nLab congruence subgroup

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 Moebius special linear group $SL(2,\mathbb{Z}$ 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

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

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

Properties

Relation to spin structures

See at level structure – relation to spin structure.

Examples

• level n subgroup of the modular group

Related concepts

• Torelli group

• modular curve

• modular equivariant elliptic cohomology

References

• Andrew Putman: The Torelli group and congruence subgroups of the mapping class group, in: Moduli Spaces of Riemann Surfaces, IAS/Park City Mathematics Series 20 (2013) [arXiv:1201.3946, doi:10.1090/pcms/020]

See also:

• Wikipedia, Congruence subgroup

A list of congruence subgroups is provided in

• Sebastian Pauli, Congruence subgroups of PSL(2,Z)