Arithmetic subgroups, finite index subgroups of \(\SL_2(\ZZ)\)

class sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup

Bases: Group

Base class for arithmetic subgroups of \(\SL_2(\ZZ)\). Not intended to be used directly, but still includes quite a few general-purpose routines which compute data about an arithmetic subgroup assuming that it has a working element testing routine.

Element

alias of ArithmeticSubgroupElement

are_equivalent(x, y, trans=False)

Test whether or not cusps x and y are equivalent modulo self.

If self has a reduce_cusp() method, use that; otherwise do a slow explicit test.

If trans = False, returns True or False. If trans = True, then return either False or an element of self mapping x onto y.

EXAMPLES:

sage: Gamma0(7).are_equivalent(Cusp(1/3), Cusp(0), trans=True)
[  3  -1]
[-14   5]
sage: Gamma0(7).are_equivalent(Cusp(1/3), Cusp(1/7))
False

as_permutation_group()

Return self as an arithmetic subgroup defined in terms of the permutation action of \(SL(2,\ZZ)\) on its right cosets.

This method uses Todd-Coxeter enumeration (via the method todd_coxeter()) which can be extremely slow for arithmetic subgroups with relatively large index in \(SL(2,\ZZ)\).

EXAMPLES:

sage: G = Gamma(3)
sage: P = G.as_permutation_group(); P
Arithmetic subgroup of index 24
sage: G.ncusps() == P.ncusps()
True
sage: G.nu2() == P.nu2()
True
sage: G.nu3() == P.nu3()
True
sage: G.an_element() in P
True
sage: P.an_element() in G
True

coset_reps(G=None)

Return right coset representatives for self \ G, where G is another arithmetic subgroup that contains self. If G = None, default to G = SL2Z.

For generic arithmetic subgroups G this is carried out by Todd-Coxeter enumeration; here G is treated as a black box, implementing nothing but membership testing.

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().coset_reps()
Traceback (most recent call last):
...
NotImplementedError: Please implement _contains_sl2 for <class 'sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup_with_category'>
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.coset_reps(Gamma0(3))
[
[1 0]  [ 0 -1]  [ 0 -1]  [ 0 -1]
[0 1], [ 1  0], [ 1  1], [ 1  2]
]

cusp_data(c)

Return a triple (g, w, t) where g is an element of self generating the stabiliser of the given cusp, w is the width of the cusp, and t is 1 if the cusp is regular and -1 if not.

EXAMPLES:

sage: Gamma1(4).cusp_data(Cusps(1/2))
(
[ 1 -1]
[ 4 -3], 1, -1
)

cusp_width(c)

Return the width of the orbit of cusps represented by c.

EXAMPLES:

sage: Gamma0(11).cusp_width(Cusps(oo))
1
sage: Gamma0(11).cusp_width(0)
11
sage: [Gamma0(100).cusp_width(c) for c in Gamma0(100).cusps()]
[100, 1, 4, 1, 1, 1, 4, 25, 1, 1, 4, 1, 25, 4, 1, 4, 1, 1]

cusps(algorithm='default')

Return a sorted list of inequivalent cusps for self, i.e. a set of representatives for the orbits of self on \(\mathbb{P}^1(\QQ)\).

These should be returned in a reduced form where this makes sense.

INPUT:

• algorithm – which algorithm to use to compute the cusps of self. 'default' finds representatives for a known complete set of cusps. 'modsym' computes the boundary map on the space of weight two modular symbols associated to self, which finds the cusps for self in the process.

EXAMPLES:

sage: Gamma0(36).cusps()
[0, 1/18, 1/12, 1/9, 1/6, 1/4, 1/3, 5/12, 1/2, 2/3, 5/6, Infinity]
sage: Gamma0(36).cusps(algorithm='modsym') == Gamma0(36).cusps()
True
sage: GammaH(36, [19,29]).cusps() == Gamma0(36).cusps()
True
sage: Gamma0(1).cusps()
[Infinity]

dimension_cusp_forms(k=2)

Return the dimension of the space of weight k cusp forms for this group. For \(k \ge 2\), this is given by a standard formula in terms of k and various invariants of the group; see Diamond + Shurman, “A First Course in Modular Forms”, section 3.5 and 3.6. If k is not given, default to k = 2.

For dimensions of spaces of cusp forms with character for Gamma1, use the dimension_cusp_forms method of the Gamma1 class, or the standalone function dimension_cusp_forms().

For weight 1 cusp forms this generic implementation only works in cases where one can prove solely via Riemann-Roch theory that there aren’t any cusp forms (i.e. when the number of regular cusps is strictly greater than the degree of the canonical divisor). Otherwise a NotImplementedError is raised.

EXAMPLES:

sage: Gamma1(31).dimension_cusp_forms(2)
26
sage: Gamma(5).dimension_cusp_forms(1)
0
sage: Gamma1(4).dimension_cusp_forms(1) # irregular cusp
0
sage: Gamma(13).dimension_cusp_forms(1)
Traceback (most recent call last):
...
NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general

dimension_eis(k=2)

Return the dimension of the space of weight k Eisenstein series for this group, which is a subspace of the space of modular forms complementary to the space of cusp forms.

INPUT:

• k - an integer (default 2).

EXAMPLES:

sage: GammaH(33,[2]).dimension_eis()
7
sage: GammaH(33,[2]).dimension_eis(3)
0
sage: GammaH(33, [2,5]).dimension_eis(2)
3
sage: GammaH(33, [4]).dimension_eis(1)
4

dimension_modular_forms(k=2)

Return the dimension of the space of weight k modular forms for this group. This is given by a standard formula in terms of k and various invariants of the group; see Diamond + Shurman, “A First Course in Modular Forms”, section 3.5 and 3.6. If k is not given, defaults to k = 2.

For dimensions of spaces of modular forms with character for Gamma1, use the dimension_modular_forms method of the Gamma1 class, or the standalone function dimension_modular_forms().

For weight 1 modular forms this generic implementation only works in cases where one can prove solely via Riemann-Roch theory that there aren’t any cusp forms (i.e. when the number of regular cusps is strictly greater than the degree of the canonical divisor). Otherwise a NotImplementedError is raised.

EXAMPLES:

sage: Gamma1(31).dimension_modular_forms(2)
55
sage: Gamma1(3).dimension_modular_forms(1)
1
sage: Gamma1(4).dimension_modular_forms(1) # irregular cusp
1
sage: Gamma(13).dimension_modular_forms(1)
Traceback (most recent call last):
...
NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general

farey_symbol()

Return the Farey symbol associated to this subgroup. See the farey_symbol module for more information.

EXAMPLES:

sage: Gamma1(4).farey_symbol()
FareySymbol(Congruence Subgroup Gamma1(4))

gen(i)

Return the i-th generator of self, i.e. the i-th element of the tuple self.gens().

EXAMPLES:

sage: SL2Z.gen(1)
[1 1]
[0 1]

generalised_level()

Return the generalised level of self, i.e., the least common multiple of the widths of all cusps.

If self is even, Wohlfart’s theorem tells us that this is equal to the (conventional) level of self when self is a congruence subgroup. This can fail if self is odd, but the actual level is at most twice the generalised level. See the paper by Kiming, Schuett and Verrill for more examples.

EXAMPLES:

sage: Gamma0(18).generalised_level()
18
sage: from sage.modular.arithgroup.arithgroup_perm import HsuExample18
sage: HsuExample18().generalised_level()
24

In the following example, the actual level is twice the generalised level. This is the group \(G_2\) from Example 17 of K-S-V.

sage: G = CongruenceSubgroup(8, [ [1,1,0,1], [3,-1,4,-1] ])
sage: G.level()
8
sage: G.generalised_level()
4

generators(algorithm='farey')

Return a list of generators for this congruence subgroup. The result is cached.

INPUT:

• algorithm (string): either farey or todd-coxeter.

If algorithm is set to "farey", then the generators will be calculated using Farey symbols, which will always return a minimal generating set. See farey_symbol for more information.

If algorithm is set to "todd-coxeter", a simpler algorithm based on Todd-Coxeter enumeration will be used. This is exceedingly slow for general subgroups, and the list of generators will be far from minimal (indeed it may contain repetitions).

EXAMPLES:

sage: Gamma(2).generators()
[
[1 2]  [ 3 -2]  [-1  0]
[0 1], [ 2 -1], [ 0 -1]
]
sage: Gamma(2).generators(algorithm="todd-coxeter")
[
[1 2]  [-1  0]  [ 1  0]  [-1  0]  [-1  2]  [-1  0]  [1 0]
[0 1], [ 0 -1], [-2  1], [ 0 -1], [-2  3], [ 2 -1], [2 1]
]

gens(*args, **kwds)

Return a tuple of generators for this congruence subgroup.

The generators need not be minimal. For arguments, see generators().

EXAMPLES:

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

genus()

Return the genus of the modular curve of self.

EXAMPLES:

sage: Gamma1(5).genus()
0
sage: Gamma1(31).genus()
26
sage: from sage.modular.dims import dimension_cusp_forms
sage: Gamma1(157).genus() == dimension_cusp_forms(Gamma1(157), 2)
True
sage: GammaH(7, [2]).genus()
0
sage: [Gamma0(n).genus() for n in [1..23]]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2]
sage: [n for n in [1..200] if Gamma0(n).genus() == 1]
[11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49]

index()

Return the index of self in the full modular group.

EXAMPLES:

sage: Gamma0(17).index()
18
sage: sage.modular.arithgroup.congroup_generic.CongruenceSubgroup(5).index()
Traceback (most recent call last):
...
NotImplementedError

is_abelian()

Return True if this arithmetic subgroup is abelian.

Since arithmetic subgroups are always nonabelian, this always returns False.

EXAMPLES:

sage: SL2Z.is_abelian()
False
sage: Gamma0(3).is_abelian()
False
sage: Gamma1(12).is_abelian()
False
sage: GammaH(4, [3]).is_abelian()
False

is_congruence()

Return True if self is a congruence subgroup.

EXAMPLES:

sage: Gamma0(5).is_congruence()
True
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().is_congruence()
Traceback (most recent call last):
...
NotImplementedError

is_even()

Return True precisely if this subgroup contains the matrix -1.

EXAMPLES:

sage: SL2Z.is_even()
True
sage: Gamma0(20).is_even()
True
sage: Gamma1(5).is_even()
False
sage: GammaH(11, [3]).is_even()
False

is_finite()

Return True if this arithmetic subgroup is finite.

Since arithmetic subgroups are always infinite, this always returns False.

EXAMPLES:

sage: SL2Z.is_finite()
False
sage: Gamma0(3).is_finite()
False
sage: Gamma1(12).is_finite()
False
sage: GammaH(4, [3]).is_finite()
False

is_normal()

Return True precisely if this subgroup is a normal subgroup of SL2Z.

EXAMPLES:

sage: Gamma(3).is_normal()
True
sage: Gamma1(3).is_normal()
False

is_odd()

Return True precisely if this subgroup does not contain the matrix -1.

EXAMPLES:

sage: SL2Z.is_odd()
False
sage: Gamma0(20).is_odd()
False
sage: Gamma1(5).is_odd()
True
sage: GammaH(11, [3]).is_odd()
True

is_parent_of(x)

Check whether this group is a valid parent for the element x. Required by Sage’s testing framework.

EXAMPLES:

sage: Gamma(3).is_parent_of(ZZ(1))
False
sage: Gamma(3).is_parent_of([1,0,0,1])
False
sage: Gamma(3).is_parent_of(SL2Z([1,1,0,1]))
False
sage: Gamma(3).is_parent_of(SL2Z(1))
True

is_regular_cusp(c)

Return True if the orbit of the given cusp is a regular cusp for self, otherwise False. This is automatically true if -1 is in self.

EXAMPLES:

sage: Gamma1(4).is_regular_cusp(Cusps(1/2))
False
sage: Gamma1(4).is_regular_cusp(Cusps(oo))
True

is_subgroup(right)

Return True if self is a subgroup of right, and False otherwise. For generic arithmetic subgroups this is done by the absurdly slow algorithm of checking all of the generators of self to see if they are in right.

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().is_subgroup(SL2Z)
Traceback (most recent call last):
...
NotImplementedError: Please implement _contains_sl2 for <class 'sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup_with_category'>
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.is_subgroup(Gamma1(18), Gamma0(6))
True

matrix_space()

Return the parent space of the matrices, which is always MatrixSpace(ZZ, 2).

EXAMPLES:

sage: Gamma(3).matrix_space()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

ncusps()

Return the number of cusps of this arithmetic subgroup. This is provided as a separate function since for dimension formulae in even weight all we need to know is the number of cusps, and this can be calculated very quickly, while enumerating all cusps is much slower.

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.ncusps(Gamma0(7))
2

ngens()

Return the size of the minimal generating set of self returned by generators().

EXAMPLES:

sage: Gamma0(22).ngens()
8
sage: Gamma1(14).ngens()
13
sage: GammaH(11, [3]).ngens()
3
sage: SL2Z.ngens()
2

nirregcusps()

Return the number of cusps of self that are “irregular”, i.e. their stabiliser can only be generated by elements with both eigenvalues -1 rather than +1. If the group contains -1, every cusp is clearly regular.

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nirregcusps(Gamma1(4))
1

nregcusps()

Return the number of cusps of self that are “regular”, i.e. their stabiliser has a generator with both eigenvalues +1 rather than -1. If the group contains -1, every cusp is clearly regular.

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nregcusps(Gamma1(4))
2

nu2()

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

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().nu2()
Traceback (most recent call last):
...
NotImplementedError: Please implement _contains_sl2 for <class 'sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup_with_category'>
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu2(Gamma0(1105)) == 8
True

nu3()

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

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().nu3()
Traceback (most recent call last):
...
NotImplementedError: Please implement _contains_sl2 for <class 'sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup_with_category'>
sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu3(Gamma0(1729)) == 8
True

We test that a bug in handling of subgroups not containing -1 is fixed:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup.nu3(GammaH(7, [2]))
2

order()

Return the number of elements in this arithmetic subgroup.

Since arithmetic subgroups are always infinite, this always returns infinity.

EXAMPLES:

sage: SL2Z.order()
+Infinity
sage: Gamma0(5).order()
+Infinity
sage: Gamma1(2).order()
+Infinity
sage: GammaH(12, [5]).order()
+Infinity

projective_index()

Return the index of the image of self in \(\PSL_2(\ZZ)\). This is equal to the index of self if self contains -1, and half of this otherwise.

This is equal to the degree of the natural map from the modular curve of self to the \(j\)-line.

EXAMPLES:

sage: Gamma0(5).projective_index()
6
sage: Gamma1(5).projective_index()
12

reduce_cusp(c)

Given a cusp \(c \in \mathbb{P}^1(\QQ)\), return the unique reduced cusp equivalent to c under the action of self, where a reduced cusp is an element \(\tfrac{r}{s}\) with r,s coprime non-negative integers, s as small as possible, and r as small as possible for that s.

NOTE: This function should be overridden by all subclasses.

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().reduce_cusp(1/4)
Traceback (most recent call last):
...
NotImplementedError

sturm_bound(weight=2)

Returns the Sturm bound for modular forms of the given weight and level this subgroup.

INPUT:

• weight - an integer \(\geq 2\) (default: 2)

EXAMPLES:

sage: Gamma0(11).sturm_bound(2)
2
sage: Gamma0(389).sturm_bound(2)
65
sage: Gamma0(1).sturm_bound(12)
1
sage: Gamma0(100).sturm_bound(2)
30
sage: Gamma0(1).sturm_bound(36)
3
sage: Gamma0(11).sturm_bound()
2
sage: Gamma0(13).sturm_bound()
3
sage: Gamma0(16).sturm_bound()
4
sage: GammaH(16,[13]).sturm_bound()
8
sage: GammaH(16,[15]).sturm_bound()
16
sage: Gamma1(16).sturm_bound()
32
sage: Gamma1(13).sturm_bound()
28
sage: Gamma1(13).sturm_bound(5)
70

FURTHER DETAILS: This function returns a positive integer \(n\) such that the Hecke operators \(T_1,\ldots, T_n\) acting on cusp forms generate the Hecke algebra as a \(\ZZ\)-module when the character is trivial or quadratic. Otherwise, \(T_1,\ldots,T_n\) generate the Hecke algebra at least as a \(\ZZ[\varepsilon]\)-module, where \(\ZZ[\varepsilon]\) is the ring generated by the values of the Dirichlet character \(\varepsilon\). Alternatively, this is a bound such that if two cusp forms associated to this space of modular symbols are congruent modulo \((\lambda, q^n)\), then they are congruent modulo \(\lambda\).

REFERENCES:

• See the Agashe-Stein appendix to Lario and Schoof, Some computations with Hecke rings and deformation rings, Experimental Math., 11 (2002), no. 2, 303-311.

• This result originated in the paper Sturm, On the congruence of modular forms, Springer LNM 1240, 275-280, 1987.

REMARK: Kevin Buzzard pointed out to me (William Stein) in Fall 2002 that the above bound is fine for \(\Gamma_1(N)\) with character, as one sees by taking a power of \(f\). More precisely, if \(f \cong 0 \pmod{p}\) for first \(s\) coefficients, then \(f^r \cong 0 \pmod{p}\) for first \(sr\) coefficients. Since the weight of \(f^r\) is \(r\cdot k(f)\), it follows that if \(s \geq b\), where \(b\) is the Sturm bound for \(\Gamma_0(N)\) at weight \(k(f)\), then \(f^r\) has valuation large enough to be forced to be \(0\) at \(r*k(f)\) by Sturm bound (which is valid if we choose \(r\) correctly). Thus \(f \cong 0 \pmod{p}\). Conclusion: For \(\Gamma_1(N)\) with fixed character, the Sturm bound is exactly the same as for \(\Gamma_0(N)\).

A key point is that we are finding \(\ZZ[\varepsilon]\) generators for the Hecke algebra here, not \(\ZZ\)-generators. So if one wants generators for the Hecke algebra over \(\ZZ\), this bound must be suitably modified (and I’m not sure what the modification is).

AUTHORS:

• William Stein

to_even_subgroup()

Return the smallest even subgroup of \(SL(2, \ZZ)\) containing self.

EXAMPLES:

sage: sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup().to_even_subgroup()
Traceback (most recent call last):
...
NotImplementedError: Please implement _contains_sl2 for <class 'sage.modular.arithgroup.arithgroup_generic.ArithmeticSubgroup_with_category'>

todd_coxeter(G=None, on_right=True)

Compute coset representatives for self \ G and action of standard generators on them via Todd-Coxeter enumeration.

If G is None, default to SL2Z. The method also computes generators of the subgroup at same time.

INPUT:

• G - intermediate subgroup (currently not implemented if different from SL(2,Z))

• on_right - boolean (default: True) - if True return right coset enumeration, if False return left one.

This is extremely slow in general.

OUTPUT:

• a list of coset representatives

• a list of generators for the group

• l - list of integers that correspond to the action of the standard parabolic element [[1,1],[0,1]] of \(SL(2,\ZZ)\) on the cosets of self.

• s - list of integers that correspond to the action of the standard element of order \(2\) [[0,-1],[1,0]] on the cosets of self.

EXAMPLES:

sage: L = SL2Z([1,1,0,1])
sage: S = SL2Z([0,-1,1,0])

sage: G = Gamma(2)
sage: reps, gens, l, s = G.todd_coxeter()
sage: len(reps) == G.index()
True
sage: all(reps[i] * L * ~reps[l[i]] in G for i in range(6))
True
sage: all(reps[i] * S * ~reps[s[i]] in G for i in range(6))
True

sage: G = Gamma0(7)
sage: reps, gens, l, s = G.todd_coxeter()
sage: len(reps) == G.index()
True
sage: all(reps[i] * L * ~reps[l[i]] in G for i in range(8))
True
sage: all(reps[i] * S * ~reps[s[i]] in G for i in range(8))
True

sage: G = Gamma1(3)
sage: reps, gens, l, s = G.todd_coxeter(on_right=False)
sage: len(reps) == G.index()
True
sage: all(~reps[l[i]] * L * reps[i] in G for i in range(8))
True
sage: all(~reps[s[i]] * S * reps[i] in G for i in range(8))
True

sage: G = Gamma0(5)
sage: reps, gens, l, s = G.todd_coxeter(on_right=False)
sage: len(reps) == G.index()
True
sage: all(~reps[l[i]] * L * reps[i] in G for i in range(6))
True
sage: all(~reps[s[i]] * S * reps[i] in G for i in range(6))
True

sage.modular.arithgroup.arithgroup_generic.is_ArithmeticSubgroup(x)

Return True if x is of type ArithmeticSubgroup.

EXAMPLES:

sage: from sage.modular.arithgroup.all import is_ArithmeticSubgroup
sage: is_ArithmeticSubgroup(GL(2, GF(7)))
False
sage: is_ArithmeticSubgroup(Gamma0(4))
True