^{\prime}), \kappa^{\prime}] = [(\lambda + \lambda^{\prime}, \mu + \mu^{\prime}), \kappa + \kappa^{\prime} + \lambda {\mu^{\prime}}^{\ast} - \mu {\lambda^{\prime}}^{\ast}] \text{.} \end{align*} $$

Note that the condition $\kappa + \mu \lambda ^{\ast } \in \mathrm {Herm}_l ({\mathcal {O}}_E)$ is equivalent to $\kappa - \lambda \mu ^{\ast } \in \mathrm {Herm}_l ({\mathcal {O}}_E)$ . There is a natural action of the Hermitian modular group $\mathrm {U} (g, g) (\mathbb {Z})$ on the Heisenberg group $H_{{\mathcal {O}}_E}^{(g, l)}$ via matrix multiplication from the right, and we define the Hermitian Jacobi group $\Gamma ^{(g, l)}$ of genus g and cogenus l to be the semidirect product $\Gamma ^{(g, l)} = \mathrm {U} (g, g) (\mathbb {Z}) \ltimes H_{{\mathcal {O}}_E}^{(g, l)}$ with the associated group law. More explicitly, the group law of the Jacobi group $\Gamma ^{(g, l)}$ reads

$$ \begin{align*} \big( \gamma, [(\lambda, \mu), \kappa] \big) \big( \gamma^{\prime}, [(\lambda^{\prime}, \mu^{\prime}), \kappa^{\prime}] \big) = \big( \gamma \gamma^{\prime}, [(\lambda + \lambda^{\prime}, \mu + \mu^{\prime}), \kappa + \kappa^{\prime} + \lambda \gamma^{\prime} {\mu^{\prime}}^{\ast} - \mu \gamma^{\prime} {\lambda^{\prime}}^{\ast}] \big) \text{.} \end{align*} $$

Based on the standard embeddings of Heisenberg groups, we define an embedding

(2.6)

of the Jacobi group into the Hermitian modular group via the formula

(2.7) $$ \begin{align} \Bigg( \begin{pmatrix} a & b \\ c & d \end{pmatrix}, [(\lambda, \mu), \kappa] \Bigg) \mapsto \begin{pmatrix} a & 0 & b & 0 \\ 0 & I_l & 0 & 0 \\ c & 0 & d & 0\\ 0 & 0 & 0 & I_l \end{pmatrix} \begin{pmatrix} I_g & 0 & 0 & \mu^{\ast} \\ \lambda & I_l & \mu & \kappa \\ 0 & 0 & I_g & - \lambda^{\ast}\\ 0 & 0 & 0 & I_l \end{pmatrix} \text{.} \end{align} $$

To relate the embedded image of the Jacobi group to the whole Hermitian modular group, we record the following fact. First we define an embedding $\mathrm {rot} : \mathrm {GL}_{g}({\mathcal {O}}_E) \longrightarrow \mathrm {U} (g, g) (\mathbb {Z})$ of groups via $u \longmapsto \mathrm {rot} (u) := \left (\begin {smallmatrix} u & 0 \\ 0 & {u^{\ast }}^{-1} \end {smallmatrix}\right )$ .

Proof. By Theorem 2.1 and Corollary 2.3 in [Reference Dern37] (for an English version, see (1.13) and (1.14) in the thesis of Wernz [Reference Wernz38]), the group $\mathrm {U} (g, g) (\mathbb {Z})$ is generated by the embedded subgroup $\mathrm {rot} \big ( \mathrm {GL}_{g}({\mathcal {O}}_E) \big )$ , the matrix $J_{g, g}$ defined in equation (2.2) and the parabolic subgroup

$$\begin{align*}P := \bigg\{\left(\begin{smallmatrix} I_g & H \\ 0 & I_g \end{smallmatrix}\right): H \in \mathrm{Herm}_{g} ({\mathcal{O}}_E)\bigg\}\text{.}\end{align*}$$

First note that the parabolic subgroup P is already in the embedded Jacobi group $\Gamma ^{(g - 1, 1)}$ by putting $l = 1$ , $a = d = I_{g - 1}$ , $c = 0_{g - 1}$ , $\lambda = 0_{1 \times (g - 1)}$ in the embedding in equation (2.7) and noticing that the condition $\kappa + \mu \lambda ^{\ast } \in \mathrm {Herm}_{l} ({\mathcal {O}}_E)$ in the definition of the Jacobi group becomes $\kappa \in \mathbb {Z}$ in this case.

Then we consider the matrix $J_{g, g}$ . For each integer j such that $1 \leq j \leq g$ , let

$$\begin{align*}\iota^{(j)}: \mathrm{U} (1, 1) (\mathbb{Z}) \longrightarrow \mathrm{U} (g, g) (\mathbb{Z})\end{align*}$$

be the jth diagonal embedding sending an element $\gamma = \left (\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right )$ to $\iota ^{(j)} (\gamma )$ whose $2 \times 2$ submatrix indexed by $(j, j), (j, j + g), (j + g, j), (j + g, j + g)$ is equal to $\gamma $ , and the remaining entries of $\iota ^{(j)} (\gamma )$ are those of the identity matrix $I_{2 g}$ . Note that the matrix $J_{g, g}$ can be decomposed into

$$\begin{align*}J_{g, g} = \iota^{(g)} (\gamma) \iota^{(g - 1)} (\gamma) \cdots \iota^{(1)} (\gamma)\end{align*}$$

for $\gamma = \left (\begin {smallmatrix} 0 & 1 \\ -1 & 0 \end {smallmatrix}\right ) \in \mathrm {U} (1, 1) (\mathbb {Z})$ . Furthermore, for any $\gamma \in \mathrm {U} (1, 1) (\mathbb {Z})$ and any $j \in \{2, 3, \ldots , g\}$ , we have the relations

$$\begin{align*}\iota^{(j)}(\gamma) = \mathrm{rot} (u_j) \iota^{(1)}(\gamma) \mathrm{rot} (u_j)\end{align*}$$

for $u_j \in \mathrm {GL}_{g}({\mathcal {O}}_E)$ defined by putting its $2 \times 2$ submatrix indexed by $(1, 1), (1, j), (j, 1), (j, j)$ to be $\left (\begin {smallmatrix} 0 & 1\\ 1 & 0 \end {smallmatrix}\right )$ , and the remaining entries to be those of the identity matrix $I_{g}$ . Finally, for any $\gamma \in \mathrm {U} (1, 1) (\mathbb {Z})$ , it is clear that $\iota ^{(1)}(\gamma )$ is an element in the embedded Jacobi group $\Gamma ^{(g - 1, 1)}$ (in fact, this element lies in the embedded Jacobi group of arbitrary cogenus, not only the one of cogenus $1$ ), as desired.

2.4 Hermitian Jacobi forms

Jacobi forms are closely related to Fourier–Jacobi expansions of automorphic forms. In the case of degree $(1, 1)$ associated to Siegel modular forms, Eichler–Zagier developed a theory of Jacobi forms in their monograph [Reference Eichler and Zagier39]. The case of degree $(1, 1)$ associated to Hermitian modular forms over imaginary quadratic fields was treated by Haverkamp [Reference Haverkamp40, Reference Haverkamp41]. Cases of higher degrees were investigated in [Reference Murase42, Reference Shimura43, Reference Yamazaki44], and in the spirit of Eichler–Zagier a theory was built up by Ziegler [Reference Ziegler45]. To fit the presentation of this paper and to simplify the expressions, we define Hermitian Jacobi forms in terms of Hermitian modular forms via the embedding of Jacobi groups into Hermitian modular groups.

Let $g, l$ be positive integers. We define the Hermitian Jacobi upper half-space $\mathbb {H}_{g, l}$ of genus g and cogenus l to be $\mathbb {H}_{g, l} := {\mathbb {H}}_g \times \mathrm {Mat}_{g,l} (\mathbb {C}) \times \mathrm {Mat}_{l, g} (\mathbb {C})$ , and there is a (biholomorphic) action of the Jacobi group $\Gamma ^{(g, l)}$ on $\mathbb {H}_{g, l}$ via the formula

$$ \begin{align*} &\Bigg( \begin{pmatrix} a & b \\ c & d \end{pmatrix}, [(\lambda, \mu), \kappa] \Bigg) (\tau, w, z)\\ =& \bigg( (a \tau + b)(c \tau + d)^{-1}, \big(a - (a \tau + b) (c \tau + d)^{-1} c\big) (w + \tau \lambda^{\ast} + \mu^{\ast}), (z + \lambda \tau + \mu) (c \tau + d)^{-1} \bigg) \text{.} \end{align*} $$

Note that the Hermitian upper half-space ${\mathbb H}_{g + l}$ projects to the Hermitian Jacobi upper half-space $\mathbb {H}_{g, l}$ by restricting to the corresponding matrix blocks.

For an imaginary quadratic field $E/\mathbb {Q}$ , we recall the dual lattice ${\mathcal {O}}_E^{\#}$ of the lattice of integers ${\mathcal {O}}_E$ , also known as the inverse different ideal of E, which in our case can be explicitly written as ${\mathcal {O}}_E^{\#} = \frac {1}{\sqrt {D_E}} {\mathcal {O}}_E$ , where $D_E (< 0)$ is the discriminant of E. We say a Hermitian matrix m is semi-integral (over E) if the diagonal entries of m are in the ring $\mathbb {Z}$ and the off-diagonal entries are in the dual lattice ${\mathcal {O}}_E^{\#}$ .

Let k be an integer, and let $m \in \mathrm {Herm}_{l} (E)$ be an $l \times l$ Hermitian matrix with entries in E. Let $\rho $ be a complex representation of the Jacobi group $\Gamma ^{(g, l)}$ . Recall that we use the notation $e (x) = \exp \big (2 \pi i \cdot \mathop {\mathrm {Tr}}(x) \big )$ for a square matrix x throughout this paper. We say a holomorphic vector-valued function $\phi : \mathbb {H}_{g, l} \longrightarrow V(\rho )$ is a Hermitian Jacobi form of genus g, weight k, index m and type $\rho $ , if the function $f: {\mathbb H}_{g + l} \longrightarrow V (\rho )$ defined via

$$ \begin{align*} f \Bigg( \begin{pmatrix} \tau & w\\ z & \tau^{\prime} \end{pmatrix} \Bigg) = \phi (\tau, w, z) e (m \tau^{\prime}) \end{align*} $$

transforms as a Hermitian modular form for the image of $\Gamma ^{(g, l)}$ under the embedding in equation (2.6), of degree $g + l$ , weight k and type $\rho $ , and if $g = 1$ , the function $\phi (\tau , \tau r + r^{\prime }, s \tau + s^{\prime })$ also being bounded (under any norm as $\dim V(\rho ) < \infty $ ) at the cusp for all row vectors $r, r^{\prime }$ and column vectors $s, s^{\prime }$ of dimension l over E. The space of Hermitian Jacobi forms of genus g, weight k, index m and type $\rho $ is denoted by $\mathrm {J}_{k, m}^{(g)} (\rho )$ . To see the similarity between Hermitian Jacobi forms and Siegel Jacobi forms, we spell out the transformation formulae for Hermitian Jacobi forms $\phi (\tau , w, z)$ explicitly using the embedding in equation (2.7). For any element $\big (\left (\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\right ), [(\lambda , \mu ), \kappa ] \big ) \in \Gamma ^{(g, l)}$ in the Jacobi group and any point $(\tau , w, z) \in \mathbb {H}_{g, l}$ in the Hermitian Jacobi upper half-space, the equation for invariance under the embedded Hermitian modular subgroup

$$ \begin{align*} \phi \bigg((a\tau + b) (c \tau + d)^{-1}),& \big(a - (a \tau + b) (c \tau + d)^{-1} c\big) w, z (c \tau + d)^{-1} \bigg)\\ &= \mathrm{det}\ (c\tau + d)^k e \big( m z (c \tau + d)^{-1} c w\big) \rho \Big(\big(\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right), [(0, 0), 0] \big)\Big) \phi (\tau, w, z) \end{align*} $$

and the equation for invariance under the embedded discrete Heisenberg group

$$ \begin{align*} \phi \big(\tau, w + \tau \lambda^{\ast} + \mu^{\ast}, z + \lambda \tau + \mu \big) = e \big( - m ( \lambda \tau \lambda^{\ast} + \lambda w + z \lambda^{\ast} + \mu \lambda^{\ast} + \kappa )\big) \rho \big(I_{2g}, [(\lambda, \mu), \kappa] \big) \phi (\tau, w, z) \end{align*} $$

hold for a Jacobi form $\phi \in \mathrm {J}_{k, m}^{(g)} (\rho )$ .

When the context is clear, for a complex representation $\rho $ of the Hermitian modular group $\mathrm {U} (g + l, g + l) (\mathbb {Z})$ , we also write $\rho $ for its restriction to the image of the Jacobi group $\Gamma ^{(g, l)}$ . Conversely, for a complex representation $\rho ^{\prime }$ of the Jacobi group $\Gamma ^{(g, l)}$ , we also write $\rho ^{\prime }$ for its restriction to the subgroup $\mathrm {U} (g, g) (\mathbb {Z})$ .

2.5 Fourier–Jacobi expansions

Let $g, l$ be integers such that $1 \leq l \leq g - 1$ . Fourier–Jacobi expansions of Hermitian modular forms are partial Fourier expansions with respect to the matrix block $\tau _2$ of the variable $\tau = \left (\begin {smallmatrix} \tau _1 & z\\ w & \tau _2 \end {smallmatrix}\right )$ . More precisely, for a Hermitian modular form f of degree g, weight k and a type $\rho $ factoring through a finite quotient of $\mathrm {U} (g, g) (\mathbb {Z})$ , there is a Fourier expansion ([Reference Braun11]) in the form of

(2.8) $$ \begin{align} f (\tau) = \sum_{t \in \mathrm{Herm}_{g} (E)_{\geq 0}} c (f; t) e (t \tau)\text{} \end{align} $$

for $c (f; t) \in V(\rho )$ , where the matrices t appearing in the Fourier expansion depend on the type $\rho $ . In particular, when $\rho $ is the trivial representation, the sum is supported on semi-integral positive semidefinite Hermitian matrices t. We write

$$ \begin{align*} \tau = \left(\begin{smallmatrix} \tau_1 & w\\ z & \tau_2 \end{smallmatrix}\right) \in {\mathbb{H}}_g \quad \text{and} \quad t = \left(\begin{smallmatrix} n & r\\ r^{\ast} & m \end{smallmatrix}\right) \in \mathrm{Herm}_{g} (E)_{\geq 0} \end{align*} $$

for $\tau _1 \in {\mathbb H}_{g - l}, \tau _2 \in {\mathbb H}_{l}, w \in \mathrm {Mat}_{g - l, l} (\mathbb {C}), z \in \mathrm {Mat}_{l, g - l} (\mathbb {C}), n \in \mathrm {Herm}_{g - l} (E)_{\geq 0}, r \in \mathrm {Mat}_{g - l, l} (E)$ and $m \in \mathrm {Herm}_{l} (E)_{\geq 0}$ . Consequently, we arrange the sum in equation (2.8) into the form

(2.9) $$ \begin{align} f (\tau) = \sum_{m \in \mathrm{Herm}_{l} (E)_{\geq 0}} \phi_m (\tau_1, w, z) e (m \tau_2) \end{align} $$

for functions $\phi _m : \mathbb {H}_{g - l, l} \longrightarrow V(\rho )$ with the ordinary Fourier expansion

(2.10) $$ \begin{align} \phi_m (\tau_1, w, z) = \sum_{\substack{n \in \mathrm{Herm}_{g - l} (E)_{\geq 0}\\ r \in \mathrm{Mat}_{g - l, l} (E)}} c (\phi_m; n, r) e (n \tau_1 + r z) e (r^{\ast} w) \text{,} \end{align} $$

whose Fourier coefficients $c (\phi _m; n, r)$ are determined by the Fourier coefficients of f via the equation

(2.11) $$ \begin{align} c (\phi_m; n, r) = c \Bigg(f; \begin{pmatrix} n & r \\ r^{\ast} & m \end{pmatrix} \Bigg) \text{.} \end{align} $$

We say equation (2.9) is the Fourier–Jacobi expansion of f of cogenus l and $\phi _m$ is the mth Fourier–Jacobi coefficient of f, or the Fourier–Jacobi coefficient of index m. Furthermore, it follows from the definition that any Jacobi form $\phi \in \mathrm {J}_{k, m}^{(g)} (\rho )$ has the Fourier expansion in the form of equation (2.10).

2.6 Symmetric formal Fourier–Jacobi series

We motivate in this subsection the notion of symmetric formal Fourier–Jacobi series for the Hermitian modular group, by combining two features of the Fourier–Jacobi expansion of a Hermitian modular form: each Fourier–Jacobi coefficient is a Hermitian Jacobi form, and they satisfy a $\mathrm {GL}_{g} ({\mathcal {O}}_E)$ -symmetry condition. A priori, for formal series, these two features ‘almost’ characterise ‘formal modularity’: that is, invariance under the action of the Hermitian modular group without assuming absolute convergence. The group-theoretic Lemma 2.2 amounts to saying that formal modularity can be reformulated in terms of two conditions: invariance under the embedded Jacobi group of cogenus $1$ , and invariance under the embedded subgroup $\mathrm {rot} \big (\mathrm {GL}_{g} ({\mathcal {O}}_E) \big )$ .

It is clear that the invariance of a formal Fourier series f in the form of equation (2.8) under the weight k, type $\rho $ -slash action of $\mathrm {rot} \big (\mathrm {GL}_{g} ({\mathcal {O}}_E)\big )$ is equivalent to a symmetry condition for the formal Fourier coefficients of f, which reads

(2.12) $$ \begin{align} \rho \Bigg( \begin{pmatrix} u & 0 \\ 0 & {u^{\ast}}^{-1} \end{pmatrix} \Bigg) c (f; u^{\ast} t u) = \big(\mathrm{det}\ u^{\ast} \big)^k c (f; t) \end{align} $$

for all $u \in \mathrm {GL}_{g} ({\mathcal {O}}_E)$ .

Moreover, the invariance of f under the weight k, type $\rho $ -slash action of the embedded Jacobi group $\Gamma ^{(g - l, l)}$ under equation (2.6) amounts to saying that f can be rearranged into a formal series in the form of equation (2.9), such that each coefficient $\phi _m$ is a formal Hermitian Jacobi form of genus $g - l$ , weight k, index k and type $\rho $ .

Combining these two aspects, for a complex representation $\rho $ of $\mathrm {U} (g, g) (\mathbb {Z})$ , we define a symmetric formal Fourier–Jacobi series f of degree g, cogenus l, weight k and type $\rho $ to be a formal series of Hermitian Jacobi forms $\phi _m \in \mathrm {J}_{k, m}^{(g - l)} (\rho )$ in the form of equation (2.9), such that its Fourier coefficients $c (f; t)$ defined by the equation

$$ \begin{align*} c \Bigg(f; \begin{pmatrix} n & r \\ r^{\ast} & m \end{pmatrix} \Bigg) = c (\phi_m; n, r) \end{align*} $$

satisfy the symmetry condition in equation (2.12).

We write $\mathrm {FM}_k^{(g, l)} (\rho )$ for the vector space of such symmetric formal Fourier–Jacobi series, and when the cogenus $l = 1$ or $\rho $ is the $1$ -dimensional trivial representation, we suppress them in the notation. Furthermore, we let

$$ \begin{align*} \mathrm{FM}_{\bullet}^{(g)} (\rho) := \bigoplus_{k \in \mathbb{Z}} \mathrm{FM}_k^{(g)}(\rho) \end{align*} $$

denote the graded module over the graded ring $\mathrm {M}_{\bullet }^{(g)}$ of classical Hermitian modular forms, and note that $\mathrm {FM}_{\bullet }^{(g)}$ is actually a graded algebra over $\mathrm {M}_{\bullet }^{(g)}$ .

2.7 Formal Fourier–Jacobi coefficients

Just as Fourier–Jacobi coefficients of arbitrary cogenus are attached to a Hermitian modular form, we associate formal Fourier–Jacobi coefficients of various cogenus to a formal Fourier–Jacobi series.

Let $1 \leq l^{\prime } < l < g$ be positive integers, and let $f \in \mathrm {M}_k^{(g)}(\rho )$ be a Hermitian modular form. For $\tau = \left (\begin {smallmatrix} \tau _1 & w \\ z & \tau _2 \end {smallmatrix}\right ) \in {\mathbb {H}}_g$ , we refine the decomposition of the matrix $\tau $ further into blocks

$$ \begin{align*} \tau = \begin{pmatrix} \tau_{11} & w_{11} & w_{12}\\ z_{11} & \tau_{12} & w_{22}\\ z_{12} & z_{22} & \tau_2 \end{pmatrix} \text{,} \end{align*} $$

so that the sizes of $\tau _{11}, \tau _{12}$ and $\tau _2$ are $(g - l) \times (g - l), (l - l^{\prime }) \times (l - l^{\prime })$ and $l^{\prime } \times l^{\prime }$ , respectively, and the sizes of the off-diagonal blocks are determined in the ordinary way. We write $w_1$ for the $(g - l)\times l$ matrix $(w_{11}\, w_{12})$ and $z_1$ for the $l \times (g - l)$ matrix $\left (\begin {smallmatrix}z_{11} \\ z_{12}\end {smallmatrix}\right )$ .

For a formal Fourier series f in the form of equation (2.8), the formal Fourier–Jacobi expansion of cogenus l is given by equation (2.9), with the mth formal Fourier–Jacobi coefficient $\psi _m$ defined in equations (2.10) and (2.11), where we replace the notation $\phi _m$ by $\psi _m$ to emphasise that it is a formal series. In particular, for a symmetric formal Fourier–Jacobi series $f \in \mathrm {FM}_k^{(g, l)} (\rho )$ with Fourier–Jacobi coefficients $\phi _m$ , and for each Hermitian matrix $m^{\prime } \in \mathrm {Herm}_{l^{\prime }} (E)_{\geq 0}$ , the formal Fourier–Jacobi coefficient $\psi _{m^{\prime }}$ of index $m^{\prime }$ is related to $\phi _m$ via an equation of formal Fourier-series, namely

(2.13) $$ \begin{align} \psi_{m^{\prime}} (\tau_1, w, z) = \sum_{\substack{n^{\prime} \in \mathrm{Herm}_{l - l^{\prime}} (E)_{\geq 0}\\ r^{\prime} \in \mathrm{Mat}_{l - l^{\prime}, l^{\prime} } (E)}} \phi_{\left(\begin{smallmatrix} n^{\prime} & r^{\prime} \\ {r^{\prime}}^{\ast} & m^{\prime} \end{smallmatrix}\right)} (\tau_{11}, w_1, z_1) e (n^{\prime} \tau_{12} + r^{\prime} z_{22}) e ({r^{\prime}}^{\ast} w_{22}) \text{.} \end{align} $$

When $l^{\prime } = 1$ , we need to use the following fact about the formal Fourier–Jacobi coefficient $\psi _0$ of index $0 \in \mathrm {Herm}_{1} (E)_{\geq 0}$ . Note that the following fact also holds for arbitrary $l^{\prime }$ , but we only need the case $l^{\prime } = 1$ in the proof of Proposition 5.5.

Proof. For any Hermitian matrix $\left (\begin {smallmatrix} n^{\prime } & r^{\prime } \\ {r^{\prime }}^{\ast } & 0 \end {smallmatrix}\right )$ that occurs in equation (2.13), it is positive semidefinite, hence $r^{\prime } = 0$ . Moreover, for any index $n^{\prime } \in \mathrm {Herm}_{l - 1} (E)_{\geq 0}$ , as $\phi _{\left (\begin {smallmatrix} n^{\prime } & 0 \\ 0 & 0 \end {smallmatrix}\right )}$ is a Hermitian Jacobi form, by the transformation law it must be a constant function in $w_{12}$ and $z_{12}$ . In particular, $\phi _{\left (\begin {smallmatrix} n^{\prime } & 0 \\ 0 & 0 \end {smallmatrix}\right )}$ can be identified as a Jacobi form of genus $g - l$ and index $n^{\prime }$ , which we write as $\phi _{n^{\prime }}$ . Therefore, $\psi _0$ can be identified as a formal Fourier–Jacobi series

$$ \begin{align*} \psi_0 (\tau_1) = \sum_{n^{\prime} \in \mathrm{Herm}_{l - 1} (E)_{\geq 0} } \phi_{n^{\prime}} (\tau_{11}, w_{11}, z_{11}) e (n^{\prime} \tau_{12}) \text{.} \end{align*} $$

The symmetry condition of the formal Fourier–Jacobi series $\psi _0$ under the action of the embedded subgroup $\mathrm {GL}_{g - 1} ({\mathcal {O}}_E)$ follows from the one of f under the action of $\mathrm {GL}_{g} ({\mathcal {O}}_E)$ .

2.8 Formal theta decomposition

We show in this subsection that the classical theta decomposition of Jacobi forms has a formal analogue, namely the formal theta decomposition of formal Fourier–Jacobi coefficients.

Let $g, l$ be arbitrary positive integers, and let $m \in \mathrm {Herm}_l (E)_{> 0}$ be a semi-integral positive definite Hermitian matrix, which we also view as an integral positive definite quadratic form. Let $\Delta _{g} (m)$ denote the finite abelian group

$$ \begin{align*} \Delta_{g} (m) = \mathrm{Mat}_{g,l} ({{\mathcal{O}}_E}^{\#}) / \mathrm{Mat}_{g,l} ({\mathcal{O}}_E) m \text{,} \end{align*} $$

and let $\rho _m^{(g)}$ be the Weil representation of $\mathrm {U} (g, g) (\mathbb {Z})$ associated with $\Delta _{g} (m)$ on $\mathbb {C} [\Delta _{g} (m)]$ . Recall that the theta series $\theta _{m, s}^{(g)}: \mathbb {H}_{g, l} \longrightarrow \mathbb {C}$ of genus g, integral quadratic form m and shift $s \in \Delta _{g} (m)$ is defined by

(2.14) $$ \begin{align} \theta_{m, s}^{(g)} (\tau, w, z) = \sum_{r \in s + \mathrm{Mat}_{g,l} ({\mathcal{O}}_E) m} e (r m^{-1} r^{\ast} \tau + r z) e(r^{\ast} w) \text{.} \end{align} $$

It is a classical result (Proposition 3.5 of [Reference Shimura43]; see also Theorem 5.1 of [Reference Eichler and Zagier39]) that the vector-valued theta series ${\big (\theta _{m, s}^{(g)}\big )}_{s \in \Delta _{g} (m)}$ transforms as a vector-valued Hermitian Jacobi form of weight l and type ${\big (\rho _m^{(g)}\big )}^{\vee }$ . Furthermore, any Hermitian Jacobi form $\phi \in \mathrm {J}_{k, m}^{(g)}$ can be uniquely written as a sum

(2.15) $$ \begin{align} \phi (\tau, w, z) = \sum_{s \in \Delta_{g} (m)} h_s (\tau) \theta_{m, s}^{(g)} (\tau, w, z) \end{align} $$

for some holomorphic functions $h_s$ , and these functions are the components of a vector-valued Hermitian modular form of type $\rho _m^{(g)}$ .

Let $1 \leq l^{\prime } < l < g$ be positive integers. For a symmetric formal Fourier–Jacobi series $f \in \mathrm {FM}_k^{(g, l)}$ with Fourier–Jacobi coefficients $\phi _m$ for $m \in \mathrm {Herm}_{l}(E)_{\geq 0}$ , we construct the formal theta decomposition of its formal Fourier–Jacobi coefficients. Let $m^{\prime } \in \mathrm {Herm}_{l^{\prime }} (E)_{> 0}$ be a positive definite Hermitian matrix, and let $\psi _{m^{\prime }}$ be the $m^{\prime }$ th formal Fourier–Jacobi coefficient of f. Since the formal Fourier coefficients of f satisfy by definition the symmetry condition in equation (2.12), setting $u = \left (\begin {smallmatrix}I_{g - l^{\prime }} & 0 \\ \lambda ^{\ast } & I_{l^{\prime }} \end {smallmatrix}\right )$ for $\lambda \in \mathrm {Mat}_{g - l^{\prime }, l^{\prime }} ({\mathcal {O}}_E)$ in equation (2.12), we find a symmetry condition for the formal Fourier coefficients of $\psi _{m^{\prime }}$ , namely

(2.16) $$ \begin{align} c (\psi_{m^{\prime}}; n^{\prime} + r^{\prime} {m^{\prime}}^{-1} {r^{\prime}}^{\ast}, r^{\prime}) = c (\psi_{m^{\prime}}; n^{\prime} + r^{\prime\prime} {m^{\prime}}^{-1} {r^{\prime\prime}}^{\ast}, r^{\prime\prime}) \end{align} $$

for all $n^{\prime } \in \mathrm {Herm}_{g - l^{\prime }} (E)_{\geq 0}$ and all $r^{\prime }, r^{\prime \prime } \in \mathrm {Mat}_{g - l^{\prime }, l^{\prime }} (E)$ such that $r^{\prime \prime } = r^{\prime } + \lambda m^{\prime }$ for some $\lambda \in \mathrm {Mat}_{g - l^{\prime }, l^{\prime }} ({\mathcal {O}}_E)$ . In particular, for each $s^{\prime } \in \Delta _{g - l^{\prime }} (m^{\prime })$ , equation (2.16) defines a formal Fourier series $h_{m^{\prime }, s^{\prime }}$ in $\tau _1$ via its formal Fourier coefficients

(2.17) $$ \begin{align} c (h_{m^{\prime}, s^{\prime}}; n^{\prime}) := c (\psi_{m^{\prime}}; n^{\prime} + r^{\prime} {m^{\prime}}^{-1} {r^{\prime}}^{\ast}, r^{\prime}) \text{,} \end{align} $$

for any choice of $r^{\prime } \in s^{\prime } + \mathrm {Mat}_{g - l^{\prime }, l^{\prime }} ({\mathcal {O}}_E) m^{\prime }$ and all $n^{\prime } \in \mathrm {Herm}_{g - l^{\prime }} (E)_{\geq 0}$ . It then follows that

(2.18) $$ \begin{align} \psi_{m^{\prime}} (\tau_1, w, z) = \sum_{s^{\prime} \in \Delta_{g - l^{\prime}} (m^{\prime})} h_{m^{\prime}, s^{\prime}} (\tau_1) \theta_{m^{\prime}, s^{\prime}}^{(g - l^{\prime})} (\tau_1, w, z) \end{align} $$

holds as an equation of formal Fourier series.

Our aim is to show that the vector-valued formal Fourier series $(h_{m^{\prime }, s^{\prime }})_{s^{\prime }}$ is in fact a symmetric formal Fourier–Jacobi series. This fact can be proved for arbitrary cogenus $l^{\prime }$ ; for simplicity, we only do the case $l^{\prime } = l - 1$ , which is the only case that we need in Section 5. We separate our proof into the following two lemmas and state the conclusion after that.

Proof. Setting $u = \left (\begin {smallmatrix} u^{\prime } & 0\\ 0 & I_{l^{\prime }} \end {smallmatrix}\right )$ for $u^{\prime } \in \mathrm {GL}_{g - l^{\prime }} ({\mathcal {O}}_E)$ in the symmetry condition in equation (2.12) of $f \in \mathrm {FM}_k^{(g, l)}$ , we find a symmetry condition

(2.19) $$ \begin{align} c (\psi_{m^{\prime}}; {u^{\prime}}^{\ast} n^{\prime} u^{\prime}, {u^{\prime}}^{\ast} r^{\prime}) = (\mathrm{det}\ {u^{\prime}}^{\ast})^k c (\psi_{m^{\prime}}; n^{\prime}, r^{\prime}) \end{align} $$

for all $n^{\prime } \in \mathrm {Herm}_{g - l^{\prime }} (E)_{\geq 0}$ , $r^{\prime } \in \mathrm {Mat}_{g - l^{\prime }, l^{\prime }} ({\mathcal {O}}_E)$ and $u^{\prime } \in \mathrm {GL}_{g - l^{\prime }} ({\mathcal {O}}_E)$ . Combining the definition in equation (2.17) of $h_{m^{\prime }, s^{\prime }}$ and equation (2.19), we see that

$$ \begin{align*} c (h_{m^{\prime}, {u^{\prime}}^{\ast} s^{\prime}}; {u^{\prime}}^{\ast} n^{\prime} u^{\prime}) = (\mathrm{det}\ {u^{\prime}}^{\ast})^k c (h_{m^{\prime}, s^{\prime}}; n^{\prime}) \end{align*} $$

holds for all $n^{\prime } \in \mathrm {Herm}_{g - l^{\prime }} (E)_{\geq 0}$ , $s^{\prime } \in \Delta _{g - l^{\prime }} (m^{\prime })$ and $u^{\prime } \in \mathrm {GL}_{g - l^{\prime }} ({\mathcal {O}}_E)$ , as desired.

Next, for the formal Fourier series $h_{m^{\prime }, s^{\prime }}$ defined by equation (2.17), in line with our goal to prove Proposition 2.8, we only need to consider their cogenus $1$ formal Fourier–Jacobi expansions

(2.20) $$ \begin{align} h_{m^{\prime}, s^{\prime}}(\tau_1) = \sum_{n \in \mathbb{Q}_{\geq 0}} \psi_{m^{\prime}, s^{\prime}, n} (\tau_{11}, w_{11}, z_{11}) e(n \tau_{12}) \text{,} \end{align} $$

and relate their formal Fourier–Jacobi coefficients $\psi _{m^{\prime }, s^{\prime }, n}$ to the Fourier–Jacobi coefficients $\phi _m$ of $f \in \mathrm {FM}_k^{(g, l)}$ . In fact, equations (2.13) and (2.18) provide such a relation. Inserting equation (2.20) in equation (2.18), and writing $s^{\prime } \in \Delta _{g - l^{\prime }} (m^{\prime })$ in the form $\left (\begin {smallmatrix}s_1^{\prime } \\ s_2^{\prime }\end {smallmatrix}\right )$ for $s_1^{\prime } \in \Delta _{g - l} (m^{\prime })$ and $s_2^{\prime } \in \Delta _{1} (m^{\prime })$ , we obtain the following result after comparing the formal Fourier coefficient at $e (n^{\prime } \tau _{12} + r^{\prime } z_{22}) e({r^{\prime }}^{\ast } w_{22})$ with equation (2.13) and simplifying the expression.

Lemma 2.7. Let equation (2.20) be the formal Fourier–Jacobi expansion of cogenus $1$ of $h_{m^{\prime }, s^{\prime }}$ . Then for any shift $s_2^{\prime } \in \Delta _{1} (m^{\prime })$ , every representative $r^{\prime } \in s_2^{\prime } + \mathrm {Mat}_{1, l^{\prime }} ({\mathcal {O}}_E) m^{\prime }$ and every integer $n^{\prime } \geq 0$ , the partial theta decomposition of Fourier–Jacobi coefficients

(2.21) $$ \begin{align} \begin{aligned} &\phi_{\left(\begin{smallmatrix}n^{\prime} & r^{\prime} \\ {r^{\prime}}^{\ast} & m^{\prime} \end{smallmatrix}\right)} (\tau_{11}, w_1, z_1) = \\ &\sum_{s_1^{\prime} \in \Delta_{g - l} (m^{\prime})} \psi_{m^{\prime}, \left(\begin{smallmatrix}s_1^{\prime} \\ s_2^{\prime}\end{smallmatrix}\right), n^{\prime} - r^{\prime} {m^{\prime}}^{-1} {r^{\prime}}^{\ast}} (\tau_{11}, w_{11}, z_{11}) \theta_{m^{\prime}, s_1^{\prime}}^{(g - l)} (\tau_{11}, w_{12} + w_{11} r^{\prime} {m^{\prime}}^{-1}, z_{12} + {m^{\prime}}^{-1} {r^{\prime}}^{\ast} z_{11}) \end{aligned} \end{align} $$

holds as an equation of formal Fourier series. In particular, every formal Fourier–Jacobi coefficient $\psi _{m^{\prime }, s^{\prime }, n}$ in equation (2.20) is a holomorphic function in $\tau _{11}\text {,}\, w_{11}$ and $z_{11}$ .

Proposition 2.8. Let $g> l$ be positive integers, and fix $l^{\prime } = l - 1$ . Let $f \in \mathrm {FM}_k^{(g, l)}$ be a symmetric formal Fourier–Jacobi series, and let $m^{\prime } \in \mathrm {Herm}_{l^{\prime }} (E)_{> 0}$ be a semi-integral positive definite Hermitian matrix. Then the formal Fourier–Jacobi coefficient $\psi _{m^{\prime }}$ of f has a formal theta expansion

$$ \begin{align*} \psi_{m^{\prime}} (\tau_1, w, z) = \sum_{s^{\prime} \in \Delta_{g - l^{\prime}} (m^{\prime})} h_{m^{\prime}, s^{\prime}} (\tau_1) \theta_{m^{\prime}, s^{\prime}}^{(g - l^{\prime})} (\tau_1, w, z) \text{,} \end{align*} $$

where the coefficients are vector-valued symmetric formal Fourier–Jacobi series, more precisely

$$ \begin{align*} (h_{m^{\prime}, s^{\prime}})_{s^{\prime}} \in \mathrm{FM}_{k - l^{\prime}}^{(g - l^{\prime})} (\rho_{m^{\prime}}^{(g - l^{\prime})}) \text{.} \end{align*} $$

Proof. By Lemmas 2.6 and 2.7, it suffices to prove the modularity of the vector-valued function

$$ \begin{align*} (\psi_{m^{\prime}, s^{\prime}, n})_{s^{\prime} \in \Delta_{g - l^{\prime}} (m^{\prime})} \text{.} \end{align*} $$

We analyse both sides of equation (2.21) in Lemma 2.7. On the left-hand side, by the assumption $f \in \mathrm {FM}_k^{(g, l)}$ , the function transforms with weight k and index $\left (\begin {smallmatrix}n^{\prime } & r^{\prime } \\ {r^{\prime }}^{\ast } & m^{\prime } \end {smallmatrix}\right )$ under the action of the Hermitian Jacobi group $\Gamma ^{(g - l, l)}$ . On the right-hand side, the vector-valued theta function transforms by the restriction of $(\rho _{m^{\prime }}^{(g)})^{\vee }$ to $\Gamma ^{(g - l, l^{\prime })}$ , of weight $l^{\prime }$ . Arguing as in Section 3 of [Reference Ziegler45], we conclude that the vector-valued holomorphic function $(\psi _{m^{\prime }, s^{\prime }, n})_{s^{\prime } \in \Delta _{g - l^{\prime }} (m^{\prime })}$ transforms by the restriction of $\rho _{m^{\prime }}^{(g)}$ to $\Gamma ^{(g - l, 1)}$ , of weight $k - l^{\prime }$ , which completes the proof.

Finally, we state the classical result mentioned from the beginning. Let $\phi \in \mathrm {J}_{k, m}^{(g)}(\rho )$ be a Hermitian Jacobi form; then its Fourier coefficients $c (\phi; n, r)$ satisfy a symmetry condition similar to equation (2.16), which then defines a formal Fourier series $h_s$ via

$$ \begin{align*} c (h_s; n) := c (\phi; n + r m^{-1} r^{\ast}, r) \end{align*} $$

for each $s \in \Delta _{g} (m)$ and any choice of $r \in s + \frac {1}{N} \mathrm {Mat}_{g, l} ({\mathcal {O}}_E) m$ for a suitable positive integer $N = N (\rho )$ . Then we have an equation of formal Fourier series similar to equation (2.15), and the vector-valued holomorphic function $(h_s)_{s \in \Delta _{g} (m)}$ satisfying the equation is unique. Similar to the formal theta decomposition, we follow the lines of [Reference Ziegler45], Section 3 and obtain the classical theta decomposition in the Hermitian case.

Proposition 2.10. Let g and l be positive integers. Let $m \in \mathrm {Herm}_{l} (E)_{>0}$ be a positive definite Hermitian matrix. Let $\rho _m^{(g)}$ be the Weil representation of $\mathrm {U} (g, g) (\mathbb {Z})$ on $\mathbb {C} [\Delta _{g} (m)]$ , and let $\rho $ be a complex finite-dimensional representation of the Jacobi group $\Gamma ^{(g, l)}$ that factors through a finite quotient. We also write $\rho $ for its restriction to the subgroup $\mathrm {U} (g, g) (\mathbb {Z})$ . Then there is an isomorphism

$$ \begin{align*} \mathrm{J}_{k, m}^{(g)}(\rho) &\cong \mathrm{M}_{k - l}^{(g)} (\rho_m^{(g)} \otimes \rho)\\ \phi &\longmapsto (h_s)_{s \in \Delta_{g} (m)} \text{,} \end{align*} $$

sending a Hermitian Jacobi form $\phi $ to the components of its theta decomposition.

3.1 Vanishing orders and slope bounds

Let $t \in \mathrm {Herm}_{g} (E)$ . We say that t represents a rational number h if there is a nonzero integral element $\omega \in {\mathcal {O}}_E^g \setminus \{0\}$ such that $\omega ^{\ast } t \omega = h$ . For a nonzero vector-valued formal series

(3.1) $$ \begin{align} f (\tau) = \sum_{t \in \mathrm{Herm}_{g} (E)} c (f; t) e (t \tau) \end{align} $$

for $\tau \in {\mathbb {H}}_g$ , we define its vanishing order to be

$$ \begin{align*} \mathrm{ord} f:= \inf \big\{h \in \mathbb{Q} : h\, \text{can be represented by some}\, t \in \mathrm{Herm}_{g} (E)\, \text{such that}\, c (f; t) \neq 0\big\} \text{.} \end{align*} $$

If $f = 0$ , we set $\mathrm {ord} f := \infty $ by convention.

Lemma 3.1. Let $f_1, f_2$ be any two formal series in the form of equation (3.1). Then their vanishing orders satisfy

$$\begin{align*}\mathrm{ord} \big(f_1 \otimes f_2 \big) \geq \mathrm{ord} f_1 + \mathrm{ord} f_2 \text{.}\end{align*}$$

Proof. The assertion is clear if $f_1 = 0$ or $f_2 = 0$ . Assume now $f_1$ and $f_2$ are both nonzero, and let $t \in \mathrm {Herm}_{g} (E)$ be a Hermitian matrix such that $c (f_1 \otimes f_2; t) \neq 0$ . Since

$$\begin{align*}c (f_1 \otimes f_2; t) = \sum_{\substack {0 \leq t_1, t_2 \in \mathrm{Herm}_{g} (E) \text{,} \\ t_1 + t_2 = t}} c (f_1; t_1) \otimes c(f_2; t_2)\text{,}\end{align*}$$

there must be some pair $(t_1, t_2)$ such that $t_1 + t_2 = t$ , $c (f_1; t_1) \neq 0$ and $c (f_2; t_2) \neq 0$ , as desired.

Similarly, we define the vanishing order of a nonzero vector-valued formal series

(3.2) $$ \begin{align} \phi (\tau, w, z) = \sum_{\substack{t \in \mathrm{Herm}_{g} (E)\text{,}\\ r \in \mathrm{Mat}_{g, l} (E)}} c (\phi; t, r) e (t \tau + r z) e (r^{\ast} w) \end{align} $$

for $(\tau , w, z) \in \mathbb {H}_{g, l}$ to be

$$ \begin{align*} \mathrm{ord}\, \phi:= \inf \big\{h \in \mathbb{Q} : h\, \text{can be represented by some}&\, t \in \mathrm{Herm}_{g} (E)\, \text{such that}\\ & c (\phi; t, r) \neq 0 \,\, \text{for some}\, r \in \mathrm{Mat}_{g,l} (E) \big\} \text{.} \end{align*} $$

If $\phi = 0$ , we set $\mathrm {ord}\, \phi := \infty $ by convention. Spaces of (vector-valued) Hermitian modular (respectively, Hermitian Jacobi) forms of genus g, weight k, (index m,) and type $\rho $ with vanishing order at least $o \in \mathbb {Q}$ in their Fourier expansions are denoted by $\mathrm {M}_k^{(g)}(\rho ) [o]$ (respectively, $\mathrm {J}_{k, m}^{(g)} (\rho ) [o]$ ).

Let f be a classical (scalar-valued) Hermitian modular form of weight k. If $\mathrm {ord} f \neq 0$ , the slope of f is defined as

$$\begin{align*}\omega(f) := \frac{k}{\mathrm{ord} f}\text{,}\end{align*}$$

and if $\mathrm {ord} f = 0$ , we define $\omega (f) = + \infty $ . The lower slope bound for classical Hermitian modular forms of degree g is defined as

$$\begin{align*}\omega_g := \inf_{\substack{k\in \mathbb{Z}\text{,} \\ f \in \mathrm{M}_k^{(g)} \setminus \{0\}}} \omega (f)\text{.}\end{align*}$$

It is well known that the lower slope bound $\omega _g$ for classical Hermitian modular forms is strictly positive, which is also shown in Corollary 3.7.

3.2 Embedding and vanishing

Lemma 3.2. Let $g \geq 2$ be an integer, and let l be an integer such that $1 \leq l \leq g - 1$ and d be a rational number. For every complex finite-dimensional representation $\rho $ of $\mathrm {U} (g, g) (\mathbb {Z})$ factoring through a finite quotient, the linear map

$$ \begin{align*} \mathrm{M}_k^{(g)} (\rho) [d] &\longrightarrow \mathrm{FM}_k^{(g, l)} (\rho) [d] \\ f & \longmapsto \sum_{0 \leq m \in \mathrm{Herm}_{l} (E)} \phi_m (\tau_1, w, z) e (m \tau_2) \end{align*} $$

given by the Fourier–Jacobi expansion of cogenus l is an embedding.

Lemma 3.3. Let $g \geq 2$ be an integer, and let $\rho $ be a complex finite-dimensional representation of $\mathrm {U} (g, g) (\mathbb {Z})$ that factors through a finite quotient. Let $N = N (\rho )$ be a positive integer such that the set of vanishing orders $\big \{\mathrm {ord} f: f\in \mathrm {FM}_k^{(g)}(\rho ) \big \}$ is contained in $\frac {1}{N}\mathbb {Z}_{\geq 0}$ . Then for every rational number $d \in \frac {1}{N} \mathbb {Z}_{\geq 0}$ , the dimension of symmetric formal Fourier–Jacobi series of vanishing order at least d can be bounded above by

$$\begin{align*}\dim \mathrm{FM}_k^{(g)} (\rho) [d] \leq \sum_{m \in {\big(\frac{1}{N} \mathbb{Z}\big)}_{\geq d}} \dim \mathrm{J}_{k, m}^{(g - 1)} (\rho) [m]\text{.}\end{align*}$$

Proof. For any $d \in \frac {1}{N} \mathbb {Z}_{\geq 0}$ and every $f \in \mathrm {FM}_k^{(g)} (\rho ) [d]$ , by picking a (noncanonical) $\mathbb {C}$ -linear section

$$\begin{align*}l_m: \mathrm{FM}_k^{(g)} (\rho) [m] / \mathrm{FM}_k^{(g)} (\rho) \Big[m + \frac{1}{N}\Big] \longrightarrow \mathrm{FM}_k^{(g)} (\rho) [m]\end{align*}$$

for each $m \in {\big (\frac {1}{N} \mathbb {Z}\big )}_{\geq d}$ , we define recursively $f_m \in \mathrm {FM}_k^{(g)} (\rho ) [m]$ for all $m \in {\big (\frac {1}{N} \mathbb {Z}\big )}_{\geq d}$ by putting $f_d := f$ and

$$ \begin{align*} f_{m + \frac{1}{N}} := f_{m} - l_{m} ([f_{m}]) \in \mathrm{FM}_k^{(g)} (\rho) \Big[m + \frac{1}{N}\Big] \text{.} \end{align*} $$

Then we define a $\mathbb {C}$ -linear map

$$ \begin{align*} i: \mathrm{FM}_k^{(g)} (\rho) [d] &\longrightarrow \prod_{m \in {\big(\frac{1}{N} \mathbb{Z}\big)}_{\geq d}} \mathrm{FM}_k^{(g)} (\rho) [m] / \mathrm{FM}_k^{(g)} (\rho) \Big[m + \frac{1}{N}\Big]\\ f &\longmapsto \big([f_m]\big)_{m} \, \text{,} \end{align*} $$

which we claim to be injective. Indeed, for any $f \in \mathop {\mathrm {ker}} i$ , since $[f_m] = [0]$ implies $l_m ([f_m]) = 0$ for each m, we have $f = f_n \in \mathrm {FM}_k^{(g)} (\rho ) [n]$ for all $n \geq d$ . In particular, the vanishing order of f is not finite, hence $f = 0$ and we find the map i is an embedding. Moreover, by sending $f \in \mathrm {FM}_k^{(g)} (\rho ) [m]$ to its mth Fourier–Jacobi coefficient, we obtain a linear map

(3.3) $$ \begin{align} \mathrm{FM}_k^{(g)} (\rho) [m] \longrightarrow \mathrm{J}_{k, m}^{(g - 1)} (\rho) [m] \end{align} $$

whose kernel is $\mathrm {FM}_k^{(g)} (\rho ) \big [m + \frac {1}{N}\big ]$ . To see this fact, recall that a symmetric formal Fourier–Jacobi series $f \in \mathrm {FM}_k^{(g)} (\rho ) [m]$ can be written as

$$\begin{align*}f(\tau) = \sum_{m^{\prime} \in {\big(\frac{1}{N} \mathbb{Z}\big)}_{\geq m}} \phi_{m^{\prime}} (\tau_1, w, z) e (m^{\prime} \tau_2) \end{align*}$$

for $\tau = \left (\begin {smallmatrix} \tau _1 & w\\ z & \tau _2 \end {smallmatrix}\right ) \in {\mathbb {H}}_g$ . If f is in the kernel of the map in equation (3.3), then its mth Fourier–Jacobi coefficient $\phi _m$ vanishes, and by the symmetry relation in equation (2.12) for symmetric formal Fourier–Jacobi series we find that

$$\begin{align*}c \Bigg(f; u^{\ast} \begin{pmatrix} n & r \\ r^{\ast} & m \end{pmatrix} u\Bigg) = 0 \end{align*}$$

for all $u \in \mathrm {GL}_{g}({\mathcal {O}}_E)$ , $n \in \mathrm {Herm}_{g - 1} (E)_{\geq 0}$ and $r \in \mathrm {Mat}_{g - 1, 1} (E)$ . This implies that for any index $t \in \mathrm {Herm}_{g} (E)_{\geq 0}$ such that t can represent the rational number m, the Fourier coefficient $c (f; t)$ vanishes, hence the vanishing order of f satisfies $\mathrm {ord} f \geq m + \frac {1}{N}$ , and the kernel of the map in equation (3.3) is contained in $\mathrm {FM}_k^{(g)} (\rho ) \big [m + \frac {1}{N}\big ]$ . The other direction is clear.

Finally, the map in equation (3.3) induces an embedding

The desired inequality follows by composing the embeddings i, j and the $\mathbb {C}$ -dimension function.

Proposition 3.4. Let g be a positive integer, and let $\rho $ be a complex finite-dimensional representation of $\mathrm {U} (g, g) (\mathbb {Z})$ that factors through a finite quotient. For every rational number $o> \frac {k}{\omega _g}$ , we have

$$ \begin{align*} \mathrm{M}_k^{(g)}(\rho) [o] = \{0\} \text{.} \end{align*} $$

Proof. If $\omega _g = 0$ , the statement holds automatically. We assume $\omega _g> 0$ . If $\rho $ is the trivial representation of dimension $1$ , the assertion is clear from the definition of $\omega _g$ . In the general case, it suffices to show that for any $f \in \mathrm {M}_k^{(g)}(\rho ) [o]$ and $v \in V(\rho )^{\vee }$ , we have $v \circ f = 0$ . For this we consider a finite product

(3.4) $$ \begin{align} f_v := \prod_{\gamma \in \mathop{\mathrm{ker}} (\rho) \backslash \mathrm{U} (g, g) (\mathbb{Z})} v \circ f|_k \gamma\text{.} \end{align} $$

It is clear that $f_v \in M_{|\rho | k}^{(g)}$ , where $|\rho |$ is the index of $\mathop {\mathrm {ker}} (\rho )$ in $\mathrm {U} (g, g) (\mathbb {Z})$ . Note that each factor $v \circ f|_k \gamma = v \circ \rho (\gamma ) f$ in equation (3.4) is a linear combination of components of the vector-valued function f and $\mathrm {ord} f \geq o$ ; therefore $f_v \in M_{|\rho | k}^{(g)} \big [|\rho | o\big ]$ by Lemma 3.1. But $|\rho | o> \frac {|\rho | k}{\omega _g}$ by the assumption $o> \frac {k}{\omega _g}$ , as shown in the trivial case $M_{|\rho | k}^{(g)} \big [|\rho | o\big ]$ vanishes, so $v \circ f|_k \gamma = 0$ for some $\gamma $ . Taking the slash action of $\gamma ^{-1}$ on each side, we find $v \circ f = 0$ , as desired.

Recall that there are exactly $5$ norm-Euclidean imaginary quadratic fields $E = \mathbb {Q}(\sqrt {d})$ , which are given by $d \in \big \{-1, -2, -3, -7, -11\big \}$ . For each of these fields E, there is a maximal uniform constant $c_E> 0$ , such that for any $\beta \in E$ , there is an integer $\alpha \in {\mathcal {O}}_E$ satisfying $\big (N_{E/\mathbb {Q}}(\beta - \alpha )\big )^2 \leq 1 - c_E$ . In particular, we have the following corollary.

Let g be a positive integer and $r \in E^g$ be a column vector of dimension g. In analogy to the ordinary vanishing order for Jacobi forms, we define the rth vanishing order $\mathrm {ord}_r \phi $ for a formal series $\phi $ in the form of equation (3.2) to be the infimum of rational numbers $m \in \mathbb {Q}$ such that there is some Hermitian positive semidefinite matrix $t \in \mathrm {Herm}_{g} (E)_{\geq 0}$ satisfying $c (\phi; t, r) \neq 0$ and the $(g, g)$ th entry $t_{g, g} = m$ . It is clear that $\mathrm {ord}\, \phi \leq \mathrm {ord}_r \phi $ for every $r \in E^g$ , and the equation $\mathrm {ord}_r\, (f \psi ) = \mathrm {ord} f + \mathrm {ord}_r \psi $ holds for every component f of a vector-valued Hermitian modular form of degree g and every component $\psi $ of a vector-valued Hermitian Jacobi form of degree g, arbitrary cogenus and any type that factors through a finite quotient of the Jacobi group.

Recall that our goal in this section is to estimate the dimension of the space of symmetric formal Fourier–Jacobi series $\mathrm {FM}_k^{(g)}(\rho )$ of degree g and cogenus $1$ . Using the rth vanishing order, we deduce the following analogous result on vanishing orders for Hermitian Jacobi forms of cogenus $1$ from Hermitian modular forms.

Proposition 3.6. Let g be a positive integer, and let $\rho $ be a complex finite-dimensional representation of $\Gamma ^{(g, 1)}$ , which factors through a finite quotient. For every rational number $m> c_E^{-1} \frac {k}{\omega _g}$ , we have

$$\begin{align*}\mathrm{J}_{k, m}^{(g)} (\rho) [m] = 0\text{.}\end{align*}$$

Proof. If $\omega _g = 0$ , the statement is automatically true. We assume $\omega _g> 0$ . Let $\phi \in \mathrm {J}_{k, m}^{(g)} (\rho ) [m]$ be a vector-valued Jacobi form of cogenus $1$ . By Proposition 2.10, this Jacobi form $\phi $ has a unique theta decomposition

(3.5) $$ \begin{align} \phi (\tau, w, z) = \sum_{s \in \Delta_g (m)} h_s (\tau) \theta_{m, s}^{(g)} (\tau, w, z) \text{.} \end{align} $$

For every $s \in \Delta _g (m)$ and every $r \in s + m {\mathcal {O}}_E^g$ , by equation (3.5) and the definition and properties of the rth vanishing order of Jacobi forms, we have

(3.6) $$ \begin{align} m \leq \mathrm{ord}\, \phi \leq \mathrm{ord}_r \phi = \mathrm{ord}_r \big(h_s \theta_{m, s}^{(g)}\big) = \mathrm{ord} (h_s) + \mathrm{ord}_r \big(\theta_{m, s}^{(g)}\big) \text{,} \end{align} $$

where the first equality is due to the fact that the only summand in the sum in equation (3.5) that yields the vector r in the Fourier expansion of $\phi $ is $h_s \theta _{m, s}^{(g)}$ .

On the other hand, by Lemma 3.5, for each $s \in \Delta _g (m)$ , there is some element $r(s) \in s + m {\mathcal {O}}_E^g$ whose entries $r_i$ satisfy $|r_i|^2 \leq (1 - c_E) m^2$ for each i. It then follows from the definition that $\mathrm {ord}_r (\theta _{m, s}^{(g)}) \leq (1 - c_E) m$ for $r = r(s)$ , and hence $\mathrm {ord} (h_s) \geq c_E m$ by the inequality in equation (3.6). Since this holds for every s, by the assumption $m> c_E^{-1} \frac {k}{\omega _g}$ , we have

$$\begin{align*}h = (h_s)_{s\in \Delta_g (m)} \in \mathrm{M}_{k - 1}^{(g)} (\rho_m^{(g)} \otimes \rho) [o]\end{align*}$$

for some $o> \frac {k}{\omega _g} > \frac {k - 1}{\omega _g}$ , which implies $h = 0$ by Proposition 3.4 and hence $\phi = 0$ , as desired.

Proof. We argue by induction. For $g = 1$ , the upper half-space ${\mathbb H}_1$ is just the Poincaré upper half plane, and the modular group $\mathrm {U}(1, 1)(\mathbb {Z})$ can be identified with $(\mathrm {U}(1) \cap {\mathcal {O}}_E) \times \mathrm {SL}_{2}(\mathbb {Z})$ . Hence $\mathrm {M}_k^{(1)}$ is just a subspace of the elliptic modular forms for $\mathrm {SL}_{2} (\mathbb {Z})$ of weight k, and it follows from [Reference Bruinier, van der Geer, Harder and Zagier46], Page 9, Proposition 2 that $\omega _1 \geq 12$ .

Now assume the assertion holds for $g = n - 1$ with $n \in \mathbb {Z}_{\geq 2}$ , and we have to show that $\omega _n \geq c_E \omega _{n - 1}$ . Combining Lemma 3.2, Lemma 3.3, the induction hypothesis and Proposition 3.6 for $g = n - 1$ , we see that the inequalities

$$\begin{align*}\dim \mathrm{M}_k^{(n)} [d] \leq \dim \mathrm{FM}_k^{(n)} [d] \leq \sum_{m = d}^{\infty} \dim \mathrm{J}_{k, m}^{(n - 1)}[m] = \sum_{m = d}^{\lfloor c_E^{-1} \frac{k}{\omega_{n - 1}} \rfloor} \dim \mathrm{J}_{k, m}^{(n - 1)}[m]\end{align*}$$

hold for every positive integer d and every weight k. In particular, if $d> c_E^{-1} \frac {k}{\omega _{n - 1}}$ , then the space $\mathrm {M}_k^{(g)} [d]$ vanishes, which implies $\omega _n \geq c_E \omega _{n - 1}$ .

3.3 Asymptotic dimensions and algebraicity of $\mathrm {FM}_{\bullet }^{(g)}$ over $\mathrm {M}_{\bullet }^{(g)}$

Lemma 3.8. Let $\rho $ be a complex finite-dimensional representation of $\mathrm {U} (1, 1) (\mathbb {Z})$ that factors through a finite quotient. Then

$$\begin{align*}\dim \mathrm{M}_k^{(1)} (\rho) \ll \dim V(\rho) k \text{.}\end{align*}$$

Lemma 3.9. Let $\rho $ be a complex finite-dimensional representation of the Jacobi group $\Gamma ^{(1, 1)}$ that factors through a finite quotient. Then

$$\begin{align*}\dim \mathrm{J}_{k, m}^{(1)} (\rho) \ll \dim V(\rho) k m^2 \text{.}\end{align*}$$

Proposition 3.11. Let $g \geq 2$ be an integer and $\rho $ be a complex finite-dimensional representation of $\mathrm {U} (g, g) (\mathbb {Z})$ that factors through a finite quotient. Then

$$\begin{align*}\dim \mathrm{FM}_k^{(g)}(\rho) \ll_{g, \rho} \dim V (\rho) k^{g^2} \text{.} \end{align*}$$

Proof. Let $N = N (\rho )$ be a positive integer such that the set of vanishing orders $\big \{\mathrm {ord} f: f\in \mathrm {FM}_k^{(g)}(\rho ) \big \}$ is contained in $\frac {1}{N}\mathbb {Z}_{\geq 0}$ . We prove the assertion by induction. For $g = 2$ , combining Lemma 3.3, Proposition 3.6 and Lemma 3.9, we have asymptotic inequalities

$$ \begin{align*} \dim \mathrm{FM}_k^{(2)} (\rho) \leq \sum_{\substack{0 \leq m \leq c_E^{-1} \frac{k}{\omega_{1}} \\ m \in \frac{1}{N} \mathbb{Z}}} \dim \mathrm{J}_{k, m}^{(1)} (\rho) [m] \ll \dim V(\rho) k \sum_{\substack{0 \leq m \leq c_E^{-1} \frac{k}{\omega_{1}} \\ m \in \frac{1}{N} \mathbb{Z}}} m^2 \ll_{\rho} \dim V(\rho) k^4 \text{.} \end{align*} $$

Assume the assertion is true for $g = n - 1$ with $n \geq 3$ , and we show it for $g = n$ . Applying Lemma 3.3, Proposition 3.6, Proposition 2.10 and Lemma 3.2 successively, we have

(3.7) $$ \begin{align} \dim \mathrm{FM}_k^{(n)}(\rho) \leq \sum_{\substack{0 \leq m \leq c_E^{-1} \frac{k}{\omega_{n - 1}} \\ m \in \frac{1}{N} \mathbb{Z}}} \dim \mathrm{J}_{k, m}^{(n - 1)} (\rho)[m] \leq \sum_{\substack{0 \leq m \leq c_E^{-1} \frac{k}{\omega_{n - 1}} \\ m \in \frac{1}{N} \mathbb{Z}}} \dim \mathrm{FM}_{k - 1}^{(n - 1)} (\rho_m^{(n - 1)} \otimes \rho) \text{.} \end{align} $$

Furthermore, the induction hypothesis and the rank of $\Delta _{n - 1} (m)$ for the Weil representation imply that

(3.8) $$ \begin{align} \dim \mathrm{FM}_{k - 1}^{(n - 1)} (\rho_m^{(n - 1)} \otimes \rho) \ll_{n - 1} \dim V(\rho) m^{2 (n - 1)} k^{(n - 1)^2} \text{.} \end{align} $$

Finally, we combine the inequalities in equation (3.7) with the one in equation (3.8) and apply Corollary 3.7 for $g = n - 1$ to see that the sums in equation (3.7) has ${\mathcal {O}}(k)$ terms, whence the desired assertion for $g = n$ .

Corollary 3.12. Let g be a positive integer. Then

$$\begin{align*}\dim \mathrm{FM}_{\leq k}^{(g)} \ll_g k^{g^2 + 1}\text{.} \end{align*}$$

To give an asymptotic lower bound for the dimension of Hermitian modular forms $\mathrm {M}_k^{(g)}$ of degree g and weight k, we use a dimension formula for Hermitian cusp forms obtained from the Selberg trace formula. For $k \geq 2g$ , we define the following positive constant:

$$\begin{align*}C (k, g) := 2^{- g^2 - g} \pi^{- g^2} \prod_{0 \leq i, j \leq g - 1} (k - 2g + 1 + i + j)\text{.}\end{align*}$$

Let $Y_g := \mathrm {U} (g, g) (\mathbb {Z}) \backslash {\mathbb {H}}_g$ denote the Hermitian modular variety. For $\tau \in Y_g$ we introduce a Bergman kernel function

$$\begin{align*}S(k, g, \tau):= \sum_{\gamma \in \mathrm{U} (g, g) (\mathbb{Z}) / Z \big(\mathrm{U} (g, g) (\mathbb{Z}) \big)} \overline{j (\gamma, \tau)}^{\, - k} \Big(\mathrm{det}\ \big(\frac{1}{2i} (\tau - (\gamma\tau)^{\ast} ) \big) \Big)^{- k}\text{,}\end{align*}$$

where $Z \big (\mathrm {U} (g, g) (\mathbb {Z}) \big )$ is the centre of $\mathrm {U} (g, g) (\mathbb {Z})$ . The sum is well-defined and absolutely convergent by general theory of Bergman kernel functions, see [Reference Eie48] for detailed discussions. We define the (weight-k normalised) hyperbolic volume form to be

$$\begin{align*}\mathrm{d} \tau := \Big(\mathrm{det}\ \big(\frac{1}{2i} (\tau - \tau^{\ast}) \big) \Big)^{k - 2g} \mathrm{d}_E \tau,\end{align*}$$

where $\mathrm {d}_E \tau $ is the (complex) Euclidean measure.

Proposition 3.13 [Reference Eie49], Page 62

Let $k> 4 g - 2$ be an even integer. Then we have the following dimension formula for the space of Hermitian cusp forms:

$$\begin{align*}\dim S_k^{(g)}(\mathrm{U} (g, g) (\mathbb{Z})) = C (k, g) \int_{Y_g} S(k, g, \tau) \mathrm{d} \tau \text{.} \end{align*}$$

We observe the asymptotic behaviour of $C(k, g)$ and the fact that the values of $\int _{Y_g} S(k, g, \tau ) \mathrm {d} \tau $ when k varies are bounded from below by some positive constant independent of k. Since the space of Hermitian cusp forms $S_k^{(g)}(\mathrm {U} (g, g) (\mathbb {Z}))$ is contained in the space of Hermitian modular forms $\mathrm {M}_k^{(g)}$ , we deduce an asymptotic lower bound for the dimension of Hermitian modular forms $\mathrm {M}_{\leq k}^{(g)}$ of degree g and weight at most k.

Corollary 3.14. Let g be a positive integer. Then

$$\begin{align*}\dim \mathrm{M}_{\leq k}^{(g)} \gg_g k^{g^2 + 1}\text{.} \end{align*}$$

Proof. It suffices to show that for an arbitrary $f \in \mathrm {FM}_{k_0}^{(g)}$ of weight $k_0$ , the set $\big \{f^i : i \in \mathbb {N} \big \}$ is of finite rank over the graded algebra $\mathrm {M}_{\bullet }^{(g)}$ . Indeed, for any positive integers $k, n$ such that $k \geq n k_0$ , by the fact that $\sum _{i = 0}^n \mathrm {M}_{\leq k}^{(g)} f^i \subseteq \mathrm {FM}_{\leq k + n k_0}^{(g)}$ combined with Corollary 3.12 and Corollary 3.14, we have the following asymptotic inequalities

$$\begin{align*}\dim_{\mathbb{C}} \big( \sum_{i = 0}^n \mathrm{M}_{\leq k}^{(g)} f^i \big) \leq \dim_{\mathbb{C}} \mathrm{FM}_{\leq k + n k_0}^{(g)} \ll_g (k + n k_0)^{g^2 + 1} \ll_g k^{g^2 + 1} \ll_g \dim_{\mathbb{C}} \mathrm{M}_{\leq k}^{(g)} \text{,}\end{align*}$$

which imply that there is a constant $C = C(g)$ such that for any large n and k satisfying $k \geq n k_0$ , we have $\dim _{\mathbb {C}} \big ( \sum _{i = 0}^n \mathrm {M}_{\leq k}^{(g)} f^i \big ) \leq C \dim _{\mathbb {C}} \mathrm {M}_{\leq k}^{(g)}$ . In particular, the rank of the set $\big \{f^i : i \in \mathbb {N} \big \}$ over the graded algebra $\mathrm {M}_{\bullet }^{(g)}$ is bounded above by C, as desired.

4.1 Toroidal boundaries of Hermitian modular varieties

The aim of this subsection is to prove Theorem 4.6 based on the following strategy: the analogous result in the Siegel case for the Satake boundary, Lemma 4.5, was proven in the work of Bruinier–Raum; to reduce to the Siegel case, we embed the Siegel upper half-space into the Hermitian upper half-space both in a generic way and over $\mathbb {Q}$ via a density argument; finally, we compare the behaviour of various boundaries under the constructed maps. We also point out a cohomological approach to the theorem at the end of this subsection.

The theory of toroidal compactifications was developed by Ash–Mumford–Rapoport–Tai [Reference Ash, Mumford, Rapoport and Tai50]. A brief discussion of the construction of partial compacitifications of Siegel modular varieties can be found, for instance, in [Reference Hulek and Sankaran51] and [Reference Bruinier, van der Geer, Harder and Zagier46], and the book [Reference Namikawa52] contains more details in an accessible way. Let $E/\mathbb {Q}$ be a fixed imaginary quadratic field with an embedding . The unitary group $\mathrm {U} (g, g) := \mathrm {U}_{E/\mathbb {Q}} (g, g)$ is a semi-simple algebraic group defined over $\mathbb {Q}$ , and the group $\mathrm {U} (g, g) (\mathbb {R})^o$ is the connected component of $\mathrm {Aut} (\mathbb D_g)$ , for the Hermitian symmetric domain $\mathbb D_g = \big \{\sigma \in \mathrm {Mat}_g (\mathbb {C}): \sigma ^{\ast } \sigma < I_g \big \} \cong {\mathbb H}_g$ . Here we define a certain toroidal compactification of the Hermitian modular variety $Y_g = \mathrm {U} (g, g) (\mathbb {Z}) \backslash {\mathbb {H}}_g$ associated to an admissible collection of polyhedra for a $0$ -dimensional cusp, since this determines the cases for the other cusps as well. For the arithmetric group $\mathrm {GL}_{g} ({\mathcal {O}}_E) \subseteq \mathrm {U} (g, g) (\mathbb {R})^o$ , let $\{\sigma _{\alpha }\}_{\alpha }$ be a $\mathrm {GL}_{g} ({\mathcal {O}}_E)$ -admissible collection of polyhedra (see, for instance, (7.3) in [Reference Namikawa52]). Then we obtain a toroidal compactification $X_g$ of the Hermitian modular variety $Y_g = \mathrm {U} (g, g) (\mathbb {Z}) \backslash {\mathbb {H}}_g$ associated to this set of data. We choose a collection of polyhedra $\{\sigma _{\alpha }\}_{\alpha }$ such that the resulting toroidal compactification $X_g$ is nonsingular in the orbifold sense. We write $\partial Y_g := X_g \setminus Y_g$ for the toroidal boundary.

Let $\overline {Y_g}$ be the Satake compactification of $Y_g$ , and let

$$\begin{align*}\pi : X_g \longrightarrow \overline{Y_g}\end{align*}$$

be the natural map, which exists by the definition of the minimal compactification $\overline {Y_g}$ . Let $\phi : {\mathbb H}_g \longrightarrow Y_g$ be the quotient map. Recall that $\mathrm {U} (g, g) (\mathbb {R})$ acts transitively on the Hermitian upper half-space ${\mathbb H}_g$ via Möbius transformations. In particular, for each $\gamma \in \mathrm {U} (g, g) (\mathbb {R})$ , we obtain an embedding of the Siegel upper half-space ${{\mathcal {H}}}_g$ into ${\mathbb H}_g$ by sending $\tau \in {{\mathcal {H}}}_g$ to $\gamma \tau $ , and the images $\gamma {\mathcal {H}}_g \subseteq {\mathbb H}_g$ for $\gamma \in \mathrm {U} (g, g) (\mathbb {R})$ cover ${\mathbb H}_g$ . Let ${{\mathcal {Y}}}_g := \mathrm {Sp}_{g}(\mathbb Z) \backslash {\mathcal {H}}_g$ and $\overline {{\mathcal {Y}}_g}$ denote the Siegel modular variety of degree g and its Satake compactification, respectively.

We say that a point $\tau $ in the Siegel upper half-space ${\mathcal {H}}_g$ is an elliptic fixed point if its stabiliser under the action of $\mathrm {Sp}_{g}(\mathbb {Z})$ on ${\mathcal {H}}_g$ is larger than the centre of $\mathrm {Sp}_{g}(\mathbb {Z})$ .

Lemma 4.3. For every $\gamma \in \mathrm {U} (g, g) (\mathbb {Q})$ , there is a positive integer $N = N(\gamma )$ such that

$$ \begin{align*} [\gamma]: {\mathcal{Y}}_g(N) = \Gamma(N) \backslash {\mathcal{H}}_g &\longrightarrow Y_g = \mathrm{U} (g, g) (\mathbb{Z}) \backslash {\mathbb H}_g \\ [\tau] &\longmapsto [\gamma \tau] \end{align*} $$

is a well-defined continuous map and induces a covering map onto the image with possible ramification at the orbits of elliptic fixed points. Furthermore, the map can be continuously extended to $\overline {[\gamma ]}: \overline {{\mathcal {Y}}_g (N)} \longrightarrow \overline {Y_g}$ on the Satake compactification, and for every open neighbourhood V of the Satake boundary $\overline {Y_g} \setminus Y_g$ , there is an open neighbourhood U of the Satake boundary $\overline {{\mathcal {Y}}_g (N)} \setminus {\mathcal {Y}}_g (N)$ , such that $\overline {[\gamma ]} (U) \subseteq V$ .

Proof. As $\gamma \in \mathrm {U} (g, g) (\mathbb {Q})$ , there are positive integers $N_1, N_2$ such that $N_1 \gamma $ and $N_2 \gamma ^{-1}$ are integral. Let $N := N_1 N_2$ . Then for any $\gamma _N \in \Gamma (N)$ , we have

$$\begin{align*}\gamma \gamma_N \gamma^{-1} - I_{2g} = \gamma (\gamma_N - I_{2g}) \gamma^{-1} \in \gamma N \mathrm{Sp}_{g} (\mathbb Z) \gamma^{-1} \subseteq \mathrm{U} (g, g) (\mathbb{Z})\text{,}\end{align*}$$

hence $\gamma \Gamma (N) \gamma ^{-1} \subseteq \mathrm {U} (g, g) (\mathbb {Z})$ . In other words, the map $[\gamma ]$ is well defined. It is clear that the induced surjective map from the complement of the orbits $[\tau ]$ of elliptic fixed points $\tau $ onto the image is a covering map (whose degree is equal to the index of the projective group of $\Gamma (N)$ in the projective group of $\gamma ^{-1} \mathrm {U} (g, g) (\mathbb {Z}) \gamma \cap \mathrm {Sp}_{g} (\mathbb Z)$ ). Furthermore, the continuous extension can be defined with respect to the topology of the Satake compactification, and the boundary is mapped to the boundary under $\overline {[\gamma ]}$ . We put $U := \overline {[\gamma ]}^{\, -1} (V)$ , which satisfies the desired property.

Lemma 4.4. For each positive integer N, the natural map

$$ \begin{align*} \phi_N: {\mathcal{Y}}_g (N) = \Gamma(N) \backslash {\mathcal{H}}_g \longrightarrow {\mathcal{Y}}_g = \mathrm{Sp}_{g}(\mathbb Z) \backslash {\mathcal{H}}_g \end{align*} $$

can be continuously extended to $\overline {\phi _N}: \overline {{\mathcal {Y}}_g (N)} \longrightarrow \overline {{\mathcal {Y}}_g}$ on the Satake compactification. For every open neighbourhood U of the Satake boundary $\overline {{\mathcal {Y}}_g (N)} \setminus {\mathcal {Y}}_g (N)$ , there is an open neighbourhood V of the Satake boundary $\overline {{\mathcal {Y}}_g} \setminus {\mathcal {Y}}_g$ such that $\overline {\phi _N}^{\, -1} (V) \subseteq U$ .

Proof. Recall the maps defined in this subsection:

$$ \begin{align*} X_g \overset{\pi} \longrightarrow \overline{Y_g} \underset{\overline{[\gamma]}} \longleftarrow \overline{{\mathcal{Y}}_g (N)} \overset{\overline{\phi_N}} \longrightarrow \overline{{\mathcal{Y}}_g} \end{align*} $$

for $\gamma \in \mathrm {U} (g, g) (\mathbb {Q})$ . If D is contained in the boundary $\partial Y_g = X_g \setminus Y_g$ , the assertion is clear. Otherwise D intersects $Y_g$ , so the pushforward $D_1 := \pi _{\ast } (D) = \overline {\pi (D)}$ is a prime divisor on $\overline {Y_g}$ and $D_1^{\prime } := D_1 \cap Y_g$ is a prime divisor on $Y_g$ .

By Lemma 4.1, there is $\gamma \in \mathrm {U} (g, g) (\mathbb {Q})$ , which we fix from now on, such that $\phi (\gamma {\mathcal {H}}_g)$ intersects $D_1^{\prime }$ transversally at some point. In particular, the intersection has codimension $1$ in $\phi (\gamma {\mathcal {H}}_g)$ at some point. Let $N := N (\gamma )$ . By Lemma 4.3, the continuous map $[\gamma ] : [\tau ] \longmapsto \phi (\gamma \tau )$ induces a covering map onto the image with possible ramification at the orbits of elliptic fixed points. By Lemma 4.2, since irreducible components are equidimensional, $\overline {[\gamma ]}^{-1} (D_1)$ contains a prime divisor $D_2$ on $\overline {{\mathcal {Y}}_g (N)}$ .

Since the Satake boundary has codimension at least $2$ for $g \geq 2$ , $D_2$ must intersect ${\mathcal {Y}}_g (N)$ . As $\phi _N$ induces a covering map onto the image with possible ramification at the orbits of elliptic fixed points, it follows that the pushforward $D_3 := \overline {\phi _N}_{\ast } (D_2)$ is a prime divisor on $\overline {{\mathcal {Y}}_g }$ .

Finally we analyse the boundary intersection. Given an arbitrary open neighbourhood $U \subseteq X_g$ of the toroidal boundary $\partial Y_g$ , the open neighbourhood $U_1 = \overline {Y_g} \setminus (X_g \setminus U) \subseteq \overline {Y_g}$ of the Satake boundary $\overline {Y_g} \setminus Y_g$ satisfies $\pi ^{-1} (U_1) = U$ . By Lemma 4.3, there is an open neighbourhood $U_2 \subseteq \overline {{\mathcal {Y}}_g (N)}$ of the boundary $\overline {{\mathcal {Y}}_g (N)} \setminus {\mathcal {Y}}_g (N)$ such that $\overline {[\gamma ]} (U_2) \subseteq U_1$ . Finally, by Lemma 4.4, there is an open neighbourhood $U_3 \subseteq \overline {{\mathcal {Y}}_g}$ of the boundary $\overline {{\mathcal {Y}}_g} \setminus {\mathcal {Y}}_g$ such that $\pi ^{-1} (U_3) \subseteq U_2$ . By Lemma 4.5, $D_3$ intersects $U_3$ , and it follows from the aforementioned choice of open neighbourhoods that $D_2$ intersects $U_2$ , $D_1$ intersects $U_1$ and D intersects U, as desired.

Remark 4.7. There is another approach to prove Theorem 4.6 using results from stable cohomology, at least for large degree g. We sketch a proof here for reference. Following the strategy of the proof of Proposition 4.1 in [Reference Bruinier and Westerholt-Raum7], it suffices to show that the Picard group of $\overline {Y_g}$ has rank at most $1$ . Since $\overline {Y_g} \setminus Y_g$ has codimension at least $2$ in $\overline {Y_g}$ for $g \geq 2$ , we have $\mathrm {Pic} (\overline {Y_g}) = \mathrm {Pic} (Y_g)$ . By the exponential sheaf sequence, we have ([Reference Hartshorne53] Appendix B) an exact sequence

(4.1) $$ \begin{align} 0 \longrightarrow H^1 (Y_g, \mathbb Z) \longrightarrow H^1 (Y_g, {\mathcal{O}}_{Y_{g}}) \longrightarrow \mathrm{Pic} (Y_{g}) \longrightarrow H^2 (Y_g, \mathbb Z) \longrightarrow \cdots \text{.} \end{align} $$

Since ${\mathbb H}_g$ is simply connected, by a standard argument we find that $\pi _1 (Y_g)$ is the projective group of $\mathrm {U} (g, g) (\mathbb {Z})$ . Hence by the Hurewicz theorem and the universal coefficient theorem, we find that $H^1 (Y_g, \mathbb {Z})$ and $H^1 (Y_g, {\mathcal {O}}_{Y_g})$ vanish. By equation (4.1), we only need to show that $H^2 (Y_g, \mathbb Z)$ has rank at most $1$ . Applying results in [Reference Borel and Wallach54] (Chapter VII), this amounts to computing the rank for the degree- $2$ part in the graded ring $H^{\ast } (\mathrm {U} (g, g) (\mathbb {Z}), \mathbb Z)$ , which is canonically isomorphic to $H^{\ast } (A ({\mathbb H}_g; \mathbb Z)^{\mathrm {U} (g, g) (\mathbb {Z})} )$ , where $A^q(M; E)$ denotes the space of smooth E-valued differential q-forms on M for $q \geq 0$ . Alternatively, we can compute the lower degree parts of $H^{\ast } (\mathrm {U} (g, g) (\mathbb {Z}), \mathbb Z)$ using Theorem 7.5, the table under 10.6 and Theorem 11.1 in [Reference Borel55]. In particular, we can apply these results to deduce Theorem 4.6 for sufficiently large g. For small g, one needs more explicit results of computing the second degree information in $H^{\ast } (A ({\mathbb H}_g; \mathbb Z)^{\mathrm {U} (g, g) (\mathbb {Z})} )$ , which we skip here.

4.2 Local convergence at the boundary

4.3 Analytic continuation and algebraic closedness of $\mathrm {M}_{\bullet }^{(g)}$ in $\mathrm {FM}_{\bullet }^{(g)}$