def __init__(self, f, p, base_extend=True):
r"""
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: example_type_space() # indirect doctest
6-dimensional type space at prime 7 of form q + q^2 + (-1/2*a1 + 1/2)*q^3 + q^4 + (a1 - 1)*q^5 + O(q^6)
"""
self._p = p
self._f = f
if f.level() % p:
raise ValueError("p must divide level")
amb = ModularSymbols(self.group(), f.weight())
self.e_space = find_in_space(f, amb,
base_extend=base_extend).sign_submodule(1)
R = self.e_space.base_ring()
mat = amb._action_on_modular_symbols([p**self.u(), 1, 0, p**self.u()])
V = amb.free_module().base_extend(R)
bvecs = []
for v in self.e_space.free_module().basis():
bvecs += mat.maxspin(v)
T = V.submodule(bvecs)
self._unipmat = mat.change_ring(R).restrict(T).transpose() / ZZ(
p**(self.u() * (f.weight() - 2)))
self.t_space = amb.base_extend(R).submodule(T, check=False)
def __init__(self, f, p, base_extend=True):
r"""
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: example_type_space() # indirect doctest
6-dimensional type space at prime 7 of form q + q^2 + (-1/2*a1 + 1/2)*q^3 + q^4 + (a1 - 1)*q^5 + O(q^6)
"""
self._p = p
self._f = f
if f.level() % p:
raise ValueError( "p must divide level" )
amb = ModularSymbols(self.group(), f.weight())
self.e_space = find_in_space(f, amb, base_extend=base_extend).sign_submodule(1)
R = self.e_space.base_ring()
mat = amb._action_on_modular_symbols([p**self.u(), 1, 0, p**self.u()])
V = amb.free_module().base_extend(R)
bvecs = []
for v in self.e_space.free_module().basis():
bvecs += mat.maxspin(v)
T = V.submodule(bvecs)
self._unipmat = mat.change_ring(R).restrict(T).transpose() / ZZ(p ** (self.u() * (f.weight() - 2)))
self.t_space = amb.base_extend(R).submodule(T, check=False)
def modular_symbols(self, sign=0, weight=2, base_ring=QQ):
"""
Return the space of modular symbols of the specified weight and sign
on the congruence subgroup self.
EXAMPLES::
sage: G = Gamma0(23)
sage: G.modular_symbols()
Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Rational Field
sage: G.modular_symbols(weight=4)
Modular Symbols space of dimension 12 for Gamma_0(23) of weight 4 with sign 0 over Rational Field
sage: G.modular_symbols(base_ring=GF(7))
Modular Symbols space of dimension 5 for Gamma_0(23) of weight 2 with sign 0 over Finite Field of size 7
sage: G.modular_symbols(sign=1)
Modular Symbols space of dimension 3 for Gamma_0(23) of weight 2 with sign 1 over Rational Field
"""
from sage.modular.modsym.modsym import ModularSymbols
return ModularSymbols(self, sign=sign, weight=weight, base_ring=base_ring)
def _modular_symbols(self):
"""
Return the modular symbols space associated to this ambient
Jacobian.
OUTPUT: modular symbols space
EXAMPLES::
sage: M = J0(33)._modular_symbols(); M
Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field
sage: J0(33)._modular_symbols() is M
True
"""
try:
return self.__modsym
except AttributeError:
self.__modsym = ModularSymbols(self.__group,
weight=2).cuspidal_submodule()
return self.__modsym
def dimension_of_ordinary_subspace(self, p=None, cusp=False):
"""
If ``cusp`` is ``True``, return dimension of cuspidal ordinary
subspace. This does a weight 2 computation with sage's ModularSymbols.
EXAMPLES::
sage: M = OverconvergentModularSymbols(11, 0, sign=-1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
2
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M.dimension_of_ordinary_subspace()
4
sage: M = OverconvergentModularSymbols(11, 0, sign=0, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
6
sage: M.dimension_of_ordinary_subspace(cusp=True)
4
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
sage: M = OverconvergentModularSymbols(11, 2, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
0
sage: M.dimension_of_ordinary_subspace()
1
sage: M = OverconvergentModularSymbols(11, 10, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
An example with odd weight and hence non-trivial character::
sage: K = Qp(11, 6)
sage: DG = DirichletGroup(11, K)
sage: chi = DG([K(378703)])
sage: MM = FamiliesOfOMS(chi, 1, p=11, prec_cap=[4, 4], base_coeffs=ZpCA(11, 4), sign=-1)
sage: MM.dimension_of_ordinary_subspace()
1
"""
try:
p = self.prime()
except AttributeError:
if p is None:
raise ValueError(
"If self doesn't have a prime, must specify p.")
try:
return self._ord_dim_dict[(p, cusp)]
except AttributeError:
self._ord_dim_dict = {}
except KeyError:
pass
from sage.modular.dirichlet import DirichletGroup
from sage.rings.finite_rings.constructor import GF
try:
chi = self.character()
except AttributeError:
chi = DirichletGroup(self.level(), GF(p))[0]
if chi is None:
chi = DirichletGroup(self.level(), GF(p))[0]
from sage.modular.modsym.modsym import ModularSymbols
r = self.weight() % (p - 1)
if chi.is_trivial():
N = chi.modulus()
if N % p != 0:
N *= p
else:
e = N.valuation(p)
N.divide_knowing_divisible_by(p**(e - 1))
chi = DirichletGroup(N, GF(p))[0]
elif chi.modulus() % p != 0:
chi = DirichletGroup(chi.modulus() * p, GF(p))(chi)
DG = DirichletGroup(chi.modulus(), GF(p))
if r == 0:
from sage.modular.arithgroup.congroup_gamma0 import Gamma0_constructor as Gamma0
verbose("in dim: %s, %s, %s" % (self.sign(), chi, p))
M = ModularSymbols(DG(chi), 2, self.sign(), GF(p))
else:
psi = [GF(p)(u)**r for u in DG.unit_gens()] #mod p Teichmuller^r
psi = DG(psi)
M = ModularSymbols(DG(chi) * psi, 2, self.sign(), GF(p))
if cusp:
M = M.cuspidal_subspace()
hecke_poly = M.hecke_polynomial(p)
verbose("in dim: %s" % (hecke_poly))
x = hecke_poly.parent().gen()
d = hecke_poly.degree() - hecke_poly.ord(x)
self._ord_dim_dict[(p, cusp)] = d
return d
def dimension_of_ordinary_subspace(self, p=None, cusp=False):
"""
If ``cusp`` is ``True``, return dimension of cuspidal ordinary
subspace. This does a weight 2 computation with sage's ModularSymbols.
EXAMPLES::
sage: M = OverconvergentModularSymbols(11, 0, sign=-1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
2
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M.dimension_of_ordinary_subspace()
4
sage: M = OverconvergentModularSymbols(11, 0, sign=0, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
6
sage: M.dimension_of_ordinary_subspace(cusp=True)
4
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
sage: M = OverconvergentModularSymbols(11, 2, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
0
sage: M.dimension_of_ordinary_subspace()
1
sage: M = OverconvergentModularSymbols(11, 10, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
An example with odd weight and hence non-trivial character::
sage: K = Qp(11, 6)
sage: DG = DirichletGroup(11, K)
sage: chi = DG([K(378703)])
sage: MM = FamiliesOfOMS(chi, 1, p=11, prec_cap=[4, 4], base_coeffs=ZpCA(11, 4), sign=-1)
sage: MM.dimension_of_ordinary_subspace()
1
"""
try:
p = self.prime()
except AttributeError:
if p is None:
raise ValueError("If self doesn't have a prime, must specify p.")
try:
return self._ord_dim_dict[(p, cusp)]
except AttributeError:
self._ord_dim_dict = {}
except KeyError:
pass
from sage.modular.dirichlet import DirichletGroup
from sage.rings.finite_rings.constructor import GF
try:
chi = self.character()
except AttributeError:
chi = DirichletGroup(self.level(), GF(p))[0]
if chi is None:
chi = DirichletGroup(self.level(), GF(p))[0]
from sage.modular.modsym.modsym import ModularSymbols
r = self.weight() % (p-1)
if chi.is_trivial():
N = chi.modulus()
if N % p != 0:
N *= p
else:
e = N.valuation(p)
N.divide_knowing_divisible_by(p ** (e-1))
chi = DirichletGroup(N, GF(p))[0]
elif chi.modulus() % p != 0:
chi = DirichletGroup(chi.modulus() * p, GF(p))(chi)
DG = DirichletGroup(chi.modulus(), GF(p))
if r == 0:
from sage.modular.arithgroup.congroup_gamma0 import Gamma0_constructor as Gamma0
verbose("in dim: %s, %s, %s"%(self.sign(), chi, p))
M = ModularSymbols(DG(chi), 2, self.sign(), GF(p))
else:
psi = [GF(p)(u) ** r for u in DG.unit_gens()] #mod p Teichmuller^r
psi = DG(psi)
M = ModularSymbols(DG(chi) * psi, 2, self.sign(), GF(p))
if cusp:
M = M.cuspidal_subspace()
hecke_poly = M.hecke_polynomial(p)
verbose("in dim: %s"%(hecke_poly))
x = hecke_poly.parent().gen()
d = hecke_poly.degree() - hecke_poly.ord(x)
self._ord_dim_dict[(p, cusp)] = d
return d