## Return to Question

Hi,

I've a question regarding the congruence subgroup.

We have the following congruence subgroup

```$\Gamma^* := \left\{ M \in \Gamma | XM \equiv X(mod \; \mathbb{Z}^2),\; m \cdot det|\begin
{smallmatrix} X \\ XM \end{smallmatrix}| \in \mathbb{Z} \right\}$```

of $\Gamma$.

Where

$\Gamma$ is a subgroup of $SL_2(\mathbb{Z})$,

`$M = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma$`, `$X = \begin{pmatrix} \lambda & \mu \end{pmatrix}\in \mathbb{Q}^2$`, $m \in \mathbb{Z}$

This subgroup can also be written as

`$\Gamma^* = \left\{ \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma | (a-1)\lambda + c \mu ,\; b \lambda + (d-1)\mu,\; m(c \mu^2 + (d-a)\lambda\mu-b\lambda^2) \in \mathbb{Z} \right\}$`

and hence contains $\Gamma \cap \Gamma \left(\frac{N^2}{(N,m)}\right)$ if `$NX \in \mathbb{Z}^2$`.

My question is, why is $\Gamma^* \supset \Gamma \cap \Gamma \left(\frac{N^2}{(N,m)}\right)$ true? What does $\Gamma \left(\frac{N^2}{(N,m)}\right)$ mean exactly? Actually I only have found references to the main congruence subgroup $\Gamma(N)$ (but never anything to a more complex form).

Thanks, Jan

2 improved formatting

Hi,

I've a question regarding the congruence subgroup.

We have the following congruence subgroup

```$\Gamma^* := \left\{ M \in \Gamma | XM \equiv X(mod \; \mathbb{Z}^2),\; m \cdot det|\begin
{smallmatrix} X \\ XM \end{smallmatrix}| \in \mathbb{Z} \right\}$```

of $\Gamma$.

Where

$\Gamma$ is a subgroup of $SL_2(\mathbb{Z})$,

`$M = \{ \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma$`, `$X = \begin{pmatrix} \lambda & \mu \end{pmatrix}\in \mathbb{Q}^2$`

This subgroup can also be written as

`$\Gamma^* = \left\{ \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma | (a-1)\lambda + c \mu ,\; b \lambda + (d-1)\mu,\; m(c \mu^2 + (d-a)\lambda\mu-b\lambda^2) \in \mathbb{Z} \right\}$`

and hence contains $\Gamma \cap \Gamma \left(\frac{N^2}{(N,m)}\right)$ if `$NX \in \mathbb{Z}^2$`.

My question is, why is $\Gamma^* \supset \Gamma \cap \Gamma \left(\frac{N^2}{(N,m)}\right)$ true? What does $\Gamma \left(\frac{N^2}{(N,m)}\right)$ mean exactly? Actually I only have found references to the main congruence subgroup $\Gamma(N)$ (but never anything to a more complex form).

Thanks, Jan

1

# congruence subgroup

Hi,

I've a question regarding the congruence subgroup.

We have the following congruence subgroup

```$\Gamma^* := \left\{ M \in \Gamma | XM \equiv X(mod \; \mathbb{Z}^2),\; m \cdot det|\begin
{smallmatrix} X \\ XM \end{smallmatrix}| \in \mathbb{Z} \right\}$``` of $\Gamma$.

Where

$\Gamma$ is a subgroup of $SL_2(\mathbb{Z})$,

`$M = \{ \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma$`, `$X = \begin{pmatrix} \lambda & \mu \end{pmatrix}\in \mathbb{Q}^2$`

This subgroup can also be written as

`$\Gamma^* = \left\{ \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma | (a-1)\lambda + c \mu ,\; b \lambda + (d-1)\mu,\; m(c \mu^2 + (d-a)\lambda\mu-b\lambda^2) \in \mathbb{Z} \right\}$`

and hence contains $\Gamma \cap \Gamma \left(\frac{N^2}{(N,m)}\right)$ if `$NX \in \mathbb{Z}^2$`.

My question is, why is $\Gamma^* \supset \Gamma \cap \Gamma \left(\frac{N^2}{(N,m)}\right)$ true? What does $\Gamma \left(\frac{N^2}{(N,m)}\right)$ mean exactly? Actually I only have found references to the main congruence subgroup $\Gamma(N)$ (but never anything to a more complex form).

Thanks, Jan
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