Some small finite groups of linear transformations and

数理解析研究所講究録 1420 巻 2005 年 83-93

83

Some small finite groups of linear transformations and their rings of invariants and from which the automorphic forms belonging to congruence subgroups are determined Michio Ozeki Department of Mathematical Sciences Faculty of Science Yamagata University 1-4-12, Koshirakawa-chou, Yamagata Japan email address : [email protected] 9. Nov. 2004

1

Introduction

It is known that from a certain finite group of linear transformations and its invariant ring one can derivc automorphic form . However so far the construction of automorphic forms is mainly focussed on forms for subgroups of the full modular group. In the present talk we try to extend the idea to the construction of Jacobi forms. The trial is only beginning. We want to cover as far as possible cases, but here are small instances for that. $\mathrm{s}$

2 Let

Finite groups of linear transformations $G_{1}$

be the group generated by the linear transformations: $(\begin{array}{ll} 00 1\end{array})$

The group of

$G_{1}$

$\mathbb{C}[x,y]^{G_{1}}$

,

$(\begin{array}{ll}1 00 i\end{array})$

,

$(\begin{array}{ll}0 i1 \bigcap_{-}\end{array})$

acts linearly on the polynomial ring

$\mathbb{C}[x, y]$

invariant under the action of $\Phi_{G_{1}}(t)$

$=$

$=$

,

$\mathbb{C}[x,y]$

$G_{1}$

$(\begin{array}{ll}0 1i 0\end{array})$

.

. Molien series

$\Phi_{G_{1}}(t)$

of the subring

is computed to be

$\sum_{i}^{\infty}\dim((\mathbb{C}[x, y]^{G_{1}})_{i})t^{i}$

$\frac{1}{(1-t^{4})(1-t^{8})}$

.

can be regarded as Ch group of invariance for tl le weigh enumerator uf the class of doubly even binary self-orthogonal codes containing all one vector

Note: The group

$1\mathrm{t}$

$G_{1}$

$\iota \mathrm{e}$

84 For instances, the polynomial $x^{4}+y^{4}$ corresponds to the code with the generator matrix (1, 1, 1, 1) and the polynomial $x^{S}+14x^{4}y^{4}+y^{8}$ corresponds to the Hamming [8, 4, code. is proved to be $\mathbb{C}[x^{4}+y^{4}, x^{4}y^{4}]=\mathbb{C}[x^{4}+y^{4},x^{8}+14x^{4}y^{4}+y^{8}]$ . But we do The ring and below. not know such interpretation for the groups $4^{1}\rfloor$

$\mathbb{C}[x, y]^{G_{1}}$

$G_{3}$

$G_{2}$

$G_{2}$

a group generated by the linear transformations: $(\begin{array}{ll}i 00 1\end{array})$

Molien series

, (t)

$\Phi_{G}.$

of the invariant ring

$G_{3}$

$\mathbb{C}[x, y]^{G_{2}}$

is proved to be

$\mathbb{C}\lceil.x^{4}$

,

$y^{4}$

$=$

.

$(\begin{array}{ll}1 0\backslash 0 i\end{array})$

$\mathbb{C}[x, y]^{G_{2}}$

$\Phi_{G_{2}}(t)$

The ring

,

is computed to be

$\frac{1}{(1-t^{4})^{2}}$

.

].

: group generated by the linear transformations represented by the matrices: $(\begin{array}{lll}-1 0 0 1 \prime\end{array})$

Molien series

$\Phi_{G_{3}}(t)$

of the invariant ring

$\mathbb{C}[x, y]^{G_{3}}$

of

$G_{3}$

$(\begin{array}{l}100-1\end{array})$

$\mathbb{C}[x_{7}y]^{G_{3}}$

$\Phi_{G\epsilon_{\backslash }^{(t)}}$

The invariant ring

,

$=$

is computed to be

$\frac{1}{(1-t^{2})^{2}}$

is proved to be

.

.

$\mathbb{C}[x^{2}, y^{2}]$

.

One observes that $G_{3}\subset G_{2}\subseteq G_{1}$

,

and $\mathbb{C}[x, y]^{G_{S}}\supset \mathbb{C}[x, y]^{G_{2}}\supset \mathbb{C}[x,y]^{G_{1}}$

3 3.1

.

Jacobi’s theta functions Definition

Let $\mathbb{H}=\{\tau|{\rm Im}\tau>0\}$ be the upper half plane, and function are defined by

$\tau$

be a variable on

$1\mathrm{S}$

$\theta_{0}(\tau, z)$

$= \sum_{n\in \mathbb{Z}}(-1)^{n}e^{\pi in^{2}\tau+2\pi inz}$

$\theta_{1}(\tau, z)$

$=$

$\frac{1}{\mathrm{i}}\sum_{n\in \mathbb{Z}}(-1)^{n}e^{\pi i(n+1/2)^{2}\tau+\pi i\zeta 2n+1)z}$

$\theta_{2}(\tau, z)$

$= \sum_{n\in \mathbb{Z}}e^{\pi i(n+1/2\rangle^{2}\tau+\pi i(2n+1)z}$

$\theta_{3}(\tau,z)$ $= \sum_{n\in \mathbb{Z}}e^{\pi in^{2}\tau+2\pi inz}$

IH[

. Jacobi’s theta

85 Next we put

for

$0\leq i\leq 3$

3.2

,

$\varphi_{i}(\tau, z)$

$=$

$\theta_{i}(2\tau, 2z)$

$\varphi_{i}(\tau)$

$=$

$\theta_{i}(2\tau)=\theta_{i}(2\tau, 0)$

.

Properties of Jacobi’s theta functions

We present many properties of Jacobi’s theta functions that are reproduction of well-known formulas.

Proposition 1 It holds that $\varphi_{2}(-\frac{1}{\tau}, \frac{z}{\tau})$

$=$

$\sqrt{\frac{\tau}{\underline{9}i}}e^{2\pi iz/\tau}(\varphi_{3}(\tau, z)-\varphi_{2}(\tau, z))$

$\varphi_{3}(-, \frac{z}{\tau})’\}^{-}\underline{1}$

$=$

$\sqrt{-_{\mathrm{i}}^{\overline{J}}9}.e^{2\pi z/\tau}.(\varphi_{3}(\tau, z)+\varphi_{2}(\tau, z_{/}^{\backslash })$

$\varphi_{2}(-\frac{1}{\tau})$

$=$

$\mathrm{v}_{\overline{2\mathrm{i}}}^{\Gamma^{\tau}}(\varphi_{3}(\tau)-\varphi_{2}(\tau))$

$\varphi_{3}(-\frac{1}{\tau})$

$=$

$\sqrt{\frac{\hat{/}}{2i}}$

$\epsilon$

$\delta$

03

(

$\varphi_{3}(\tau)+$

$=e^{-\pi i(z+\tau/4)}$

’ $=e^{-\pi i(_{\sim}\approx+\tau)}$

$\theta_{3}(\tau, z+\frac{1}{2})$

$=\theta_{0}(\tau, z)$

23

$=\epsilon\theta_{2}(\tau, z)$

$(\tau_{7}z+\underline{.\frac{\tau}{)}})$

$( \tau, z+\frac{\tau}{2}\frac{1}{2}+)$

,

$z)$

$\theta_{3}(\tau, z+1)$

$=\theta_{3}(\tau, z)$

$\theta_{3}(\tau, z+\tau)$

$=\mathit{5}\theta_{3}(\tau, z)$

,

$\theta_{2_{\backslash }^{(_{\mathcal{T}.Z+\frac{1}{2})}}}$

$\theta_{2}\{\tau$

$\theta_{3}(\tau,$

$=i\epsilon\theta_{1(\mathcal{T}}’$

,

$z+ \frac{\tau}{2^{\backslash }})$

$=$

$-\theta_{3}(\tau, z)$

$=\epsilon\theta_{3}(\tau,$

$z\grave{)}$

$z+ \frac{1}{?}.+\frac{1}{2}\grave{J}$

$=$

$-\mathrm{i}\epsilon\theta_{0}(\tau, z)$

$\theta_{2_{\backslash }^{(\tau,z+1)}}$

$=$

$-\theta_{2}(\tau, z)$

$\theta_{2}(\tau, z+\tau)$

$=\delta\theta_{2}(\tau, z)$

$\theta_{2}(\tau, z+1+\tau)$

$=$

$-\delta\theta_{2}(\tau, z)$

$\varphi_{3}(\tau, z+1)$

$=\varphi_{3}(\tau, z)$

$\varphi_{3}(\tau, z+\tau)$

$=\delta\varphi_{3}(\tau, z)$

$\varphi_{2}(\tau, z+1)$

$=$

$\varphi_{2}(\tau, z)$

W2 (r))

88 4 4.1

Modular group and the congruence subgroups modular groups

Let $N$ be a positive integer. The principal congruence subgroup group is defined to be

$\Gamma(N)$

of level $N$ of the modular

$SL(\underline{?}, \mathbb{Z})$

$\Gamma(N)=\{\sigma=(\begin{array}{ll}a bc d\end{array})$

$\in SL(2, \mathbb{Z})|$

$(\mathrm{m}\mathrm{o}\mathrm{d} N)\}$

$(\begin{array}{ll}a bc d\end{array})\equiv(\begin{array}{ll}1 00 1\end{array})$

Another important classes of congruence subgroups are $\Gamma_{0}(N)=\{\sigma=(\begin{array}{lll}a b c d \prime\end{array})$

$\in SL(2,\mathbb{Z})|c\equiv 0$

$(\mathrm{m}\mathrm{o}\mathrm{d} N)\}$

,

and $\Gamma_{1}(N)=\{_{\mathrm{t}}\sigma$

$=(\begin{array}{ll}a bc d\end{array})$

(mod

$\in \mathrm{r}\mathrm{o}(2)\mathbb{Z})|a\equiv d\equiv 1$

$N$

),

(mod

$c\equiv 0$

$N$

) , $\}$

We are interested in the groups , , , and . The picture below describes the relation among some congruence subgroups. Here a downward line implies the subgroup relation with the relative index attached. $\mathrm{r}\mathrm{o}(2)$

$\Gamma_{0}(4)$

$\mathrm{r}\mathrm{o}(4)\cap\Gamma(2)$

$\Gamma(2)$

$\Gamma\acute{(}1)=SL(2,\mathbb{Z})$

$|3$

index

$\Gamma_{0}(2)$

$|2$

$\Gamma(4)$

The generators of some of these groups are computed to be $\Gamma(1)$

$=$

$\{$

$(\begin{array}{ll}1 10 1\end{array})$

$(\begin{array}{l}0-11\mathrm{O}\end{array})$

$\}$

$(\begin{array}{ll}1 -12 -1\end{array})$

,

$f$

$\Gamma_{0}(2)$

$=$

$\{$

$(\begin{array}{ll}1 10 1\end{array})$

,

$=$

$\{$

$(\begin{array}{ll}1 10 1\end{array})$

,

$(\begin{array}{ll}1 02 1\end{array})$

,

$(\begin{array}{ll}-1 00 -1\end{array})$

$(\begin{array}{ll}-1 00 -1\end{array})$

$\}$

$\}$

87 $\Gamma_{0}(4)$

$\Gamma(2)$

$\Gamma_{1}(4)\cap\Gamma(21,$

4.2

,

,

$(\begin{array}{l}1-14-3\end{array})$

$(\begin{array}{ll}1 10 \mathrm{l}\end{array})$

,

$(\begin{array}{ll}1 04 1\end{array})$

$\{$

$(\begin{array}{ll}1 2\backslash 0 1\end{array})$

,

$(\begin{array}{ll}1 02 1\end{array})$

$=$

$\{$

$(\begin{array}{ll}1 0\underline{.)}1 \end{array})$

,

$(\begin{array}{l}-3-2\underline{?}1\end{array})$

$=$

$\{$

$=$

$\{$

$(\begin{array}{ll}1 \mathrm{l}0 1\end{array})$

$=$

$\{$

$=$

$(\begin{array}{ll}-1 \mathrm{U}0 -1\end{array})$

,

$(\begin{array}{ll}-1 00 -1\end{array})$

$\}$

,

$(\begin{array}{ll}-1 00 -1\end{array})$

$\}$

$\}$

$(\begin{array}{ll}\vee-1 00 -1\end{array})$ $\}$ $\}$

,

$(\begin{array}{l}3-28-5\end{array})$

$(\begin{array}{ll}1 04 1\end{array})$

.

$(\begin{array}{ll}-7 2-4 1\end{array})$

$\}$

A definition of modular forms

on is called a modular form of weight A holomorphic function if it satisfies the conditions: , (1) $f( \frac{atau+b}{c\tau+d})=(c\tau+d)^{k}.f(\tau)$ for $f(\tau)$

$\mathbb{H}$

$k$

for a group

$\Gamma\subset SL(2, \mathbb{Z})$

$(\begin{array}{ll}a bc d\end{array})\in\Gamma$

and (2) holomorphic at each cusp of . the linear space of modular forms of weight We denote by $\Gamma$

$M_{k}(\Gamma f$

4.3

$k$

belonging to F.

A summary of known results

Proposition 2 We have the following isomorphisms between the rings .

$arrow iso$

$\mathbb{C}[x, y]^{G_{1}}=\mathbb{C}[x^{4}+y^{4}, x^{4}y^{4}]$

$. \bigoplus_{k\in \mathbb{Z}_{>0}}M_{h}(\Gamma_{0}(_{\backslash }\underline{?}))$

$\cap$

$\cap$

$\underline{ieo.}$

$\mathbb{C}[x, y]^{G_{2}}=\mathbb{C}[x^{4}, y^{4}]$

$\bigoplus_{k\in \mathbb{Z}_{>0}}M_{h}(\Gamma_{0}(4))$

$\cap$

$\cap$

.

$arrow i\epsilon \mathit{0}$

$\mathbb{C}[x, y]^{G_{3}}=\mathbb{C}[x^{2}, y^{2}]$

$\bigoplus_{\mathrm{k}\in \mathbb{Z}_{>0}}M_{k}$

where the isomorphism is realized by sending

$x$

to

$\varphi_{3}(\tau)$

and

$(\Gamma_{0}(4)\cap \Gamma(2))$

$y$

to

$\varphi_{2}(\tau)$

,

respectively.

These are not new results. Actually the first and second lines are due to Maher[9], and the third line is due to Hiramatsu [8].

Yet another group

4.4 Let

$G_{4}$

be the group generated by a linear transformation

ily proved that the invariant ring

$\mathbb{C}[x,y]^{G_{4}}$

equals

$\mathbb{C}[x,y^{4}]$

.

$(\begin{array}{ll}1 00 \dot{f}\end{array})$

. Then it can be eas-

We set a map from

$\mathbb{C}[x,y^{4}]$

to

88 by substituting and , where space of modular for ms discussed in H. Cohen [5]. Cohen showed that $\oplus_{k\in \mathbb{Z}}M_{k}$

(

$\Gamma_{0}(4)$

,

,

$1_{k}1$

$x=\varphi_{3}(\tau)$

$y=\varphi_{2}(\tau)$

$M_{k}(\Gamma_{0}(4), 1_{k})\cong \mathbb{C}[\varphi \mathrm{a}\zeta\tau),\mathcal{F}_{\mathrm{Z}}(\tau)]$

where

$\mathcal{F}_{2}(\tau)$

$= \frac{\eta 8(4\tau)}{\eta^{4}(2\tau)}$

with Dedekind’s eta function

$M_{k}(\Gamma_{0}(4))1_{k}$

) is a linear

,

. We show

$\eta(\tau)\backslash$

Proposition 3 It holds that $\varphi_{2}^{4}(\tau)=9^{4}\lrcorner \mathcal{F}_{2}(\tau)$

.

A proof is done by using the infinite product expansions of both

5 Let

$\eta(\tau)$

$=$

$q^{\frac{1}{24}} \prod_{m=1}^{\infty}(1-q^{m})q=e^{2ni\tau}$

$\varphi_{2}(\tau)$

$=$

2

$\varphi_{2}(\tau)$

$q^{\frac{1}{4}} \prod_{m=1}^{\infty}(1-q^{2m})(_{\backslash }1+q^{2m})^{2}$

and

$\eta(\tau)$

:

.

Diagonalized groups $G_{1}\oplus G_{1}$

be the group generated by the linear transformations:

,

$(\begin{array}{llll}i 0 0 00 1 0 00 0 i 00 0 0 1\end{array})$

$(\begin{array}{llll}\mathrm{O} i 0 01 0 0 00 0 0 i0 0 1 \mathrm{O}\end{array})$

The ring of invariants for

$G_{1}$

@

$G_{1}$

$(\begin{array}{llll}1 0 0 00 i 0 00 0 1 0\mathrm{O} 0 0 i\end{array})$

,

$(\begin{array}{llll}0 1 0 0i 0 0 \mathrm{O}0 0 0 10 0 i 0\end{array})$

is denoted by $\mathcal{R}_{1}=\mathbb{C}[x, y, u_{\gamma}\prime v]^{G_{1}\oplus G_{1}}$

Molicn scries for

$G_{1}$

%

$G_{1}$

,

.

is computed to bc $\Phi_{G_{1}\oplus G_{1}}(t)=\frac{1+3t^{4}+13t^{8}+9t^{12}+6b^{1\overline{\mathrm{b}}}}{(1-t^{4})^{2}(1-t^{8})^{2}}$

Primary invariants ( given by

$\mathrm{i}.\mathrm{c}$

.

. the polynomials implied by the numerator of

$x^{4}+y^{4}$ $u^{4}+v^{4}$ $x^{8}+y^{8}$ $u^{8}+v^{8}$

,

$\Phi_{G_{1}\oplus G_{1}}(t)$

) of

$\mathcal{R}_{1}$

arc

8a Secondary invariants (i.e. the polynomials implied by the denom inator of given by $a_{0}$

$a_{4,2}$

$a_{4,1}$

$a_{4_{\}}3}$

$=$

1,

$=x^{2}u^{2}+y^{2}v^{2}$ $=x^{3}\alpha+y^{3}v$

,

$=xu^{3\theta}+y\tau J$

,

,

$=x^{4}u^{4}+2x^{2}y^{2}u^{2}v^{2}+y^{4}v^{4}$

$a_{8,4}$

$b_{8,3}$

,

$x^{3}u^{5}+x^{2}yu^{2}v^{3}+xy^{2}u^{3}v^{2}+y^{3}v^{5}$

,

$=$

$\mathrm{a}8)8$

$=x^{6}u^{2}+2x^{3}y^{3}uv+y^{6}v^{2}$

$a_{8,2}$

,

$=x^{4}u^{4}+x^{3}yuv^{3}+xy^{3}u^{3}v+y^{4}v^{4}$

$a_{8,4}$

$=x^{2}u^{6}+2xyu^{3}v^{3}+y^{2}v^{6}$

$\mathrm{a}8|8$

$=x^{4}u^{4}+y^{4}v^{4}$

$a_{8_{1}4}$

$=x^{4}y^{2}v^{2}+x^{2}y^{4}u^{2}$

$b_{8,6}$

$=x^{2}u^{2}v^{4}+y^{2}u^{4}v^{2}$

$=x^{4}y^{3}v+x^{3}y^{4}u$

$a_{8,1}$

,

,

,

$b_{8,2}$

, ,

,

, , $=xu^{3}v^{4}+yu^{4}v^{3}$ , $=x^{3}uv^{4}+y^{3}u^{4}v$

$\mathrm{a}8|8$

$a_{3,\theta}$

$a_{8,7}$

$a_{12,6}$

$a_{12,3}$

$a_{12,7}$

$a_{12,5}$

$b_{12,5}$

$=$

$x^{4}y\tau t^{3}+xy^{4}u^{3}$

$=$

$x^{6}u^{6}+x^{4}y^{2}u^{4}v^{2}+x^{2}y^{4}u^{2}v^{4}+y^{6}v^{6}$

,

$=x^{5}u^{3}v^{4}+x^{3}y^{2}u\prime u^{6}+x^{2}y^{3}u^{6}v+y^{5}u^{4}v^{3}$

,

$=x^{6}yu^{2}v^{3}+x^{4}y^{3}v^{\mathrm{S}}+x^{3}y^{4}u^{\mathrm{S}}+xy^{6}u^{3}v^{2}$

,

$=x^{3}u^{5}v^{4}+x^{2}yu^{6}v^{3}+xy^{2}u^{3}v^{6}+y^{3}u^{4}v^{5}$

,

$=$

$b_{12,7}$

,

$a_{12,8}$

$a_{16,8}$

$=x^{7}yuv^{3}+x^{4}y^{4}u^{4}+x^{4}y^{4}v^{4}+xy’ u^{3}v$

,

$=x^{8}u^{8}+2x^{6}y^{2}u^{6}v^{2}+2x^{4}y^{4}u^{4}v^{4}+$

,

$=x^{7}u^{6}\iota^{4}’+2x^{5}y^{2}u^{3}v^{6}+x^{4}y^{3}u^{8}v+$

,

$=x^{8}yu^{4}v^{3}+2x^{6}y^{3}u^{2}v^{5}+x^{5}y^{4}u^{7}+x^{4}y^{5}v^{7}$ $+2x^{3}y^{6}u^{5}v^{2}+xy^{8}u^{3}v^{4}$

$a_{16,6}$

,

$=x^{4}u^{4}v^{4}+x^{3}yu^{\mathrm{S}}v^{3}+xy^{3}u^{3}v^{5}+y^{4}u^{4}v^{4}$

$x^{3}y^{4}uv^{8}\neq 2x^{2}y^{5}u^{6}v^{3}+y^{7}u^{4}v^{5}$

$a_{16,7}$

,

$x^{5}u^{3}v^{4}+x^{3}y^{2}u^{5}v^{2}+x^{2}y^{3}u^{2}v^{5}+y^{5}u^{4}v^{3}$

$2x^{2}y^{6}u^{2}v^{6}+y^{8}v^{8}$

$a_{16,9}$

,

$=x^{6}y^{3}u^{2}v+x^{5}y^{4}u^{3}+x^{4}y^{5}v^{3}+x^{3}y^{6}uv^{2}$

$=x^{7}u^{5}+x^{4}y^{3}u^{4}v+x^{3}y^{4}uv^{4}+y^{7}v^{5}$

$a_{12,5}$

$b_{12.4},$

,

$=x^{5}u^{3}+x^{3}y^{2}uv^{2}+x^{2}y^{3}u^{2}v+y^{5}v^{3}$

,

$=x^{9}yu^{3}v^{3}+x^{7}y^{3}uv^{5}+x^{6}y^{4}u^{6}+x^{6}y^{4}u^{2}v^{4}$ $+x^{4}y^{6}u^{4}v^{2}+x^{4}y^{6}v^{6}+x^{3}y^{7}u^{5}v+xy^{9}u^{3}v^{3}$

$a_{1\mathrm{B},10}$

,

$=x^{6}u^{6}v^{4}+x^{5}yu^{7_{I}}v^{3}+x^{4}y^{2}u^{4}v^{6}+$

2 $x^{3}y^{3}u^{5}v^{5}+x^{2}y^{4}u^{6}v^{4}+xy^{5}u^{3}v^{7}+y^{6}u^{4}v^{6}$, $a_{16,8}$

$=x^{8}u^{4}v^{4}+x^{6}y^{2}u^{6}n^{2}+2x^{5}y^{3}u^{3}v^{5}\neq$

$\Phi_{G_{1}\oplus G_{1}}(t)$

) are

90 $2x^{3}y^{5}u^{5}v^{3}+x^{2}y^{6}u^{2}v^{6}+y^{8}u^{4}v^{4}$

Let

$G_{2}\oplus$

G2 be the group generated by the linear transformations:

$(\begin{array}{lll}i0 \mathrm{O} 00100 00i0 0001 \end{array})$

The ring of invariants for

$G_{2}\oplus G_{2}$

,

$(\begin{array}{lll}1 \mathrm{O} 000i00 001 0000i \end{array})$

is denoted by $\mathcal{R}_{2}=\mathbb{C}[x,y, u, v]^{G_{2}\oplus G_{2}}$

Molien series for

$G_{2}\oplus G_{2}$

.

is $\Phi_{G_{2}\oplus G_{2}}(t)=\frac{1+6t^{4}+9t^{8}}{(1-t^{4})^{4}}$

$\mathrm{P}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{a}1^{\tau}\mathrm{y}\mathrm{I}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{s}$

of the group

and secondary invariants of

$G_{2}\oplus G_{2}$

$G_{2}\oplus G_{2}$

arc $=$

$x^{4}$

$c_{4,0}$

$d_{4,0}$

$=$

$y^{4}$

$=$

$u^{4}$

$c_{4,4}$

$d_{4,4}$

$=$

$v^{4}$

,

, ,

are $c_{0,0}$

$=$

$c_{4,2}$

$=$

$1$

,

$x^{2}u^{2}$

,

$y^{2}v_{7}^{2}$

$d_{4,2}$

$=$

$c_{4,1}$

$=$

$x^{3}u$

$c_{4,3}$

$=$

$xu^{3}$

$d_{4,1}$

$=$

$y^{3}v$

$d_{4,3}$

$=$

$yv^{3}$

$c_{8,4}$

$=$

$x^{2}y^{2}u^{2}v^{2}$

$=$

$x^{2}y^{3}u^{2}v$

$c_{8,5}$

$=$

$x^{2}yu^{2}v^{3}$

$d_{8,3}$

$=$

$x^{3}y^{\mathrm{q}}.uv^{2}$

$d_{8,5}$

$=$

$xy^{2}u^{3}v^{2}$

$c_{8,2}$

$=$

$x^{3}y^{3}uv$

,

$d_{8,4}$

$=$

$x^{3}yuv^{3}$

,

$e_{8,4}$

$=$

$xy^{3}u^{3}v$

,

$=$

$xyu^{3}v^{3}$

.

$\mathrm{c}_{8,3}$

$c_{8,6}$

,

, ,

,

, , ,

,

,

.

.

91 Let

$G_{3}\oplus G_{3}$

be the group generated by the linear transformations;

(

$00$

01 01 $\frac{00}{0}1$

The ring of invariants for

$G_{3}\oplus G_{3}$

$\ovalbox{\tt\small REJECT}$

,

$\ovalbox{\tt\small REJECT}_{0}^{1}00\frac{0}{0,0’}10001-1000$

is denoted by $\mathcal{R}_{3}=\mathbb{C}[x,y,u, v]^{G_{\}\oplus G\mathrm{a}}$

Molien series for

$G_{3}\oplus G_{3}$

).

.

is computed to be $\Phi_{G_{3}\oplus G_{3}}(t)=\frac{1+2t^{2}+t^{4}}{(1-t^{2})^{4}}$

.

Prim ary invarian ts of the group are given by; $f_{2,0}$

$=x^{2}$

$g_{2,0}$

$=$

$y^{2}$

$f_{2,2}$

$=$

$u^{2}$

$g_{2_{1}2}$

, , ,

$=v^{2}$

and secondary invariants are $f_{0,0}$

$=$

1,

$f_{2,1}$

$=$

$xu$

,

$g_{2,1}$

$=$

$yv$

,

$f_{4,2}$

$=$

xyuv

From finite groups to Jacobi forms

6

Definition of Jacobi form

6,1

defined on . A holomorphic function Let be a subgroup of it satisfies if $m$ group a Jacobi form of weight and index for $k$

(3)

$f(\tau, z)$

$SL(2_{1}\mathbb{Z})$

$\Gamma$

$f( \frac{a\tau+b}{c\tau+d},$

$\frac{z}{c\tau+d})=(c\tau+d)^{k}e^{2\pi\dot{\mathrm{a}}mcz/(\mathrm{c}\tau+d)}f(\tau,z)$

$\Gamma \mathrm{x}$

(or

and (4) $f(\tau,z+\lambda\tau+\mu)=e^{-2\pi m(\lambda^{2}+2\lambda z)}f(\tau,z)$ for any modular group we let For a subgroup of weight and index $m$ associated with the group $\Gamma$

$k$

$\mathrm{t}1_{1}\mathrm{e}$

$\{$

$acdJb\backslash \in\Gamma$

is called a

$(\lambda, \mu).\in \mathbb{Z}^{2}$

,

.

$Jac(k, m,\Gamma \mathrm{x} \mathbb{Z}^{2})$

$\Gamma \mathrm{x}$

$\mathbb{H}\mathrm{x}\mathbb{C}$

$\mathbb{Z}^{2}$

$\mathbb{Z}^{2}$

.

to denote Jacobi forms of

92 6,2

A result

Proposition 4 We have the following injective isomorphisms berween the rings $\mathbb{C}[x, y, u,v]^{G_{1}\oplus G_{1}}$

$\mathrm{c}arrow$

$k\in \mathbb{Z}_{>0},m\geq 0\oplus Jac(k, m, \Gamma_{\mathrm{D}}(\underline{?})\mathrm{x} \mathbb{Z}^{2})$

$\cap$

$\cap$

$\mathbb{C}[x, y, u,\cdot v]^{G_{2}\oplus G_{2}}$

$\mapsto$

$k\in \mathbb{Z}_{>},m\geq 0\oplus_{0}Jac(k, m, \Gamma_{0}(4)\mathrm{x} \mathbb{Z}^{2})$

$\cap$

$\cap$

y, u,

$\mathbb{C}[x,$

$v]^{G\mathrm{a}\oplus G_{3}}$

$\prec$ $h\in \mathbb{Z}_{>0}\oplus$

Jac

$(\ ,$

where the isomorphism is realized by sending x to z) respectively.

m,

$(\Gamma_{0}(4)\cap\Gamma(2))\mathrm{x} \mathbb{Z}^{2})$

to

$\varphi_{3}(\tau\grave{)}fy$

$\varphi_{2}(\tau)$

, u to

,

$\varphi_{3}(\tau,$

z) and v to

$\varphi_{2}(\tau,$

7 Let

Jacobi forms associated with $G_{4}\oplus G_{4}$

.

$\Gamma_{0}(4)$

bc the group generated by the linear transformation:

$(\begin{array}{lll}10 \mathrm{O} \mathrm{O}0i00 0010 000i \end{array})$

,

then we know that $\mathbb{C}[x, y, u, v]^{G_{4}\oplus G_{4}}=\mathcal{R}_{2}$

where and index 1 for

$v^{4}$

$u$

,

$\varphi_{3}(\tau, z)$

$\frac{1}{2}$

$1_{k}$

c.obi form of weight 2 and in dex 4 Jacobi forms of weight 2 and index 2 Jacobi forms of weight 2 and index 1 Jacobi forms of weight 2 and index 3

$\}arrow$

$\varphi_{2}(\tau, z)^{4}\mathrm{J}.\mathrm{a}$

$\vdasharrow$

$\varphi_{2}(\tau)^{2}\varphi_{2}(\tau, z)^{2}$

$y^{3}v$

$\succarrow$

$\varphi_{2}(\tau)^{3}\varphi_{2}(\tau, z)$

$yv^{3}$

$\vdash+$

$\varphi_{2}(\tau)\varphi_{2}(\tau_{\gamma}z)^{3}$

$y^{2}v^{2}$

$y^{2}v^{2}\mathcal{R}_{2}\oplus vy^{3}\mathcal{R}_{2}\oplus yv^{3}\mathcal{R}_{2}$

To we associate , and it gives Jacobi fo rms of weight and character . In the same way we have

$\mathcal{R}_{2}=\mathbb{C}[x, y^{4}, u, v^{4}]$

$\Gamma_{0}(4)$

.

%

$1\mathrm{S}$

For the proof of these statements we need algebraic properties of the automorphic factor of the Jacobi forms.

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$\mathrm{B}\mathrm{i}\mathrm{r}^{1}\kappa$

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