A ring of symmetric Hermitian modular forms of degree $2$ with integral Fourier coefficients

TL;DR

This paper characterizes the structure of the ring of symmetric Hermitian modular forms of degree 2 over the integers, providing generators, and establishes Sturm bounds for certain weights and primes.

Contribution

It explicitly determines the generators of the integral symmetric Hermitian modular forms ring of degree 2 and derives Sturm bounds for specific primes and weights.

Findings

Identified 24 generators for the modular forms ring.

Established Sturm bounds for weights divisible by 4 at primes 2 and 3.

Provided structure theorem for the ring over integers.

Abstract

We determine the structure over $\mathbb{Z}$ of the ring of symmetric Hermitian modular forms with respect to $\mathbb{Q}(\sqrt{-1})$ of degree $2$ (with a character), whose Fourier coefficients are integers. Namely, we give a set of generators consisting of $24$ modular forms. As an application of our structure theorem, we give the Sturm bounds of such the modular forms of weight $k$ with $4\mid k$, in the case $p=2$, $3$. We remark that the bounds for $p\ge 5$ are already known.

Equations

$$A({\Gamma }_1;\mathbb{Z}):=\underset{k\in \mathbb{Z}}{⨁}{M}_k({\Gamma }_1;\mathbb{Z}).$$

$$A({\Gamma }_1;\mathbb{Z})=\mathbb{Z}[e_4,e_6,\delta ].$$

$$4,8,12,12,12,16,16,20,24,24,28,28,32,$$

$$36,36,36,40,40,48,48,52,60,60,72,84.$$

$${\mathbb{H}}_2:=\{Z\in M_2(\mathbb{C})|\frac{1}{2i}(Z-{}^t\overline{Z})>0\}$$

$$U_2(\mathcal{O}):=\left\{M\in M_4(\mathcal{O})|{}^t\overline{M}{J}_{2}M=J_2,J_2=(\genfrac{}{}{0ex}{}{0_2-1_2}{1_2{0}_2})\right\}$$

$${\mathbb{H}}_2\ni Z⟼M⟨Z⟩:=(AZ+B)(CZ+D{)}^{-1},M=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in U_2(\mathcal{O}).$$

$$F{\mid }_{k}M(Z):=\det (CZ+D{)}^{-k}F(M⟨Z⟩)={\det (M)}^k\cdot F(Z),$$

$$\Phi\Big{(}F\mid_{k}\binom{\!{}^{t}\overline{U}\;0}{0\;\;U}\Big{)}\equiv 0\quad\text{for}\;\text{all}\;U\in GL_{2}(\mathbb{Q}(\sqrt{-1}))$$

$$F(Z+B)=F(Z)\text{for}\text{all}B\in He{r}_2(\mathcal{O}),$$

$$F(Z)=\sum _{0\le H\in {\Lambda }_2(\mathbf{K})}{a}_F(H)e^{2\pi i\text{tr}(HZ)},$$

$${\Lambda }_2(\mathbf{K}):=\{H=(h_{ij})\in He{r}_2(\mathbf{K})|h_{ii}\in \mathbb{Z},2h_{ij}\in \mathcal{O}\}.$$

$${\dot{q}}_{11}:=\exp (2\pi i{z}_{11}),{\dot{q}}_{22}:=\exp (2\pi i{z}_{22}),$$

$${\dot{q}}_{12}:=\exp \left(2\pi i\frac{z_{12}-z_{21}}{-2i}\right),{\ddot{q}}_{12}:=\exp \left(2\pi i\frac{z_{12}+z_{21}}{2}\right).$$

$$e^{2\pi i\text{tr}(HZ)}={\dot{q_1}}^m{\dot{q_{12}}}^r{\ddot{q_{12}}}^s{\dot{q_2}}^n.$$

$$R[[\dot{\mathbf{q}}]]:=R[{\dot{q}}_{12}^{\pm 1},{\ddot{q}}_{12}^{\pm }][[\dot{q_1},\dot{q_2}]].$$

$$E_k(Z):=\sum _{M=\left(\begin{array}{cc}\ast & \ast \\ C& D\end{array}\right)}(\det M{)}^{\frac{k}{2}}\det (CZ+D{)}^{-k},Z\in {\mathbb{H}}_2,$$

$$E_k\in M_k(U_2(\mathcal{O})).$$

$$a_{E_k}(H)$$

$$=\{\begin{array}{ll}1& \text{if}H=0_2,\\ -\frac{2k}{B_k}{\sigma }_{k-1}(\epsilon (H))& \text{if}\mathrm{rank}(H)=1,\\ \frac{4k(k-1)}{B_k\cdot B_{k-1,{\chi }_{-4}}}\sum _{0<d|\epsilon (H)}{d}^{k-1}{G}_{\mathbf{K}}(k-2,4\det (H)/d^2)& \text{if}\mathrm{rank}(H)=2.\end{array}$$

$$\begin{array}{rl}& G_{\mathbf{K}}(m,N):=\frac{1}{1+|{\chi }_{-4}(N)|}({\sigma }_{m,{\chi }_{-4}}(N)-{\sigma }_{m,{\chi }_{-4}}^{\ast }(N))\\ & {\sigma }_{m,{\chi }_{-4}}(N):=\sum _{0<d|N}{\chi }_{-4}(d)d^m,{\sigma }_{m,{\chi }_{-4}}^{\ast }(N):=\sum _{0<d|N}{\chi }_{-4}(N/d)d^m.\end{array}$$

$$E_{10}-E_4{E}_6\in S_{10}(U_2(\mathcal{O})),$$

$$E_{12}-\frac{441}{691}{E}_4^3-\frac{250}{691}{E}_6^2\in S_{12}(U_2(\mathcal{O})).$$

$$F(Z)=\sum _{0\le T\in {\Lambda }_2}{a}_F(T)e^{2\pi i\text{tr}(TZ)},$$

$${\Lambda }_2=Sy{m}_2^{\ast }(\mathbb{Z}):=\{T=(t_{ij})\in Sy{m}_2(\mathbb{Q})|t_{ii},2t_{ij}\in \mathbb{Z}\}$$

$$e^{2\pi i\text{tr}(TZ)}=q_{11}^m{q}_{12}^r{q}_{22}^n.$$

$$M_k({\Gamma }_2;R):=\{F=\sum _{T\in {\Lambda }_n}{a}_F(T)q^T|a_F(T)\in R(\forall T\in {\Lambda }_2)\},$$

$$S_k({\Gamma }_n;R):=M_k({\Gamma }_2)\cap S_k({\Gamma }_2).$$

$$R[[\mathbf{q}]]:=R[q_{12}^{-1},q_{12}][[q_{11},q_{22}]].$$

$${\mathbb{S}}_2:={\mathbb{H}}_2\cap Sy{m}_2(\mathbb{C}).$$

$$R[[\dot{\mathbf{q}}]]⟶R[[\mathbf{q}]]$$

$$G_k(Z):=\sum _{M=\left(\begin{array}{cc}\ast & \ast \\ C& D\end{array}\right)}\det (CZ+D{)}^{-k},Z\in {\mathbb{S}}_2$$

$$\begin{array}{rl}{X}_{10}:& =-\frac{43867}{2^{10}\cdot 3^5\cdot 5^2\cdot 7\cdot 53}(G_{10}-G_4{G}_6),\\ X_{12}:& =-\frac{691\cdot 1847}{2^{13}\cdot 3^6\cdot 5^3\cdot 7^2}(G_{12}-\frac{441}{691}{G}_4^3-\frac{250}{691}{G}_6^2).\end{array}$$

Full text

2010 Mathematics subject classification: Primary 11F30 $\cdot$ Secondary 11F55

Key words: ring of modular forms, Hermitian modular forms, generators.

1 Introduction

Let $e_{4}$, $e_{6}$ be the normalized Eisenstein series of respective weight $4$, $6$ for $\Gamma_{1}:=SL_{2}(\mathbb{Z})$ and $\delta$ the Ramanujan delta function defined by $\delta=2^{-6}\cdot 3^{-3}(e_{4}^{3}-e_{6}^{2})$. For the $\mathbb{Z}$-module $M_{k}(\Gamma_{1};\mathbb{Z})$ consisting of modular forms of weight $k$ for $\Gamma_{1}$ whose Fourier coefficients are in $\mathbb{Z}$, we define an algebra over $\mathbb{Z}$ as

[TABLE]

Then it is well-known as a classical result that all the Fourier coefficients of the modular forms $e_{4}$, $e_{6}$, $\delta$ are integers and they generate $A(\Gamma_{1};\mathbb{Z})$. Namely we have

[TABLE]

In the case of Siegel modular forms for the symplectic group $\Gamma_{2}:=Sp_{2}(\mathbb{Z})$ of degree $2$, there is a famous result of Igusa. He showed such the ring over $\mathbb{Z}$ are generated by $15$ modular forms. He showed also that its set of generators are minimal.

In this paper, we consider the ring of symmetric Hermitian modular forms of degree $2$ with respect to $\mathbb{Q}(\sqrt{-1})$ whose Fourier coefficients are in $\mathbb{Z}$. Since it seems to be difficult to give generators of the full space of them, we restrict our selves to the case where the weights are multiples of $4$. We remark that, the ring of Siegel modular forms whose weights are multiples of $4$ are generated over $\mathbb{Z}$ by $23$ modular forms. This is an easy conclusion of Igusa’s result.

In our case, there exists a set of generators consisting of $24$ modular forms whose weights are

[TABLE]

The precise statement can be found in Theorem 3.8. We construct explicitly these generators in Subsection 3.1.

As an application of this result, we can obtain the Sturm bounds for $p=2$, $3$, in the Hermitian modular forms whose weights are multiples of $4$ (Theorem 3.10). We remark that the Sturm bounds for $p\geq 5$ are already known in [6].

2 Preliminaries

2.1 Hermitian modular forms of degree $2$

We deal with Hermitian modular forms of degree $2$ only for ${\boldsymbol{K}}:=\mathbb{Q}(\sqrt{-1})$. Let ${\mathcal{O}}$ be the ring of Gauss integers, i.e. ${\mathcal{O}}=\mathbb{Z}[\sqrt{-1}]$.

Let $\mathbb{H}_{2}$ be the Hermitian upper half-space of degree $2$ defined as

[TABLE]

where ${}^{t}\overline{Z}$ is the transposed complex conjugate of $Z$.

The Hermitian modular group of degree $2$

[TABLE]

acts on $\mathbb{H}_{2}$ by fractional transformation

[TABLE]

We denote by $M_{k}(U_{2}({\mathcal{O}}))=M_{k}(U_{2}({\mathcal{O}}),{\det}^{k})$ the space of symmetric Hermitian modular forms of weight $k$ and character ${\det}^{k}$ with respect to $U_{2}({\mathcal{O}})$. (We deal with modular forms with character ${\det}^{k}$, but we drop this in the notation). Namely, it consists of holomorphic functions $F:\mathbb{H}_{2}\longrightarrow\mathbb{C}$ satisfying

[TABLE]

for all $M=\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in U_{2}({\mathcal{O}})$ and $F({}^{t}Z)=F(Z)$. Note that one has $M_{k}(U_{2}({\mathcal{O}}))=\{0\}$ if $k$ is odd.

The cusp forms are characterized by the condition

[TABLE]

where $\Phi$ is the Siegel $\Phi$-operator. We denote by $S_{k}(U_{2}({\mathcal{O}}))$ the subspace consisting of all cusp forms in $M_{k}(U_{2}({\mathcal{O}}))$.

2.2 Fourier expansion

Since all of $F$ in $M_{k}(U_{2}({\mathcal{O}}))$ satisfies the condition

[TABLE]

it has a Fourier expansion of the form

[TABLE]

where

[TABLE]

We write $H=(m,r,s,n)$ for $H=\begin{pmatrix}m&\frac{r+si}{2}\\ \frac{r-si}{2}&n\end{pmatrix}\in\Lambda_{2}({\mathcal{O}})$ and also $a_{F}(m,r,s,n)$ for $a_{F}\begin{pmatrix}m&\frac{r+si}{2}\\ \frac{r-si}{2}&n\end{pmatrix}$ simply. Let $R$ be a subring of $\mathbb{C}$, we define $M_{k}(U_{2}({\mathcal{O}});R)$ as an $R$-module of all of $F\in M_{k}(U_{2}({\mathcal{O}}))$ such that $a_{F}(H)\in R$ for any $H\in\Lambda_{2}({\mathcal{O}})$. We put also $S_{k}(U_{2}({\mathcal{O}});R):=M_{k}(U_{2}({\mathcal{O}});R)\cap S_{k}(U_{2}({\mathcal{O}}))$.

We put

[TABLE]

Then for $H=(m,r,s,n)$ we have

[TABLE]

Then any element $F\in M_{k}(U_{2}({\mathcal{O}});R)$ can be regarded as an element of

[TABLE]

This notation is useful to calculate the Fourier expansion of Hermitian modular forms.

We consider the Hermitian Eisenstein series of degree $2$

[TABLE]

where $k>4$ is even and $M=\begin{pmatrix}*&*\\ C&D\end{pmatrix}$ runs over a set of representatives of $\left\{\begin{pmatrix}*&*\\ 0_{2}&*\end{pmatrix}\right\}\backslash U_{2}(\mathcal{O})$. Then we have

[TABLE]

Moreover $E_{4}\in M_{4}(U_{2}(\mathcal{O}))$ is constructed by the Maass lift ([9]). The Fourier coefficient of $E_{k}$ is given by the following formula:

Theorem 2.1** (Krieg [9], Dern [2]).**

The Fourier coefficient $a_{E_{k}}(H)$ of $E_{k}$ is given as follows.

[TABLE]

where

 $B_{m}$ is the $m$-th Bernoulli number,

 $B_{m,\chi_{-4}}$ is the $m$-th generalized Bernoulli number associated with the Kronecker character $\chi_{-4}=\left(\frac{-4}{*}\right)$,

 $\varepsilon(H):={\rm max}\{l\in\mathbb{N}\,|\,l^{-1}H\in\Lambda_{2}(\boldsymbol{K})\,\}$,

and

[TABLE]

We can construct cusp forms by the Hermitian Eisenstein series (cf. [3], Corollary 2);

[TABLE]

2.3 Siegel modular forms of degree $2$

Let $M_{k}(\Gamma_{2})$ denote the space of Siegel modular forms of weight $k$ $(\in\mathbb{Z})$ for the Siegel modular group $\Gamma_{2}:=Sp_{2}(\mathbb{Z})$ and $S_{k}(\Gamma_{2})$ the subspace of cusp forms.

Any Siegel modular form $F$ in $M_{k}(\Gamma_{2})$ has a Fourier expansion of the form

[TABLE]

where

[TABLE]

(the lattice in $Sym_{2}(\mathbb{R})$ of half-integral, symmetric matrices). We write $T=(m,r,n)$ for $T=\begin{pmatrix}m&\frac{r}{2}\\ \frac{r}{2}&n\end{pmatrix}$ and also $a_{F}(m,r,n)$ for $a_{F}\begin{pmatrix}m&\frac{r}{2}\\ \frac{r}{2}&n\end{pmatrix}$.

Taking $q_{ij}:=\text{exp}(2\pi iz_{ij})$ with $Z=(z_{ij})\in\mathbb{H}_{2}$, we have for $T=(m,r,n)$

[TABLE]

For any subring $R\subset\mathbb{C}$, we adopt the notation,

[TABLE]

Any element $F\in M_{k}(\Gamma_{2};R)$ can be regarded as an element of

[TABLE]

The space $\mathbb{H}_{2}$ contains the Siegel upper half-space of degree $2$

[TABLE]

Hence we can define the restriction map

[TABLE]

via the correspondence $F\mapsto F|_{\mathbb{S}_{2}}:=F(z_{ij})|_{z_{21}=z_{12}}$ (this means $\dot{q}_{12}\mapsto 1$, $\ddot{q}_{12}\mapsto q_{12}$). In particular, if $F\in M_{k}(U_{2}({\mathcal{O}});R)\subset R[\![\dot{\boldsymbol{q}}]\!]$ then we have $F|_{\mathbb{S}_{2}}\in M_{k}(\Gamma_{2};R)\subset R[\![\boldsymbol{q}]\!]$. This fact comes from each condition of the modularity.

2.4 Igusa’s generators over $\mathbb{Z}$

Let $k$ be an even integer with $k\geq 4$. The Siegel Eisenstein series

[TABLE]

defies an element of $M_{k}(\Gamma_{2};\mathbb{Q})$. Here $M=\begin{pmatrix}*&*\\ C&D\end{pmatrix}$ runs over a set of representatives $\left\{\begin{pmatrix}*&*\\ 0_{2}&*\end{pmatrix}\right\}\backslash\Gamma_{2}$. We write $X_{4}:=G_{4}$ and $X_{6}:=G_{6}$. We set

[TABLE]

Then we have $X_{k}\in S_{k}(\Gamma_{2};\mathbb{Z})$ $(k=10,12)$ and $a_{X_{10}}(1,1,1)=a_{X_{12}}(1,1,1)=1$.

Let $k$ be an even integer with $k\geq 4$ and $G_{k}$ the normalized Siegel Eisenstein series of weight $k$. We set

[TABLE]

We write

[TABLE]

The following structure theorem is due to Igusa.

Theorem 2.2** (Igusa [4]).**

One has $X_{k}\in M_{k}(\Gamma_{2};\mathbb{Z})$ ($k=4$, $6$, $\cdots$, $48$) and $Y_{12}\in M_{12}(\Gamma_{2};\mathbb{Z})$ and the graded ring $A^{(2)}(\Gamma_{2};\mathbb{Z})$ is generated over $\mathbb{Z}$ by them. Moreover, the set of $14$ generators are minimal.

Remark 2.3**.**

Actually, he determined the structure of the full space $A^{(1)}(\Gamma_{2};\mathbb{Z})$ by using the cusp form of weight $35$. However, since we do not use this result we do not mention its detail.

From his result, we have immediately the following property.

Corollary 2.4**.**

The ring $A^{(4)}(\Gamma_{2};\mathbb{Z})$ is generated over $\mathbb{Z}$ by the following $23$ generators;

[TABLE]

For later use, we introduce the Sturm bounds for Siegel modular forms of degree $2$.

Theorem 2.5** (Choi-Choie-Kikuta [1], Kikuta-Takemori [7]).**

Let $k$ be a positive integer and $p$ an any prime. Let $F\in M_{k}(\Gamma_{2};\mathbb{Z}_{(p)})$. Suppose that $a_{F}(m,r,n)\equiv 0$ mod $p$ for any $m$, $r$, $n$ with

[TABLE]

and $4mn-r^{2}\geq 0$. Then we have $F\equiv 0$ mod $p$.

2.5 Structure over $\mathbb{Z}[1/2,1/3]$

We set $H_{4}:=E_{4}$ and

[TABLE]

We define the graded ring $A^{(m)}(U_{2}({\mathcal{O}});R)$ over $R$ by

[TABLE]

Theorem 2.6** (Kikuta-Nagaoka [5] (cf. Dern-Krieg [3])).**

Then all of $H_{4}$, $E_{6}$, $H_{8}$, $F_{10}$, $H_{12}$ have Fourier coefficients in $\mathbb{Z}$ and they generate the graded ring

[TABLE]

Moreover, these $5$ generators are algebraically independent over $\mathbb{C}$ and we have

[TABLE]

Remark 2.7**.**

The ring $A^{(2)}(U_{2}({\mathcal{O}});R)$ coincides with the ring of the full space $A^{(1)}(U_{2}({\mathcal{O}});R)$ of symmetric Hermitian modular forms, because of $M_{k}(U_{2}({\mathcal{O}}))=\{0\}$ for odd $k$.

Let $p$ be a prime and $\mathbb{Z}_{(p)}$ the localization of $\mathbb{Z}$ at the prime ideal $(p)=p\mathbb{Z}$, namely, $\mathbb{Z}_{(p)}:=\mathbb{Q}\cap\mathbb{Z}_{p}$. The following lemma will be needed in later sections. For a formal Fourier series of the form $F=\sum a_{F}(H)e^{2\pi i{\rm tr}(HZ)}$, we define $v_{p}(F)\in\mathbb{Z}$ as usual by

[TABLE]

Lemma 2.8**.**

For any $F_{i}=\sum a_{F_{i}}(H)e^{2\pi i{\rm tr}(HZ)}$ ($i=1$, $2$) with $v_{p}(F_{i})<\infty$, we have

[TABLE]

Proof.

We can easily prove this property, if we define an order for two elements of $\Lambda_{2}({\mathcal{O}})$ in the same way as in [6]. ∎

We will need the Sturm bounds in the later sections.

Theorem 2.9** (Kikuta-Nagaoka [6]).**

Let $p$ be a prime with $p\geq 5$. Suppose that $F\in M_{k}(U_{2}({\mathcal{O}});\mathbb{Z}_{(p)})$ satisfies that $a_{F}(m,r,s,n)\equiv 0$ mod $p$ for all $m$, $n\leq[k/8]$. Then we have $F\equiv 0$ mod $p$.

Remark 2.10**.**

In [6] Theorem 2, we obtained the similar type bounds as this statement, but they are not same. We can modify the proof in the similar way as in [8] Proposition 4.5.

In general, the Sturm bounds imply the ordinary vanishing conditions.

Corollary 2.11**.**

Suppose that $F\in M_{k}(U_{2}({\mathcal{O}});\mathbb{Q})$ satisfies that $a_{F}(m,r,s,n)=0$ for all $m$, $n\leq[k/8]$. Then we have $F=0$.

Proof.

We may apply Theorem 2.9 to $F$ for infinitely many primes $p\geq 5$. ∎

3 Structure over $\mathbb{Z}$

3.1 Construction of generators

We set

[TABLE]

In order to construct further generators, we use temporarily the alphabets $K$, $L$.

[TABLE]

From these definition and Theorem 2.6, it is easy to see that

[TABLE]

Finally we put

[TABLE]

By the definition of them and from Theorem 2.6, we can easily confirm the following property.

Proposition 3.1**.**

We have

[TABLE]

for each $k_{1}$, $k_{2}$, $k_{3}$ with

[TABLE]

3.2 Integralities of generators

First our purpose is to prove that, all Fourier coefficients of the modular forms constructed in the previous subsection are integers. We start with proving several lemmas.

We put $H_{4}=1+2^{4}\cdot 3S$, $E_{6}=1+2^{3}\cdot 3^{2}T$ with $S$, $T\in\mathbb{Z}[\![\dot{\boldsymbol{q}}]\!]$.

Lemma 3.2**.**

We have $S\equiv T$ mod $2^{2}\cdot 3$.

Proof.

For $H\in\Lambda_{2}({\mathcal{O}})$ with ${\rm rank}(H)=1$, we have

[TABLE]

The assertion (for ${\rm rank}(H)=1$) follows from $5\equiv-7$ mod $2^{2}\cdot 3$ and an application of the Euler congruence

[TABLE]

Let $H\in\Lambda_{2}({\mathcal{O}})$ with ${\rm rank}(H)=2$. Then

[TABLE]

The Euler congruence implies that

[TABLE]

On the other hand, we have

[TABLE]

Therefore the assertion holds. ∎

By this lemma, we can put $T=S+2^{2}\cdot 3U$ with $U\in\mathbb{Z}[\![\dot{\boldsymbol{q}}]\!]$. Then we have

[TABLE]

This is the one of important fact for our arguments on integralities of generators.

Forms of weight $\mathbb{12}$

We remark that $J_{12}=E_{6}^{2}\in M_{12}(U_{2}({\mathcal{O}});\mathbb{Z})$ follows from $E_{6}\in M_{6}(U_{2}({\mathcal{O}});\mathbb{Z})$.

Lemma 3.3**.**

We have $I_{12}\in M_{12}(U_{2}({\mathcal{O}});\mathbb{Z})$.

Proof.

We know by Theorem 2.6 that $H_{12}\in M_{12}(U_{2}({\mathcal{O}});\mathbb{Z})$. Hence, it suffices to prove that $2^{-6}\cdot 3^{-3}(H_{4}^{3}-E_{6}^{2})\in M_{12}(U_{2}({\mathcal{O}});\mathbb{Z})$. This can be confirmed by the expansion

[TABLE]

where $S$, $U$ is defined as above. ∎

Forms of weight $\mathbb{14}$, $\mathbb{16}$, $\mathbb{18}$

For the proof of their integralities, we use (as in [5]) the correspondence between the Maass space and the Kohnen plus subspace which given by Krieg [9]. We review it briefly.

We define the congruence subgroup of $\Gamma_{1}=SL_{2}(\mathbb{Z})$ with level $N$ ($N\in\mathbb{N}$) as

[TABLE]

Let $M_{k}(\Gamma_{0}^{(1)}(4),\chi_{-4}^{k})$ be the space of elliptic modular forms with character $\chi_{-4}^{k}$ for $\Gamma_{0}^{(1)}(4)$. Let ${\mathcal{M}}_{k}(U_{2}({\mathcal{O}}))$ be the Maass space consisting of all of $F\in M_{k}(U_{2}({\mathcal{O}}))$ satisfying the Maass relation. For the precise definition, see [9], p.676.

The Hermitian modular forms version of the Kohnen plus subspace is defined as

[TABLE]

Krieg [9] gave the isomorphism as vector spaces

[TABLE]

Let

[TABLE]

with $\sigma_{1}(n):=\sum_{0<d\mid n}d$ and $q:=e^{2\pi i\tau}$, $\tau\in\mathbb{H}_{1}:=\{\tau=x+iy\;|\;y>0\}$. Then it is known that $\theta^{2}\in M_{1}(\Gamma_{0}^{(1)}(4),\chi_{-4})$ and $f_{2}\in M_{2}(\Gamma_{0}^{(1)}(4),1)$ and they generate the graded ring

[TABLE]

Hence we can construct a Hermitian modular form ${\rm Lift}(h)\in M_{k}(U_{2}({\mathcal{O}}))$ from a polynomial $h\in\mathbb{C}[\theta^{2},f_{2}]$ (such that $h\in M_{k-1}^{+}(\Gamma_{0}^{(1)}(4),\chi_{-4}^{k-1})$), by the relation between their Fourier coefficients

[TABLE]

Lemma 3.4**.**

We have $I_{16}\in M_{16}(U_{2}({\mathcal{O}});\mathbb{Z})$ and $K_{k}\in M_{k}(U_{2}({\mathcal{O}});\mathbb{Z})$ for $k=14$, $18$.

Proof.

We set

[TABLE]

Then we have $h_{15}\in M_{15}(\Gamma_{0}^{(1)}(4),\chi_{-4})$. By an easy numerical experiments, we can confirm that $a_{h_{15}}(n)=0$ for all $n$ with $n\leq 500$ and $n\equiv 1$ mod $4$. In fact, we can prove $h_{15}\in M^{+}_{15}(\Gamma_{0}^{(1)}(4),\chi_{-4})$ as follows. We consider

[TABLE]

where $U(l)$, $V(l)$ is the usual operators and $T_{\chi}$ is the twisting operator of the Dirichlet character $\chi$ given in Shimura [10]. Namely, their action for $f=\sum_{n=0}^{\infty}a_{f}(n)q^{n}$ is described as

[TABLE]

We remark that we have (at least) $f|U(l)\in M_{k}(\Gamma_{0}^{(1)}(Nl),\psi)$, $f|V(l)\in M_{k}(\Gamma_{0}^{(1)}(Nl^{2}),\psi)$ and $f|T_{\chi}\in M_{k}(\Gamma_{0}^{(1)}(Nl^{2}),\psi\chi^{2})$ when $f\in M_{k}(\Gamma^{(1)}_{0}(N),\psi)$.

Since the Sturm bound for $M_{15}(\Gamma_{0}^{(1)}(32),\chi_{-4})$ is

[TABLE]

our numerical experiment for $n\leq 500$ is sufficient. Namely this shows that $\sum_{n\equiv 1\bmod{4}}a_{h_{15}(n)}q^{n}=0$ and hence $h_{15}\in M_{15}^{+}(\Gamma_{0}^{(4)},\chi_{-4})$. Therefore we can apply the isomorphism constructed by Krieg, there exists ${\rm Lift}(h_{15})\in M_{16}(U_{2}({\mathcal{O}}))$ satisfying that

[TABLE]

By the definition of $h_{15}$, we see that $h_{15}\equiv f_{2}^{4}$ mod $2$ because of $\theta\equiv 1$ mod $2$. This implies immediately

[TABLE]

for each $d$. Namely ${\rm Lift}(h_{15})\in M_{16}(U_{2}({\mathcal{O}});\mathbb{Z})$ follows.

By a direct calculation, we see that

[TABLE]

for all $(m,r,s,n)\in\Lambda_{2}({\mathcal{O}})$ with $m$, $n\leq 2=[16/8]$. Applying Corollary 2.11, we obtain

[TABLE]

Since ${\rm Lift}(h_{15})-56H_{8}^{2}\in M_{16}(U_{2}({\mathcal{O}});\mathbb{Z})$, we have the assertion $I_{16}\in M_{16}(U_{2}({\mathcal{O}});\mathbb{Z})$.

Similarly, if we set

[TABLE]

then we can prove the following equalities

[TABLE]

The assertions for $K_{14}$, $K_{18}$ follow from these fact immediately.

We will give numerical data we used in the proofs, in Subsection 4.1 ∎

Lemma 3.5**.**

We have

 (1) $H_{16}\in M_{16}(U_{2}({\mathcal{O}});\mathbb{Z})$,

 (2) $6H_{4}H_{12}-E_{6}H_{10}+H_{4}^{2}H_{8}\equiv 0$ mod $2^{3}\cdot 3^{3}$.

Proof.

(1) By the definition of $I_{16}$, we have

[TABLE]

Since $2^{2}\cdot 3I_{16}\equiv 0$ mod $2^{2}\cdot 3$ because of $I_{16}\in M_{16}(U_{2}({\mathcal{O}});\mathbb{Z})$, we have $H_{16}\in M_{16}(U_{2}{(\mathcal{O}});\mathbb{Z})$.

(2) By the definition of $H_{16}$, we have

[TABLE]

Hence we can write as

[TABLE]

Since $I_{16}\in M_{16}(U_{2}({\mathcal{O}});\mathbb{Z})$, we have $6H_{4}H_{12}-E_{6}F_{10}+H_{4}^{2}H_{8}\equiv 0$ mod $2^{3}\cdot 3^{2}$. Using the fact that $H_{4}\equiv 1$ mod $2^{4}\cdot 3$, $E_{6}\equiv 1$ mod $2^{3}\cdot 3^{2}$, we get

[TABLE]

∎

From (2) in this lemma, we may write as

[TABLE]

with $V\in\mathbb{Z}[\![\dot{\boldsymbol{q}}]\!]$. This description is another important thing for our arguments.

Forms of weight $\boldsymbol{k}$ with $\boldsymbol{k\geq 20}$

First we remark that $I_{24}\in M_{24}(U_{2}({\mathcal{O}});\mathbb{Z})$ is trivial because of $I_{24}=E_{6}K_{18}$ and $E_{6}\in M_{6}(U_{2}({\mathcal{O}});\mathbb{Z})$, $K_{18}\in M_{18}(U_{2}({\mathcal{O}});\mathbb{Z})$. Similarly, the integralities of $I_{36}=K_{18}^{2}$, $J_{36}=E_{6}K_{30}$, $I_{48}=K_{18}K_{30}$, $H_{60}=K_{30}^{2}$, $I_{60}=K_{18}K_{42}$, $H_{72}=K_{32}K_{42}$, $H_{84}=K_{42}^{2}$ follow from that of $E_{6}$, $K_{18}$, $K_{30}$, $K_{32}$, $K_{42}$.

Lemma 3.6**.**

We have the integralities of all the generators constructed in Section LABEL:Subsec:gen.

Proof.

By the definition of $H_{20}$, we can write as

[TABLE]

If we use the descriptions

[TABLE]

then we have

[TABLE]

This shows $H_{20}\in M_{20}(U_{2}({\mathcal{O}});\mathbb{Z})$.

Similarly, we can prove the integralities of all the generators. In fact we can confirm that, all the generators have descriptions as polynomials of $H_{12}$, $H_{8}$, $S$, $U$, $V\in\mathbb{Z}[\![\dot{\boldsymbol{q}}]\!]$ with integral coefficients (see Subsection 4.2). ∎

Now we could prove the integralities of our generators:

Theorem 3.7**.**

All of the modular forms

[TABLE]

and also

[TABLE]

are elements of $\mathbb{Z}[\![\dot{\boldsymbol{q}}]\!]$.

3.3 Proof of the structure theorem

We are now in a position to prove the following main result.

Theorem 3.8**.**

The graded ring $A^{(4)}(U_{2}({\mathcal{O}});\mathbb{Z})$ over $\mathbb{Z}$ is generated by $24$ modular forms

[TABLE]

In other words, for any $F\in M_{k}(U_{2}({\mathcal{O}});\mathbb{Z})$ with $4\mid k$, there exists a polynomial with $24$ variables having coefficients in $\mathbb{Z}$ such that $F=P(H_{4},H_{8},H_{12},\cdots,H_{84})$.

Proof.

We prove it by an induction on the weight.

For $k=4$, the statement is true clearly. Suppose that the statement is true for all $k$ with $k<k_{0}$. Let $F\in M_{k_{0}}(U_{2}({\mathcal{O}});\mathbb{Z})$. Then there exists a polynomial $P$ with $23$ variables having coefficients in $\mathbb{Z}$ such that $F|_{\mathbb{S}_{2}}=P(S_{4},S_{12},T_{12},\cdots,S_{84})$ because of Corollary 2.4. Then we have $F-P(H_{4},H_{12},I_{12},\cdots,H_{84})\in M_{k_{0}}(U_{2}({\mathcal{O}};\mathbb{Z}))$ and $(F-P(H_{4},H_{12},I_{12},\cdots,H_{84}))|_{\mathbb{S}_{2}}=0$. Therefore there exists $F^{\prime}\in M_{k_{0}-8}(U_{2}({\mathcal{O}});\mathbb{Q})$ such that $F-P(H_{4},H_{12},I_{12},\cdots,H_{84})=H_{8}F^{\prime}$. Since all Fourier coefficients of $P(H_{4},H_{12},I_{12},\cdots,H_{84})$ are in $\mathbb{Z}$, we have $H_{8}F^{\prime}\in M_{k}(U_{2}({\mathcal{O}});\mathbb{Z})$. By $v_{p}(H_{8})=0$ for any prime $p$, we have $F^{\prime}\in M_{k_{0}-8}(U_{2}({\mathcal{O}});\mathbb{Z})$ because of Lemma 2.8. By the induction hypothesis, there exists a polynomial $P^{\prime}$ such that $F^{\prime}=P^{\prime}(H_{4},H_{8},H_{12},\cdots,H_{84})$. Therefore we have

[TABLE]

This completes the proof of Theorem 3.8. ∎

Remark 3.9**.**

To determine the structure of $A^{(2)}(U_{2}({\mathcal{O}});\mathbb{Z})$ by our method, we need $K_{46}\in M_{46}(U_{2}({\mathcal{O}});\mathbb{Z})$ such that $K_{46}|_{\mathbb{S}_{2}}=X_{10}X_{36}$. However, we predict that there does not exist such $K_{46}$, due to the leading terms of Fourier expansions. This is a main reason why we restricted our selves to the case where the weights are multiples of $4$. We remark also that we can construct $K_{46}^{\prime}\in M_{46}(U_{2}({\mathcal{O}});\mathbb{Z})$ such that $K_{46}^{\prime}|_{\mathbb{S}_{2}}=3X_{10}X_{36}$.

3.4 An Application

As an application, we have the following Sturm bounds for any $k$ with $4\mid k$.

Theorem 3.10**.**

Let $p$ be an any prime and $k$ an integer with $4\mid k$. Suppose that $F\in M_{k}(U_{2}({\mathcal{O}});\mathbb{Z})$ satisfies that $a_{F}(m,r,s,n)\equiv 0$ mod $p$ for all $m$, $n\in\mathbb{Z}$ with

[TABLE]

Then we have $F\equiv 0$ mod $p$.

For the primes $p\geq 5$, we can prove the statement in the similar way. Hence we prove the essential case $p=2$, $3$ only.

Lemma 3.11**.**

Let $p=2$, $3$ and $k$ be an even integer with $4\mid k$. Suppose that $F\in M_{k}(U_{2}({\mathcal{O}});\mathbb{Z})$ satisfies $F|_{\mathbb{S}_{2}}\equiv 0$ mod $p$, then there exists $F^{\prime}\in M_{k-8}(U_{2}({\mathcal{O}});\mathbb{Z})$ such that $F\equiv H_{8}F^{\prime}$ mod $p$.

Proof.

For $k=4$, $8$, we have as free $\mathbb{Z}$-modules

[TABLE]

If $k\neq 8$ and $F\not\equiv 0$ mod $p$, then $F|_{\mathbb{S}_{2}}\equiv 0$ mod $p$ is impossible. If $k=8$, then $F|_{\mathbb{S}_{2}}\equiv 0$ mod $p$ is possible only if $F\equiv cH_{8}$ mod $p$ for some $c\in\mathbb{Z}$. Therefore the statements for $k=4$, $8$ are true.

We prove the case $k\geq 12$ with $4\mid k$. Since $F|_{\mathbb{S}_{2}}\equiv 0$ mod $p$, we have $\frac{1}{p}F|_{\mathbb{S}_{2}}\in M_{k}(\Gamma_{2};\mathbb{Z})$. By Corollary 2.4, there exists an isobaric polynomial $P$ with coefficients in $\mathbb{Z}$ such that $\frac{1}{p}F|_{\mathbb{S}_{2}}=P(S_{4},S_{12},\cdots,S_{84})$. If we put

[TABLE]

then we have $G\in M_{k}(U_{2}({\mathcal{O}});\mathbb{Z})$ and $(F-pG)|_{\mathbb{S}_{2}}=0$. By the result of Dern-Krieg [3], there exists $F^{\prime}\in M_{k-8}(U_{2}({\mathcal{O}});\mathbb{Q})$ such that $F-pG=H_{8}F^{\prime}$. Since $v_{p}(F-pG)\geq 0$ and $v_{p}(H_{8})=0$ for any $p\geq 2$, it should be that $F^{\prime}\in M_{k-8}(U_{2}({\mathcal{O}});\mathbb{Z})$. Then we have $F\equiv H_{8}F^{\prime}$ mod $p$.

This competes the proof of Lemma 3.11. ∎

We prove Theorem 3.10.

Proof of Theorem 3.10.

For $k=4$, $8$, we have as free $\mathbb{Z}$-modules

[TABLE]

Since $H_{4}\equiv 1$ mod $p$ and $H_{8}\not\equiv c$ mod $p$ for any $c\in\mathbb{Z}$, the statements for $k=4$, $8$ are trivial.

Let $k\geq 12$. From $[k/8]\geq[k/10]$, we can apply the Sturm bound in Theorem 2.5 to $F|_{\mathbb{S}_{2}}$ and then we have $F|_{\mathbb{S}_{2}}\equiv 0$ mod $p$. By Lemma 3.11, there exists $F^{\prime}\in M_{k-8}(U_{2}({\mathcal{O}});\mathbb{Z})$ such that $F\equiv H_{8}F^{\prime}$ mod $p$. Then $F^{\prime}$ has the property that $a_{F^{\prime}}(m,r,s,n)\equiv 0$ mod $p$ for any $m$, $n\in\mathbb{Z}$ with

[TABLE]

This is due to the explicit form of the Fourier expansion of $H_{8}$ (the same reason as in [7] Lemma 5.1);

[TABLE]

Note here that $4\mid k-8$ and we can apply the above argument to $F^{\prime}$.

If we apply this argument repeatedly, we have $F\equiv 0$ mod $p$. This completes the proof of Theorem 3.10. ∎

4 Completion of the proofs by numerical data

4.1 Fourier expansions of $\boldsymbol{h_{13}}$, $\boldsymbol{h_{15}}$, $\boldsymbol{h_{17}}$

In the proof of Lemma 3.4, we relied on the numerical data. Hence we give its data here.

Let

[TABLE]

be the Sturm bounds we mentioned in the proof of Lemma 3.4. Then we have $b_{13}=52$, $b_{15}=60$, $b_{17}=68$. Therefore the following numerical data are sufficient for our purpose.

[TABLE]

[TABLE]

[TABLE]

4.2 Proof of integralities of the generators

In this subsection, we list the descriptions of our generators as polynomials with variables $H_{12}$, $H_{8}$, $S$, $U$, $V\in\mathbb{Z}[\![\dot{\boldsymbol{q}}]\!]$, where $S$, $U$, $V$ are defined by

[TABLE]

The list below shows that the integralities of corresponding generators as in Subsection 3.2. Namely we prove that our generators are elements of the ring $\mathbb{Z}[H_{12},H_{8},S,U,V]$ in the following.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Acknowledgement

The idea of proof of $h_{15}\in M_{15}^{+}(\Gamma_{0}^{(1)}(4),\chi_{-4})$ using the twisting operator is due to Professor S. Böcherer. This makes it possible to prove Lemma 3.4. The author is supported by JSPS KAKENHI Grant Number JP18K03229.