nLab congruence subgroup




Of the modular group

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

p n:SL 2()SL 2(/n) 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 /n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} to the integers modulo n.

The mod-nn congruence subgroups of the special linear group SL 2()SL_2(\mathbb{Z}) (essentially the modular group) are the preimages under p np_n of subgroups of SL 2(/n)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:

Γ(n)ker(p n)=p n 1({1 0 0 1}). \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 Γ(n)\Gamma(n) are congruent modulo nn to the identity.

The other two congruence subgroups having special symbols are

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


Relation to spin structures

See at level structure – relation to spin structure.



A list of congruence subgroups is provided in

Last revised on December 8, 2022 at 23:18:42. See the history of this page for a list of all contributions to it.