"""
Space of modular symbols (base class)
All the spaces of modular symbols derive from this class. This class is an
abstract base class.
"""
import sage.modules.free_module as free_module
import sage.matrix.matrix_space as matrix_space
from sage.modules.free_module_element import is_FreeModuleElement
import sage.misc.misc as misc
import sage.modular.hecke.all as hecke
import sage.rings.arith as arith
import sage.rings.fast_arith as fast_arith
from sage.rings.all import PowerSeriesRing, Integer, O, QQ, ZZ, is_NumberField, infinity, Zmod
from sage.structure.all import Sequence, SageObject
import sage.modular.modsym.ambient
from sage.modular.arithgroup.all import Gamma0, is_Gamma0
import hecke_operator
from sage.misc.cachefunc import cached_method
def is_ModularSymbolsSpace(x):
r"""
Return True if x is a space of modular symbols.
EXAMPLES::
sage: M = ModularForms(3, 2)
sage: sage.modular.modsym.space.is_ModularSymbolsSpace(M)
False
sage: sage.modular.modsym.space.is_ModularSymbolsSpace(M.modular_symbols(sign=1))
True
"""
return isinstance(x, ModularSymbolsSpace)
class ModularSymbolsSpace(hecke.HeckeModule_free_module):
r"""
Base class for spaces of modular symbols.
"""
def __init__(self, group, weight, character, sign, base_ring):
"""
Create a space of modular symbols.
EXAMPLES::
sage: M = ModularSymbols(22,6) ; M
Modular Symbols space of dimension 30 for Gamma_0(22) of weight 6 with sign 0 over Rational Field
sage: M == loads(dumps(M))
True
"""
self.__group = group
self.__character = character
self.__sign = sign
hecke.HeckeModule_free_module.__init__(self, base_ring, group.level(), weight)
def __cmp__(self, other):
"""
Compare self and other.
When spaces are in a common ambient space, we order
lexicographically by the sequence of traces of Hecke operators
`T_p`, for all primes `p`. In general we order
first by the group, then the weight, then the character, then the
sign then the base ring, then the dimension.
EXAMPLES::
sage: M = ModularSymbols(21,4) ; N = ModularSymbols(Gamma1(5),6)
sage: M.cuspidal_submodule().__cmp__(N)
1
sage: M.cuspidal_submodule() == N
False
"""
if not isinstance(other, ModularSymbolsSpace):
return cmp(type(self), type(other))
c = cmp(self.__group,other.__group)
if c: return c
c = cmp(self.weight(), other.weight())
if c: return c
c = cmp(self.__character, other.__character)
if c: return c
c = cmp(self.__sign, other.__sign)
if c: return c
c = cmp(self.base_ring(), other.base_ring())
if c: return c
if self.is_ambient() or other.is_ambient():
return cmp(self.dimension(), other.dimension())
c = cmp(self.ambient_hecke_module(), other.ambient_hecke_module())
if c: return c
c = cmp(self.dimension(), other.dimension())
if c: return c
d = cmp(self.free_module(), other.free_module())
if d == 0:
return d
try:
for p in fast_arith.prime_range(self.hecke_bound()):
ap = self.hecke_matrix(p).trace()
bp = other.hecke_matrix(p).trace()
c = cmp(ap, bp)
if c: return c
except ArithmeticError:
pass
return d
def _hecke_operator_class(self):
"""
Return the class to be used for instantiating Hecke operators
acting on self.
EXAMPLES::
sage: ModularSymbols(81,2)._hecke_operator_class()
<class 'sage.modular.modsym.hecke_operator.HeckeOperator'>
"""
return hecke_operator.HeckeOperator
def compact_system_of_eigenvalues(self, v, names='alpha', nz=None):
r"""
Return a compact system of eigenvalues `a_n` for
`n\in v`. This should only be called on simple factors of
modular symbols spaces.
INPUT:
- ``v`` - a list of positive integers
- ``nz`` - (default: None); if given specifies a column index
such that the dual module has that column nonzero.
OUTPUT:
- ``E`` - matrix such that E\*v is a vector with components
the eigenvalues `a_n` for `n \in v`.
- ``v`` - a vector over a number field
EXAMPLES::
sage: M = ModularSymbols(43,2,1)[2]; M
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1 over Rational Field
sage: E, v = M.compact_system_of_eigenvalues(prime_range(10))
sage: E
[ 3 -2]
[-3 2]
[-1 2]
[ 1 -2]
sage: v
(1, -1/2*alpha + 3/2)
sage: E*v
(alpha, -alpha, -alpha + 2, alpha - 2)
TESTS:
Verify that Trac #12772 is fixed::
sage: M = ModularSymbols(1,12,sign=1).cuspidal_subspace().new_subspace()
sage: A = M.decomposition()[0]
sage: A.compact_system_of_eigenvalues(prime_range(10))
(
[ -24]
[ 252]
[ 4830]
[-16744], (1)
)
"""
if nz is None:
nz = self._eigen_nonzero()
M = self.ambient()
try:
E = M.hecke_images(nz, v) * self.dual_free_module().basis_matrix().transpose()
except AttributeError:
raise NotImplementedError, "ambient space must implement hecke_images but doesn't yet"
v = self.dual_eigenvector(names=names, lift=False, nz=nz)
return E, v
def character(self):
"""
Return the character associated to self.
EXAMPLES::
sage: ModularSymbols(12,8).character()
Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1
sage: ModularSymbols(DirichletGroup(25).0, 4).character()
Dirichlet character modulo 25 of conductor 25 mapping 2 |--> zeta20
"""
return self.__character
def cuspidal_submodule(self):
"""
Return the cuspidal submodule of self.
.. note::
This should be overridden by all derived classes.
EXAMPLES::
sage: sage.modular.modsym.space.ModularSymbolsSpace(Gamma0(11),2,DirichletGroup(11).gens()[0]**10,0,QQ).cuspidal_submodule()
Traceback (most recent call last):
...
NotImplementedError: computation of cuspidal submodule not yet implemented for this class
sage: ModularSymbols(Gamma0(11),2).cuspidal_submodule()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
"""
raise NotImplementedError, "computation of cuspidal submodule not yet implemented for this class"
def cuspidal_subspace(self):
"""
Synonym for cuspidal_submodule.
EXAMPLES::
sage: m = ModularSymbols(Gamma1(3),12); m.dimension()
8
sage: m.cuspidal_subspace().new_subspace().dimension()
2
"""
return self.cuspidal_submodule()
def new_subspace(self, p=None):
"""
Synonym for new_submodule.
EXAMPLES::
sage: m = ModularSymbols(Gamma0(5),12); m.dimension()
12
sage: m.new_subspace().dimension()
6
sage: m = ModularSymbols(Gamma1(3),12); m.dimension()
8
sage: m.new_subspace().dimension()
2
"""
return self.new_submodule(p)
def old_subspace(self, p=None):
"""
Synonym for old_submodule.
EXAMPLES::
sage: m = ModularSymbols(Gamma1(3),12); m.dimension()
8
sage: m.old_subspace().dimension()
6
"""
return self.old_submodule(p)
def eisenstein_subspace(self):
"""
Synonym for eisenstein_submodule.
EXAMPLES::
sage: m = ModularSymbols(Gamma1(3),12); m.dimension()
8
sage: m.eisenstein_subspace().dimension()
2
sage: m.cuspidal_subspace().dimension()
6
"""
return self.eisenstein_submodule()
def dimension_of_associated_cuspform_space(self):
"""
Return the dimension of the corresponding space of cusp forms.
The input space must be cuspidal, otherwise there is no
corresponding space of cusp forms.
EXAMPLES::
sage: m = ModularSymbols(Gamma0(389),2).cuspidal_subspace(); m.dimension()
64
sage: m.dimension_of_associated_cuspform_space()
32
sage: m = ModularSymbols(Gamma0(389),2,sign=1).cuspidal_subspace(); m.dimension()
32
sage: m.dimension_of_associated_cuspform_space()
32
"""
if not self.is_cuspidal():
raise ArithmeticError, "space must be cuspidal"
if self.sign() == 0:
return self.dimension() // 2
return self.dimension()
def dual_star_involution_matrix(self):
"""
Return the matrix of the dual star involution, which is induced by
complex conjugation on the linear dual of modular symbols.
.. note::
This should be overridden in all derived classes.
EXAMPLES::
sage: sage.modular.modsym.space.ModularSymbolsSpace(Gamma0(11),2,DirichletGroup(11).gens()[0]**10,0,QQ).dual_star_involution_matrix()
Traceback (most recent call last):
...
NotImplementedError: computation of dual star involution matrix not yet implemented for this class
sage: ModularSymbols(Gamma0(11),2).dual_star_involution_matrix()
[ 1 0 0]
[ 0 -1 0]
[ 0 1 1]
"""
raise NotImplementedError, "computation of dual star involution matrix not yet implemented for this class"
def group(self):
"""
Returns the group of this modular symbols space.
INPUT:
- ``ModularSymbols self`` - an arbitrary space of
modular symbols
OUTPUT:
- ``CongruenceSubgroup`` - the congruence subgroup
that this is a space of modular symbols for.
ALGORITHM: The group is recorded when this space is created.
EXAMPLES::
sage: m = ModularSymbols(20)
sage: m.group()
Congruence Subgroup Gamma0(20)
"""
return self.__group
def is_ambient(self):
"""
Return True if self is an ambient space of modular symbols.
EXAMPLES::
sage: ModularSymbols(21,4).is_ambient()
True
sage: ModularSymbols(21,4).cuspidal_submodule().is_ambient()
False
"""
return isinstance(self, sage.modular.modsym.ambient.ModularSymbolsAmbient)
def is_cuspidal(self):
"""
Return True if self is a cuspidal space of modular symbols.
.. note::
This should be overridden in all derived classes.
EXAMPLES::
sage: sage.modular.modsym.space.ModularSymbolsSpace(Gamma0(11),2,DirichletGroup(11).gens()[0]**10,0,QQ).is_cuspidal()
Traceback (most recent call last):
...
NotImplementedError: computation of cuspidal subspace not yet implemented for this class
sage: ModularSymbols(Gamma0(11),2).is_cuspidal()
False
"""
raise NotImplementedError, "computation of cuspidal subspace not yet implemented for this class"
def is_simple(self):
"""
Return whether not this modular symbols space is simple as a module
over the anemic Hecke algebra adjoin \*.
EXAMPLES::
sage: m = ModularSymbols(Gamma0(33),2,sign=1)
sage: m.is_simple()
False
sage: o = m.old_subspace()
sage: o.decomposition()
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 6 for Gamma_0(33) of weight 2 with sign 1 over Rational Field,
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 6 for Gamma_0(33) of weight 2 with sign 1 over Rational Field
]
sage: C=ModularSymbols(1,14,0,GF(5)).cuspidal_submodule()
sage: C
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(1) of weight 14 with sign 0 over Finite Field of size 5
sage: C.is_simple()
True
"""
try:
return self._is_simple
except AttributeError:
D = self.factorization()
if len(D) == 0 or len(D) == 1 and D[0][1] == 1:
self._is_simple = True
else:
self._is_simple = False
return self._is_simple
def multiplicity(self, S, check_simple=True):
"""
Return the multiplicity of the simple modular symbols space S in
self. S must be a simple anemic Hecke module.
ASSUMPTION: self is an anemic Hecke module with the same weight and
group as S, and S is simple.
EXAMPLES::
sage: M = ModularSymbols(11,2,sign=1)
sage: N1, N2 = M.decomposition()
sage: N1.multiplicity(N2)
0
sage: M.multiplicity(N1)
1
sage: M.multiplicity(ModularSymbols(14,2))
0
"""
if self.level() % S.level() != 0 or S.weight() != self.weight():
return 0
if check_simple and not S.is_simple():
raise ArithmeticError, "S must be simple"
A = self.ambient_hecke_module()
B = A.submodule_generated_by_images(S)
C = self.intersection(B)
d = C.rank()
n = S.rank()
assert d % n == 0, "the dimension of intersection must be a multiple of dimension of simple space. bug!"
return d//n
def ngens(self):
"""
The number of generators of self.
INPUT:
- ``ModularSymbols self`` - arbitrary space of modular symbols.
OUTPUT:
- ``int`` - the number of generators, which is the same as the
dimension of self.
ALGORITHM: Call the dimension function.
EXAMPLES::
sage: m = ModularSymbols(33)
sage: m.ngens()
9
sage: m.rank()
9
sage: ModularSymbols(100, weight=2, sign=1).ngens()
18
"""
return self.rank()
def default_prec(self):
r"""
Get the default precision for computation of `q`-expansion
associated to the ambient space of this space of modular symbols
(and all subspaces). Use ``set_default_prec`` to
change the default precision.
EXAMPLES::
sage: M = ModularSymbols(15)
sage: M.cuspidal_submodule().q_expansion_basis()
[
q - q^2 - q^3 - q^4 + q^5 + q^6 + O(q^8)
]
sage: M.set_default_prec(20)
Notice that setting the default precision of the ambient space
affects the subspaces.
::
sage: M.cuspidal_submodule().q_expansion_basis()
[
q - q^2 - q^3 - q^4 + q^5 + q^6 + 3*q^8 + q^9 - q^10 - 4*q^11 + q^12 - 2*q^13 - q^15 - q^16 + 2*q^17 - q^18 + 4*q^19 + O(q^20)
]
sage: M.cuspidal_submodule().default_prec()
20
"""
if not self.is_ambient():
return self.ambient_hecke_module().default_prec()
try:
return self.__default_prec
except AttributeError:
self.__default_prec = Integer(8)
return self.__default_prec
def set_default_prec(self, prec):
"""
Set the default precision for computation of `q`-expansion
associated to the ambient space of this space of modular symbols
(and all subspaces).
EXAMPLES::
sage: M = ModularSymbols(Gamma1(13),2)
sage: M.set_default_prec(5)
sage: M.cuspidal_submodule().q_expansion_basis()
[
q - 4*q^3 - q^4 + O(q^5),
q^2 - 2*q^3 - q^4 + O(q^5)
]
"""
if not self.is_ambient():
return self.ambient_hecke_module().set_default_prec(prec)
else:
self.__default_prec = Integer(prec)
def set_precision(self, prec):
"""
Same as self.set_default_prec(prec).
EXAMPLES::
sage: M = ModularSymbols(17,2)
sage: M.cuspidal_submodule().q_expansion_basis()
[
q - q^2 - q^4 - 2*q^5 + 4*q^7 + O(q^8)
]
sage: M.set_precision(10)
sage: M.cuspidal_submodule().q_expansion_basis()
[
q - q^2 - q^4 - 2*q^5 + 4*q^7 + 3*q^8 - 3*q^9 + O(q^10)
]
"""
self.set_default_prec(prec)
def q_expansion_basis(self, prec=None, algorithm='default'):
r"""
Returns a basis of q-expansions (as power series) to precision prec
of the space of modular forms associated to self. The q-expansions
are defined over the same base ring as self, and a put in echelon
form.
INPUT:
- ``self`` - a space of CUSPIDAL modular symbols
- ``prec`` - an integer
- ``algorithm`` - string:
- ``'default' (default)`` - decide which algorithm to
use based on heuristics
- ``'hecke'`` - compute basis by computing
homomorphisms T - K, where T is the Hecke algebra
- ``'eigen'`` - compute basis using eigenvectors for
the Hecke action and Atkin-Lehner-Li theory to patch them together
- ``'all'`` - compute using hecke_dual and eigen
algorithms and verify that the results are the same.
The computed basis is *not* cached, though of course Hecke
operators used in computing the basis are cached.
EXAMPLES::
sage: M = ModularSymbols(1, 12).cuspidal_submodule()
sage: M.q_expansion_basis(8)
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + O(q^8)
]
::
sage: M.q_expansion_basis(8, algorithm='eigen')
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + O(q^8)
]
::
sage: M = ModularSymbols(1, 24).cuspidal_submodule()
sage: M.q_expansion_basis(8, algorithm='eigen')
[
q + 195660*q^3 + 12080128*q^4 + 44656110*q^5 - 982499328*q^6 - 147247240*q^7 + O(q^8),
q^2 - 48*q^3 + 1080*q^4 - 15040*q^5 + 143820*q^6 - 985824*q^7 + O(q^8)
]
::
sage: M = ModularSymbols(11, 2, sign=-1).cuspidal_submodule()
sage: M.q_expansion_basis(8, algorithm='eigen')
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
]
::
sage: M = ModularSymbols(Gamma1(13), 2, sign=1).cuspidal_submodule()
sage: M.q_expansion_basis(8, algorithm='eigen')
[
q - 4*q^3 - q^4 + 3*q^5 + 6*q^6 + O(q^8),
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 + O(q^8)
]
::
sage: M = ModularSymbols(Gamma1(5), 3, sign=-1).cuspidal_submodule()
sage: M.q_expansion_basis(8, algorithm='eigen') # dimension is 0
[]
::
sage: M = ModularSymbols(Gamma1(7), 3, sign=-1).cuspidal_submodule()
sage: M.q_expansion_basis(8)
[
q - 3*q^2 + 5*q^4 - 7*q^7 + O(q^8)
]
::
sage: M = ModularSymbols(43, 2, sign=0).cuspidal_submodule()
sage: M[0]
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0 over Rational Field
sage: M[0].q_expansion_basis()
[
q - 2*q^2 - 2*q^3 + 2*q^4 - 4*q^5 + 4*q^6 + O(q^8)
]
sage: M[1]
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0 over Rational Field
sage: M[1].q_expansion_basis()
[
q + 2*q^5 - 2*q^6 - 2*q^7 + O(q^8),
q^2 - q^3 - q^5 + q^7 + O(q^8)
]
"""
if prec is None:
prec = self.default_prec()
else:
prec = Integer(prec)
if prec < 1:
raise ValueError, "prec (=%s) must be >= 1"%prec
if not self.is_cuspidal():
raise ArithmeticError, "space must be cuspidal"
if self.sign() == 0:
P = self.plus_submodule(compute_dual=True)
return Sequence(P.q_expansion_basis(prec=prec, algorithm=algorithm), cr=True)
if self.dimension() == 0:
return Sequence([])
if algorithm == 'default':
algorithm = 'hecke'
if algorithm == 'hecke':
return Sequence(self._q_expansion_basis_hecke_dual(prec), cr=True)
elif algorithm == 'eigen':
return Sequence(self._q_expansion_basis_eigen(prec, 'alpha'), cr=True)
elif algorithm == 'all':
B1 = self._q_expansion_basis_hecke_dual(prec)
B2 = self._q_expansion_basis_eigen(prec, 'alpha')
if B1 != B2:
raise RuntimeError, "There is a bug in q_expansion_basis -- basis computed differently with two algorithms:\n%s\n%s\n"%(B1, B2,)
return Sequence(B1, cr=True)
else:
raise ValueError, "no algorithm '%s'"%algorithm
def q_expansion_module(self, prec = None, R=None):
r"""
Return a basis over R for the space spanned by the coefficient
vectors of the `q`-expansions corresponding to self. If R
is not the base ring of self, returns the restriction of scalars
down to R (for this, self must have base ring `\QQ`
or a number field).
INPUT:
- ``self`` - must be cuspidal
- ``prec`` - an integer (default:
self.default_prec())
- ``R`` - either ZZ, QQ, or the base_ring of self
(which is the default)
OUTPUT: A free module over R.
TODO - extend to more general R (though that is fairly easy for the
user to get by just doing base_extend or change_ring on the
output of this function).
Note that the prec needed to distinguish elements of the
restricted-down-to-R basis may be bigger than ``self.hecke_bound()``,
since one must use the Sturm bound for modular forms on `\Gamma_H(N)`.
EXAMPLES WITH SIGN 1 and R=QQ:
Basic example with sign 1::
sage: M = ModularSymbols(11, sign=1).cuspidal_submodule()
sage: M.q_expansion_module(5, QQ)
Vector space of degree 5 and dimension 1 over Rational Field
Basis matrix:
[ 0 1 -2 -1 2]
Same example with sign -1::
sage: M = ModularSymbols(11, sign=-1).cuspidal_submodule()
sage: M.q_expansion_module(5, QQ)
Vector space of degree 5 and dimension 1 over Rational Field
Basis matrix:
[ 0 1 -2 -1 2]
An example involving old forms::
sage: M = ModularSymbols(22, sign=1).cuspidal_submodule()
sage: M.q_expansion_module(5, QQ)
Vector space of degree 5 and dimension 2 over Rational Field
Basis matrix:
[ 0 1 0 -1 -2]
[ 0 0 1 0 -2]
An example that (somewhat spuriously) is over a number field::
sage: x = polygen(QQ)
sage: k = NumberField(x^2+1, 'a')
sage: M = ModularSymbols(11, base_ring=k, sign=1).cuspidal_submodule()
sage: M.q_expansion_module(5, QQ)
Vector space of degree 5 and dimension 1 over Rational Field
Basis matrix:
[ 0 1 -2 -1 2]
An example that involves an eigenform with coefficients in a number
field::
sage: M = ModularSymbols(23, sign=1).cuspidal_submodule()
sage: M.q_eigenform(4, 'gamma')
q + gamma*q^2 + (-2*gamma - 1)*q^3 + O(q^4)
sage: M.q_expansion_module(11, QQ)
Vector space of degree 11 and dimension 2 over Rational Field
Basis matrix:
[ 0 1 0 -1 -1 0 -2 2 -1 2 2]
[ 0 0 1 -2 -1 2 1 2 -2 0 -2]
An example that is genuinely over a base field besides QQ.
::
sage: eps = DirichletGroup(11).0
sage: M = ModularSymbols(eps,3,sign=1).cuspidal_submodule(); M
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 and level 11, weight 3, character [zeta10], sign 1, over Cyclotomic Field of order 10 and degree 4
sage: M.q_eigenform(4, 'beta')
q + (-zeta10^3 + 2*zeta10^2 - 2*zeta10)*q^2 + (2*zeta10^3 - 3*zeta10^2 + 3*zeta10 - 2)*q^3 + O(q^4)
sage: M.q_expansion_module(7, QQ)
Vector space of degree 7 and dimension 4 over Rational Field
Basis matrix:
[ 0 1 0 0 0 -40 64]
[ 0 0 1 0 0 -24 41]
[ 0 0 0 1 0 -12 21]
[ 0 0 0 0 1 -4 4]
An example involving an eigenform rational over the base, but the
base is not QQ.
::
sage: k.<a> = NumberField(x^2-5)
sage: M = ModularSymbols(23, base_ring=k, sign=1).cuspidal_submodule()
sage: D = M.decomposition(); D
[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(23) of weight 2 with sign 1 over Number Field in a with defining polynomial x^2 - 5,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(23) of weight 2 with sign 1 over Number Field in a with defining polynomial x^2 - 5
]
sage: M.q_expansion_module(8, QQ)
Vector space of degree 8 and dimension 2 over Rational Field
Basis matrix:
[ 0 1 0 -1 -1 0 -2 2]
[ 0 0 1 -2 -1 2 1 2]
An example involving an eigenform not rational over the base and
for which the base is not QQ.
::
sage: eps = DirichletGroup(25).0^2
sage: M = ModularSymbols(eps,2,sign=1).cuspidal_submodule(); M
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 and level 25, weight 2, character [zeta10], sign 1, over Cyclotomic Field of order 10 and degree 4
sage: D = M.decomposition(); D
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 and level 25, weight 2, character [zeta10], sign 1, over Cyclotomic Field of order 10 and degree 4
]
sage: D[0].q_eigenform(4, 'mu')
q + mu*q^2 + ((zeta10^3 + zeta10 - 1)*mu + zeta10^2 - 1)*q^3 + O(q^4)
sage: D[0].q_expansion_module(11, QQ)
Vector space of degree 11 and dimension 8 over Rational Field
Basis matrix:
[ 0 1 0 0 0 0 0 0 -20 -3 0]
[ 0 0 1 0 0 0 0 0 -16 -1 0]
[ 0 0 0 1 0 0 0 0 -11 -2 0]
[ 0 0 0 0 1 0 0 0 -8 -1 0]
[ 0 0 0 0 0 1 0 0 -5 -1 0]
[ 0 0 0 0 0 0 1 0 -3 -1 0]
[ 0 0 0 0 0 0 0 1 -2 0 0]
[ 0 0 0 0 0 0 0 0 0 0 1]
sage: D[0].q_expansion_module(11)
Vector space of degree 11 and dimension 2 over Cyclotomic Field of order 10 and degree 4
Basis matrix:
[ 0 1 0 zeta10^2 - 1 -zeta10^2 - 1 -zeta10^3 - zeta10^2 zeta10^2 - zeta10 2*zeta10^3 + 2*zeta10 - 1 zeta10^3 - zeta10^2 - zeta10 + 1 zeta10^3 - zeta10^2 + zeta10 -2*zeta10^3 + 2*zeta10^2 - zeta10]
[ 0 0 1 zeta10^3 + zeta10 - 1 -zeta10 - 1 -zeta10^3 - zeta10^2 -2*zeta10^3 + zeta10^2 - zeta10 + 1 zeta10^2 0 zeta10^3 + 1 2*zeta10^3 - zeta10^2 + zeta10 - 1]
EXAMPLES WITH SIGN 0 and R=QQ:
TODO - this doesn't work yet as it's not implemented!!
::
sage: M = ModularSymbols(11,2).cuspidal_submodule() #not tested
sage: M.q_expansion_module() #not tested
... boom ...
EXAMPLES WITH SIGN 1 and R=ZZ (computes saturation)::
sage: M = ModularSymbols(43,2, sign=1).cuspidal_submodule()
sage: M.q_expansion_module(8, QQ)
Vector space of degree 8 and dimension 3 over Rational Field
Basis matrix:
[ 0 1 0 0 0 2 -2 -2]
[ 0 0 1 0 -1/2 1 -3/2 0]
[ 0 0 0 1 -1/2 2 -3/2 -1]
sage: M.q_expansion_module(8, ZZ)
Free module of degree 8 and rank 3 over Integer Ring
Echelon basis matrix:
[ 0 1 0 0 0 2 -2 -2]
[ 0 0 1 1 -1 3 -3 -1]
[ 0 0 0 2 -1 4 -3 -2]
"""
if prec is None:
prec = self.default_prec()
if R == ZZ and self.base_ring() == QQ:
return self._q_expansion_module_integral(prec)
elif R == QQ:
return self._q_expansion_module_rational(prec)
elif R is None or R == self.base_ring():
return self._q_expansion_module(prec)
else:
raise NotImplementedError, "R must be ZZ, QQ, or the base ring of the modular symbols space."
def _q_eigenform_images(self, A, prec, names):
"""
Return list of images in space corresponding to self of eigenform
corresponding to A under the degeneracy maps. This is mainly a
helper function for other internal functions.
INPUT:
- ``self`` - space of modular symbols
- ``A`` - cuspidal simple space of level dividing the
level of self and the same weight
- ``prec`` - a positive integer
EXAMPLES::
sage: M = ModularSymbols(33,2,sign=1)
sage: A = M.modular_symbols_of_level(11).cuspidal_submodule()
sage: M._q_eigenform_images(A, 10, names='a')
[q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 + O(q^10),
q^3 - 2*q^6 - q^9 + 2*q^12 + q^15 + 2*q^18 - 2*q^21 - 2*q^27 + O(q^30)]
"""
f = A.q_eigenform(prec, names)
if A.level() == self.level():
return [f]
D = arith.divisors(self.level() // A.level())
q = f.parent().gen()
return [f] + [f(q**d) for d in D if d > 1]
def _q_expansion_module(self, prec, algorithm='hecke'):
"""
Return module spanned by the `q`-expansions corresponding to self. See
the docstring for ``q_expansion_module`` (without underscore) for
further details. Note that this will not work if ``algorithm=eigen``
and the sign is 0.
EXAMPLE::
sage: ModularSymbols(11, 2, base_ring=GF(4,'a')).cuspidal_submodule()._q_expansion_module(prec=4, algorithm="hecke")
Vector space of degree 4 and dimension 1 over Finite Field in a of size 2^2
Basis matrix:
[0 1 0 1]
sage: ModularSymbols(11, 2, base_ring=QuadraticField(-7,'b'), sign=1).cuspidal_submodule()._q_expansion_module(prec=4, algorithm="eigen")
Vector space of degree 4 and dimension 1 over Number Field in b with defining polynomial x^2 + 7
Basis matrix:
[ 0 1 -2 -1]
"""
if not self.is_cuspidal():
raise ValueError, "self must be cuspidal"
R = self.base_ring()
if not R.is_field():
if R == ZZ:
return self._q_expansion_module_integral(prec)
raise NotImplementedError, "base ring must be a field (or ZZ)."
if algorithm == 'hecke' or algorithm == 'default':
A = R ** prec
return A.span([f.padded_list(prec) for f in self.q_expansion_basis(prec, algorithm)])
if algorithm != 'eigen':
raise ValueError, "unknown algorithm '%s'"%algorithm
V = R ** prec
def q_eigen_gens(f):
r""" Temporary function for internal use.
EXAMPLE::
sage: ModularSymbols(11, 4, base_ring=QuadraticField(-7,'b'),sign=1).cuspidal_submodule()._q_expansion_module(prec=5, algorithm="eigen") # indirect doctest
Vector space of degree 5 and dimension 2 over Number Field in b with defining polynomial x^2 + 7
Basis matrix:
[ 0 1 0 3 -6]
[ 0 0 1 -4 2]
"""
X = f.padded_list(prec)
d = A.dimension()
if d == 1:
return [X]
else:
return [[X[i][j] for i in xrange(prec)] for j in xrange(d)]
if self.sign() == 0:
X = self.plus_submodule(compute_dual=True)
else:
X = self
B = [sum([q_eigen_gens(f) for f in self._q_eigenform_images(A, prec, 'zeta')], []) for A, _ in X.factorization()]
A = R ** prec
return A.span(sum(B, []))
def _q_expansion_module_rational(self, prec):
"""
Return a vector space over `\QQ` for the space spanned by the
`q`-expansions corresponding to self. The base ring of self must be
`\QQ` or a number field, and self must be cuspidal. The returned space
is a `\QQ`-vector space, where the coordinates are the coefficients of
`q`-expansions.
INPUT:
- ``prec`` (integer) - number of q-expansion terms to calculate.
EXAMPLE::
sage: ModularSymbols(11, 4).cuspidal_submodule()._q_expansion_module_rational(5)
Vector space of degree 5 and dimension 2 over Rational Field
Basis matrix:
[ 0 1 0 3 -6]
[ 0 0 1 -4 2]
"""
if not self.is_cuspidal():
raise ValueError, "self must be cuspidal"
K = self.base_ring()
if not is_NumberField(K):
raise TypeError, "self must be over QQ or a number field."
n = K.degree()
if n == 1:
return self._q_expansion_module(prec)
V = QQ**prec
def q_eigen_gens(f):
r"""
Temporary function for internal use.
EXAMPLE::
sage: ModularSymbols(13, 6).cuspidal_submodule()._q_expansion_module_rational(4) # indirect doctest
Vector space of degree 4 and dimension 3 over Rational Field
Basis matrix:
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
"""
X = f.padded_list(prec)
d = A.dimension()
if d == 1:
return [[X[i][j] for i in xrange(prec)] for j in xrange(n)]
else:
return [[(X[i].list())[j][k] for i in xrange(prec)] for j in xrange(d) for k in range(n)]
if self.sign() == 0:
X = self.plus_submodule(compute_dual=True)
else:
X = self
B = [sum([q_eigen_gens(f) for f in self._q_eigenform_images(A, prec, 'alpha')], []) for A, _ in X.factorization()]
A = QQ**prec
W = A.span(sum(B, []))
return W
def _q_expansion_module_integral(self, prec):
r"""
Return module over `\ZZ` for the space spanned by
the `q`-expansions corresponding to self. The base ring of
self must be `\QQ` or a number field, and self must
be cuspidal. The returned space is a `\ZZ`-module,
where the coordinates are the coefficients of
`q`-expansions.
EXAMPLES::
sage: M = ModularSymbols(11, sign=1).cuspidal_submodule()
sage: M._q_expansion_module_integral(5)
Free module of degree 5 and rank 1 over Integer Ring
Echelon basis matrix:
[ 0 1 -2 -1 2]
sage: V = M.complement().cuspidal_submodule()
sage: V._q_expansion_module_integral(5)
Free module of degree 5 and rank 0 over Integer Ring
Echelon basis matrix:
[]
"""
V = self.q_expansion_module(prec, QQ)
return free_module.FreeModule(ZZ, V.degree()).span(V.basis()).saturation()
def congruence_number(self, other, prec=None):
r"""
Given two cuspidal spaces of modular symbols, compute the
congruence number, using prec terms of the `q`-expansions.
The congruence number is defined as follows. If `V` is the
submodule of integral cusp forms corresponding to self (saturated in
`\ZZ[[q]]`, by definition) and `W` is the
submodule corresponding to other, each computed to precision prec,
the congruence number is the index of `V+W` in its
saturation in `\ZZ[[q]]`.
If prec is not given it is set equal to the max of the
``hecke_bound`` function called on each space.
EXAMPLES:
sage: A, B = ModularSymbols(48, 2).cuspidal_submodule().decomposition()
sage: A.congruence_number(B)
2
"""
if not self.is_cuspidal():
raise ValueError, "self must be cuspidal"
if not other.is_cuspidal():
raise ValueError, "right must be cuspidal"
if prec is None:
prec = max(self.hecke_bound(), other.hecke_bound())
prec = int(prec)
V = self.q_expansion_module(prec, ZZ)
W = other.q_expansion_module(prec, ZZ)
K = V+W
return K.index_in_saturation()
@cached_method
def q_eigenform_character(self, names=None):
"""
Return the Dirichlet character associated to the specific
choice of `q`-eigenform attached to this simple cuspidal
modular symbols space.
INPUT::
- ``names` -- string, name of the variable.
OUTPUT:
- a Dirichlet character taking values in the Hecke eigenvalue
field, where the indeterminant of that field is determined
by the given variable name.
EXAMPLES::
sage: f = ModularSymbols(Gamma1(13),2,sign=1).cuspidal_subspace().decomposition()[0]
sage: eps = f.q_eigenform_character('a'); eps
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -a - 1
sage: parent(eps)
Group of Dirichlet characters of modulus 13 over Number Field in a with defining polynomial x^2 + 3*x + 3
sage: eps(3)
a + 1
The modular symbols space must be simple.::
sage: ModularSymbols(Gamma1(17),2,sign=1).cuspidal_submodule().q_eigenform_character('a')
Traceback (most recent call last):
...
ArithmeticError: self must be simple
If the character is specified when making the modular symbols
space, then names need not be given and the returned character
is just the character of the space.::
sage: f = ModularSymbols(kronecker_character(19),2,sign=1).cuspidal_subspace().decomposition()[0]
sage: f
Modular Symbols subspace of dimension 8 of Modular Symbols space of dimension 10 and level 76, weight 2, character [-1, -1], sign 1, over Rational Field
sage: f.q_eigenform_character()
Dirichlet character modulo 76 of conductor 76 mapping 39 |--> -1, 21 |--> -1
sage: f.q_eigenform_character() is f.character()
True
The input space need not be cuspidal::
sage: M = ModularSymbols(Gamma1(13),2,sign=1).eisenstein_submodule()[0]
sage: M.q_eigenform_character('a')
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -1
The modular symbols space does not have to come from a decomposition::
sage: ModularSymbols(Gamma1(16),2,sign=1).cuspidal_submodule().q_eigenform_character('a')
Dirichlet character modulo 16 of conductor 16 mapping 15 |--> 1, 5 |--> -a - 1
"""
eps = self.character()
if eps is not None:
return eps
v = self.dual_eigenvector(names=names)
i = v.nonzero_positions()[0]
K = v.base_ring()
from sage.modular.dirichlet import DirichletGroup
G = DirichletGroup(self.level(), K)
M = self.ambient_module()
b = [(M.diamond_bracket_matrix(u)*v)[i] / v[i] for u in G.unit_gens()]
return G(b)
def q_eigenform(self, prec, names=None):
"""
Returns the q-expansion to precision prec of a new eigenform
associated to self, where self must be new, cuspidal, and simple.
EXAMPLES::
sage: ModularSymbols(2, 8)[1].q_eigenform(5, 'a')
q - 8*q^2 + 12*q^3 + 64*q^4 + O(q^5)
sage: ModularSymbols(2, 8)[0].q_eigenform(5,'a')
Traceback (most recent call last):
...
ArithmeticError: self must be cuspidal.
"""
if self.dimension() > 1 and names is None:
raise ValueError, "please specify a name to use for the field of eigenvalues"
if prec is None:
prec = self.default_prec()
try:
f = self._q_expansion_dict[names]
except (AttributeError, KeyError):
self._q_expansion_dict = {}
if not self.is_cuspidal():
raise ArithmeticError, "self must be cuspidal."
if not self.is_simple():
if self.sign() == 0:
return self.plus_submodule(compute_dual=True).q_eigenform(prec, names)
raise ArithmeticError, "self must be simple."
a2 = self.eigenvalue(2, names)
R = PowerSeriesRing(a2.parent(), "q")
q = R.gen(0)
f = q + a2*q**2 + O(q**3)
if f.prec() < prec:
R = f.parent()
ext = [self.eigenvalue(n, names) for n in range(f.prec(),prec)]
f = R(f.padded_list(f.prec()) + ext)
self._q_expansion_dict[names] = f.add_bigoh(prec)
return self._q_expansion_dict[names]
else:
return f.O(prec)
def _q_expansion_basis_eigen(self, prec, names):
r"""
Return a basis of eigenforms corresponding to this space, which must be
new, cuspidal and simple.
EXAMPLE::
sage: ModularSymbols(17, 2,sign=1).cuspidal_submodule()._q_expansion_basis_eigen(2, "a")
[q + O(q^2)]
"""
if self.is_simple():
f = self.q_eigenform(prec, names)
R = PowerSeriesRing(self.base_ring(), 'q')
B = [R([f[i][j] for i in xrange(prec)],prec) for j in range(self.rank())]
return B
else:
raise NotImplementedError
def q_expansion_cuspforms(self, prec=None):
"""
Returns a function f(i,j) such that each value f(i,j) is the
q-expansion, to the given precision, of an element of the
corresponding space `S` of cusp forms. Together these
functions span `S`. Here `i,j` are integers with
`0\leq i,j < d`, where `d` is the dimension of
self.
For a reduced echelon basis, use the function
``q_expansion_basis`` instead.
More precisely, this function returns the `q`-expansions
obtained by taking the `ij` entry of the matrices of the
Hecke operators `T_n` acting on the subspace of the linear
dual of modular symbols corresponding to self.
EXAMPLES::
sage: S = ModularSymbols(11,2, sign=1).cuspidal_submodule()
sage: f = S.q_expansion_cuspforms(8)
sage: f(0,0)
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
::
sage: S = ModularSymbols(37,2).cuspidal_submodule()
sage: f = S.q_expansion_cuspforms(8)
sage: f(0,0)
q + q^3 - 2*q^4 - q^7 + O(q^8)
sage: f(3,3)
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + O(q^8)
sage: f(1,2)
q^2 + 2*q^3 - 2*q^4 + q^5 - 3*q^6 + O(q^8)
::
sage: S = ModularSymbols(Gamma1(13),2,sign=-1).cuspidal_submodule()
sage: f = S.q_expansion_cuspforms(8)
sage: f(0,0)
q - 2*q^2 + q^4 - q^5 + 2*q^6 + O(q^8)
sage: f(0,1)
q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 + O(q^8)
::
sage: S = ModularSymbols(1,12,sign=-1).cuspidal_submodule()
sage: f = S.q_expansion_cuspforms(8)
sage: f(0,0)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + O(q^8)
"""
if prec is None:
prec = self.default_prec()
if not self.is_cuspidal():
raise ArithmeticError, "self must be cuspidal"
K = self.base_ring()
M = matrix_space.MatrixSpace(K, prec-1, self.dimension())
T = [self.dual_hecke_matrix(n) for n in range(1,prec)]
R = PowerSeriesRing(self.base_ring(), 'q')
return lambda i, j: R([0] + [t[i,j] for t in T], prec)
def _q_expansion_basis_hecke_dual(self, prec):
r"""
Compute a basis of q-expansions for the associated space of cusp forms
to the given precision, by using linear functionals on the Hecke
algebra as described in William Stein's book (Algorithm 3.26, page 56)
EXAMPLES::
sage: ModularSymbols(37, 2).cuspidal_submodule()._q_expansion_basis_hecke_dual(12)
[q + q^3 - 2*q^4 - q^7 - 2*q^9 + 3*q^11 + O(q^12),
q^2 + 2*q^3 - 2*q^4 + q^5 - 3*q^6 - 4*q^9 - 2*q^10 + 4*q^11 + O(q^12)]
sage: ModularSymbols(37, 2).cuspidal_submodule()._q_expansion_basis_hecke_dual(2)
[q + O(q^2)]
"""
d = self.dimension_of_associated_cuspform_space()
prec = Integer(prec)
if prec < 1:
raise ValueError, "prec (=%s) must be >= 1"%prec
if d >= prec-1:
d = prec-1
K = self.base_ring()
A = free_module.VectorSpace(K, prec-1)
M = matrix_space.MatrixSpace(K, prec-1, self.dimension())
V = A.zero_submodule()
i = self.dimension()-1
j = 0
t = misc.verbose('computing basis to precision %s'%prec)
while V.dimension() < d and i >= 0:
v = [self.dual_hecke_matrix(n).column(i) for n in range(1,prec)]
t = misc.verbose('iteration: %s'%j,t)
X = M(v).transpose()
V += X.row_space()
t = misc.verbose('addition of row space: %s'%j,t)
i -= 1
j += 1
R = PowerSeriesRing(K, 'q')
B = V.basis()
if len(B) < d:
B += [V(0)] * (d-len(B))
return [R([0] + b.list(), prec) for b in B]
def hecke_module_of_level(self, level):
r"""
Alias for ``self.modular_symbols_of_level(level)``.
EXAMPLE::
sage: ModularSymbols(11, 2).hecke_module_of_level(22)
Modular Symbols space of dimension 7 for Gamma_0(22) of weight 2 with sign 0 over Rational Field
"""
return self.modular_symbols_of_level(Integer(level))
def sign(self):
"""
Returns the sign of self.
For efficiency reasons, it is often useful to compute in the
(largest) quotient of modular symbols where the \* involution acts
as +1, or where it acts as -1.
INPUT:
- ``ModularSymbols self`` - arbitrary space of modular
symbols.
OUTPUT:
- ``int`` - the sign of self, either -1, 0, or 1.
- ``-1`` - if this is factor of quotient where \* acts
as -1,
- ``+1`` - if this is factor of quotient where \* acts
as +1,
- ``0`` - if this is full space of modular symbols (no
quotient).
EXAMPLES::
sage: m = ModularSymbols(33)
sage: m.rank()
9
sage: m.sign()
0
sage: m = ModularSymbols(33, sign=0)
sage: m.sign()
0
sage: m.rank()
9
sage: m = ModularSymbols(33, sign=-1)
sage: m.sign()
-1
sage: m.rank()
3
"""
return self.__sign
def simple_factors(self):
"""
Returns a list modular symbols spaces `S` where `S`
is simple spaces of modular symbols (for the anemic Hecke algebra)
and self is isomorphic to the direct sum of the `S` with
some multiplicities, as a module over the *anemic* Hecke algebra.
For the multiplicities use factorization() instead.
ASSUMPTION: self is a module over the anemic Hecke algebra.
EXAMPLES::
sage: ModularSymbols(1,100,sign=-1).simple_factors()
[Modular Symbols subspace of dimension 8 of Modular Symbols space of dimension 8 for Gamma_0(1) of weight 100 with sign -1 over Rational Field]
sage: ModularSymbols(1,16,0,GF(5)).simple_factors()
[Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(1) of weight 16 with sign 0 over Finite Field of size 5,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(1) of weight 16 with sign 0 over Finite Field of size 5,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(1) of weight 16 with sign 0 over Finite Field of size 5]
"""
return [S for S,_ in self.factorization()]
def star_eigenvalues(self):
"""
Returns the eigenvalues of the star involution acting on self.
EXAMPLES::
sage: M = ModularSymbols(11)
sage: D = M.decomposition()
sage: M.star_eigenvalues()
[1, -1]
sage: D[0].star_eigenvalues()
[1]
sage: D[1].star_eigenvalues()
[1, -1]
sage: D[1].plus_submodule().star_eigenvalues()
[1]
sage: D[1].minus_submodule().star_eigenvalues()
[-1]
"""
try:
return self.__star_eigenvalues
except AttributeError:
pass
if self.sign() != 0:
return [self.sign()]
M = self.star_involution().matrix()
R = self.base_ring()
if M == R(1):
self.__star_eigenvalues = [R(1)]
elif M == R(-1):
self.__star_eigenvalues = [R(-1)]
else:
self.__star_eigenvalues = [R(1), R(-1)]
return self.__star_eigenvalues
def star_decomposition(self):
r"""
Decompose self into subspaces which are eigenspaces for the star
involution.
EXAMPLE::
sage: ModularSymbols(Gamma1(19), 2).cuspidal_submodule().star_decomposition()
[
Modular Symbols subspace of dimension 7 of Modular Symbols space of dimension 31 for Gamma_1(19) of weight 2 with sign 0 and over Rational Field,
Modular Symbols subspace of dimension 7 of Modular Symbols space of dimension 31 for Gamma_1(19) of weight 2 with sign 0 and over Rational Field
]
"""
S = self.star_involution()
return S.decomposition()
def integral_structure(self):
r"""
Return the `\ZZ`-structure of this modular symbols
spaces generated by all integral modular symbols.
EXAMPLES::
sage: M = ModularSymbols(11,4)
sage: M.integral_structure()
Free module of degree 6 and rank 6 over Integer Ring
Echelon basis matrix:
[ 1 0 0 0 0 0]
[ 0 1/14 1/7 5/14 1/2 13/14]
[ 0 0 1/2 0 0 1/2]
[ 0 0 0 1 0 0]
[ 0 0 0 0 1 0]
[ 0 0 0 0 0 1]
sage: M.cuspidal_submodule().integral_structure()
Free module of degree 6 and rank 4 over Integer Ring
Echelon basis matrix:
[ 0 1/14 1/7 5/14 1/2 -15/14]
[ 0 0 1/2 0 0 -1/2]
[ 0 0 0 1 0 -1]
[ 0 0 0 0 1 -1]
"""
try:
return self.__integral_structure
except AttributeError:
pass
A = self.ambient_hecke_module()
I = A.integral_structure()
if I.echelonized_basis_matrix().is_one() and (self.free_module().denominator() == 1):
J = I.submodule(self.free_module().basis(), check=False, already_echelonized=True)
else:
J = self.free_module().intersection(I)
self.__integral_structure = J
return J
def intersection_number(self, M):
"""
Given modular symbols spaces self and M in some common ambient
space, returns the intersection number of these two spaces. This is
the index in their saturation of the sum of their underlying
integral structures.
If self and M are of weight two and defined over QQ, and correspond
to newforms f and g, then this number equals the order of the
intersection of the modular abelian varieties attached to f and g.
EXAMPLES::
sage: m = ModularSymbols(389,2)
sage: d = m.decomposition(2)
sage: eis = d[0]
sage: ell = d[1]
sage: af = d[-1]
sage: af.intersection_number(eis)
97
sage: af.intersection_number(ell)
400
"""
if not isinstance(M, ModularSymbolsSpace):
raise TypeError, "M must be a modular symbols space"
if M.ambient() != self.ambient():
raise ValueError, "self and M must be in the same ambient space."
A = self.integral_structure()
B = M.integral_structure()
return (A+B).index_in_saturation()
def integral_basis(self):
r"""
Return a basis for the `\ZZ`-submodule of this
modular symbols space spanned by the generators.
Modular symbols spaces for congruence subgroups have a
`\ZZ`-structure. Computing this
`\ZZ`-structure is expensive, so by default modular
symbols spaces for congruence subgroups in Sage are defined over
`\QQ`. This function returns a tuple of independent
elements in this modular symbols space whose
`\ZZ`-span is the corresponding space of modular
symbols over `\ZZ`.
EXAMPLES::
sage: M = ModularSymbols(11)
sage: M.basis()
((1,0), (1,8), (1,9))
sage: M.integral_basis()
((1,0), (1,8), (1,9))
sage: S = M.cuspidal_submodule()
sage: S.basis()
((1,8), (1,9))
sage: S.integral_basis()
((1,8), (1,9))
::
sage: M = ModularSymbols(13,4)
sage: M.basis()
([X^2,(0,1)], [X^2,(1,4)], [X^2,(1,5)], [X^2,(1,7)], [X^2,(1,9)], [X^2,(1,10)], [X^2,(1,11)], [X^2,(1,12)])
sage: M.integral_basis()
([X^2,(0,1)], 1/28*[X^2,(1,4)] + 2/7*[X^2,(1,5)] + 3/28*[X^2,(1,7)] + 11/14*[X^2,(1,9)] + 2/7*[X^2,(1,10)] + 11/28*[X^2,(1,11)] + 3/28*[X^2,(1,12)], [X^2,(1,5)], 1/2*[X^2,(1,7)] + 1/2*[X^2,(1,9)], [X^2,(1,9)], [X^2,(1,10)], [X^2,(1,11)], [X^2,(1,12)])
sage: S = M.cuspidal_submodule()
sage: S.basis()
([X^2,(1,4)] - [X^2,(1,12)], [X^2,(1,5)] - [X^2,(1,12)], [X^2,(1,7)] - [X^2,(1,12)], [X^2,(1,9)] - [X^2,(1,12)], [X^2,(1,10)] - [X^2,(1,12)], [X^2,(1,11)] - [X^2,(1,12)])
sage: S.integral_basis()
(1/28*[X^2,(1,4)] + 2/7*[X^2,(1,5)] + 3/28*[X^2,(1,7)] + 11/14*[X^2,(1,9)] + 2/7*[X^2,(1,10)] + 11/28*[X^2,(1,11)] - 53/28*[X^2,(1,12)], [X^2,(1,5)] - [X^2,(1,12)], 1/2*[X^2,(1,7)] + 1/2*[X^2,(1,9)] - [X^2,(1,12)], [X^2,(1,9)] - [X^2,(1,12)], [X^2,(1,10)] - [X^2,(1,12)], [X^2,(1,11)] - [X^2,(1,12)])
This function currently raises a NotImplementedError on modular
symbols spaces with character of order bigger than `2`:
EXAMPLES::
sage: M = ModularSymbols(DirichletGroup(13).0^2, 2); M
Modular Symbols space of dimension 4 and level 13, weight 2, character [zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2
sage: M.basis()
((1,0), (1,5), (1,10), (1,11))
sage: M.integral_basis()
Traceback (most recent call last):
...
NotImplementedError
"""
try:
return self.__integral_basis
except AttributeError:
pass
B = self.integral_structure().basis()
self.__integral_basis = tuple([self(b) for b in B])
return self.__integral_basis
def integral_hecke_matrix(self, n):
r"""
Return the matrix of the `n`th Hecke operator acting on the integral
structure on self (as returned by ``self.integral_structure()``). This
is often (but not always) different from the matrix returned by
``self.hecke_matrix``, even if the latter has integral entries.
EXAMPLE::
sage: M = ModularSymbols(6,4)
sage: M.hecke_matrix(3)
[27 0 0 0 6 -6]
[ 0 1 -4 4 8 10]
[18 0 1 0 6 -6]
[18 0 4 -3 6 -6]
[ 0 0 0 0 9 18]
[ 0 0 0 0 12 15]
sage: M.integral_hecke_matrix(3)
[ 27 0 0 0 6 -6]
[ 0 1 -8 8 12 14]
[ 18 0 5 -4 14 8]
[ 18 0 8 -7 2 -10]
[ 0 0 0 0 9 18]
[ 0 0 0 0 12 15]
"""
n = int(n)
try:
return self.__integral_hecke_matrix[n]
except AttributeError:
self.__integral_hecke_matrix = {}
except KeyError:
pass
A = self.ambient_hecke_module()
T = A.hecke_matrix(n)
S = T.restrict(self.integral_structure()).change_ring(ZZ)
self.__integral_hecke_matrix[n] = S
return S
def sturm_bound(self):
r"""
Returns the Sturm bound for this space of modular symbols.
Type ``sturm_bound?`` for more details.
EXAMPLES::
sage: ModularSymbols(11,2).sturm_bound()
2
sage: ModularSymbols(389,2).sturm_bound()
65
sage: ModularSymbols(1,12).sturm_bound()
1
sage: ModularSymbols(1,36).sturm_bound()
3
sage: ModularSymbols(DirichletGroup(31).0^2).sturm_bound()
6
sage: ModularSymbols(Gamma1(31)).sturm_bound()
160
"""
try:
return self.__sturm_bound
except:
if self.character() is not None:
self.__sturm_bound = Gamma0(self.level()).sturm_bound(self.weight())
else:
self.__sturm_bound = self.group().sturm_bound(self.weight())
return self.__sturm_bound
def plus_submodule(self, compute_dual=True):
"""
Return the subspace of self on which the star involution acts as
+1.
INPUT:
- ``compute_dual`` - bool (default: True) also
compute dual subspace. This are useful for many algorithms.
OUTPUT: subspace of modular symbols
EXAMPLES::
sage: ModularSymbols(17,2)
Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field
sage: ModularSymbols(17,2).plus_submodule()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field
"""
return self.sign_submodule(+1, compute_dual)
def minus_submodule(self, compute_dual=True):
"""
Return the subspace of self on which the star involution acts as
-1.
INPUT:
- ``compute_dual`` - bool (default: True) also
compute dual subspace. This are useful for many algorithms.
OUTPUT: subspace of modular symbols
EXAMPLES::
sage: ModularSymbols(14,4)
Modular Symbols space of dimension 12 for Gamma_0(14) of weight 4 with sign 0 over Rational Field
sage: ModularSymbols(14,4).minus_submodule()
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 12 for Gamma_0(14) of weight 4 with sign 0 over Rational Field
"""
return self.sign_submodule(-1, compute_dual)
def _compute_sign_submodule(self, sign, compute_dual=True):
r"""
Compute the submodule of self which is an eigenspace for the star involution with the given sign.
INPUT:
- sign (integer): 1 or -1
- compute_dual (True or False, default True): also compute the dual submodule (useful for some algorithms)
OUTPUT: a submodule of self
EXAMPLE::
sage: ModularSymbols(Gamma1(11), 3)._compute_sign_submodule(-1)
Modular Symbols subspace of dimension 10 of Modular Symbols space of dimension 20 for Gamma_1(11) of weight 3 with sign 0 and over Rational Field
"""
A = self.ambient()
S = A.sign_submodule(sign, compute_dual=compute_dual)
V = S.free_module().intersection(self.free_module())
if compute_dual:
W = S.dual_free_module()
Y = self.dual_free_module()
D = W.intersection(Y)
M = A.submodule(V, D, check=False)
else:
M = A.submodule(V, check=False)
M._set_sign(sign)
return M
def _set_sign(self, sign):
r"""
Store the sign of this module (used by various initialisation
routines).
INPUT:
- (integer) sign (must be -1, 0 or 1)
OUTPUT: none
EXAMPLES::
sage: ModularSymbols(11, 2)._set_sign(123)
Traceback (most recent call last):
...
ValueError: sign (=123) must be -1, 0, or 1
sage: M = ModularSymbols(11, 2); M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: M._set_sign(-1); M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign -1 over Rational Field
sage: M._set_sign(0)
"""
sign = int(sign)
if not (sign in [-1,0,1]):
raise ValueError, "sign (=%s) must be -1, 0, or 1"%sign
self.__sign = sign
def sign_submodule(self, sign, compute_dual=True):
"""
Return the subspace of self that is fixed under the star
involution.
INPUT:
- ``sign`` - int (either -1, 0 or +1)
- ``compute_dual`` - bool (default: True) also
compute dual subspace. This are useful for many algorithms.
OUTPUT: subspace of modular symbols
EXAMPLES::
sage: M = ModularSymbols(29,2)
sage: M.sign_submodule(1)
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 5 for Gamma_0(29) of weight 2 with sign 0 over Rational Field
sage: M.sign_submodule(-1)
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(29) of weight 2 with sign 0 over Rational Field
sage: M.sign_submodule(-1).sign()
-1
"""
sign = int(sign)
if not sign in [-1, 0, 1]:
raise ValueError, "sign must be -1, 0 or 1"
if self.sign() == sign:
return self
if self.sign() == -sign:
return self.zero_submodule()
if sign == 0:
raise ArithmeticError, "There is no sign 0 subspace of a space of modular symbols with nonzero sign."
try:
return self.__plus_submodule[(sign, compute_dual)]
except AttributeError:
self.__plus_submodule = {}
except KeyError:
pass
P = self._compute_sign_submodule(sign, compute_dual)
P.__star_eigenvalue = sign
self.__plus_submodule[(sign,compute_dual)] = P
return P
def star_involution(self):
"""
Return the star involution on self, which is induced by complex
conjugation on modular symbols. Not implemented in this abstract base
class.
EXAMPLES::
sage: M = ModularSymbols(11, 2); sage.modular.modsym.space.ModularSymbolsSpace.star_involution(M)
Traceback (most recent call last):
...
NotImplementedError
"""
raise NotImplementedError
def abelian_variety(self):
"""
Return the corresponding abelian variety.
INPUT:
- ``self`` - modular symbols space of weight 2 for a
congruence subgroup such as Gamma0, Gamma1 or GammaH.
EXAMPLES::
sage: ModularSymbols(Gamma0(11)).cuspidal_submodule().abelian_variety()
Abelian variety J0(11) of dimension 1
sage: ModularSymbols(Gamma1(11)).cuspidal_submodule().abelian_variety()
Abelian variety J1(11) of dimension 1
sage: ModularSymbols(GammaH(11,[3])).cuspidal_submodule().abelian_variety()
Abelian variety JH(11,[3]) of dimension 1
The abelian variety command only works on cuspidal modular symbols
spaces::
sage: M = ModularSymbols(37)
sage: M[0].abelian_variety()
Traceback (most recent call last):
...
ValueError: self must be cuspidal
sage: M[1].abelian_variety()
Abelian subvariety of dimension 1 of J0(37)
sage: M[2].abelian_variety()
Abelian subvariety of dimension 1 of J0(37)
"""
try:
return self.__modular_abelian_variety
except AttributeError:
if not self.is_cuspidal():
raise ValueError, "self must be cuspidal"
from sage.modular.abvar.abvar import ModularAbelianVariety_modsym
A = ModularAbelianVariety_modsym(self, check=False)
self.__modular_abelian_variety = A
return A
def rational_period_mapping(self):
r"""
Return the rational period mapping associated to self.
This is a homomorphism to a vector space whose kernel is the same as
the kernel of the period mapping associated to self. For this to exist,
self must be Hecke equivariant.
Use ``integral_period_mapping`` to obtain a homomorphism to a
`\ZZ`-module, normalized so the image of integral modular symbols is
exactly `\ZZ^n`.
EXAMPLES::
sage: M = ModularSymbols(37)
sage: A = M[1]; A
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field
sage: r = A.rational_period_mapping(); r
Rational period mapping associated to Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field
sage: r(M.0)
(0, 0)
sage: r(M.1)
(1, 0)
sage: r.matrix()
[ 0 0]
[ 1 0]
[ 0 1]
[-1 -1]
[ 0 0]
sage: r.domain()
Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field
sage: r.codomain()
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]
"""
try:
return self.__rational_period_mapping
except AttributeError:
pass
V = self.dual_free_module()
A = V.basis_matrix().transpose()
A.set_immutable()
R = RationalPeriodMapping(self, A)
self.__rational_period_mapping = R
return R
def integral_period_mapping(self):
r"""
Return the integral period mapping associated to self.
This is a homomorphism to a vector space whose kernel is the same
as the kernel of the period mapping associated to self, normalized
so the image of integral modular symbols is exactly
`\ZZ^n`.
EXAMPLES::
sage: m = ModularSymbols(23).cuspidal_submodule()
sage: i = m.integral_period_mapping()
sage: i
Integral period mapping associated to Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Rational Field
sage: i.matrix()
[-1/11 1/11 0 3/11]
[ 1 0 0 0]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 1]
sage: [i(b) for b in m.integral_structure().basis()]
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]
sage: [i(b) for b in m.ambient_module().basis()]
[(-1/11, 1/11, 0, 3/11),
(1, 0, 0, 0),
(0, 1, 0, 0),
(0, 0, 1, 0),
(0, 0, 0, 1)]
We compute the image of the winding element::
sage: m = ModularSymbols(37,sign=1)
sage: a = m[1]
sage: f = a.integral_period_mapping()
sage: e = m([0,oo])
sage: f(e)
(-2/3)
The input space must be cuspidal::
sage: m = ModularSymbols(37,2,sign=1)
sage: m.integral_period_mapping()
Traceback (most recent call last):
...
ValueError: integral mapping only defined for cuspidal spaces
"""
try:
return self.__integral_period_mapping
except AttributeError:
pass
if self.base_ring() != QQ:
raise ValueError, "integral mapping only defined for spaces over QQ"
if not self.is_cuspidal():
raise ValueError, "integral mapping only defined for cuspidal spaces"
D = self.dual_free_module().basis_matrix().transpose()
I = self.ambient_module().cuspidal_submodule().integral_structure().basis_matrix()
C = I * D
if not C.is_one():
if not C.is_square():
C = (ZZ**C.ncols()).span(C.rows()).basis_matrix()
D = D * C**(-1)
D.set_immutable()
R = IntegralPeriodMapping(self, D)
self.__integral_period_mapping = R
return R
def modular_symbols_of_sign(self, sign, bound=None):
"""
Returns a space of modular symbols with the same defining
properties (weight, level, etc.) and Hecke eigenvalues as this
space except with given sign.
INPUT:
- ``self`` - a cuspidal space of modular symbols
- ``sign`` - an integer, one of -1, 0, or 1
- ``bound`` - integer (default: None); if specified
only use Hecke operators up to the given bound.
EXAMPLES::
sage: S = ModularSymbols(Gamma0(11),2,sign=0).cuspidal_subspace()
sage: S
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: S.modular_symbols_of_sign(-1)
Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field
::
sage: S = ModularSymbols(43,2,sign=1)[2]; S
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 for Gamma_0(43) of weight 2 with sign 1 over Rational Field
sage: S.modular_symbols_of_sign(-1)
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(43) of weight 2 with sign -1 over Rational Field
::
sage: S.modular_symbols_of_sign(0)
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 7 for Gamma_0(43) of weight 2 with sign 0 over Rational Field
::
sage: S = ModularSymbols(389,sign=1)[3]; S
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 33 for Gamma_0(389) of weight 2 with sign 1 over Rational Field
sage: S.modular_symbols_of_sign(-1)
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 32 for Gamma_0(389) of weight 2 with sign -1 over Rational Field
sage: S.modular_symbols_of_sign(0)
Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 65 for Gamma_0(389) of weight 2 with sign 0 over Rational Field
::
sage: S = ModularSymbols(23,sign=1,weight=4)[2]; S
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 7 for Gamma_0(23) of weight 4 with sign 1 over Rational Field
sage: S.modular_symbols_of_sign(1) is S
True
sage: S.modular_symbols_of_sign(0)
Modular Symbols subspace of dimension 8 of Modular Symbols space of dimension 12 for Gamma_0(23) of weight 4 with sign 0 over Rational Field
sage: S.modular_symbols_of_sign(-1)
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(23) of weight 4 with sign -1 over Rational Field
"""
if sign == self.sign():
return self
if not self.is_cuspidal():
raise ValueError, "self must be cuspidal for modular symbols space with given sign to be defined."
d = self.dimension()
if d == 0:
return self
if sign != 0:
if self.sign() == 0:
d = d//2
elif sign == 0:
d = 2*d
B = self.ambient_module().modular_symbols_of_sign(sign)
p = 2
if bound is None:
bound = self.hecke_bound()
while B.dimension() > d and p <= bound:
while self.level() % p == 0:
p = arith.next_prime(p)
f = self.hecke_polynomial(p)
g = misc.prod(g for g,_ in f.factor())
t = B.hecke_operator(p)
s = g(t)
B = s.kernel()
p = arith.next_prime(p)
return B
def abvarquo_cuspidal_subgroup(self):
"""
Compute the rational subgroup of the cuspidal subgroup (as an
abstract abelian group) of the abelian variety quotient A of
the relevant modular Jacobian attached to this modular symbols
space.
We assume that self is defined over QQ and has weight 2. If
the sign of self is not 0, then the power of 2 may be wrong.
EXAMPLES::
sage: D = ModularSymbols(66,2,sign=0).cuspidal_subspace().new_subspace().decomposition()
sage: D[0].abvarquo_cuspidal_subgroup()
Finitely generated module V/W over Integer Ring with invariants (3)
sage: [A.abvarquo_cuspidal_subgroup().invariants() for A in D]
[(3,), (2,), ()]
sage: D = ModularSymbols(66,2,sign=1).cuspidal_subspace().new_subspace().decomposition()
sage: [A.abvarquo_cuspidal_subgroup().invariants() for A in D]
[(3,), (2,), ()]
sage: D = ModularSymbols(66,2,sign=-1).cuspidal_subspace().new_subspace().decomposition()
sage: [A.abvarquo_cuspidal_subgroup().invariants() for A in D]
[(), (), ()]
"""
try: return self.__abvarquo_cuspidal_subgroup
except AttributeError: pass
if self.base_ring() != QQ:
raise ValueError, "base ring must be QQ"
if self.weight() != 2:
raise NotImplementedError, "only implemented when weight is 2"
M = self.ambient_module()
phi = self.integral_period_mapping()
P = [c for c in M.cusps() if not c.is_infinity()]
ims = [phi(M([c,infinity])) for c in P]
A = phi.codomain().span(ims, ZZ)
C = (A.ambient_module() + A)/A.ambient_module()
self.__abvarquo_cuspidal_subgroup = C
return C
def abvarquo_rational_cuspidal_subgroup(self):
r"""
Compute the rational subgroup of the cuspidal subgroup (as an
abstract abelian group) of the abelian variety quotient A of
the relevant modular Jacobian attached to this modular symbols
space. If C is the subgroup of A generated by differences of
cusps, then C is equipped with an action of Gal(Qbar/Q), and
this function computes the fixed subgroup, i.e., C(Q).
We assume that self is defined over QQ and has weight 2. If
the sign of self is not 0, then the power of 2 may be wrong.
EXAMPLES:
First we consider the fairly straightforward level 37 case,
where the torsion subgroup of the optimal quotients (which are
all elliptic curves) are all cuspidal::
sage: M = ModularSymbols(37).cuspidal_subspace().new_subspace()
sage: D = M.decomposition()
sage: [(A.abvarquo_rational_cuspidal_subgroup().invariants(), A.T(19)[0,0]) for A in D]
[((), 0), ((3,), 2)]
sage: [(E.torsion_subgroup().invariants(),E.ap(19)) for E in cremona_optimal_curves([37])]
[((), 0), ((3,), 2)]
Next we consider level 54, where the rational cuspidal
subgroups of the quotients are also cuspidal::
sage: M = ModularSymbols(54).cuspidal_subspace().new_subspace()
sage: D = M.decomposition()
sage: [A.abvarquo_rational_cuspidal_subgroup().invariants() for A in D]
[(3,), (3,)]
sage: [E.torsion_subgroup().invariants() for E in cremona_optimal_curves([54])]
[(3,), (3,)]
Level 66 is interesting, since not all torsion of the quotient
is rational. In fact, for each elliptic curve quotient, the
`\QQ`-rational subgroup of the image of the cuspidal subgroup
in the quotient is a nontrivial subgroup of `E(\QQ)_{tor}`.
Thus not all torsion in the quotient is cuspidal!
sage: M = ModularSymbols(66).cuspidal_subspace().new_subspace()
sage: D = M.decomposition()
sage: [(A.abvarquo_rational_cuspidal_subgroup().invariants(), A.T(19)[0,0]) for A in D]
[((3,), -4), ((2,), 4), ((), 0)]
sage: [(E.torsion_subgroup().invariants(),E.ap(19)) for E in cremona_optimal_curves([66])]
[((6,), -4), ((4,), 4), ((10,), 0)]
sage: [A.abelian_variety().rational_cuspidal_subgroup().invariants() for A in D]
[[6], [4], [10]]
In this example, the abelian varieties involved all having
dimension bigger than 1 (unlike above). We find that all torsion
in the quotient in each of these cases is cuspidal::
sage: M = ModularSymbols(125).cuspidal_subspace().new_subspace()
sage: D = M.decomposition()
sage: [A.abvarquo_rational_cuspidal_subgroup().invariants() for A in D]
[(), (5,), (5,)]
sage: [A.abelian_variety().rational_torsion_subgroup().multiple_of_order() for A in D]
[1, 5, 5]
"""
try: return self.__abvarquo_rational_cuspidal_subgroup
except AttributeError: pass
if self.base_ring() != QQ:
raise ValueError, "base ring must be QQ"
if self.weight() != 2:
raise NotImplementedError, "only implemented when weight is 2"
if not is_Gamma0(self.group()):
raise NotImplementedError, "only implemented when group is Gamma0"
N = self.level()
if N.is_squarefree():
return self.abvarquo_cuspidal_subgroup()
M = self.ambient_module()
phi = self.integral_period_mapping()
P = [c for c in M.cusps() if not c.is_infinity()]
V = ZZ**len(P)
ims = [phi(M([c,infinity])) for c in P]
A = phi.codomain().span(ims, ZZ)
C = (A.ambient_module() + A)/A.ambient_module()
D = V/V.zero_submodule()
psi = D.hom([C(x) for x in ims])
G = Zmod(N).unit_gens()
CQ = C
for t in G:
T = self._matrix_of_galois_action(t, P) - 1
if not T: continue
im_gens = [psi(psi.lift(g).lift() * T) for g in CQ.gens()]
h = CQ.hom(im_gens)
CQ = h.kernel()
if CQ.cardinality() == 1:
break
self.__abvarquo_rational_cuspidal_subgroup = CQ
return CQ
def _matrix_of_galois_action(self, t, P):
"""
Compute the matrix of the action of the element of the
cyclotomic Galois group defined by t on the set of cusps in P
(which is the set of finite cusps). This function is used
internally by the (rational) cuspidal subgroup and quotient
functions.
INPUT:
- `t` -- integer
- `P` -- list of cusps
EXAMPLES::
We compute the matrix of the element of the Galois group
associated to 5 and 31 for level 32. In the first case the
Galois action is trivial, and in the second it is
nontrivial.::
sage: M = ModularSymbols(32)
sage: P = [c for c in Gamma0(32).cusps() if not c.is_infinity()]
sage: M._matrix_of_galois_action(5, P)
[1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 0 0 1]
sage: z = M._matrix_of_galois_action(31, P); z
[1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 0 1]
[0 0 1 0 0 0 0]
[0 0 0 0 0 1 0]
[0 0 0 1 0 0 0]
sage: z.charpoly().factor()
(x + 1)^2 * (x - 1)^5
"""
N = self.level()
from sage.matrix.constructor import matrix
A = matrix(ZZ, len(P))
for i, c in enumerate(P):
d = c.galois_action(t, N)
for j, e in enumerate(P):
if d.is_gamma0_equiv(e, N, False):
A[i,j] = 1
A.set_immutable()
return A
class PeriodMapping(SageObject):
r"""
Base class for representing a period mapping attached to a space of modular
symbols. To be used via the derived classes RationalPeriodMapping and
IntegralPeriodMapping.
"""
def __init__(self, modsym, A):
r"""
Standard initialisation function.
INPUT:
- ``modsym`` - a space of modular symbols
- ``A`` - matrix of the associated period map
EXAMPLE::
sage: ModularSymbols(2, 8).cuspidal_submodule().integral_period_mapping() # indirect doctest
Integral period mapping associated to Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 for Gamma_0(2) of weight 8 with sign 0 over Rational Field
"""
self.__modsym = modsym
self.__domain = modsym.ambient_module()
self.__A = A
A.set_immutable()
def modular_symbols_space(self):
r"""
Return the space of modular symbols to which this period mapping
corresponds.
EXAMPLES::
sage: ModularSymbols(17, 2).rational_period_mapping().modular_symbols_space()
Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field
"""
return self.__modsym
def __call__(self, x):
r"""
Evaluate this mapping at an element of the domain.
EXAMPLE::
sage: M = ModularSymbols(17, 2).cuspidal_submodule().integral_period_mapping()
sage: M(vector([1,0,2]))
(0, 9/4)
"""
if is_FreeModuleElement(x):
v = x
else:
v = self.__domain(x).element()
return v*self.__A
def matrix(self):
r"""
Return the matrix of this period mapping.
EXAMPLE::
sage: ModularSymbols(11, 2).cuspidal_submodule().integral_period_mapping().matrix()
[ 0 1/5]
[ 1 0]
[ 0 1]
"""
return self.__A
def domain(self):
r"""
Return the domain of this mapping (which is the ambient space of the
corresponding modular symbols space).
EXAMPLE::
sage: ModularSymbols(17, 2).cuspidal_submodule().integral_period_mapping().domain()
Modular Symbols space of dimension 3 for Gamma_0(17) of weight 2 with sign 0 over Rational Field
"""
return self.__domain
def codomain(self):
r"""
Return the codomain of this mapping.
EXAMPLE:
Note that this presently returns the wrong answer, as a consequence of
various bugs in the free module routines::
sage: ModularSymbols(11, 2).cuspidal_submodule().integral_period_mapping().codomain()
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]
"""
return self.__A.row_module()
class RationalPeriodMapping(PeriodMapping):
def _repr_(self):
"""
Return the string representation of self.
EXAMPLES::
sage: ModularSymbols(40,2).rational_period_mapping()._repr_()
'Rational period mapping associated to Modular Symbols space of dimension 13 for Gamma_0(40) of weight 2 with sign 0 over Rational Field'
"""
return "Rational period mapping associated to %s"%self.modular_symbols_space()
class IntegralPeriodMapping(PeriodMapping):
def _repr_(self):
"""
Return the string representation of self.
EXAMPLES::
sage: ModularSymbols(40,2).cuspidal_submodule().integral_period_mapping()._repr_()
'Integral period mapping associated to Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 13 for Gamma_0(40) of weight 2 with sign 0 over Rational Field'
"""
return "Integral period mapping associated to %s"%self.modular_symbols_space()
