A note on the Sturm bound for Siegel modular forms of type (k, 2)

Abstract

We study analogues of Sturm’s bounds for vector valued Siegel modular forms of type (k, 2), which was already studied by Sturm in the case of an elliptic modular form and by Choi–Choie–Kikuta, Poor–Yuen and Raum–Richter in the case of scalar valued Siegel modular forms.

1 Introduction

In this paper, we consider a question about congruences for the Fourier coefficients of vector valued Siegel modular forms of type (k, 2), which was answered by Sturm [11] in the case of an elliptic modular form and by Choi–Choie–Kikuta [2], Poor–Yuen [8] in the case of a scalar valued Siegel modular form and Raum–Richter [9] in the case of a scalar valued Siegel modular form and of some, but not of type (k, 2), vector valued Siegel modular forms.

Let p be a prime number and ${\mathbb {Z}}_{(p)}$ be the local ring of the p-integral rational numbers. Suppose that ${f=\sum\nolimits _{n=0}^{\infty }a(n;f)q^n}$ is an elliptic modular form of weight k with integral coefficients. In [11] Sturm proved that if $a(n;f)\equiv 0\pmod {p}$ for $0\le n\le \frac{k}{12}$, then $a(n;f)\equiv 0\pmod {p}$ for every $n\ge 0$. This bound is called a Sturm bound. In this paper we study the Sturm bound of the vector-valued Siegel modular forms of type (k, 2) and degree 2 such that all Fourier coefficients lie in ${\mathrm {Sym}}_2^{*}({\mathbb {Z}}_{(p)}):=\{T=(t_{ij})\in {\mathrm {Sym}}_2({\mathbb {Q}})\mid t_{ii},2t_{ij}\in {\mathbb {Z}}_{(p)}\}$. Here p is a prime with $p\ge 5$ and k is an even integer.

We state our result more precisely. A Siegel modular form of type (k, 2) is a holomorphic function f on the Siegel upper-half plane ${\mathbb {H}}_2$ with values in ${\mathrm {Sym}}_2({\mathbb {C}})$, satisfying

$$\begin{aligned} f(M\langle Z\rangle )={\mathrm {det}}(CZ+D)^k(CZ+D)f(Z){}^t\!(CZ+D) \end{aligned}$$

for all $M=\left( {\begin{array}{c}A\;B\\ C\;D\end{array}}\right)$ in the Siegel modular group $\varGamma _2=Sp_2({\mathbb {Z}})$ and for all $Z\in {\mathbb {H}}_2$. Here (k, 2) comes from the fact that the automorphy factor is the one of representatives in the equivalence class of the representation ${\mathrm {det}}^k\otimes{ \mathrm {Sym}}(2)$. We denote by $M_{k,2}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}$ the module consisting of all such f whose Fourier coefficients are in ${\mathrm {Sym}}_2^{*}({\mathbb {Z}}_{(p)})$. The following theorem is our main result.

Theorem 1

For each even integer k and each prime $p\ge 5$, suppose that F is a Siegel modular form in $M_{k, 2}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}$ having the form

$$\begin{aligned} F(\tau , \tau ^\prime , \omega )=\sum \limits_{\begin{subarray}{c} m,n\in {\mathbb {Z}}, \ r\in \frac{1}{2}{\mathbb {Z}} \\ m,n,mn-r^2\ge 0 \end{subarray}}A(m,n,r)q_\tau ^mq_{\tau ^\prime }^nq_{\omega }^{2r} \end{aligned}$$

with $q_\tau =e^{2\pi i\tau }, q_{\tau ^\prime }=e^{2\pi i\tau ^\prime }, q_{\omega }=e^{2\pi i\omega }$ and $\left( {\begin{array}{c}\tau \ \omega \\ \omega \ \tau ^\prime \end{array}}\right)$ in the Siegel upper half space ${\mathbb {H}}_2$ of degree 2, where ${\mathbb {H}}_2:=\left\{ \left( {\begin{array}{c}\tau \ \omega \\ \omega \ \tau ^\prime \end{array}}\right) \in M_2({\mathbb {C}})\mid Im \left( {\begin{array}{c}\tau \ \omega \\ \omega \ \tau ^\prime \end{array}}\right) >0\right\}$. If $A(m, n, r)\equiv 0\pmod {p}$, i.e. all the elements of A(m, n, r) are congruent to 0 mod p, for every m, n such that

$$\begin{aligned} 0\le m\le \left[ \frac{k}{10}\right] ,\ \ \ \ 0\le n\le \left[ \frac{k}{10}\right] , \end{aligned}$$

then $F\equiv 0\pmod {p}$.

The proof of the theorem is due to an inductive argument on the determinant weight k and our main tool is the Witt operator and Theorem 2 (Sect. 2.5).

2 Preliminary

2.1 Siegel modular forms of type (k, 2) and degree 2

The Siegel upper-half space of degree 2 is defined as

$$\begin{aligned} {\mathbb {H}}_2:= \{Z=X+iY\in {\mathrm {Sym}}_2({\mathbb {C}})\;|\; Y>0\;(\text {positive definite})\;\}. \end{aligned}$$

The real symplectic group ${\mathrm {Sp}}_2({\mathbb {R}})$ acts on ${\mathbb {H}}_{2}$ in the following way:

$$\begin{aligned}&Z\longrightarrow M\langle Z\rangle :=(AZ+B)(CZ+D)^{-1},\\&Z\in {\mathbb {H}}_2,\ M=\begin{pmatrix}A&{}B\\ C&{}D\end{pmatrix} \in {\mathrm {Sp}}_2({\mathbb {R}}). \end{aligned}$$

A Siegel modular form of type (k, 2) on $\varGamma _2$ with character $\nu$ is a holomorphic function f on ${\mathbb {H}}_2$ with values in ${\mathrm {Sym}}_2({\mathbb {C}})$, satisfying

$$\begin{aligned} f(M\langle Z\rangle )=\nu (M){\mathrm {det}}(CZ+D)^k(CZ+D)f(Z){}^t\!(CZ+D) \end{aligned}$$

for all $M=\left( {\begin{array}{c}A\;B\\ C\;D\end{array}}\right) \in \varGamma _2$ and for all $Z\in {\mathbb {H}}_2$. We denote by $M_{k,2}(\varGamma _2, \nu )$ (resp. $S_{k,2}(\varGamma _2,\nu )$) the ${\mathbb {C}}$-vector space of Siegel modular forms (resp. cusp forms) of type (k, 2) on $\varGamma _2$ with character $\nu$.

2.2 Fourier expansions

Any $F(Z)\in M_{k,2}(\varGamma _2, \nu )$ has a Fourier expansion of the form

$$\begin{aligned} F(Z)=\sum \limits_{0\le T\in \frac{1}{2}\varLambda _2}a(T;F)\exp (2\pi i{\mathrm {tr}}(TZ)),\quad a(T;F)\in {\mathrm {Sym}} _2({\mathbb {C}}), \end{aligned}$$

where T runs over all positive semi-definite elements of $\frac{1}{2}\varLambda _2$ defined as

$$\begin{aligned} \varLambda _2 :=\{T=(t_{ij})\in {\mathrm {Sym}}_2({\mathbb {Q}})\mid t_{ii},2t_{ij}\in {\mathbb {Z}}\}. \end{aligned}$$

Taking $q_{\tau }:={\mathrm {exp}}(2\pi i\tau )$, $q_{\omega }:={\mathrm {exp}}(2\pi i\omega )$ and $q_{\tau ^\prime }:={\mathrm {exp}}(2\pi i\tau ^{\prime })$ for $Z=\left( {\begin{array}{c}\tau \omega \\ \omega \tau ^{\prime }\end{array}}\right) \in {\mathbb {H}}_2$, we can write

$$\begin{aligned} q^T:={\mathrm {exp}}(2\pi i{\mathrm {tr}}(TZ))=q_{\omega }^{2t_{12}}q_{\tau }^{t_{11}}q_{\tau ^{\prime }}^{t_{22}}. \end{aligned}$$

Using this notation, we have the generalized q-expansion:

$$\begin{aligned} F&=\sum \limits_{0\le T\in \frac{1}{2}\varLambda _2}a(T;F)q^T\\&=\sum \limits_{0\le (t_{ij})\in \frac{1}{2}\varLambda _2}(a(T;F)q_{\omega }^{2t_{12}})q_{\tau }^{t_{11}}q_{\tau ^{\prime }}^{t_{22}}\in {\mathrm {Sym}}_2({\mathbb {C}})[q_{\omega }^{-\frac{1}{2}},q_{\omega }^{\frac{1}{2}}] [\![q_{\tau }^{\frac{1}{2}},q_{\tau ^{\prime }}^{\frac{1}{2}}]\!]. \end{aligned}$$

For any subring R of ${\mathbb {C}}$, we denote by $M_{k,2}(\varGamma _2, \nu )_R$ the R-module consisting of those F in $M_{k,2}(\varGamma _2,\nu )$ for which a(T; F) is in ${\mathrm {Sym}}_2^{*}(R)$ for every $T\in \frac{1}{2}\varLambda _2$ where

$$\begin{aligned} {\mathrm {Sym}}_2^{*}(R):=\{T=(t_{ij})\in {\mathrm {Sym}}_2({\mathbb {C}})\mid t_{ii},2t_{ij}\in R\}. \end{aligned}$$

2.3 Generators of scalar valued Siegel modular forms

Let $\varphi _4$, $\varphi _6$, $X_{10}$, $X_{12}$ be Igusa’s generators over ${\mathbb {Z}}$ of weight 4, 6, 10, 12, respectively given in [5]. Let $M_k(\varGamma _2,\nu )$ (resp. $S_k(\varGamma _2,\nu )$) be the ${\mathbb {C}}$-vector space consisting of the scalar valued Siegel modular forms (resp. cusp forms) of weight k on $\varGamma _2$ with the character $\nu$. We denote by $M_{k}(\varGamma _2, \nu )_{{\mathbb {Z}}_{(p)}}$ (resp. $S_k(\varGamma _2,\nu )_{{\mathbb {Z}}_{(p)}}$) the ${\mathbb {Z}}_{(p)}$-module consisting of the scalar valued Siegel modular forms in $M_{k}(\varGamma _2,\nu )$ (resp. cusp forms in $S_{k}(\varGamma _2,\nu )$) for which a(T; F) is in ${\mathbb {Z}}_{(p)}$ for every $T\in \frac{1}{2}\varLambda _2$. In the case of scalar valued Siegel modular forms, we have

$$\begin{aligned} M_*^{\mathrm {ev}}({\mathbb {Z}}_{(p)})&:=\bigoplus _{k\in 2{{\mathbb {Z}}}}M_k(\varGamma _2)_{{{\mathbb {Z}}}_{(p)}}\\&={{\mathbb {Z}}}_{(p)}[\varphi _4,\varphi _6,X_{10},X_{12}] ,\ {\mathrm{if}}\ p\ge 5. \end{aligned}$$

Let ${\mathcal {C}}{\mathrm {Sp}}_2({\mathbb {Z}})$ be the commutator subgroup of $\varGamma _2$. Let $\chi :\varGamma _2 \rightarrow \{\pm 1\}$ be a non-trivial abelian character, which is basically a character of ${\mathrm {Sp}}_2({\mathbb {Z}}/2{\mathbb {Z}})\cong \varSigma _6$, the symmetric group of 6 letters. Any Siegel modular form of weight k on ${\mathcal {C}}{\mathrm {Sp}}_2({\mathbb {Z}})$ also has a Fourier expansion of the form

$$\begin{aligned} \sum \limits_{0\le T\in \frac{1}{2}\varLambda _2}b(T;F)\exp (2\pi i{\mathrm {tr}}(TZ)),\ \ b(T;F)\in {\mathbb {C}}. \end{aligned}$$

In this case Igusa [4] showed that

$$\begin{aligned} M_{*}({\mathcal {C}}{\mathrm {Sp}}_2({\mathbb {Z}}))&:=\bigoplus _{k}M_k({\mathcal {C}}{\mathrm {Sp}}_2({\mathbb {Z}})) \\&= {\mathbb {C}}[\varphi _4,\varDelta _5,\varphi _6,X_{12},\varDelta _{30}], \end{aligned}$$

where $\varDelta _k\in M_{k}({\mathcal {C}}{\mathrm {Sp}}_2({\mathbb {Z}}))=M_k(\varGamma _2, \chi )$ is constructed by theta series with Fourier coefficients

$$\begin{aligned} b\left( \begin{pmatrix}\frac{1}{2}&{}\frac{1}{4}\\ \frac{1}{4}&{}\frac{1}{2}\end{pmatrix};\varDelta _5\right) =b\left( \begin{pmatrix}\frac{3}{2}&{}\frac{1}{4}\\ \frac{1}{4}&{}\frac{5}{2}\end{pmatrix};\varDelta _{30}\right) =1. \end{aligned}$$

The modular forms $\varphi _4, \varDelta _5, \varphi _6, X_{12}$ are algebraically independent and $\varDelta _5 ^2=X_{10}$. We remark that there exists a unique relation among the generators;

$$\begin{aligned} \varDelta _{30}^2\in {\mathbb {C}}[\varphi _4, \varDelta _5, \varphi _6, X_{12}]. \end{aligned}$$

2.4 p-order of modular forms

We shall define the p-order of modular forms. Let p be a prime with $p\ge 5$ and $\nu _p$ the additive valuation on ${\mathbb {Q}}$ normalized as $\nu _p(p)=1$.

Let F be a formal power series with bounded denominators of the form

$$\begin{aligned} F=\sum \limits_{T\in \frac{1}{N}\varLambda _2}a(T;F)q^T,\quad a(T;F)\in {\mathrm {Sym}}_2({\mathbb {Q}}). \end{aligned}$$

for some positive integer N.

In the scalar valued case, let $\nu _p$ be just as in Böcherer–Nagaoka [1] and elsewhere. Define a value $\nu _p$ for F with $a(T;F)\in {\mathrm {Sym}}_2({\mathbb {Q}})$ as

$$\begin{aligned} \nu _p (F):=\inf \left\{ \nu _p(a(T;F)) \,|\, T\in {\mathrm {Sym}}_2({\mathbb {Q}}) \right\} , \end{aligned}$$

where ${\nu _p(a(T;F)):={\mathrm {min}}_{1\le i, j\le 2}(\nu _p(a_{ij}(T; F)))}$ for $a(T; F)=\left( {\begin{array}{c}a_{11}(T; F) \ a_{12}(T; F) \\ a_{12}(T; F) \ a_{22}(T; F) \end{array}}\right)$.

The following statement and its proof are due to Kikuta:

Lemma 1

(1) Let ${ f=\sum \nolimits_{T\in \frac{1}{N}\varLambda _2} a(T;f)q^T}$ and ${ g=\sum \nolimits_{T\in \frac{1}{N}\varLambda _2} a(T;g)q^T}$ with a(T; f), $a(T;g)\in {\mathbb {Q}}$ be formal power series with bounded denominators. Then we have $\nu _p(fg)=\nu _p (f)+\nu _p (g)$.

(2) Let ${F=\sum \nolimits_{T\in \frac{1}{N}\varLambda _2}a(T;F)q^T}$ with $a(T;F)\in {\mathrm {Sym}}_2({\mathbb {Q}})$ be a formal power series with bounded denominators and g be as in (1). Then we have $\nu _p(Fg)=\nu _p (F)+\nu _p (g)$.

The proof of Lemma 1 is, for example, in [6].

We remark that, for a formal power series of the form

$$\begin{aligned} F=\sum \limits_{T\in \frac{1}{N}\varLambda _2}a(T;F)q^T,\quad a(T;F)\in {\mathrm {Sym}}_2({\mathbb {Q}}), \end{aligned}$$

we have $a(T;F)\in {\mathrm {Sym}}_2^{*}({\mathbb {Z}}_{(p)})$ for all $T\in \frac{1}{N}\varLambda _2$ if and only if $\nu _p(F)\ge 0$.

2.5 Generators of vector valued Siegel modular forms over ${\mathbb {Z}}_{(p)}$

Let R be a subring of ${{\mathbb {C}}}$ and N be 1 or 2. For a formal power series f of the form

$$\begin{aligned} f=\sum \limits_{T\in \frac{1}{N}\varLambda _2}a(T;f)q^{T}\in R[q_{\omega }^{-\frac{1}{N}},q_{\omega }^{\frac{1}{N}}][\![q_{\tau }^{\frac{1}{N}},q_{\tau ^{\prime }}^{\frac{1}{N}}]\!], \end{aligned}$$

the theta operator $\varTheta ^{[1]}$ is defined by

$$\begin{aligned} \varTheta ^{[1]}(f)=\sum \limits_{T\in \frac{1}{N}\varLambda _2}T\cdot a(T;f)q^{T} \in {\mathrm {Sym}}_2^{*}(R)[q_{\omega }^{-\frac{1}{N}},q_{\omega }^{\frac{1}{N}}][\![q_{\tau }^{\frac{1}{N}},q_{\tau ^{\prime }}^{\frac{1}{N}}]\!] . \end{aligned}$$

Let $\varGamma$ be either $\varGamma _2$ or ${\mathcal {C}}Sp_2({\mathbb {Z}})$ and $f\in M_k(\varGamma )$ and $g\in M_j(\varGamma )$. We put

$$\begin{aligned}{}[f, g]:= \frac{1}{j}f\varTheta ^{[1]}(g)-\frac{1}{k}g\varTheta ^{[1]}(f). \end{aligned}$$

Then the results of Satoh [10] states that $[f, g]\in M_{k+j, 2}(\varGamma )$.

Let $\varphi _4$, $\varphi _6$, $X_{10}$, $X_{12}$ be Igusa’s generators over ${\mathbb {Z}}$ of weight 4, 6, 10, 12, respectively given in [5]. It is known that the $M_{*}^{\text {ev}}(\varGamma _2)$-module of Siegel modular forms of type (k, 2) over ${\mathbb {Z}}_{(p)}$ has six generators:

Theorem 2

([6])

For each even integer k and each prime $p\ge 5$, ${\bigoplus _{k\in 2{\mathbb {Z}}} M_{k,2}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}}$ is a $M_{*}^{\text {ev}}({\mathbb {Z}}_{(p)})$-module generated by 6 elements whose weights are 10, 14, 16, 16, 18, 22. If we write them as $\varPhi _k\in M_{k,2}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}(k=10,14,16,18,22)$ and $\varPsi _{16}\in M_{16,2}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}$, then we have (as a ${\mathbb {Z}}_{(p)}$-module)

$$\begin{aligned} M_{k, 2}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}&=M_{k-10}(\varGamma _2)_{{\mathbb {Z}}_{(p)}} \varPhi _{10} \oplus M_{k-14}(\varGamma _2)_{{\mathbb {Z}}_{(p)}} \varPhi _{14}\\&\oplus M_{k-16}(\varGamma _2)_{{\mathbb {Z}}_{(p)}} \varPhi _{16}\oplus V_{k-16}(\varGamma _2)_{{\mathbb {Z}}_{(p)}} \varPsi _{16}\\&\oplus V_{k-18}(\varGamma _2)_{{\mathbb {Z}}_{(p)}} \varPhi _{18}\oplus W_{k-22}(\varGamma _2)_{{\mathbb {Z}}_{(p)}} \varPhi _{22},\end{aligned}$$

where

$$\begin{aligned} V_{k}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}=M_{k}(\varGamma _2)_{\mathbb Z_{(p)}}\cap {\mathbb {Z}}_{(p)}[\varphi _6, X_{10}, X_{12}], \ \ \ \ W_{k}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}=M_{k}(\varGamma _2)_{\mathbb Z_{(p)}}\cap {\mathbb {Z}}_{(p)}[X_{10}, X_{12}]. \end{aligned}$$

We construct $\varPhi _k$ ($k=10$, 14, 16, 18, 22) and $\varPsi _{16}$ by taking constant multiples of these generators:

$$\begin{aligned}&\varPhi _{10}=-\frac{1}{144}[\varphi _4, \varphi _6],&\varPhi _{14} =10[\varphi _4, X_{10}],&\varPhi _{16}=12[\varphi _4, X_{12}],&\\&\varPsi _{16} =10[\varphi _6, X_{10}],&\varPhi _{18}=12[\varphi _6, X_{12}],&\varPhi _{22}=-120[X_{10}, X_{12}].&\end{aligned}$$

Then we have

$$\begin{aligned}&a\left( \left( {\begin{array}{c}1\;0\\ 0\;0\end{array}}\right) ; \varPhi _{10}\right) =\left( {\begin{array}{c}1\;0\\ 0\;0\end{array}}\right) ,&a\left( \left( {\begin{array}{c}1\;\frac{1}{2}\\ \frac{1}{2}\;1\end{array}}\right) ;\varPhi _{14}\right) =\left( {\begin{array}{c}1\;\frac{1}{2}\\ \frac{1}{2}\;1\end{array}}\right) ,&a\left( \left( {\begin{array}{c}1\;\frac{1}{2}\\ \frac{1}{2}\;1\end{array}}\right) ;\varPhi _{16}\right) =\left( {\begin{array}{c}1\;\frac{1}{2}\\ \frac{1}{2}\;1\end{array}}\right) ,&\\&a\left( \left( {\begin{array}{c}1\;\frac{1}{2}\\ \frac{1}{2}\;1\end{array}}\right) ;\varPsi _{16}\right) =\left( {\begin{array}{c}1\;\frac{1}{2}\\ \frac{1}{2}\;1\end{array}}\right) ,&a\left( \left( {\begin{array}{c}1\;\frac{1}{2}\\ \frac{1}{2}\;1\end{array}}\right) ;\varPhi _{18}\right) =\left( {\begin{array}{c}1\;\frac{1}{2}\\ \frac{1}{2}\;1\end{array}}\right) ,&a\left( \left( {\begin{array}{c}2\;1\\ 1\;2\end{array}}\right) ;\varPhi _{22}\right) =\left( {\begin{array}{c}2\;1\\ 1\;2\end{array}}\right) . \end{aligned}$$

Moreover, we put also $\varPhi _9:=10[\varphi _4, \varDelta _{5}]$, $\varPhi _{11}:=10[\varphi _6, \varDelta _{5}]$, $\varPhi _{17}:=-120[\varDelta _{5}, X_{12}]$. We will use them in the proof of our main theorem.

Proposition 1

([6]) Let p be a prime with $p\ge 5$. Then we have $\nu _p(\varPsi _{16})\ge 0$ and $\nu _p (\varPhi _k)\ge 0$ for $k=9$, 10, 11, 14, 16, 17, 18, 22.

2.6 The Witt operator

Let F be a holomorphic function on ${\mathbb {H}}_2$. Then the Witt operator is defined by

$$\begin{aligned} W(F)(\tau , \tau ^\prime ):=F\left( {\begin{array}{c}\tau \;0\\ 0\;\tau ^\prime \end{array}}\right) , \ \ (\tau , \tau ^\prime )\in {\mathbb {H}}_1\times {\mathbb {H}}_1. \end{aligned}$$

This operator was first introduced in Witt [12]. We extend the Witt operator to the case of vector valued forms. Let $G=\left( {\begin{array}{c}G_{11}\;G_{12}\\ G_{12}\;G_{22}\end{array}}\right) \in M_{k,2}(\varGamma _2, \nu )$ be a vector valued Siegel modular form of type (k, 2) on $\varGamma _2$ with character $\nu$, then we define

$$\begin{aligned} W(G)(\tau , \tau ^\prime ):=\begin{pmatrix}W(G_{11})&{}W(G_{12})\\ W(G_{12})&{}W(G_{22})\end{pmatrix}, \ \ (\tau , \tau ^\prime )\in {\mathbb {H}}_1\times {\mathbb {H}}_1. \end{aligned}$$

For later use, we introduce some examples:

$$\begin{aligned}&W(\varphi _4)(\tau , \tau ^\prime )=E_4(\tau )E_4(\tau ^\prime ), \ \ \ \ \ \ \ \ W(\varphi _6)(\tau , \tau ^\prime )=E_6(\tau )E_6(\tau ^\prime ),\\&W(X_{10})(\tau , \tau ^\prime )\equiv 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ W(X_{12})(\tau , \tau ^\prime )=12\varDelta (\tau )\varDelta (\tau ^\prime ),&\\&W(\varDelta _5)(\tau , \tau ^\prime )\equiv 0,\\&W(\varPhi _{10})=\begin{pmatrix}\varDelta (\tau )E_4(\tau ^\prime )E_6(\tau ^\prime )&{}0\\ 0&{}E_4(\tau )E_6(\tau )\varDelta (\tau ^\prime )\end{pmatrix},&\\&W(\varPhi _{16})=12\begin{pmatrix}E_6(\tau )\varDelta (\tau )E_4(\tau ^\prime )\varDelta (\tau ^\prime )&{}0\\ 0&{}E_4(\tau )\varDelta (\tau )E_6(\tau ^\prime )\varDelta (\tau ^\prime )\end{pmatrix},&\\&W(\varPhi _{18})=12\begin{pmatrix}E_4(\tau )^2\varDelta (\tau )E_6(\tau ^\prime )\varDelta (\tau ^\prime )&{}0\\ 0&{}E_6(\tau )\varDelta (\tau )E_4(\tau ^\prime )^2\varDelta (\tau ^\prime )\end{pmatrix},&\\&W(\varPhi _{14})=W(\varPsi _{16})=W(\varPhi _{22})\equiv 0,\\&W(\varPhi _9)=2E_4(\tau )E_4(\tau ^\prime )\eta (\tau )^{12}\eta (\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) ,\\&W(\varPhi _{11})=2E_6(\tau )E_6(\tau ^\prime )\eta (\tau )^{12}\eta (\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) ,\\&W(\varPhi _{17})=288\varDelta (\tau )\varDelta (\tau ^\prime )\eta (\tau )^{12}\eta (\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) , \end{aligned}$$

where $\eta$ is the usual Dedekind eta function defined as $\eta (\tau )=q_{\tau }^{\frac{1}{24}}\prod _{m=1}^\infty (1-q_{\tau }^m)$.

Theorem 3

(Freitag [3]) If $F\in M_k(\varGamma _2)$ satisfies $W(F)\equiv 0$, then $\frac{F}{X_{10}}\in M_{k-10}(\varGamma _2)$, namely, F is divisible by $X_{10}$.

Nagaoka’s reasoning on page 416 of [7] proves the following lemma.

Lemma 2

(Nagaoka [7]) Assume that $p\ge 5$. Let $F\in {\mathbb {Q}}[\![q_{\tau },q_{\tau ^{\prime }}]\!]$ be a formal power series of the form

$$\begin{aligned} F=\sum \limits_{a,b,c\ge 0}\gamma _{abc}W(\varphi _4)^aW(\varphi _6)^bW(X_{12})^c,\quad \gamma _{abc}\in {\mathbb {Q}}. \end{aligned}$$

If $\nu _p(F)\ge 0$, then the $\gamma _{abc}$ satisfy $\nu _p(\gamma _{abc})\ge 0$ for all $a,b,c\ge 0$.

From this lemma, we get the following corollary.

Corollary 1

Assume that $p\ge 5$. Let $F\in {\mathbb {Q}}[\![q_{\tau },q_{\tau ^{\prime }}]\!]$ be a formal power series of the form

$$\begin{aligned} F=\sum \limits_{a,b,c\ge 0}\gamma _{abc}W(\varphi _4)^aW(\varphi _6)^bW(X_{12})^c,\quad \gamma _{abc}\in {\mathbb {Q}}. \end{aligned}$$

If $\nu _p(F)\ge 1$, then the $\gamma _{abc}$ satisfy $\nu _p(\gamma _{abc})\ge 1$ for all $a,b,c\ge 0$.

Proof

Since $\nu _p(F)\ge 1$, we get $\nu _p(\frac{1}{p}F)\ge 0$. Hence from Lemma 2, we can take

$$\begin{aligned} \frac{1}{p}F=\sum \limits_{a,b,c\ge 0}\frac{1}{p}\gamma _{abc}W(\varphi _4)^aW(\varphi _6)^bW(X_{12})^c,\quad \nu _p(\frac{1}{p}\gamma _{abc})\ge 0. \end{aligned}$$

Hence we can take $\nu _p(\gamma _{abc})\ge 1$ for all $a,b,c\ge 0$. $\square$

3 Proof of the main theorem (Theorem 1)

We prove it by an inductive argument on the weight. By Theorem 2 (Sect. 2.5), for any $F\in M_{k,2}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}$, we can write F in the form

$$\begin{aligned} F&=(P_1+X_{10}Q_1)\varPhi _{10}+(P_2+X_{10}Q_2)\varPhi _{14}+(P_3+X_{10}Q_3)\varPhi _{16}\\& \quad +(P_4+X_{10}Q_4)\varPsi _{16}+(P_5+X_{10}Q_5)\varPhi _{18}+(P_6+X_{10}Q_6)\varPhi _{22}, \end{aligned}$$

where $P_1\in M_{k-10}(\varGamma _2)\cap {\mathbb {Z}}_{(p)}[\varphi _4, \varphi _6, X_{12}]$, $Q_1 \in M_{k-20}(\varGamma _2)_{\mathbb Z_{(p)}}$, $P_2\in M_{k-14}(\varGamma _2)\cap {\mathbb {Z}}_{(p)}[\varphi _4, \varphi _6, X_{12}]$, $Q_2\in M_{k-24}(\varGamma _2)_{\mathbb Z_{(p)}}$, $P_3\in M_{k-16}(\varGamma _2)\cap {\mathbb {Z}}_{(p)}[\varphi _4, \varphi _6, X_{12}]$, $Q_3\in M_{k-26}(\varGamma _2)_{\mathbb Z_{(p)}}$, $P_4\in V_{k-16}(\varGamma _2)\cap {\mathbb {Z}}_{(p)}[\varphi _6, X_{12}]$, $Q_4\in V_{k-26}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}$, $P_5\in V_{k-18}(\varGamma _2)\cap \ {\mathbb {Z}}_{(p)}[\varphi _6, X_{12}]$, $Q_5\in V_{k-28}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}$, $P_6\in W_{k-22}(\varGamma _2)\cap {\mathbb {Z}}_{(p)}[X_{12}]$, $Q_6\in W_{k-32}(\varGamma _2)_{{\mathbb {Z}}_{(p)}}$.

Here we regard $P_i$ as polynomials (with coefficients in $\mathbb Z_{(p)}$) $P_1=P_1(\varphi _4, \varphi _6, X_{12})$, $P_2=P_2(\varphi _4, \varphi _6, X_{12})$, $P_3=P_3(\varphi _4, \varphi _6, X_{12})$, $P_4=P_4(\varphi _6, X_{12})$, $P_5=P_5(\varphi _6, X_{12})$, $P_6=P_6(X_{12})$.

We apply the Witt operator to F. Since $W(X_{10})=W(\varPhi _{14})=W(\varPsi _{16})=W(\varPhi _{22})=0$, we get

$$\begin{aligned}W(F) &=W(P_1)W(\varPhi _{10})+W(P_3)W(\varPhi _{16})+W(P_5)W(\varPhi _{18})\\&=\begin{pmatrix}ME_1&{}0\\ 0&{}ME_2\end{pmatrix}\\&:=\begin{pmatrix} {\sum \limits_{m,n\ge 0}B_{11}(m, n)q_{\tau }^mq_{\tau ^\prime }^n}&{}0\\ 0&{} {\sum \limits_{m,n\ge 0}B_{22}(m, n)q_{\tau }^mq_{\tau ^\prime }^n}\end{pmatrix} ,\end{aligned}$$

where

$$\begin{aligned}&ME_1= {\sum \limits_{\begin{subarray}{c} 12i+4j+6t=k+2\\ 12i^\prime +4j^\prime +6t^\prime =k\\ t, t^\prime =0,1 \end{subarray}}C_1(i, i^\prime )\varDelta (\tau )^iE_4(\tau )^jE_6(\tau )^t\varDelta (\tau ^\prime )^{i^\prime }E_4(\tau ^\prime )^{j^\prime }E_6(\tau ^\prime )^{t^\prime }}\\&ME_2= {\sum \limits_{\begin{subarray}{c} 12i+4j+6t=k\\ 12i^\prime +4j^\prime +6t^\prime =k+2\\ t, t^\prime =0,1 \end{subarray}}C_2(i, i^\prime )\varDelta (\tau )^iE_4(\tau )^jE_6(\tau )^t\varDelta (\tau ^\prime )^{i^\prime }E_4(\tau ^\prime )^{j^\prime }E_6(\tau ^\prime )^{t^\prime }}. \end{aligned}$$

The $q_{\tau }$-expansion of $\varDelta (\tau )^iE_4(\tau )^jE_6(\tau )^t$ has the form

$$\begin{aligned} \varDelta (\tau )^iE_4(\tau )^jE_6(\tau )^t=q_{\tau }^i+\cdots . \end{aligned}$$

The numbers j and t, where t is 0 or 1, are uniquely determined by choosing a value of i.

For each m, n such that $0 \le m, n \le \left[ \frac{k}{10}\right]$, $A(m,n,r)\equiv 0\pmod {p}$. We have that if $m\le \left[ \frac{k}{10}\right]$ and $n\le \left[ \frac{k}{10}\right]$, then $B_{11}(m,n)\equiv B_{22}(m,n)\equiv 0\pmod {p}$. This implies that $C_1(i,i^\prime )\equiv C_2(i,i^\prime )\equiv 0\pmod {p}$ for $i, i^\prime \le \left[ \frac{k}{10}\right]$. Note that $i, i^\prime \le \left[ \frac{k}{10}\right]$ since $12i+4j+6t=k$ or $k+2$ and $12i^\prime +4j^\prime +6t^\prime =k$ or $k+2$ and $\left[ \frac{k}{12}\right] \le \left[ \frac{k+2}{12}\right] \le \left[ \frac{k}{10}\right]$. Thus we have $W(F) \equiv 0\pmod {p}$.

Lemma 3

$P_1, P_3, P_5\equiv 0\pmod {p}$.

Proof of Lemma 3

Using fact that $W(\varDelta _5)=0$, we get

$$\begin{aligned}W(F)&=W(P_1)W(\varPhi _{10})+W(P_3)W(\varPhi _{16})+W(P_5)W(\varPhi _{18})\\&=W(P_1)\begin{pmatrix}\varDelta (\tau )E_4(\tau ^\prime )E_6(\tau ^\prime )&{}0\\ 0&{}E_4(\tau )E_6(\tau )\varDelta (\tau ^\prime )\end{pmatrix}\\&\quad +12W(P_3)\begin{pmatrix}E_6(\tau )\varDelta (\tau )E_4(\tau ^\prime )\varDelta (\tau ^\prime )&{}0\\ 0&{}E_4(\tau )\varDelta (\tau )E_6(\tau ^\prime )\varDelta (\tau ^\prime )\end{pmatrix}\\&\quad +12W(P_5)\begin{pmatrix}E_4(\tau )^2\varDelta (\tau )E_6(\tau ^\prime )\varDelta (\tau ^\prime )&{}0\\ 0&{}E_6(\tau )\varDelta (\tau )E_4(\tau ^\prime )^2\varDelta (\tau ^\prime )\end{pmatrix}\\&=\{ W(P_1)\begin{pmatrix}E_4(\tau ^\prime )E_6(\tau ^\prime )&{}0\\ 0&{}E_4(\tau )E_6(\tau )\end{pmatrix}\\&\quad +12W(P_3)\begin{pmatrix}E_6(\tau )E_4(\tau ^\prime )\varDelta (\tau ^\prime )&{}0\\ 0&{}E_4(\tau )\varDelta (\tau )E_6(\tau ^\prime )\end{pmatrix}\\&\quad +12W(P_5)\begin{pmatrix}E_4(\tau )^2E_6(\tau ^\prime )\varDelta (\tau ^\prime )&{}0\\ 0&{}E_6(\tau )\varDelta (\tau )E_4(\tau ^\prime )^2\end{pmatrix}\}\begin{pmatrix}\varDelta (\tau )&{}0\\ 0&{}\varDelta (\tau ^\prime )\end{pmatrix}\\&=\left( {\begin{array}{c}f_{11}\ 0\\ \ 0\ \ \ f_{22}\end{array}}\right) \begin{pmatrix}\varDelta (\tau )&{}0\\ 0&{}\varDelta (\tau ^\prime )\end{pmatrix}. \end{aligned}$$

where the (1, 1)-component and (2, 2)-component of W(F) are

$$\begin{aligned} f_{11}\varDelta (\tau )&=(W(P_1)E_4(\tau ^\prime )E_6(\tau ^\prime )+12W(P_3)E_6(\tau )E_4(\tau ^\prime )\varDelta (\tau ^\prime )\\&\quad +12W(P_5)E_4(\tau )^2E_6(\tau ^\prime )\varDelta (\tau ^\prime ))\varDelta (\tau ),\\ f_{22}\varDelta (\tau ^\prime )&=(W(P_1)E_4(\tau )E_6(\tau )+12W(P_3)E_4(\tau )\varDelta (\tau )E_6(\tau ^\prime )\\&\quad +12W(P_5)E_6(\tau )\varDelta (\tau )E_4(\tau ^\prime )^2)\varDelta (\tau ^\prime ). \end{aligned}$$

Since $\nu _p(W(F))\ge 1$, and $\nu _p(\varDelta (\tau ))=\nu _p(\varDelta (\tau ^\prime ))=0$, we have $\nu _p(f_{11})=\nu _p(f_{22})\ge 1$. Then we get

$$\begin{aligned}&\nu _p(-E_4(\tau )\varDelta (\tau )E_6(\tau ^\prime )f_{11}+E_6(\tau )E_4(\tau ^\prime )\varDelta (\tau ^\prime )f_{22})\\&=\nu _p((E_4(\tau )^3\varDelta (\tau ^\prime )-\varDelta (\tau )E_4(\tau ^\prime )^3 )(E_4(\tau )E_4(\tau ^\prime )W(P_1)+2^8\cdot 3^4\varDelta (\tau )\varDelta (\tau ^\prime )W(P_5)))\\ {}&\ge 1 \end{aligned}$$

and

$$\begin{aligned}&\nu _p(E_4(\tau )E_6(\tau )f_{11}-E_4(\tau ^\prime )E_6(\tau ^\prime )f_{22})\\&\nu _p(2^2\cdot 3(E_4(\tau )^3\varDelta (\tau ^\prime )-\varDelta (\tau )E_4((\tau ^\prime )^3))(E_4(\tau )E_4(\tau ^\prime )W(P_3)+E_6(\tau )E_6(\tau ^\prime )W(P_5)))\\&\ge 1. \end{aligned}$$

Since $\nu _p(E_4(\tau )^3\varDelta (\tau ^\prime )-\varDelta (\tau )E_4((\tau ^\prime )^3))=0$, we get

$$\begin{aligned}&\nu _p(E_4(\tau )E_4(\tau ^\prime )W(P_1)+2^8\cdot 3^4\varDelta (\tau )\varDelta (\tau ^\prime )W(P_5))\ge 1,\end{aligned}$$

(1)

$$\begin{aligned}&\nu _p(E_4(\tau )E_4(\tau ^\prime )W(P_3)+E_6(\tau )E_6(\tau ^\prime )W(P_5))\ge 1. \end{aligned}$$

(2)

Case ($k \not \equiv 0\pmod {6}$)

We have $W(P_5)=0$. Hence we have $\nu _p(W(P_1))\ge 1$ and $\nu _p(W(P_3))\ge 1$. We can write

$$\begin{aligned}&W(P_1)=\sum \limits_{\begin{subarray}{c} \\ 4a+12b+12c=k-10 \end{subarray}}\gamma _{abc}W(\varphi _4)^aW(\varphi _6)^bW(X_{12})^c,\\&W(P_3)=\sum \limits_{\begin{subarray}{c} 4a+12b+12c=k-16 \end{subarray}}\gamma ^{\prime }_{abc}W(\varphi _4)^aW(\varphi _6)^bW(X_{12})^c. \end{aligned}$$

From Corollary 1, we have $\nu _p(\gamma _{abc})\ge 1$ and $\nu _p(\gamma ^{\prime } _{abc})\ge 1$. Using ${W(P_1-\sum \limits_{\begin{subarray}{c} 4a+12b+12c=k-10 \end{subarray}}\gamma _{abc}\varphi _4^a\varphi _6^bX_{12}^c)=0}$, ${W(P_3-\sum \limits_{\begin{subarray}{c} 4a+12b+12c=k-16 \end{subarray}}\gamma ^{\prime }_{abc}\varphi _4^a\varphi _6^bX_{12}^c)=0}$, we have ${P_1=\sum \limits_{\begin{subarray}{c} 4a+12b+12c=k-10 \end{subarray}}\gamma _{abc}\varphi _4^a\varphi _6^bX_{12}^c}$ and

${P_3=\sum \limits_{\begin{subarray}{c} 4a+12b+12c=k-16 \end{subarray}}\gamma ^{\prime }_{abc}\varphi _4^a\varphi _6^bX_{12}^c}$ because the Witt operator is injective on ${\mathbb {C}} [\varphi _4, \varphi _6, X_{12}]$ by Theorem 3 and the fact that Igusa’s generators are algebraically independent over ${\mathbb {C}}$. Hence $\nu _p(P_1(\varphi _4, \varphi _6, X_{12}))\ge 1$ and $\nu _p(P_3(\varphi _4, \varphi _6, X_{12}))\ge 1$.

Case ( $k \equiv 0\pmod {12}$)

We can write

$$\begin{aligned}&W(P_1)=E_6(\tau )E_6(\tau ^\prime )\sum \limits_{\begin{subarray}{c} a\equiv 2\pmod {3}\\ 4a+12b+12c=k-16 \end{subarray}}\gamma _{abc}W(\varphi _4)^aW(\varphi _6^2)^bW(X_{12})^c,\\&W(P_3)=\sum \limits_{\begin{subarray}{c} a\equiv 2\pmod {3}\\ 4a+12b+12c=k-16 \end{subarray}}\gamma ^{\prime }_{abc}W(\varphi _4)^aW(\varphi _6^2)^bW(X_{12})^c,\\&W(P_5)=E_6(\tau )E_6(\tau ^\prime )\sum \limits_{12b+12c=k-24}\gamma ^{\prime \prime }_{bc}W(\varphi _{6}^2)^bW(X_{12})^c. \end{aligned}$$

Using these formulas, we can write

$$\begin{aligned}&E_4(\tau )E_4(\tau ^\prime )W(P_1)+2^8\cdot 3^4\varDelta (\tau )\varDelta (\tau ^\prime )W(P_5)\\&\quad =E_6(\tau )E_6(\tau ^\prime )\sum \limits_{\begin{subarray}{c} a\equiv 0\pmod {3},\ a\ge 3\\ 4a+12b+12c=k-12 \end{subarray}}\gamma _{a-1bc}W(\varphi _4)^aW(\varphi _6^2)^bW(X_{12})^c\\&\qquad +2^6\cdot 3^3E_6(\tau )E_6(\tau ^\prime )\sum \limits_{\begin{subarray}{c} c\ge 1\\ 12b+12c=k-12 \end{subarray}}\gamma ^{\prime \prime }_{bc-1}W(\varphi _6^2)^bW(X_{12})^c\\&\quad =E_6(\tau )E_6(\tau ^\prime )\{\sum \limits_{\begin{subarray}{c} a\equiv 0\pmod {3},\ a\ge 3\\ 4a+12b+12c=k-12 \end{subarray}}\gamma _{a-1bc}W(\varphi _4)^aW(\varphi _6)^{2b}W(X_{12})^c\\&\qquad +2^6\cdot 3^3\sum \limits_{\begin{subarray}{c} c\ge 1\\ 12b+12c=k-12 \end{subarray}}\gamma ^{\prime \prime }_{bc-1}W(\varphi _6)^{2b}W(X_{12})^c\} . \end{aligned}$$

Since $\nu _p(\mathrm{LHS})\ge 1$, we have $\nu _p(\mathrm{RHS})\ge 1$ for both of two formulas above. From Corollary 1 and Theorem 3, we get $\nu _p(P_1(\varphi _4, \varphi _6, X_{12}))\ge 1$ and $\nu _p(P_5(\varphi _4, \varphi _6, X_{12}))\ge 1$. From the formula (2), $\nu _p(P_3(\varphi _4, \varphi _6, X_{12}))\ge 1$.

Case $k\equiv 6\pmod {12}(k\equiv 2\pmod {4}\ \text {and}\ k\equiv 0\pmod {6})$: Similarly to the case of $k\equiv 0\pmod {12}$, we can prove the assertion of Lemma 3. $\square$

By the Lemma 3 above, we get

$$\begin{aligned} F&\equiv \varDelta _5\cdot \{\varDelta _5(Q_1 \varPhi _{10}+Q_3\varPhi _{16}+Q_5\varPhi _{18})+(P_2+X_{10}Q_2)\varPhi _9+(P_4+X_{10}Q_4)\varPhi _{11}\\&\quad +(P_6+X_{10}Q_6)\varPhi _{17}\} \\&:=\varDelta _5\cdot G. \end{aligned}$$

It is known that $\varDelta _5\not \equiv 0\pmod {p}$ and $q_{\tau }^{\frac{1}{2}}q_{\tau ^\prime }^{\frac{1}{2}}\mid \varDelta _5$ but $q_{\tau }q_{\tau ^\prime }\nmid \varDelta _5$. Next we apply the Witt operator to G.

$$\begin{aligned}W(G& =W(P_2)W(\varPhi _9)+W(P_4)W(\varPhi _{11})+W(P_6)W(\varPhi _{17})\\& = {\sum \limits_{\begin{subarray}{c} 12i+4j+6t=k-10\\ 12i^\prime +4j^\prime +6t^\prime =k-10\\ t, t^\prime =0,1 \end{subarray}}C_3(i, i^\prime )\varDelta (\tau )^iE_4(\tau )^jE_6(\tau )^t\varDelta (\tau ^\prime )^{i^\prime }E_4(\tau ^\prime )^{j^\prime }E_6(\tau ^\prime )^{t^\prime }}\eta (\tau )^{12}\eta _(\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) \\& =\sum \limits_{m,n\ge 0}B_{12}(m, n)q_{\tau }^mq_{\tau ^\prime }^n\eta (\tau )^{12}\eta _(\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) .\end{aligned}$$

It is known that $\eta (\tau )^{12}\not \equiv 0\pmod {p}$ and $q_{\tau }^{\frac{1}{2}}q_{\tau ^\prime }^{\frac{1}{2}}\mid \eta (\tau )^{12}$ but $q_{\tau }q_{\tau ^\prime }\nmid \eta (\tau )^{12}$. Hence we have that if $m\le \left[ \frac{k}{10}\right] -1$ and $n\le \left[ \frac{k}{10}\right] -1$, then $B_{12}(m,n)\equiv 0\pmod {p}$. This implies that $C_3(i,i^\prime )\equiv 0\pmod {p}$ for $i, i^\prime \le \left[ \frac{k}{10}\right] -1$. Note that $i, i^\prime \le \left[ \frac{k}{10}\right] -1$ since $12i+4j+6t=k-10$ and $12i^\prime +4j^\prime +6t^\prime =k-10$ and $\left[ \frac{k-10}{12}\right] \le \left[ \frac{k}{10}\right] -1$. Thus we have $W(G) \equiv 0\pmod {p}$.

Lemma 4

$P_2, P_4, P_6\equiv 0\pmod {p}$.

Proof of Lemma 4

Using fact that $W(\varDelta _5)=0$, we get

$$\begin{aligned} W(G)&=W(P_2)W(\varPhi _{9})+W(P_4)W(\varPhi _{11})+W(P_6)W(\varPhi _{17})\\&=W(P_2)\left( 2E_4(\tau )E_4(\tau ^\prime )\eta (\tau )^{12}\eta (\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) \right) \\&\quad +W(P_4)\left( 2E_6(\tau )E_6(\tau ^\prime )\eta (\tau )^{12}\eta (\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) \right) \\&\quad +W(P_6)\left( 2^5\cdot 3^2\varDelta (\tau )\varDelta (\tau ^\prime )\eta (\tau )^{12}\eta (\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) \right) . \end{aligned}$$

Case $k \not \equiv 4\pmod {6}$: In this case we have $P_4=P_6=0$ as polynomials. Therefore we get

$\nu _p(W(P_2))\ge 1$. From Corollary 1 and Theorem 3, we get $\nu _p(P_2(\varphi _4, \varphi _6, X_{12}))\ge 1$.

Case $k\equiv 4\pmod {12}$: We can write

$$\begin{aligned}&W(P_2)=E_6(\tau )E_6(\tau ^\prime )\sum \limits_{\begin{subarray}{c} a\equiv 2\pmod {3}\\ 4a+12b+12c=k-20 \end{subarray}}\gamma _{abc}W(\varphi _4)^aW(\varphi _6^2)^bW(X_{12})^c,\\&W(P_4)=\sum \limits_{12b+12c=k-16}\gamma ^{\prime }_{bc}W(\varphi _{6}^2)^bW(X_{12})^c,\\&W(P_6)=0. \end{aligned}$$

Using these formulas, we can write as

$$\begin{aligned} W(G)&=2\Big ( \sum \limits_{\begin{subarray}{c} a\equiv 0\pmod {3},\ a\ge 3\\ 4a+12b+12c=k-16 \end{subarray}}\gamma _{a-1bc}W(\varphi _4)^aW(\varphi _6)^{2b}W(X_{12})^c\\&\quad +\sum \limits_{12b+12c=k-16}\gamma ^{\prime }_{bc}W(\varphi _6)^{2b}W(X_{12})^c\Big )E_6(\tau )E_6(\tau ^\prime )\eta (\tau )^{12}\eta (\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) . \end{aligned}$$

Again from Corollary 1, we have $\nu _p(\gamma _{a-1bc})\ge 1$ and $\nu _p(\gamma ^{\prime }_{bc})\ge 1$. These mean that, from Theorem 3, $\nu _p(P_2(\varphi _4, \varphi _6, X_{12}))\ge 1$ and $\nu _p(P_4(\varphi _6, X_{12}))\ge 1$.

Case$k\equiv 10\pmod {12}$: We can write

$$\begin{aligned}&W(P_2)=\sum \limits_{\begin{subarray}{c} a\equiv 2\pmod {3}\\ 4a+12b+12c=k-14 \end{subarray}}\gamma _{abc}W(\varphi _4)^aW(\varphi _6^2)^bW(X_{12})^c,\\&W(P_4)=E_6(\tau )E_6(\tau ^\prime )\sum \limits_{12b+12c=k-22}\gamma ^{\prime }_{bc}W(\varphi _6^2)^bW(X_{12})^c,\\&W(P_6)=\gamma ^{\prime \prime }_{\frac{k-22}{12}}W(X_{12})^{\frac{k-22}{12}}. \end{aligned}$$

Using these formulas, we can write as

$$\begin{aligned} W(G)&=\Big (2\sum \limits_{\begin{subarray}{c} a\equiv 0\pmod {3},\ a\ge 3\\ 4a+12b+12c=k-10 \end{subarray}}\gamma _{a-3bc}W(\varphi _4)^aW(\varphi _6)^{2b}W(X_{12})^c\\&\quad +2\sum \limits_{\begin{subarray}{c} b\ge 1\\ 12b+12c=k-10 \end{subarray}}\gamma ^{\prime }_{b-1c}W(\varphi _6)^{2b}W(X_{12})^c\\&\quad +2^3\cdot 3\gamma ^{\prime \prime }_{\frac{k-22}{12}}W(X_{12})^{\frac{k-10}{12}}\Big )\eta (\tau )^{12}\eta (\tau ^\prime )^{12}\left( {\begin{array}{c}01\\ 10\end{array}}\right) .\end{aligned}$$

Again from Corollary 1, we have $\nu _p(2\gamma _{a-3bc})\ge 1$, $\nu _p(2\gamma ^{\prime }_{b-1c})\ge 1$ and $\nu _p(2^3\cdot 3\gamma ^{\prime \prime }_{\frac{k-22}{12}})\ge 1$. These mean that, from Theorem 3, $\nu _p(P_2(\varphi _4, \varphi _6, X_{12}))\ge 1$, $\nu _p(P_4(\varphi _6, X_{12}))\ge 1$ and $\nu _p(P_6(X_{12}))\ge 1$.

This completes the proof of Lemma 4. $\square$

From Lemma 4, we get

$$\begin{aligned} F\equiv X_{10}\cdot (Q_1\varPhi _{10}+Q_2\varPhi _{14}+Q_3\varPhi _{16}+Q_4\varPsi _{16}+Q_5\varPhi _{18}+Q_6\varPhi _{22})\pmod {p}. \end{aligned}$$

Then $H_1:=Q_1\varPhi _{10}+Q_2\varPhi _{14}+Q_3\varPhi _{16}+Q_4\varPsi _{16}+Q_5\varPhi _{18}+Q_6\varPhi _{22}\in M_{k-10, 2}(\varGamma _2)_{\mathbb Z_{(p)}}$ and $A((m,n,r);H_1)\equiv 0\pmod {p}$ for every m, n such that $0\le m\le \left[ \frac{k-10}{10}\right] ,\ \ 0\le n\le \left[ \frac{k-10}{10}\right]$. Moreover $\nu _p(F)=\nu _p(H_1)$ since $\nu _p(X_{10})=0$.

By repeating this argument, there exists the modular form $H_t$ of weight $k-10t$ and $t_0$ such that

$$\begin{aligned} F\equiv H_t\cdot X_{10}^{t}\pmod {p} \end{aligned}$$

where $1\le t\le t_0$ and

$$\begin{aligned} A((m,n,r);H_t)\equiv 0\pmod {p} \end{aligned}$$

for every m, n such that $0\le m\le \left[ \frac{k-10t}{10}\right] ,\ \ \ \ 0\le n\le \left[ \frac{k-10t}{10}\right]$. Thus we have

$$\begin{aligned} \nu _p(F)=\nu _p(H_{t_0}). \end{aligned}$$

Since the weight of $H_{t_0}\le 22$, we should check the case $k\le 22$ directly.

Case ($k \equiv 0\pmod {10}$)

$H_{t_0}\in M_{10, 2}(\varGamma _2)$ and $t_0=\frac{k-10}{10}$. Since

$$\begin{aligned} X_{10}=(-2+q_{\omega }+q_{\omega }^{-1})+\cdots , \end{aligned}$$

we have that if $n\le \left[ \frac{k}{10}\right] -t_0=1$ and $m\le \left[ \frac{k}{10}\right] -t_0=1$, then $a((m,n,r);H_{t_0})\equiv 0\pmod {p}$. With $a_1\in {\mathbb {Z}}_{(p)}$ we can write

$$\begin{aligned} H_{t_0}&=a_1\varPhi _{10}=\left( {\begin{array}{c}a_10\\ 0\ 0\end{array}}\right) q_{\tau }+\cdots . \end{aligned}$$

Hence we have $a_1\equiv 0\pmod {p}$. Hence we have $F\equiv 0\pmod {p}$.

Case ($k \equiv 4\pmod {10}$)

$H_{t_0}\in M_{14, 2}(\varGamma _2)$ and $t_0=\frac{k-14}{10}$. Then we have that if $n\le \left[ \frac{k}{10}\right] -t_0=1$ and $m\le \left[ \frac{k}{10}\right] -t_0=1$, then $a((m,n,r);H_{t_0})\equiv 0\pmod {p}$. With $a_1, a_2\in {\mathbb {Z}}_{(p)}$ we can write

$$\begin{aligned} H_{t_0}&=a_1\varphi _4\varPhi _{10}+a_2\varPhi _{14}\\&=\left( {\begin{array}{c}a_10\\ 0\ 0\end{array}}\right) q_{\tau }+\left( {\begin{array}{c}0\ \ 0\\ 0\ a_1\end{array}}\right) q_{\tau ^\prime }\\&\quad +\left\{ \begin{pmatrix}30a_1+2a_2&{}0\\ 0&{}30a_1+2a_2\end{pmatrix}+\begin{pmatrix}-28a_1-a_2&{}-14a_1-\frac{1}{2}a_2\\ -14a_1-\frac{1}{2}a_2&{}-28a_1-a_2\end{pmatrix}q_{\omega }\right. \\&\quad +\left. \begin{pmatrix}-28a_1-a_2&{}14a_1+\frac{1}{2}a_2\\ 14a_1+\frac{1}{2}a_2&{}-28a_1-a_2\end{pmatrix}q_{\omega }^{-1}+\left( {\begin{array}{c}a_1a_1\\ a_1a_1\end{array}}\right) q_{\omega }^{2}+\left( {\begin{array}{c}a_1\ -a_1\\ -a_1\ \ a_1\end{array}}\right) q_{\omega }^{-2}\right\} q_{\tau }q_{\tau ^\prime }+\cdots . \end{aligned}$$

Hence we have $a_1\equiv 0\pmod {p}$ and $-28a_1-a_2\equiv 0\pmod {p}$. Hence we get $a_2\equiv 0\pmod {p}$. Hence we have $F\equiv 0\pmod {p}$.

Case ( $k \equiv 6\pmod {10}$)

$H_{t_0}\in M_{16, 2}(\varGamma _2)$ and $t_0=\frac{k-16}{10}$. Then we have that if $n\le \left[ \frac{k}{10}\right] -t_0=1$ and $m\le \left[ \frac{k}{10}\right] -t_0=1$, then $a((m,n,r);H_{t_0})\equiv 0\pmod {p}$. With $a_1, a_2, a_3\in {\mathbb {Z}}_{(p)}$ we can write

$$\begin{aligned} H_{t_0}&=a_1\varphi _6\varPhi _{10}+a_2\varPhi _{16}+a_3\varPsi _{16}\\&=\left( {\begin{array}{c}a_10\\ 0\ 0\end{array}}\right) q_{\tau }+\left( {\begin{array}{c}0\ \ 0\\ 0\ a_1\end{array}}\right) q_{\tau ^\prime }\\&\quad +\left\{ \begin{pmatrix}-714a_1+10a_2-2a_3&{}0\\ 0&{}-714a_1+10a_2-2a_3\end{pmatrix}\right. \\&\left. \quad +\begin{pmatrix}-28a_1+a_2+a_3&{}-14a_1+\frac{1}{2}a_2+\frac{1}{2}a_3 \\ -14a_1+\frac{1}{2}a_2+\frac{1}{2}a_3&{}-28a_1+a_2+a_3\end{pmatrix}q_{\omega }\right. \\&\quad \left. +\begin{pmatrix}-28a_1+a_2+a_3&{}14a_1-\frac{1}{2}a_2-\frac{1}{2}a_3\\ 14a_1-\frac{1}{2}a_2-\frac{1}{2}a_3&{}-28a_1+a_2+a_3\end{pmatrix}q_{\omega }^{-1}+\left( {\begin{array}{c}a_1a_1\\ a_1a_1\end{array}}\right) q_{\omega }^{2}+\left( {\begin{array}{c}a_1\ -a_1\\ -a_1\ \ a_1\end{array}}\right) q_{\omega }^{-2}\right\} q_{\tau }q_{\tau ^\prime }+\cdots . \end{aligned}$$

Hence we have $a_1\equiv -714a_1+10a_2-2a_3\equiv -28a_1+a_2+a_3\equiv 0\pmod {p}$. Hence we get $a_2=\frac{1}{2^2\cdot 3}(10a_2-2a_3+2(a_2+a_3))\equiv 0\pmod {p}$ and $a_3=\frac{1}{2^2\cdot 3}(-(10a_2-2a_3)+10(a_2+a_3))\equiv 0\pmod {p}$. Hence we have $F\equiv 0\pmod {p}$.

(Case $k \equiv 8\pmod {10}$)

$H_{t_0}\in M_{18, 2}(\varGamma _2)$ and $t_0=\frac{k-18}{10}$. Then we have that if $n\le \left[ \frac{k}{10}\right] -t_0=1$ and $m\le \left[ \frac{k}{10}\right] -t_0=1$, then $a((m,n,r);H_{t_0})\equiv 0\pmod {p}$. With $a_1, a_2, a_3\in {\mathbb {Z}}_{(p)}$ we can write

$$\begin{aligned} H_{t_0}&=a_1\varphi _4^2\varPhi _{10}+a_2\varphi _4\varPhi _{14}+a_3\varPhi _{18}\\&=\left( {\begin{array}{c}a_10\\ 0\ 0\end{array}}\right) q_{\tau }+\left( {\begin{array}{c}0\ \ 0\\ 0\ a_1\end{array}}\right) q_{\tau ^\prime }\\&\quad +\left\{ \begin{pmatrix}270a_1-2a_2+10a_3&{}0\\ 0&{}270a_1-2a_2+10a_3\end{pmatrix}\right. \\&\left. \quad +\begin{pmatrix}-28a_1+a_2+a_3&{}-14a_1+\frac{1}{2}a_2+\frac{1}{2}a_3\\ -14a_1+\frac{1}{2}a_2+\frac{1}{2}a_3&{}-28a_1+a_2+a_3\end{pmatrix}q_{\omega }\right. \\&\quad \left. +\begin{pmatrix}-28a_1+a_2+a_3&{}14a_1-\frac{1}{2}a_2-\frac{1}{2}a_3\\ 14a_1-\frac{1}{2}a_2-\frac{1}{2}a_3&{}-28a_1+a_2+a_3\end{pmatrix}q_{\omega }^{-1}+\left( {\begin{array}{c}a_1a_1\\ a_1a_1\end{array}}\right) q_{\omega }^{2}+\left( {\begin{array}{c}a_1\ -a_1\\ -a_1\ \ a_1\end{array}}\right) q_{\omega }^{-2}\right\} q_{\tau }q_{\tau ^\prime }+\cdots . \end{aligned}$$

Hence we have $a_1\equiv 270a_1-2a_2+10a_3\equiv -28a_1+a_2+a_3\equiv 0\pmod {p}$. Hence we get $a_2=\frac{1}{2^2\cdot 3}(-(-2a_2+10a_3)+10(a_2+a_3))\equiv 0$ and $a_3=\frac{1}{2^2\cdot 3}(-2a_2+10a_3+2(a_2+a_3))\equiv 0\pmod {p}$. Hence we have $F\equiv 0\pmod {p}$.

(Case $k \equiv 2\pmod {10}$)

$H_{t_0}\in M_{22, 2}(\varGamma _2)$ and $t_0=\frac{k-22}{10}$. Then we have that if $n\le \left[ \frac{k}{10}\right] -t_0=2$ and $m\le \left[ \frac{k}{10}\right] -t_0=2$, then $a((m,n,r);H_{t_0})\equiv 0\pmod {p}$. With $a_1, a_2, a_3, a_4, a_5, a_6, a_7\in \mathbb {Z}_{(p)}$ we can write

$$\begin{aligned} H_{t_0}&=(a_1\varphi _4^3+a_2\varphi _6^2+a_3X_{12})\varPhi _{10}+a_4\varphi _4^2\varPhi _{14}+a_5\varphi _6\varPhi _{16}+a_6\varphi _6\varPsi _{16}+a_7\varPhi _{22}\\&=\left( {\begin{array}{c}a_1+a_2\ 0\\ 0\ \ \ \ \ \ \ \ \ \ 0\end{array}}\right) q_{\tau }+\left( {\begin{array}{c}0\ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\\ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ a_1+a_2\end{array}}\right) q_{\tau ^\prime }\\&\quad + \left\{ \begin{pmatrix}510a_1-1218a_2-2a_4+10a_5-2a_6&{}0\\ 0&{}510a_1-1218a_2-2a_4+10a_5-2a_6\end{pmatrix}\right. \\&\quad +\begin{pmatrix}-28a_1-28a_2+a_4+a_5+a_6&{}-14a_1-14a_2+\frac{1}{2}a_4+\frac{1}{2}a_5+\frac{1}{2}a_6\\ -14a_1-14a_2+\frac{1}{2}a_4+\frac{1}{2}a_5+\frac{1}{2}a_6&{}-28a_1-28a_2+a_4+a_5+a_6\end{pmatrix}q_{\omega }\\&\quad +\begin{pmatrix}-28a_1-28a_2+a_4+a_5+a_6&{}14a_1+14a_2-\frac{1}{2}a_4-\frac{1}{2}a_5-\frac{1}{2}a_6\\ 14a_1+14a_2-\frac{1}{2}a_4-\frac{1}{2}a_5-\frac{1}{2}a_6&{}-28a_1-28a_2+a_4+a_5+a_6\end{pmatrix}q_{\omega }^{-1}\\&\quad \left. +\left( {\begin{array}{c}a_1+a_2\ a_1+a_2\\ a_1+a_2\ a_1+a_2\end{array}}\right) q_{\omega }^{2}+\left( {\begin{array}{c}a_1+a_2\ -a_1-a_2\\ -a_1-a_2\ \ a_1+a_2\end{array}}\right) q_{\omega }^{-2}\right\} q_{\tau }q_{\tau ^\prime }\\&\quad +\left( {\begin{array}{c}696a_1-1032a_2\ 0\ \\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\end{array}}\right) q_{\tau }^2+\left( {\begin{array}{c}0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\\ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ 696a_1-1032a_2\end{array}}\right) q_{\tau ^\prime }^2\\&\quad +\left\{ \begin{pmatrix}-54324a_1+350028a_2-1404a_4-2772a_5+2052a_6&{}0\\ 0&{}279432a_1+1088136a_2+10a_3-168a_4-10104a_5+408a_6\end{pmatrix}\right. \\&\quad +\begin{pmatrix}-46464a_1+1920a_2+704a_4-352a_5-1024a_6&{}-23232a_1+960a_2+352a_4-176a_5-512a_6\\ -23232a_1+960a_2+352a_4-176a_5-512a_6&{}17952a_1+114720a_2+a_3+88a_4-1160a_5-200a_6\end{pmatrix} q_{\omega }\\ \end{aligned}$$

$$\begin{aligned}&\quad +\begin{pmatrix}-46464a_1+1920a_2+704a_4-352a_5-1024a_6&{}23232a_1-960a_2-352a_4+176a_5+512a_6\\ 23232a_1-960a_2-352a_4+176a_5+512a_6&{}17952a_1+114720a_2+a_3+88a_4-1160a_5-200a_6\end{pmatrix}q_{\omega }^{-1}\\&\quad +\begin{pmatrix}510a_1-1218a_2-2a_4+10a_5-2a_6&{}510a_1-1218a_2-2a_4+10a_5-2a_6\\ 510a_1-1218a_2-2a_4+10a_5-2a_6&{}1020a_1-2436a_2-4a_4+20a_5-4a_6\end{pmatrix}q_{\omega }^2\\&\quad \left. +\begin{pmatrix}510a_1-1218a_2-2a_4+10a_5-2a_6&{}-510a_1+1218a_2+2a_4-10a_5+2a_6\\ -510a_1+1218a_2+2a_4-10a_5+2a_6&{}1020a_1-2436a_2-4a_4+20a_5-4a_6\end{pmatrix}q_{\omega }^{-2}\right\} q_{\tau }q_{\tau ^\prime }^2\\&\quad +\left\{ \begin{pmatrix}A_1&{}0\\ 0&{}B_1\end{pmatrix} +\begin{pmatrix}A_2&{}C_2\\ C_2&{}B_2\end{pmatrix}q_{\omega } +\begin{pmatrix}A_2&{}-C_2\\ -C_2&{}B_2\end{pmatrix}q_{\omega }^{-1}\right. \end{aligned}$$

$$\begin{aligned}&\quad +\begin{pmatrix}-688416a_1+217056a_2-8a_3+3840a_4-1920a_5+19968a_6+2a_7&{}-344208a_1+108528a_2-4a_3+1920a_4-960a_5+9984a_6+a_7\\ -344208a_1+108528a_2-4a_3+1920a_4-960a_5+9984a_6+a_7&{}-688416a_1+217056a_2-8a_3+3840a_4-1920a_5+19968a_6+2a_7\end{pmatrix}q_{\omega }^2\\ &\quad +\begin{pmatrix}-688416a_1+217056a_2-8a_3+3840a_4-1920a_5+19968a_6+2a_7&{}344208a_1-108528a_2+4a_3-1920a_4+960a_5-9984a_6-a_7\\ 344208a_1-108528a_2+4a_3-1920a_4+960a_5-9984a_6-a_7&{}-688416a_1+217056a_2-8a_3+3840a_4-1920a_5+19968a_6+2a_7\end{pmatrix}q_{\omega }^{-2}\end{aligned}$$

$$\begin{aligned}&\quad +\begin{pmatrix}17952a_1+114720a_2+a_3+88a_4-1160a_5-200a_6&{}41184a_1+113760a_2+a_3-264a_4-984a_5+312a_6\\ 41184a_1+113760a_2+a_3-264a_4-984a_5++312a_6&{}17952a_1+114720a_2+a_3+88a_4-1160a_5-200a_6\end{pmatrix}q_{\omega }^3\\&\quad +\begin{pmatrix}17952a_1+114720a_2+a_3+88a_4-1160a_5-200a_6&{}-41184a_1-113760a_2-a_3+264a_4+984a_5-312a_6\\ -41184a_1-113760a_2-a_3+264a_4+984a_5-312a_6&{}17952a_1+114720a_2+a_3+88a_4-1160a_5-200a_6\end{pmatrix}q_{\omega }^{-3}\\&\quad \left. +\begin{pmatrix}696a_1-1032a_2&{}696a_1-1032a_2\\ 696a_1-1032a_2&{}696a_1-1032a_2\end{pmatrix}q_{\omega }^4+\begin{pmatrix}696a_1-1032a_2&{}-696a_1+1032a_2\\ -696a_1+1032a_2&{}696a_1-1032a_2\end{pmatrix}q_{\omega }^{-4}\right\} q_{\tau }^2q_{\tau ^\prime }^2+\cdots , \end{aligned}$$

where

$$\begin{aligned}&A_1=B_1=-59095920a_1-257328624a_2-2288a_3-75776a_4+2537728a_5-329216a_6-36a_7\\&A_2=B_2=-20671008a_1-53005344a_2-577a_3+33960a_4+517512a_5+144840a_6+16a_7\\&C_2=-24544032a_1+15272544a_2-139a_3+239160a_4-205656a_5+511608a_6+70a_7\\ \end{aligned}$$

Hence we get

$$\begin{aligned}&a_1=\frac{1}{2^6\cdot 3^3}(1032(a_1+a_2)+(696a_1-1032a_2))\equiv 0\pmod {p},\\&a_2=\frac{1}{2^6\cdot 3^3}(696(a_1+a_2)-(696a_1-1032a_2))\equiv 0\pmod {p}. \end{aligned}$$

Then we get

$$\begin{aligned} a_5=\frac{1}{2^2\cdot 3}(2(a_4+a_5+a_6)+(-2a_4+10a_5-2a_6))\equiv 0\pmod {p}. \end{aligned}$$

Hence we get $a_4+a_6\equiv 0\pmod {p}$ and $704a_4-1024a_6\equiv 0\pmod {p}$. Hence we get

$$\begin{aligned} a_4=\frac{1}{2^6\cdot 3^3}(1024(a_4+a_6)+(704a_4-1024a_6))\equiv 0\pmod {p} \end{aligned}$$

and

$$\begin{aligned} a_6=\frac{1}{2^6\cdot 3^3}(704(a_4+a_6)-(704a_4-1024a_6))\equiv 0\pmod {p} \end{aligned}$$

. Hence we get $a_3 \equiv a_7\equiv 0\pmod {p}$. Hence we have $F\equiv 0\pmod {p}$. $\square$