Farey Symbol for arithmetic subgroups of \({\rm PSL}_2(\ZZ)\)

AUTHORS:

  • Hartmut Monien (08 - 2011)

based on the KFarey package by Chris Kurth. Implemented as C++ module for speed.

class sage.modular.arithgroup.farey_symbol.Farey

Bases: object

A class for calculating Farey symbols of arithmetics subgroups of \({\rm PSL}_2(\ZZ)\).

The arithmetic subgroup can be either any of the congruence subgroups implemented in Sage, i.e. Gamma, Gamma0, Gamma1 and GammaH or a subgroup of \({\rm PSL}_2(\ZZ)\) which is given by a user written helper class defining membership in that group.

REFERENCES:

INPUT:

  • \(G\) - an arithmetic subgroup of \({\rm PSL}_2(\ZZ)\)

EXAMPLES:

Create a Farey symbol for the group \(\Gamma_0(11)\):

sage: f = FareySymbol(Gamma0(11)); f
FareySymbol(Congruence Subgroup Gamma0(11))

Calculate the generators:

sage: f.generators()
[
[1 1]  [ 7 -2]  [ 8 -3]  [-1  0]
[0 1], [11 -3], [11 -4], [ 0 -1]
]

Pickling the FareySymbol and recovering it:

sage: f == loads(dumps(f))
True

Calculate the index of \(\Gamma_H(33, [2, 5])\) in \({\rm PSL}_2(\ZZ)\) via FareySymbol:

sage: FareySymbol(GammaH(33, [2, 5])).index()
48

Calculate the generators of \(\Gamma_1(4)\):

sage: FareySymbol(Gamma1(4)).generators()
[
[1 1]  [-3  1]
[0 1], [-4  1]
]

Calculate the generators of the example of an index 10 arithmetic subgroup given by Tim Hsu:

sage: from sage.modular.arithgroup.arithgroup_perm import HsuExample10
sage: FareySymbol(HsuExample10()).generators()
[
[1 2]  [-2  1]  [ 4 -3]
[0 1], [-7  3], [ 3 -2]
]

Calculate the generators of the group \(\Gamma' = \Gamma_0(8)\cap\Gamma_1(4)\) using a helper class to define group membership:

sage: class GPrime:
....:   def __contains__(self, M):
....:       return M in Gamma0(8) and M in Gamma1(4)
....:

sage: FareySymbol(GPrime()).generators()
[
[1 1]  [ 5 -1]  [ 5 -2]
[0 1], [16 -3], [ 8 -3]
]

Calculate cusps of arithmetic subgroup defined via permutation group:

sage: L = SymmetricGroup(4)('(1, 2, 3)')

sage: R = SymmetricGroup(4)('(1, 2, 4)')

sage: FareySymbol(ArithmeticSubgroup_Permutation(L, R)).cusps()
[-1, Infinity]

Calculate the left coset representation of \(\Gamma_H(8, [3])\):

sage: FareySymbol(GammaH(8, [3])).coset_reps()
[
[1 0]  [ 4 -1]  [ 3 -1]  [ 2 -1]  [ 1 -1]  [ 3 -1]  [ 2 -1]  [-1  0]
[0 1], [ 1  0], [ 1  0], [ 1  0], [ 1  0], [ 4 -1], [ 3 -1], [ 3 -1],
[ 1 -1]  [-1  0]  [ 0 -1]  [-1  0]
[ 2 -1], [ 2 -1], [ 1 -1], [ 1 -1]
]
coset_reps()

Left coset of the arithmetic group of the FareySymbol.

EXAMPLES:

Calculate the left coset of \(\Gamma_0(6)\):

sage: FareySymbol(Gamma0(6)).coset_reps()
[
[1 0]  [ 3 -1]  [ 2 -1]  [ 1 -1]  [ 2 -1]  [ 3 -2]  [ 1 -1]  [-1  0]
[0 1], [ 1  0], [ 1  0], [ 1  0], [ 3 -1], [ 2 -1], [ 2 -1], [ 2 -1],
[ 1 -1]  [ 0 -1]  [-1  0]  [-2  1]
[ 3 -2], [ 1 -1], [ 1 -1], [ 1 -1]
]
cusp_class(c)

Cusp class of a cusp in the FareySymbol.

INPUT:

c – a cusp

EXAMPLES:

sage: FareySymbol(Gamma0(12)).cusp_class(Cusp(1, 12))
5
cusp_widths()

Cusps widths of the FareySymbol.

EXAMPLES:

sage: FareySymbol(Gamma0(6)).cusp_widths()
[6, 2, 3, 1]
cusps()

Cusps of the FareySymbol.

EXAMPLES:

sage: FareySymbol(Gamma0(6)).cusps()
[0, 1/3, 1/2, Infinity]
fractions()

Fractions of the FareySymbol.

EXAMPLES:

sage: FareySymbol(Gamma(4)).fractions()
[0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4]
fundamental_domain(show_pairing=True, color_even='white', color='lightgray', thickness=1, zorder=2, alpha=1, tesselation='Dedekind', fill=True, linestyle='solid', ymax=1, **options)

Plot a fundamental domain of an arithmetic subgroup of \({\rm PSL}_2(\ZZ)\) corresponding to the Farey symbol.

OPTIONS:

  • fill – boolean (default True) fill the fundamental domain
  • linestyle – string (default: ‘solid’) The style of the line, which is one of ‘dashed’, ‘dotted’, ‘solid’, ‘dashdot’, or ‘–’, ‘:’, ‘-‘, ‘-.’, respectively
  • color – (default: ‘lightgray’) fill color; fill color for odd part of Dedekind tesselation.
  • show_pairing – boolean (default: True) flag for pairing
  • tesselation – (default: ‘Dedekind’) The type of hyperbolic tesselation which is one of ‘coset’, ‘Dedekind’ or None respectively
  • color_even – fill color for even parts of Dedekind tesselation (default ‘white’); ignored for other tesselations
  • thickness – float (default: \(1\)) the thickness of the line
  • ymax – float (default: \(1\)) maximal height

EXAMPLES:

For example, to plot the fundamental domain of \(\Gamma_0(11)\) with pairings use the following command:

sage: FareySymbol(Gamma0(11)).fundamental_domain()
Graphics object consisting of 54 graphics primitives

indicating that side 1 is paired with side 3 and side 2 is paired with side 4, see also paired_sides().

To plot the fundamental domain of \(\Gamma(3)\) without pairings use the following command:

sage: FareySymbol(Gamma(3)).fundamental_domain(show_pairing=False)
Graphics object consisting of 48 graphics primitives

Plot the fundamental domain of \(\Gamma_0(23)\) showing the left coset representatives:

sage: FareySymbol(Gamma0(23)).fundamental_domain(tesselation='coset')
Graphics object consisting of 58 graphics primitives

The same as above but with a custom linestyle:

sage: FareySymbol(Gamma0(23)).fundamental_domain(tesselation='coset', linestyle=':', thickness='2')
Graphics object consisting of 58 graphics primitives
generators()

Minimal set of generators of the group of the FareySymbol.

EXAMPLES:

Calculate the generators of \(\Gamma_0(6)\):

sage: FareySymbol(Gamma0(6)).generators()
[
[1 1]  [ 5 -1]  [ 7 -3]  [-1  0]
[0 1], [ 6 -1], [12 -5], [ 0 -1]
]

Calculate the generators of \({\rm SL}_2(\ZZ)\):

sage: FareySymbol(SL2Z).generators()
[
[ 0 -1]  [ 0 -1]
[ 1  0], [ 1 -1]
]

The unique index 2 even subgroup and index 4 odd subgroup each get handled correctly:

sage: FareySymbol(ArithmeticSubgroup_Permutation(S2="(1,2)", S3="()")).generators()
[
[ 0  1]  [-1  1]
[-1 -1], [-1  0]
]
sage: FareySymbol(ArithmeticSubgroup_Permutation(S2="(1,2, 3, 4)", S3="(1,3)(2,4)")).generators()
[
[ 0  1]  [-1  1]
[-1 -1], [-1  0]
]
genus()

Return the genus of the arithmetic group of the FareySymbol.

EXAMPLES:

sage: [FareySymbol(Gamma0(n)).genus() for n in range(16, 32)]
[0, 1, 0, 1, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 2]
index()

Return the index of the arithmetic group of the FareySymbol in \({\rm PSL}_2(\ZZ)\).

EXAMPLES:

sage: [FareySymbol(Gamma0(n)).index() for n in range(1, 16)]
[1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24]
level()

Return the level of the arithmetic group of the FareySymbol.

EXAMPLES:

sage: [FareySymbol(Gamma0(n)).level() for n in range(1, 16)]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
nu2()

Return the number of elliptic points of order two.

EXAMPLES:

sage: [FareySymbol(Gamma0(n)).nu2() for n in range(1, 16)]
[1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0]
nu3()

Return the number of elliptic points of order three.

EXAMPLES:

sage: [FareySymbol(Gamma0(n)).nu3() for n in range(1, 16)]
[1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0]
paired_sides()

Pairs of index of the sides of the fundamental domain of the Farey symbol of the arithmetic group. The sides of the hyperbolic polygon are numbered 0, 1, ... from left to right.

../../../_images/pairing.png

EXAMPLES:

sage: FareySymbol(Gamma0(11)).paired_sides()
[(0, 5), (1, 3), (2, 4)]

indicating that the side 0 is paired with 5, 1 with 3 and 2 with 4.

pairing_matrices()

Pairing matrices of the sides of the fundamental domain. The sides of the hyperbolic polygon are numbered 0, 1, ... from left to right.

EXAMPLES:

sage: FareySymbol(Gamma0(6)).pairing_matrices()
[
[1 1]  [ 5 -1]  [ 7 -3]  [ 5 -3]  [ 1 -1]  [-1  1]
[0 1], [ 6 -1], [12 -5], [12 -7], [ 6 -5], [ 0 -1]
]
pairing_matrices_to_tietze_index()

Obtain the translation table from pairing matrices to generators.

The result is cached.

OUTPUT:

a list where the \(i\)-th entry is a nonzero integer \(k\), such that if \(k > 0\) then the \(i\)-th pairing matrix is (up to sign) the \((k-1)\)-th generator and, if \(k < 0\), then the \(i\)-th pairing matrix is (up to sign) the inverse of the \((-k-1)\)-th generator.

EXAMPLES:

sage: F = Gamma0(40).farey_symbol()
sage: table = F.pairing_matrices_to_tietze_index()
sage: table[12]
(-2, -1)
sage: F.pairing_matrices()[12]
[  3  -1]
[ 40 -13]
sage: F.generators()[1]**-1
[ -3   1]
[-40  13]
pairings()

Pairings of the sides of the fundamental domain of the Farey symbol of the arithmetic group.

The sides of the hyperbolic polygon are numbered 0, 1, ... from left to right. Conventions: even pairings are denoted by -2, odd pairings by -3 while free pairings are denoted by an integer number greater than zero.

EXAMPLES:

Odd pairings:

sage: FareySymbol(Gamma0(7)).pairings()
[1, -3, -3, 1]

Even and odd pairings:

FareySymbol(Gamma0(13)).pairings()
[1, -3, -2, -2, -3, 1]

Only free pairings:

sage: FareySymbol(Gamma0(23)).pairings()
[1, 2, 3, 5, 3, 4, 2, 4, 5, 1]
reduce_to_cusp(r)

Transformation of a rational number to cusp representative.

INPUT:

r – a rational number

EXAMPLES:

sage: FareySymbol(Gamma0(12)).reduce_to_cusp(5/8)
[ 5  -3]
[12  -7]

Reduce 11/17 to a cusp of for HsuExample10():

sage: from sage.modular.arithgroup.arithgroup_perm import HsuExample10
sage: f = FareySymbol(HsuExample10())
sage: f.reduce_to_cusp(11/17)
[14 -9]
[-3  2]
sage: _.acton(11/17)
1
sage: f.cusps()[f.cusp_class(11/17)]
1
word_problem(M, output='standard', check=True)

Solve the word problem (up to sign) using this Farey symbol.

INPUT:

  • M – An element \(M\) of \({\rm SL}_2(\ZZ)\).
  • output – (default: 'standard') Should be one of 'standard', 'syllables', 'gens'.
  • check – (default: True) Whether to check for correct input and output.

OUTPUT:

A solution to the word problem for the matrix \(M\). The format depends on the output parameter, as follows.

  • standard returns the so called the Tietze representation, consists of a tuple of nonzero integers \(i\), where if \(i\) > 0 then it indicates the \(i`th generator (that is, ``self.generators()[0]`\) would correspond to \(i\) = 1), and if \(i\) < 0 then it indicates the inverse of the \(i\)-th generator.
  • syllables returns a tuple of tuples of the form \((i,n)\), where \((i,n)\) represents self.generators()[i] ^ n, whose product equals \(M\) up to sign.
  • gens returns tuple of tuples of the form \((g,n)\), \((g,n)\) such that the product of the matrices \(g^n\) equals \(M\) up to sign.

EXAMPLES:

sage: F = Gamma0(30).farey_symbol()
sage: gens = F.generators()
sage: g = gens[3] * gens[10] * gens[8]^-1 * gens[5]
sage: g
[-628597   73008]
[-692130   80387]
sage: F.word_problem(g)
(4, 11, -9, 6)
sage: g = gens[3] * gens[10]^2 * gens[8]^-1 * gens[5]
sage: g
[-5048053   586303]
[-5558280   645563]
sage: F.word_problem(g, output = 'gens')
((
[109 -10]
[120 -11], 1
),
 (
[ 19  -7]
[ 30 -11], 2
),
 (
[ 49  -9]
[ 60 -11], -1
),
 (
[17 -2]
[60 -7], 1
))
sage: F.word_problem(g, output = 'syllables')
((3, 1), (10, 2), (8, -1), (5, 1))

TESTS:

Check that problem with forgotten generator is fixed:

sage: from sage.misc.misc_c import prod
sage: G = Gamma0(10)
sage: F = G.farey_symbol()
sage: g = G([-701,-137,4600,899])
sage: g1 = prod(F.generators()[i]**a for i,a in F.word_problem(g, output = 'syllables'))
sage: g == g1
True

Check that it works for GammaH as well (trac ticket #19660):

sage: G = GammaH(147, [8])
sage: G.farey_symbol().word_problem(G([1,1,0,1]))
(1,)

Check that trac ticket #20347 is solved:

sage: from sage.misc.misc_c import prod
sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)",S3="(1,2,3)")
sage: S = G.farey_symbol()
sage: g1,g2 = S.generators()
sage: g = g1^3 * g2^-2 * g1 * g2
sage: S.word_problem(g)
(2, 2, 2, 1, 1, 1, 2, 1, 2)
sage: h = prod(S.generators()[i]**a for i,a in S.word_problem(g, output = 'syllables'))
sage: g == h
True