Arithmetic subgroups defined by permutations of cosets¶
A subgroup of finite index of a finitely generated group
is completely
described by the action of a set of generators of
on the right cosets
. 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
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 used in this module are named as follows
,
,
,
which are defined by
Those generators satisfy the following relations
In particular not all four are needed to generate the whole group . Three couples which generate
are of particular
interest:
as they are also semigroup generators for the semigroup of matrices in
with non-negative entries,
as they are closely related to the continued fraction algorithm,
as the group
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 |
[Go09] | Alexey G. Gorinov, “Combinatorics of double cosets and fundamental domains for the subgroups of the modular group”, preprint Arxiv 0901.1340 |
[HL14] | (1, 2) Thomas Hamilton and David Loeffler, “Congruence testing for odd modular subgroups”, LMS J. Comput. Math. 17 (2014), no. 1, 206-208, doi:10.1112/S1461157013000338. |
[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) |
[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, doi: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”, http://wayback.archive-it.org/855/20100510123900/http://www.public.iastate.edu/~kurthc/research/index.html |
[KuLo] | Chris Kurth and Ling Long, “Computations with finite index subgroups
of ![]() |
[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
where the elements
and
are hyperbolic and
are parabolic (
) or elliptic elements (
).
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
- David Loeffler & Thomas Hamilton (2012): generalised Hsu congruence test for odd subgroups
-
sage.modular.arithgroup.arithgroup_perm.
ArithmeticSubgroup_Permutation
(L=None, R=None, S2=None, S3=None, relabel=False, check=True)¶ Construct a subgroup of
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:
An error will be raised if only one permutation is given. If no arguments are given at all, the full modular group
is returned.
INPUT:
S2
,S3
,L
,R
- permutations - action of matrices on the right cosets (each coset is identified to an element ofwhere
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
defined by the action of generators on its coset graph.
The class stores the action of generators
on right cosets
of a finite index subgroup
. In particular the action of
on the cosets is on right.
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
as a permutation of cosets.
EXAMPLES:
sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)") sage: G.L() (1,3)
-
R
()¶ Return the action of the matrix
as a permutation of cosets.
EXAMPLES:
sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)") sage: G.R() (2,3)
-
S2
()¶ Return the action of the matrix
as a permutation of cosets.
EXAMPLES:
sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)") sage: G.S2() (1,2)
-
S3
()¶ Return the action of the matrix
as a permutation of cosets.
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
be the generalised level of self. If this subgroup is even, let
, else let
. Then any congruence subgroup containing self contains
(a generalisation of Wohlfahrt’s theorem due to Kiming, Verrill and Schuett). So we compute the image of self modulo
and return the preimage of that.
Note
If you just want to know if the subgroup is congruence or not, it is much faster to use
is_congruence()
.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 graphs2_edges
- bool (default: True) - put edges associated to s2s3_edges
- bool (default: True) - put edges associated to s3l_edges
- bool (default: False) - put edges associated to lr_edges
- bool (default: False) - put edges associated to rs2_label
,s3_label
,l_label
,r_label
- the labels to put on the edges corresponding to the generators action. UseNone
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_congruence
()¶ Return
True
if this is a congruence subgroup, andFalse
otherwise.ALGORITHM:
Uses Hsu’s algorithm [Hsu96]. Adapted from Chris Kurth’s implementation in KFarey [Kur08].
For odd subgroups, Hsu’s algorithm still works with minor modifications, using the extension of Wohlfarht’s theorem due to Kiming, Schuett and Verrill [KSV11]. See [HL14] for details.
The algorithm is as follows. Let
be a finite-index subgroup of
, and let
and
be the permutations of the cosets of
given by the elements
and
. Let
be the generalized level of
(if
is even) or twice the generalized level (if
is odd). Then:
if
is odd,
is congruence if and only if the relation
holds, where
is understood as the multiplicative inverse of 2 modulo N.
if
is a power of 2, then
is congruence if and only if the relations
hold, where
,
being the inverse of 5 modulo N.
if
is neither odd nor a power of 2, seven relations (B1-7) hold, for which see [HL14], or the source code of this function.
If the Sage verbosity flag is set (using
set_verbose()
), then extra output will be produced indicating which of the relations (A1-3) or (B1-7) is not satisfied.EXAMPLES:
Test if
is congruence:
sage: a = ArithmeticSubgroup_Permutation(L='',R='') sage: a.index() 1 sage: a.is_congruence() True
This example is congruence – it is
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: 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
The following example (taken from [KSV11]) shows that a lifting of a congruence subgroup of
to a subgroup of
need not necessarily be 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
is a lifting to
of the group
:
sage: G.to_even_subgroup() == Gamma0(6) 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
(element of order two),
(element of order 3),
(parabolic element) and
(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), (1,8,3)(2,4,6)(5,7,10), (1,4)(2,5,9,10,8)(3,7,6), (1,7,9,10,6)(2,3)(4,5,8)]
-
permutation_action
(x)¶ Given an element
x
of, 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
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 range(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 with 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 optioninplace
.ALGORITHM:
We give an overview of how the canonical labels for the modular subgroup are built. The procedure only uses the permutations
and
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
and set the current vertex to be the identity.
2. Number the cycle of
the current vertex belongs to: if the current vertex is labeled
then the numbering in such way that the cycle becomes
).
3. Find a new current vertex using the permutation
. 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
-
surgroups
()¶ Return an iterator through the non-trivial intermediate groups between
and this finite index group.
EXAMPLES:
sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)(5,6)", S3="(1,2,3)(4,5,6)") sage: H = next(G.surgroups()) sage: H Arithmetic subgroup with permutations of right cosets S2=(1,2) S3=(1,2,3) L=(1,3) R=(2,3) sage: G.is_subgroup(H) True
The principal congruence group
has thirteen surgroups:
sage: G = Gamma(3).as_permutation_group() sage: G.index() 24 sage: l = [] sage: for H in G.surgroups(): ....: l.append(H.index()) ....: assert G.is_subgroup(H) and H.is_congruence() sage: l [6, 3, 4, 8, 4, 8, 4, 12, 4, 6, 6, 8, 8]
-
-
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 of
containing
, represented in terms of the right action of
on its cosets.
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: G Arithmetic subgroup of index 24 sage: G.is_congruence() True
The following is the unique index 2 even subgroup of
:
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_even
()¶ Returns True if this subgroup contains the matrix
.
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
.
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
, where
is this subgroup. (Equivalently, return the liftings of
to
.) This can take rather a long time if the index of this subgroup is large.
See also
one_odd_subgroup()
, which returns just one of the odd subgroups (which is much quicker than enumerating them all).ALGORITHM:
If
has an element of order 4, then there are no index 2 odd subgroups, so return the empty set.
If
has no elements of order 4, then the permutation
is a combination of 2-cycles with no fixed points on
. We construct the permutation
of
which has a 4-cycle
for each 2-cycle
in
S2
. Similarly, we construct a permutationwhich has a 6-cycle
for each 3-cycle
in
, and a 2-cycle
for each fixed point
of
.
Then the permutations
and
satisfy
where
is the order 2 permutation interchanging
and
for each
. So the subgroup corresponding to these permutations is an index 2 odd subgroup of
.
The other index 2 odd subgroups of
are obtained from the pairs
where
is an element of the group generated by the 2-cycles
.
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
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]
-
one_odd_subgroup
(random=False)¶ Return an odd subgroup of index 2 in
, where
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. Ifrandom
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
we get back
:
sage: G = Gamma(4).as_permutation_group() sage: G.is_odd(), G.index() (True, 48) sage: Ge = G.to_even_subgroup() sage: Go = Ge.one_odd_subgroup() sage: Go.is_odd(), Go.index() (True, 48) sage: Go == G True
Strating from
we get a different group:
sage: G = Gamma(6).as_permutation_group() sage: G.is_odd(), G.index() (True, 144) sage: Ge = G.to_even_subgroup() sage: Go = Ge.one_odd_subgroup() sage: 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)
wherecoset_reps
is a list of coset representatives of the subgroup,gens
a list of generators,l
ands2
are list that corresponds to the action of the matrixand
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 range(4)) True sage: all(reps[i]*L*~reps[l[i]] in G for i in range(4)) True
-
todd_coxeter_l_s2
()¶ Returns a 4-tuple
(coset_reps, gens, l, s2)
wherecoset_reps
is a list of coset representatives of the subgroup,gens
a list of generators,l
ands2
are list that corresponds to the action of the matrixand
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 range(4)) True sage: all(reps[i]*L*~reps[l[i]] in G for i in range(4)) True
-
todd_coxeter_s2_s3
()¶ Returns a 4-tuple
(coset_reps, gens, s2, s3)
wherecoset_reps
are coset representatives of the subgroup,gens
is a list of generators,s2
ands3
are the action of the matricesand
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 range(4)) True sage: all(reps[i]*S3*~reps[s3[i]] in G for i in range(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
An arithmetic subgroup of
not containing
, represented in terms of the right action of
on its cosets.
EXAMPLES:
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: type(G) <class 'sage.modular.arithgroup.arithgroup_perm.OddArithmeticSubgroup_Permutation_with_category'>
-
cusp_widths
(exp=False)¶ Return the list of cusp widths.
INPUT:
exp
- boolean (default: False) - if True, return a dictionary 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_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
or
. The irregular cusps are the one which are stabilised by
.
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
or
. The irregular cusps correspond to the ones which are not stabilised by
.
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
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.
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
, 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)
, wherea = 0
or1
denotingL
orR
respectively, andb
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
as a list of 2-tuples
(index,power)
and permutationsp1
andp2
return the product ofp1
andp2
that corresponds tow
.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