Contents

# Contents

## Definition

### Of the modular group

Let $n \in \mathbb{N}$ be a natural number. Write

$p_n \;\colon\; SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/n\mathbb{Z})$

for the projection from the special linear group induced by the quotient projection $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ to the integers modulo n.

The mod-$n$ congruence subgroups of the special linear group $SL_2(\mathbb{Z})$ (essentially the modular group) are the preimages under $p_n$ of subgroups of $SL_2(\mathbb{Z}/n\mathbb{Z})$.

Some of these have traditional names and symbols;

The principal congruence subgroup is the preimage of the trivial group:

$\Gamma(n) \coloneqq ker(p_n) = p_n^{-1}\left(\left\{\array{ 1 & 0 \\ 0 & 1}\right\}\right) \,.$

This is the origin of the term: the elements of $\Gamma(n)$ are congruent modulo $n$ to the identity.

The other two congruence subgroups having special symbols are

$\Gamma_0(n) \coloneqq p_n^{-1}\left(\left\{\array{ \ast & \ast \\ 0 & \ast}\right\}\right)$
$\Gamma_1(n) \coloneqq p_n^{-1}\left(\left\{\array{ 1 & \ast \\ 0 & \ast}\right\}\right)$

## Examples

A list of congruence subgroups is provided in