Arithmetic subgroups defined by permutations of cosets

A subgroup of finite index \(H\) of a finitely generated group \(G\) is completely described by the action of a set of generators of \(G\) on the right cosets \(H \backslash G = \{Hg\}_{g \in G}\). After some arbitrary choice of numbering one can identify the action of generators as elements of a symmetric group acting transitively (and satisfying the relations of the relators in G). As \({\rm SL}_2(\ZZ)\) has a very simple presentation as a central extension of a free product of cyclic groups, one can easily design algorithms from this point of view.

The generators of \({\rm SL}_2(\ZZ)\) used in this module are named as follows \(s_2\), \(s_3\), \(l\), \(r\) which are defined by

\[\begin{split}s_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},\quad s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad l = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix},\quad r = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}.\end{split}\]

Those generators satisfy the following relations

\[s_2^2 = s_3^3 = -1, \quad r = s_2^{-1}\ l^{-1}\ s_2.\]

In particular not all four are needed to generate the whole group \({\rm SL}_2(\ZZ)\). Three couples which generate \({\rm SL}_2(\ZZ)\) are of particular interest:

  • \((l,r)\) as they are also semigroup generators for the semigroup of matrices in \({\rm SL}_2(\ZZ)\) with non-negative entries,
  • \((l,s_2)\) as they are closely related to the continued fraction algorithm,
  • \((s_2,s_3)\) as the group \({\rm PSL}_2(\ZZ)\) is the free product of the finite cyclic groups generated by these two elements.

Part of these functions are based on Chris Kurth’s KFarey package [Kur08]. For tests see the file sage.modular.arithgroup.tests.

REFERENCES:

[AtSD71]A. O. L. Atkin and H. P. F. Swinnerton-Dyer, “Modular forms on noncongruence subgroups”, Proc. Symp. Pure Math., Combinatorics (T. S. Motzkin, ed.), vol. 19, AMS, Providence 1971
[Hsu96]Tim Hsu, “Identifying congruence subgroups of the modular group”, Proc. AMS 124, no. 5, 1351-1359 (1996)
[Hsu97]Tim Hsu, “Permutation techniques for coset representations of modular subgroups”, in L. Schneps (ed.), Geometric Galois Actions II: Dessins d’Enfants, Mapping Class Groups and Moduli, volume 243 of LMS Lect. Notes, 67-77, Cambridge Univ. Press (1997)
[Go09]Alexey G. Gorinov, “Combinatorics of double cosets and fundamental domains for the subgroups of the modular group”, preprint Arxiv 0901.1340
[KSV11](1, 2, 3, 4) Ian Kiming, Matthias Schuett and Helena Verrill, “Lifts of projective congruence groups”, J. London Math. Soc. (2011) 83 (1): 96-120, http://dx.doi.org/10.1112/jlms/jdq062. Arxiv version: Arxiv 0905.4798.
[Kul91]Ravi Kulkarni “An arithmetic geometric method in the study of the subgroups of the modular group”, American Journal of Mathematics 113 (1991), no 6, 1053-1133
[Kur08](1, 2) Chris Kurth, “K Farey package for Sage”
[KuLo]Chris Kurth and Ling Long, “Computations with finite index subgroups of \({\rm PSL}_2(\ZZ)\) using Farey symbols”, Arxiv 0710.1835
[Ve]Helena Verrill, “Fundamental domain drawer”, Java program, http://www.math.lsu.edu/~verrill/

TODO:

  • modular Farey symbols

  • computation of generators of a modular subgroup with a standard surface group presentation. In other words, compute a presentation of the form

    \[\langle x_i,y_i,c_j |\ \prod_i [x_i,y_i] \prod_j c_j^{\nu_j} = 1\rangle\]

    where the elements \(x_i\) and \(y_i\) are hyperbolic and \(c_j\) are parabolic (\(\nu_j=\infty\)) or elliptic elements (\(\nu_j < \infty\)).

  • computation of centralizer.

  • generation of modular (even) subgroups of fixed index.

AUTHORS:

  • Chris Kurth (2008): created KFarey package
  • David Loeffler (2009): adapted functions from KFarey for inclusion into Sage
  • Vincent Delecroix (2010): implementation for odd groups, new design, improvements, documentation
  • David Loeffler (2011): congruence testing for odd subgroups, enumeration of liftings of projective subgroups
sage.modular.arithgroup.arithgroup_perm.ArithmeticSubgroup_Permutation(L=None, R=None, S2=None, S3=None, relabel=False, check=True)

Construct a subgroup of \({\rm SL}_2(\ZZ)\) from the action of generators on its right cosets.

Return an arithmetic subgroup knowing the action, given by permutations, of at least two standard generators on the its cosets. The generators considered are the following matrices:

\[\begin{split}s_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},\quad s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad l = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix},\quad r = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}.\end{split}\]

An error will be raised if only one permutation is given. If no arguments are given at all, the full modular group \({\rm SL}(2, \ZZ)\) is returned.

INPUT:

  • S2, S3, L, R - permutations - action of matrices on the right cosets (each coset is identified to an element of \(\{1,\dots,n\}\) where \(1\) is reserved for the identity coset).
  • relabel - boolean (default: False) - if True, renumber the cosets in a canonical way.
  • check - boolean (default: True) - check that the input is valid (it may be time efficient but less safe to set it to False)

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)",S3="(1,2,3)")
sage: G
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,4)
 S3=(1,2,3)
 L=(1,4,3)
 R=(2,4,3)
sage: G.index()
4

sage: G = ArithmeticSubgroup_Permutation(); G
Arithmetic subgroup with permutations of right cosets
 S2=()
 S3=()
 L=()
 R=()
sage: G == SL2Z
True

Some invalid inputs:

sage: ArithmeticSubgroup_Permutation(S2="(1,2)")
Traceback (most recent call last):
...
ValueError: Need at least two generators
sage: ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(3,4,5)")
Traceback (most recent call last):
...
ValueError: Permutations do not generate a transitive group
sage: ArithmeticSubgroup_Permutation(L="(1,2)",R="(1,2,3)")
Traceback (most recent call last):
...
ValueError: Wrong relations between generators
sage: ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="()")
Traceback (most recent call last):
...
ValueError: S2^2 does not equal to S3^3
sage: ArithmeticSubgroup_Permutation(S2="(1,4,2,5,3)", S3="(1,3,5,2,4)")
Traceback (most recent call last):
...
ValueError: S2^2 = S3^3 must have order 1 or 2

The input checks can be disabled for speed:

sage: ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(3,4,5)", check=False) # don't do this!
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)
 S3=(3,4,5)
 L=(1,2)(3,5,4)
 R=(1,2)(3,4,5)
class sage.modular.arithgroup.arithgroup_perm.ArithmeticSubgroup_Permutation_class

Bases: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup

A subgroup of \({\rm SL}_2(\ZZ)\) defined by the action of generators on its coset graph.

The class stores the action of generators \(s_2, s_3, l, r\) on right cosets \(Hg\) of a finite index subgroup \(H < {\rm SL}_2(\ZZ)\). In particular the action of \({\rm SL}_2(\ZZ)\) on the cosets is on right.

\[\begin{split}s_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},\quad s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad l = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix},\quad r = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}.\end{split}\]

TEST:

sage: s2 = PermutationGroupElement('(1,2)(3,4)(5,6)')
sage: s3 = PermutationGroupElement('(1,3,5)(2,4,6)')
sage: G = ArithmeticSubgroup_Permutation(S2=s2, S3=s3)
sage: G.S2() == s2
True
sage: G.S3() == s3
True
sage: G == ArithmeticSubgroup_Permutation(L=G.L(), R=G.R())
True
sage: G == ArithmeticSubgroup_Permutation(L=G.L(), S2=G.S2())
True
sage: G == ArithmeticSubgroup_Permutation(L=G.L(), S3=G.S3())
True
sage: G == ArithmeticSubgroup_Permutation(R=G.R(), S2=G.S2())
True
sage: G == ArithmeticSubgroup_Permutation(R=G.R(), S3=G.S3())
True
sage: G == ArithmeticSubgroup_Permutation(S2=G.S2(), S3=G.S3())
True

sage: G = ArithmeticSubgroup_Permutation(S2='',S3='')
sage: TestSuite(G).run()

sage: S2 = '(1,2)(3,4)(5,6)'
sage: S3 = '(1,2,3)(4,5,6)'
sage: G = ArithmeticSubgroup_Permutation(S2=S2, S3=S3)
sage: TestSuite(G).run()
L()

Return the action of the matrix \(l\) as a permutation of cosets.

\[\begin{split}l = \begin{pmatrix}1&1\\0&1\end{pmatrix}\end{split}\]

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)")
sage: G.L()
(1,3)
R()

Return the action of the matrix \(r\) as a permutation of cosets.

\[\begin{split}r = \begin{pmatrix}1&0\\1&1\end{pmatrix}\end{split}\]

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)")
sage: G.R()
(2,3)
S2()

Return the action of the matrix \(s_2\) as a permutation of cosets.

\[\begin{split}s_2 = \begin{pmatrix}0&-1\\1&0\end{pmatrix}\end{split}\]

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)")
sage: G.S2()
(1,2)
S3()

Return the action of the matrix \(s_3\) as a permutation of cosets.

\[\begin{split}s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad\end{split}\]

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)")
sage: G.S3()
(1,2,3)
congruence_closure()

Returns the smallest congruence subgroup containing self. If self is congruence, this is just self, but represented as a congruence subgroup data type. If self is not congruence, then it may be larger.

In practice, we use the following criterion: let \(m\) be the generalised level of self. If this subgroup is even, let \(n = m\), else let \(n = 2m\). Then any congruence subgroup containing self contains \(\Gamma(n)\) (a generalisation of Wohlfahrt’s theorem due to Kiming, Verrill and Schuett). So we compute the image of self modulo \(n\) and return the preimage of that.

EXAMPLE:

sage: Gamma1(3).as_permutation_group().congruence_closure()
Congruence subgroup of SL(2,Z) of level 3, preimage of:
 Matrix group over Ring of integers modulo 3 with 2 generators (
[1 2]  [1 1]
[0 1], [0 1]
)
sage: sage.modular.arithgroup.arithgroup_perm.HsuExample10().congruence_closure()  # long time (11s on sage.math, 2012)
Modular Group SL(2,Z)
coset_graph(right_cosets=False, s2_edges=True, s3_edges=True, l_edges=False, r_edges=False, s2_label='s2', s3_label='s3', l_label='l', r_label='r')

Return the right (or left) coset graph.

INPUT:

  • right_cosets - bool (default: False) - right or left coset graph
  • s2_edges - bool (default: True) - put edges associated to s2
  • s3_edges - bool (default: True) - put edges associated to s3
  • l_edges - bool (default: False) - put edges associated to l
  • r_edges - bool (default: False) - put edges associated to r
  • s2_label, s3_label, l_label, r_label - the labels to put on the edges corresponding to the generators action. Use None for no label.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="()")
sage: G
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)
 S3=()
 L=(1,2)
 R=(1,2)
sage: G.index()
2
sage: G.coset_graph()
Looped multi-digraph on 2 vertices
generalised_level()

Return the generalised level of this subgroup.

The generalised level of a subgroup of the modular group is the least common multiple of the widths of the cusps. It was proven by Wohlfart that for even congruence subgroups, the (conventional) level coincides with the generalised level. For odd congruence subgroups the level is either the generalised level, or twice the generalised level [KSV11].

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.generalised_level()
2
sage: G = Gamma0(3).as_permutation_group()
sage: G.generalised_level()
3
index()

Returns the index of this modular subgroup in the full modular group.

EXAMPLES:

sage: G = Gamma(2)
sage: P = G.as_permutation_group()
sage: P.index()
6
sage: G.index() == P.index()
True

sage: G = Gamma0(8)
sage: P = G.as_permutation_group()
sage: P.index()
12
sage: G.index() == P.index()
True

sage: G = Gamma1(6)
sage: P = G.as_permutation_group()
sage: P.index()
24
sage: G.index() == P.index()
True
is_normal()

Test whether the group is normal

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.is_normal()
True

sage: G = Gamma1(2).as_permutation_group()
sage: G.is_normal()
False
perm_group()

Return the underlying permutation group.

The permutation group returned is isomorphic to the action of the generators \(s_2\) (element of order two), \(s_3\) (element of order 3), \(l\) (parabolic element) and \(r\) (parabolic element) on right cosets (the action is on the right).

EXAMPLE:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample10().perm_group()
Permutation Group with generators [(1,2)(3,4)(5,6)(7,8)(9,10), (1,4)(2,5,9,10,8)(3,7,6), (1,7,9,10,6)(2,3)(4,5,8), (1,8,3)(2,4,6)(5,7,10)]
permutation_action(x)

Given an element x of \({\rm SL}_2(\ZZ)\), compute the permutation of the cosets of self given by right multiplication by x.

EXAMPLE:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample10().permutation_action(SL2Z([32, -21, -67, 44]))
(1,4,6,2,10,5,3,7,8,9)
random_element(initial_steps=30)

Returns a random element in this subgroup.

The algorithm uses a random walk on the Cayley graph of \({\rm SL}_2(\ZZ)\) stopped at the first time it reaches the subgroup after at least initial_steps steps.

INPUT:

  • initial_steps - positive integer (default: 30)

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2='(1,3)(4,5)',S3='(1,2,5)(3,4,6)')
sage: all(G.random_element() in G for _ in xrange(10))
True
relabel(inplace=True)

Relabel the cosets of this modular subgroup in a canonical way.

The implementation of modular subgroup by action of generators on cosets depends on the choice of a numbering. This function provides canonical labels in the sense that two equal subgroups whith different labels are relabeled the same way. The default implementation relabels the group itself. If you want to get a relabeled copy of your modular subgroup, put to False the option inplace.

ALGORITHM:

We give an overview of how the canonical labels for the modular subgroup are built. The procedure only uses the permutations \(S3\) and \(S2\) that define the modular subgroup and can be used to renumber any transitive action of the symmetric group. In other words, the algorithm construct a canonical representative of a transitive subgroup in its conjugacy class in the full symmetric group.

1. The identity is still numbered \(0\) and set the curent vertex to be the identity.

2. Number the cycle of \(S3\) the current vertex belongs to: if the current vertex is labeled \(n\) then the numbering in such way that the cycle becomes \((n, n+1, \ldots, n+k)\)).

3. Find a new current vertex using the permutation \(S2\). If all vertices are relabeled then it’s done, otherwise go to step 2.

EXAMPLES:

sage: S2 = "(1,2)(3,4)(5,6)"; S3 = "(1,2,3)(4,5,6)"
sage: G1 = ArithmeticSubgroup_Permutation(S2=S2,S3=S3); G1
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,4)(5,6)
 S3=(1,2,3)(4,5,6)
 L=(1,4,5,3)
 R=(2,4,6,3)
sage: G1.relabel(); G1
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,4)(5,6)
 S3=(1,2,3)(4,5,6)
 L=(1,4,5,3)
 R=(2,4,6,3)

sage: S2 = "(1,2)(3,5)(4,6)"; S3 = "(1,2,3)(4,5,6)"
sage: G2 = ArithmeticSubgroup_Permutation(S2=S2,S3=S3); G2
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,5)(4,6)
 S3=(1,2,3)(4,5,6)
 L=(1,5,6,3)
 R=(2,5,4,3)
sage: G2.relabel(); G2
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,4)(5,6)
 S3=(1,2,3)(4,5,6)
 L=(1,4,5,3)
 R=(2,4,6,3)

sage: S2 = "(1,2)(3,6)(4,5)"; S3 = "(1,2,3)(4,5,6)"
sage: G3 = ArithmeticSubgroup_Permutation(S2=S2,S3=S3); G3
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,6)(4,5)
 S3=(1,2,3)(4,5,6)
 L=(1,6,4,3)
 R=(2,6,5,3)
sage: G4 = G3.relabel(inplace=False)
sage: G4
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,4)(5,6)
 S3=(1,2,3)(4,5,6)
 L=(1,4,5,3)
 R=(2,4,6,3)
sage: G3 is G4
False

TESTS:

sage: S2 = "(1,2)(3,6)(4,5)"
sage: S3 = "(1,2,3)(4,5,6)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: H = G.relabel(inplace=False)
sage: G is H
False
sage: G._S2 is H._S2 or G._S3 is H._S3 or G._L is H._L or G._R is H._R
False
sage: G.relabel(inplace=False) is H
True
sage: H.relabel(inplace=False) is H
True
sage: G.relabel(); G
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,4)(5,6)
 S3=(1,2,3)(4,5,6)
 L=(1,4,5,3)
 R=(2,4,6,3)
sage: G.relabel(inplace=False) is G
True
class sage.modular.arithgroup.arithgroup_perm.EvenArithmeticSubgroup_Permutation(S2, S3, L, R, canonical_labels=False)

Bases: sage.modular.arithgroup.arithgroup_perm.ArithmeticSubgroup_Permutation_class

An arithmetic subgroup \(\Gamma\) defined by two permutations, giving the action of four standard generators

\[\begin{split}s_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},\quad s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad l = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix},\quad r = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}.\end{split}\]

by right multiplication on the coset representatives \(\Gamma \backslash {\rm SL}_2(\ZZ)\).

EXAMPLES:

Construct a noncongruence subgroup of index 7 (the smallest possible):

sage: a2 = SymmetricGroup(7)([(1,2),(3,4),(6,7)]); a3 = SymmetricGroup(7)([(1,2,3),(4,5,6)])
sage: G = ArithmeticSubgroup_Permutation(S2=a2, S3=a3); G
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)(6,7)
S3=(1,2,3)(4,5,6)
L=(1,4,7,6,5,3)
R=(2,4,5,7,6,3)
sage: G.index()
7
sage: G.dimension_cusp_forms(4)
1
sage: G.is_congruence()
False

Convert some standard congruence subgroups into permutation form:

sage: G = Gamma0(8).as_permutation_group()
sage: G.index()
12
sage: G.is_congruence()
True

sage: G = Gamma0(12).as_permutation_group()
sage: print G
Arithmetic subgroup of index 24
sage: G.is_congruence()
True

The following is the unique index 2 even subgroup of \({\rm SL}_2(\ZZ)\):

sage: w = SymmetricGroup(2)([2,1])
sage: G = ArithmeticSubgroup_Permutation(L=w, R=w)
sage: G.dimension_cusp_forms(6)
1
sage: G.genus()
0
coset_reps()

Return coset representatives.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)",S3="(1,2,3)")
sage: c = G.coset_reps()
sage: len(c)
4
sage: [g in G for g in c]
[True, False, False, False]
cusp_widths(exp=False)

Return the list of cusp widths of the group.

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.cusp_widths()
[2, 2, 2]
sage: G.cusp_widths(exp=True)
{2: 3}

sage: S2 = "(1,2)(3,4)(5,6)"
sage: S3 = "(1,2,3)(4,5,6)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: G.cusp_widths()
[1, 1, 4]
sage: G.cusp_widths(exp=True)
{1: 2, 4: 1}

sage: S2 = "(1,2)(3,4)(5,6)"
sage: S3 = "(1,3,5)(2,4,6)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: G.cusp_widths()
[6]
sage: G.cusp_widths(exp=True)
{6: 1}
is_congruence()

Return True if this is a congruence subgroup.

ALGORITHM:

Uses Hsu’s algorithm [Hsu96]. Adapted from Chris Kurth’s implementation in KFarey [Kur08].

EXAMPLES:

Test if \({\rm SL}_2(\ZZ)\) is congruence:

sage: a = ArithmeticSubgroup_Permutation(L='',R='')
sage: a.index()
1
sage: a.is_congruence()
True

This example is congruence – it’s Gamma0(3) in disguise:

sage: S2 = SymmetricGroup(4)
sage: l = S2((2,3,4))
sage: r = S2((1,3,4))
sage: G = ArithmeticSubgroup_Permutation(L=l,R=r)
sage: print G
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,4)
 S3=(1,4,2)
 L=(2,3,4)
 R=(1,3,4)
sage: G.is_congruence()
True

This one is noncongruence:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample10().is_congruence()
False
is_even()

Returns True if this subgroup contains the matrix \(-Id\).

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.is_even()
True
is_odd()

Returns True if this subgroup does not contain the matrix \(-Id\).

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.is_odd()
False
ncusps()

Return the number of cusps of this arithmetic subgroup.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)(5,6)",S3="(1,2,3)(4,5,6)")
sage: G.ncusps()
3
nu2()

Returns the number of orbits of elliptic points of order 2 for this arithmetic subgroup.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,4)(2)(3)",S3="(1,2,3)(4)")
sage: G.nu2()
2
nu3()

Returns the number of orbits of elliptic points of order 3 for this arithmetic subgroup.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,4)(2)(3)",S3="(1,2,3)(4)")
sage: G.nu3()
1
odd_subgroups()

Return a list of the odd subgroups of index 2 in \(\Gamma\), where \(\Gamma\) is this subgroup. (Equivalently, return the liftings of \(\bar{\Gamma} \le {\rm PSL}(2, \ZZ)\) to \({\rm SL}(2, \ZZ)\).)

See also

one_odd_subgroup(), which returns just one of the odd subgroups (which is much quicker than enumerating them all).

ALGORITHM:

  • If \(\Gamma\) has an element of order 4, then there are no index 2 odd subgroups, so return the empty set.

  • If \(\Gamma\) has no elements of order 4, then the permutation \(S_2\) is a combination of 2-cycles with no fixed points on \(\{1, \dots, N\}\). We construct the permutation \(\tilde{S}_2\) of \(\{1, \dots, 2N\}\) which has a 4-cycle \((a, b, a+N, b+N)\) for each 2-cycle \((a,b)\) in S2. Similarly, we construct a permutation \(\tilde{S}_3\) which has a 6-cycle \((a,b,c,a+N,b+N,c+N)\) for each 3-cycle \((a,b,c)\) in \(S_3\), and a 2-cycle \((a,a+N)\) for each fixed point \(a\) of \(S_3\).

    Then the permutations \(\tilde{S}_2\) and \(\tilde{S}_3\) satisfy \(\tilde{S}_2^2 = \tilde{S}_3^3 = \iota\) where \(\iota\) is the order 2 permutation interchanging \(a\) and \(a+N\) for each \(a\). So the subgroup corresponding to these permutations is an index 2 odd subgroup of \(\Gamma\).

  • The other index 2 odd subgroups of \(\Gamma\) are obtained from the pairs \(\tilde{S}_2, \tilde{S}_3^\sigma\) where \(\sigma\) is an element of the group generated by the 2-cycles \((a, a+N)\).

Studying the permutations in the first example below gives a good illustration of the algorithm.

EXAMPLES:

sage: G = sage.modular.arithgroup.arithgroup_perm.HsuExample10()
sage: [G.S2(), G.S3()]
[(1,2)(3,4)(5,6)(7,8)(9,10), (1,8,3)(2,4,6)(5,7,10)]
sage: X = G.odd_subgroups()
sage: for u in X: print [u.S2(), u.S3()]
[(1,2,11,12)(3,4,13,14)(5,6,15,16)(7,8,17,18)(9,10,19,20), (1,8,3,11,18,13)(2,4,6,12,14,16)(5,7,10,15,17,20)(9,19)]
[(1,2,11,12)(3,4,13,14)(5,6,15,16)(7,8,17,18)(9,10,19,20), (1,18,13,11,8,3)(2,4,6,12,14,16)(5,7,10,15,17,20)(9,19)]
[(1,2,11,12)(3,4,13,14)(5,6,15,16)(7,8,17,18)(9,10,19,20), (1,8,13,11,18,3)(2,4,6,12,14,16)(5,7,10,15,17,20)(9,19)]
[(1,2,11,12)(3,4,13,14)(5,6,15,16)(7,8,17,18)(9,10,19,20), (1,18,3,11,8,13)(2,4,6,12,14,16)(5,7,10,15,17,20)(9,19)]

A projective congruence subgroup may have noncongruence liftings, as the example of \(\bar{\Gamma}_0(6)\) illustrates (see [KSV11]):

sage: X = Gamma0(6).as_permutation_group().odd_subgroups(); Sequence([[u.S2(), u.S3()] for u in X],cr=True)
[
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,2,3,13,14,15)(4,5,6,16,17,18)(7,8,9,19,20,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,14,15,13,2,3)(4,5,6,16,17,18)(7,8,9,19,20,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,2,3,13,14,15)(4,17,6,16,5,18)(7,8,9,19,20,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,14,15,13,2,3)(4,17,6,16,5,18)(7,8,9,19,20,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,2,3,13,14,15)(4,5,6,16,17,18)(7,20,9,19,8,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,14,15,13,2,3)(4,5,6,16,17,18)(7,20,9,19,8,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,2,3,13,14,15)(4,17,6,16,5,18)(7,20,9,19,8,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,14,15,13,2,3)(4,17,6,16,5,18)(7,20,9,19,8,21)(10,11,12,22,23,24)]
]
sage: [u.is_congruence() for u in X]
[True, False, False, True, True, False, False, True]

Subgroups of large index may take a very long time:

sage: len(GammaH(11,[-1]).as_permutation_group().odd_subgroups())  # long time (up to 51s on sage.math, 2012)
2048
one_odd_subgroup(random=False)

Return an odd subgroup of index 2 in \(\Gamma\), where \(\Gamma\) is this subgroup. If the optional argument random is False (the default), this returns an arbitrary but consistent choice from the set of index 2 odd subgroups. If random is True, then it will choose one of these at random.

For details of the algorithm used, see the docstring for the related function odd_subgroups(), which returns a list of all the index 2 odd subgroups.

EXAMPLES:

Starting from \(\Gamma(4)\) we get back \(\Gamma(4)\):

sage: G = Gamma(4).as_permutation_group()
sage: print G.is_odd(), G.index()
True 48
sage: Ge = G.to_even_subgroup()
sage: Go = Ge.one_odd_subgroup()
sage: print Go.is_odd(), Go.index()
True 48
sage: Go == G
True

Strating from \(\Gamma(6)\) we get a different group:

sage: G = Gamma(6).as_permutation_group()
sage: print G.is_odd(), G.index()
True 144
sage: Ge = G.to_even_subgroup()
sage: Go = Ge.one_odd_subgroup()
sage: print Go.is_odd(), Go.index()
True 144
sage: Go == G
False

An error will be raised if there are no such subgroups, which occurs if and only if the group contains an element of order 4:

sage: Gamma0(10).as_permutation_group().one_odd_subgroup()
Traceback (most recent call last):
...
ValueError: Group contains an element of order 4, hence no index 2 odd subgroups

Testing randomness:

sage: G = Gamma(6).as_permutation_group().to_even_subgroup()
sage: G1 = G.one_odd_subgroup(random=True) # random
sage: G1.is_subgroup(G)
True
to_even_subgroup(relabel=True)

Return the subgroup generated by self and -Id. Since self is even, this is just self. Provided for compatibility.

EXAMPLE:

sage: G = Gamma0(4).as_permutation_group()
sage: H = G.to_even_subgroup()
sage: H == G
True
todd_coxeter()

Returns a 4-tuple (coset_reps, gens, l, s2) where coset_reps is a list of coset representatives of the subgroup, gens a list of generators, l and s2 are list that corresponds to the action of the matrix \(S2\) and \(L\) on the cosets.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2='(1,2)(3,4)',S3='(1,2,3)')
sage: reps,gens,l,s=G.todd_coxeter_l_s2()
sage: reps
[
[1 0]  [ 0 -1]  [1 2]  [1 1]
[0 1], [ 1  0], [0 1], [0 1]
]
sage: gens
[
[1 3]  [ 1  0]  [ 2 -3]
[0 1], [-1  1], [ 1 -1]
]
sage: l
[3, 1, 0, 2]
sage: s
[1, 0, 3, 2]
sage: S2 = SL2Z([0,-1,1,0])
sage: L = SL2Z([1,1,0,1])
sage: reps[0] == SL2Z([1,0,0,1])
True
sage: all(reps[i]*S2*~reps[s[i]] in G for i in xrange(4))
True
sage: all(reps[i]*L*~reps[l[i]] in G for i in xrange(4))
True
todd_coxeter_l_s2()

Returns a 4-tuple (coset_reps, gens, l, s2) where coset_reps is a list of coset representatives of the subgroup, gens a list of generators, l and s2 are list that corresponds to the action of the matrix \(S2\) and \(L\) on the cosets.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2='(1,2)(3,4)',S3='(1,2,3)')
sage: reps,gens,l,s=G.todd_coxeter_l_s2()
sage: reps
[
[1 0]  [ 0 -1]  [1 2]  [1 1]
[0 1], [ 1  0], [0 1], [0 1]
]
sage: gens
[
[1 3]  [ 1  0]  [ 2 -3]
[0 1], [-1  1], [ 1 -1]
]
sage: l
[3, 1, 0, 2]
sage: s
[1, 0, 3, 2]
sage: S2 = SL2Z([0,-1,1,0])
sage: L = SL2Z([1,1,0,1])
sage: reps[0] == SL2Z([1,0,0,1])
True
sage: all(reps[i]*S2*~reps[s[i]] in G for i in xrange(4))
True
sage: all(reps[i]*L*~reps[l[i]] in G for i in xrange(4))
True
todd_coxeter_s2_s3()

Returns a 4-tuple (coset_reps, gens, s2, s3) where coset_reps are coset representatives of the subgroup, gens is a list of generators, s2 and s3 are the action of the matrices \(S2\) and \(S3\) on the list of cosets.

The so called Todd-Coxeter algorithm is a general method for coset enumeration for a subgroup of a group given by generators and relations.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2='(1,2)(3,4)',S3='(1,2,3)')
sage: G.genus()
0
sage: reps,gens,s2,s3=G.todd_coxeter_s2_s3()
sage: g1,g2 = gens
sage: g1 in G and g2 in G
True
sage: g1
[-1  0]
[ 1 -1]
sage: g2
[-1  3]
[-1  2]
sage: S2 = SL2Z([0,-1,1,0])
sage: S3 = SL2Z([0,1,-1,1])
sage: reps[0] == SL2Z([1,0,0,1])
True
sage: all(reps[i]*S2*~reps[s2[i]] in G for i in xrange(4))
True
sage: all(reps[i]*S3*~reps[s3[i]] in G for i in xrange(4))
True
sage.modular.arithgroup.arithgroup_perm.HsuExample10()

An example of an index 10 arithmetic subgroup studied by Tim Hsu.

EXAMPLE:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample10()
Arithmetic subgroup with permutations of right cosets
 S2=(1,2)(3,4)(5,6)(7,8)(9,10)
 S3=(1,8,3)(2,4,6)(5,7,10)
 L=(1,4)(2,5,9,10,8)(3,7,6)
 R=(1,7,9,10,6)(2,3)(4,5,8)
sage.modular.arithgroup.arithgroup_perm.HsuExample18()

An example of an index 18 arithmetic subgroup studied by Tim Hsu.

EXAMPLE:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample18()
Arithmetic subgroup with permutations of right cosets
 S2=(1,5)(2,11)(3,10)(4,15)(6,18)(7,12)(8,14)(9,16)(13,17)
 S3=(1,7,11)(2,18,5)(3,9,15)(4,14,10)(6,17,12)(8,13,16)
 L=(1,2)(3,4)(5,6,7)(8,9,10)(11,12,13,14,15,16,17,18)
 R=(1,12,18)(2,6,13,9,4,8,17,7)(3,16,14)(5,11)(10,15)
class sage.modular.arithgroup.arithgroup_perm.OddArithmeticSubgroup_Permutation(S2, S3, L, R, canonical_labels=False)

Bases: sage.modular.arithgroup.arithgroup_perm.ArithmeticSubgroup_Permutation_class

TESTS:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="(1,3)(2,4)")
sage: G
Arithmetic subgroup with permutations of right cosets
 S2=(1,2,3,4)
 S3=(1,3)(2,4)
 L=(1,2,3,4)
 R=(1,4,3,2)
sage: TestSuite(G).run()
cusp_widths(exp=False)

Return the list of cusp widths.

INPUT:

exp - boolean (default: False) - if True, return a dictionnary with keys the possible widths and with values the number of cusp with that width.

EXAMPLES:

sage: G = Gamma1(5).as_permutation_group()
sage: G.cusp_widths()
[1, 1, 5, 5]
sage: G.cusp_widths(exp=True)
{1: 2, 5: 2}
is_congruence()

Test whether self is a congruence group, i.e.~whether or not it contains the subgroup \(\Gamma(n)\) for some \(n\).

For odd groups, we first test whether the group generated by \(G\) and \(-1\) is congruence, and then use a theorem of Kiming, Schuett and Verrill [KSV11], which shows that an odd subgroup is congruence if and only if it contains \(\Gamma(N)\) where \(N\) is twice its generalised level (the least common multiple of its cusp widths). We can therefore proceed by calculating the index of the subgroup of \({\rm SL}(2, \ZZ / N\ZZ)\) generated by the gens of self, and checking whether or not it has the same index as self.

EXAMPLES:

sage: GammaH(11,[4]).as_permutation_group().is_congruence()
True

The following example (taken from [KSV11]) shows that it may be the case that G is not congruence, even if its image in \({\rm PSL}(2,\ZZ)\) is congruence:

sage: S2 = "(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23)"
sage: S3 = "(1,14,15,13,2,3)(4,5,6,16,17,18)(7,8,9,19,20,21)(10,11,12,22,23,24)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: G.is_congruence()
False
sage: G.to_even_subgroup().is_congruence()
True

In fact G is a lifting to \({\rm SL}(2,\ZZ)\) of the group \(\bar{\Gamma}_0(6)\):

sage: G.to_even_subgroup() == Gamma0(6)
True
is_even()

Test whether the group is even.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,6,4,3)(2,7,5,8)",S3="(1,2,3,4,5,6)(7,8)")
sage: G.is_even()
False
is_odd()

Test whether the group is odd.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,6,4,3)(2,7,5,8)",S3="(1,2,3,4,5,6)(7,8)")
sage: G.is_odd()
True
ncusps()

Returns the number of cusps.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="(1,3)(2,4)")
sage: G.ncusps()
1

sage: G = Gamma1(3).as_permutation_group()
sage: G.ncusps()
2
nirregcusps()

Return the number of irregular cusps.

The cusps are associated to cycles of the permutations \(L\) or \(R\). The irregular cusps are the one which are stabilised by \(-Id\).

EXAMPLES:

sage: S2 = "(1,3,2,4)(5,7,6,8)(9,11,10,12)"
sage: S3 = "(1,3,5,2,4,6)(7,9,11,8,10,12)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: G.nirregcusps()
3
nregcusps()

Return the number of regular cusps of the group.

The cusps are associated to cycles of \(L\) or \(R\). The irregular cusps correspond to the ones which are not stabilised by \(-Id\).

EXAMPLES:

sage: G = Gamma1(3).as_permutation_group()
sage: G.nregcusps()
2
nu2()

Return the number of elliptic points of order 2.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="(1,3)(2,4)")
sage: G.nu2()
0

sage: G = Gamma1(2).as_permutation_group()
sage: G.nu2()
1
nu3()

Return the number of elliptic points of order 3.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="(1,3)(2,4)")
sage: G.nu3()
2

sage: G = Gamma1(3).as_permutation_group()
sage: G.nu3()
1
to_even_subgroup(relabel=True)

Returns the group with \(-Id\) added in it.

EXAMPLES:

sage: G = Gamma1(3).as_permutation_group()
sage: G.to_even_subgroup()
Arithmetic subgroup with permutations of right cosets
 S2=(1,3)(2,4)
 S3=(1,2,3)
 L=(2,3,4)
 R=(1,4,2)

sage: H = ArithmeticSubgroup_Permutation(S2 = '(1,4,11,14)(2,7,12,17)(3,5,13,15)(6,9,16,19)(8,10,18,20)', S3 = '(1,2,3,11,12,13)(4,5,6,14,15,16)(7,8,9,17,18,19)(10,20)')
sage: G = H.to_even_subgroup(relabel=False); G
Arithmetic subgroup with permutations of right cosets
 S2=(1,4)(2,7)(3,5)(6,9)(8,10)
 S3=(1,2,3)(4,5,6)(7,8,9)
 L=(1,5)(2,4,9,10,8)(3,7,6)
 R=(1,7,10,8,6)(2,5,9)(3,4)
sage: H.is_subgroup(G)
True
sage.modular.arithgroup.arithgroup_perm.eval_sl2z_word(w)

Given a word in the format output by sl2z_word_problem(), convert it back into an element of \({\rm SL}_2(\ZZ)\).

EXAMPLES:

sage: from sage.modular.arithgroup.arithgroup_perm import eval_sl2z_word
sage: eval_sl2z_word([(0, 1), (1, -1), (0, 0), (1, 3), (0, 2), (1, 9), (0, -1)])
[ 66 -59]
[ 47 -42]
sage.modular.arithgroup.arithgroup_perm.sl2z_word_problem(A)

Given an element of \({\rm SL}_2(\ZZ)\), express it as a word in the generators L = [1,1,0,1] and R = [1,0,1,1].

The return format is a list of pairs (a,b), where a = 0 or 1 denoting L or R respectively, and b is an integer exponent.

See also the function eval_sl2z_word().

EXAMPLES:

sage: from sage.modular.arithgroup.arithgroup_perm import eval_sl2z_word, sl2z_word_problem
sage: m = SL2Z([1,0,0,1])
sage: eval_sl2z_word(sl2z_word_problem(m)) == m
True
sage: m = SL2Z([0,-1,1,0])
sage: eval_sl2z_word(sl2z_word_problem(m)) == m
True
sage: m = SL2Z([7,8,-50,-57])
sage: eval_sl2z_word(sl2z_word_problem(m)) == m
True
sage.modular.arithgroup.arithgroup_perm.word_of_perms(w, p1, p2)

Given a word \(w\) as a list of 2-tuples (index,power) and permutations p1 and p2 return the product of p1 and p2 that corresponds to w.

EXAMPLES:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: S2 = SymmetricGroup(4)
sage: p1 = S2('(1,2)(3,4)')
sage: p2 = S2('(1,2,3,4)')
sage: ap.word_of_perms([(1,1),(0,1)], p1, p2) ==  p2 * p1
True
sage: ap.word_of_perms([(0,1),(1,1)], p1, p2) == p1 * p2
True

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