Congruence Subgroups

We denote by \({\operatorname{SL}}_2({\mathbb{Z}})\) the group of \(2\) by \(2\) matrices with integer coefficients and determinant \(1\). The group \({\operatorname{PSL}}_2({\mathbb{Z}})\) is the projectivization of \({\operatorname{SL}}_2({\mathbb{Z}})\). For any integer \(N\) we have groups

\[\begin{split}\Gamma_0(N)=\left\{\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in{\operatorname{SL}}_2({\mathbb{Z}}) \ \Big| \begin{pmatrix}a & b\\ c & d\end{pmatrix} \equiv \begin{pmatrix}* & *\\ 0 & *\end{pmatrix} \bmod N\right\}\end{split}\]

\[\begin{split}\Gamma_1(N)=\left\{\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in{\operatorname{SL}}_2({\mathbb{Z}}) \ \Big| \begin{pmatrix}a & b\\ c & d\end{pmatrix} \equiv \begin{pmatrix}1 & *\\ 0 & 1\end{pmatrix} \bmod N\right\}\end{split}\]

\[\begin{split}\Gamma(N)=\left\{\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in{\operatorname{SL}}_2({\mathbb{Z}}) \ \Big| \begin{pmatrix}a & b\\ c & d\end{pmatrix} \equiv \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix} \bmod N\right\}\end{split}\]

\[\begin{split}\Gamma^1(N)=\left\{\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in{\operatorname{SL}}_2({\mathbb{Z}}) \ \Big| \begin{pmatrix}a & b\\ c & d\end{pmatrix} \equiv \begin{pmatrix}1 & 0\\ * & 1\end{pmatrix} \bmod N\right\}\end{split}\]

\[\begin{split}\Gamma^0(N)=\left\{\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in{\operatorname{SL}}_2({\mathbb{Z}}) \ \Big| \begin{pmatrix}a & b\\ c & d\end{pmatrix} \equiv \begin{pmatrix}* & 0\\ * & *\end{pmatrix} \bmod N\right\}\end{split}\]

A congruence subgroup is any discrete subgroup \(\Gamma\) of \({\operatorname{SL}}_2({\mathbb{R}})\) which is commensurable with \({\operatorname{SL}}_2({\mathbb{Z}})\), that is, \(\Gamma\cap{\operatorname{SL}}_2({\mathbb{Z}})\) has finite index in \(\Gamma\) and in \({\operatorname{SL}}_2({\mathbb{Z}})\), and such that \(\Gamma(N)\) is contained in \(G\) for some \(N\). The level \(N\) of a congruence subgroup \(G\) is the greatest integer \(N\) such that \(\Gamma(N)\) is contained in \(\Gamma\). We will abuse notation and also refer to the projectivizations of these groups by the same names.

Creation of Subgroups of \({\operatorname{PSL}}_2({\mathbb{R}})\)

PSL2(R): Rng -> GrpPSL2

Returns \({\operatorname{PSL}}_2(R)\), the projective linear group over the ring \(R\).

Gamma0(N): RngIntElt -> GrpPSL2

The group \(\Gamma_0(N)\) for any positive integer \(N\).

Gamma1(N): RngIntElt -> GrpPSL2

The group \(\Gamma_1(N)\) for any positive integer \(N\).

GammaUpper0(N): RngIntElt -> GrpPSL2

The group \(\Gamma^0(N)\) for any positive integer \(N\).

GammaUpper1(N): RngIntElt -> GrpPSL2

The group \(\Gamma^1(N)\) for any positive integer \(N\).

CongruenceSubgroup(N): RngIntElt -> GrpPSL2

The group \(\Gamma(N)\) for any positive integer \(N\).

CongruenceSubgroup(i, N): RngIntElt, RngIntElt -> GrpPSL2

For a positive integer \(N\) and \(i=0,1,2,3,\) or \(4\), this is the group \(\Gamma_0(N)\), \(\Gamma_1(N)\), \(\Gamma(N)\), \(\Gamma^1(N)\), or \(\Gamma^0(N)\) respectively.

CongruenceSubgroup([N,M,P]): SeqEnum -> GrpPSL2

This is the congruence subgroup consisting of 2 by 2 matrices with integer coefficients \([a,b,c,d]\) with \(b = 0 \pmod P\), \(c = 0 \pmod N\), and \(a = d = 1 \pmod M\). It is required that \(M\) divides \(NP\).

Intersection(G, H): GrpPSL2, GrpPSL2 -> GrpPSL2

G meet H: GrpPSL2, GrpPSL2 -> GrpPSL2

The intersection of congruence subgroups \(G\) and \(H\).

Example: Defining Congruence Subgroups (ex-5fb4fb)

Examples of defining different congruence subgroups:

> G := PSL2(Integers());
> H := CongruenceSubgroup([2,3,6]);
> H;
Gamma_0(2) intersection Gamma^1(3) intersection Gamma^0(2)
> K := CongruenceSubgroup(0,5);
> K meet H;
Gamma_0(10) intersection Gamma^1(3) intersection Gamma^0(2)

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Relations

G eq H: GrpPSL2, GrpPSL2 -> BoolElt

Returns true if and only if the congruence subgroups \(G\) and \(H\) are equal.

H subset G: GrpPSL2, GrpPSL2 -> BoolElt

For congruence subgroups \(G\) and \(H\) contained in \({\operatorname{PSL}}_2({\mathbb{Z}})\), returns true if and only if \(H\) is a subgroup of \(G\).

Index(G, H): GrpPSL2, GrpPSL2 -> RngIntElt

For congruence subgroups \(G\) and \(H\), returns the index of \(G\) in \(H\) provided \(G\) is a subgroup of \(H\).

Index(G): GrpPSL2 -> RngIntElt

For \(G\) a congruence subgroup in \({\operatorname{PSL}}_2({\mathbb{Z}})\), returns the index in \({\operatorname{PSL}}_2({\mathbb{Z}})\).

Basic Attributes

Level(G): GrpPSL2 -> RngIntElt

The level of a congruence subgroup \(G\).

IsCongruence(G): GrpPSL2 -> BoolElt

Returns true if and only if \(G\) is a congruence subgroup.

IsGamma0(G): GrpPSL2 -> BoolElt

Returns true if and only if \(G\) is equal to \(\Gamma_0(N)\) for some integer \(N\).

IsGamma1(G): GrpPSL2 -> BoolElt

Returns true if and only if \(G\) is equal to \(\Gamma_1(N)\) for some integer \(N\).

BaseRing(G): GrpPSL2 -> Rng

Returns the base ring over which matrices of the congruence subgroup \(G\) are defined.

Identity(G): GrpPSL2 -> GrpPSL2Elt

Returns the identity matrix in the congruence subgroup \(G\).