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abstract_ring.py
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abstract_ring.py
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r"""
Graded rings of modular forms for Hecke triangle groups
AUTHORS:
- Jonas Jermann (2013): initial version
"""
#*****************************************************************************
# Copyright (C) 2013-2014 Jonas Jermann <[email protected]>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.rings.all import FractionField, PolynomialRing, ZZ, QQ, infinity
from sage.algebras.free_algebra import FreeAlgebra
from sage.structure.parent import Parent
from sage.misc.cachefunc import cached_method
from hecke_triangle_groups import HeckeTriangleGroup
from constructor import FormsRing, FormsSpace, rational_type
# Maybe replace Parent by just SageObject?
class FormsRing_abstract(Parent):
r"""
Abstract (Hecke) forms ring.
This should never be called directly. Instead one should
instantiate one of the derived classes of this class.
"""
from graded_ring_element import FormsRingElement
Element = FormsRingElement
from analytic_type import AnalyticType
AT = AnalyticType()
def __init__(self, group, base_ring, red_hom):
r"""
Abstract (Hecke) forms ring.
INPUT:
- ``group`` - The Hecke triangle group (default: ``HeckeTriangleGroup(3)``)
- ``base_ring`` - The base_ring (default: ``ZZ``).
- ``red_hom`` - If True then results of binary operations are considered
homogeneous whenever it makes sense (default: False).
This is mainly used by the (Hecke) forms.
OUTPUT:
The corresponding abstract (Hecke) forms ring.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: MR = ModularFormsRing(group=5, base_ring=ZZ, red_hom=True)
sage: MR
ModularFormsRing(n=5) over Integer Ring
sage: MR.group()
Hecke triangle group for n = 5
sage: MR.base_ring()
Integer Ring
sage: MR.has_reduce_hom()
True
sage: MR.is_homogeneous()
False
"""
from graded_ring import canonical_parameters
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
if (group == infinity):
raise NotImplementedError
#if (not group.is_arithmetic() and base_ring.characteristic()>0):
# raise NotImplementedError
#if (base_ring.characteristic().divides(2*group.n()*(group.n()-2))):
# raise NotImplementedError
if (base_ring.characteristic() > 0):
raise NotImplementedError
self._group = group
self._red_hom = red_hom
self._base_ring = base_ring
self._coeff_ring = FractionField(PolynomialRing(base_ring,'d'))
self._pol_ring = PolynomialRing(base_ring,'x,y,z,d')
self._rat_field = FractionField(self._pol_ring)
# default values
self._weight = None
self._ep = None
self._analytic_type = self.AT(["quasi", "mero"])
self.default_prec(10)
self.disp_prec(5)
self.default_num_prec(53)
#super(FormsRing_abstract, self).__init__(self.coeff_ring())
def _repr_(self):
r"""
Return the string representation of ``self``.
EXAMPLES::
sage: from graded_ring import QModularFormsRing
sage: QModularFormsRing(group=4)
QuasiModularFormsRing(n=4) over Integer Ring
"""
return "{}FormsRing(n={}) over {}".format(self._analytic_type.analytic_space_name(), self._group.n(), self._base_ring)
def _latex_(self):
r"""
Return the LaTeX representation of ``self``.
EXAMPLES::
sage: from graded_ring import QWeakModularFormsRing
sage: latex(QWeakModularFormsRing())
\mathcal{ QM^! }_{n=3}(\Bold{Z})
"""
from sage.misc.latex import latex
return "\\mathcal{{ {} }}_{{n={}}}({})".format(self._analytic_type.latex_space_name(), self._group.n(), latex(self._base_ring))
def _element_constructor_(self, x):
r"""
Return ``x`` coerced/converted into this forms ring.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: MR = ModularFormsRing()
sage: (x,y,z,d) = MR.pol_ring().gens()
sage: MR(x^3)
f_rho^3
sage: el = MR.Delta().full_reduce()
sage: MR(el)
f_rho^3*d - f_i^2*d
sage: el.parent() == MR
False
sage: MR(el).parent() == MR
True
"""
from graded_ring_element import FormsRingElement
if isinstance(x, FormsRingElement):
x = self._rat_field(x._rat)
else:
x = self._rat_field(x)
return self.element_class(self, x)
def _coerce_map_from_(self, S):
r"""
Return whether or not there exists a coercion from ``S`` to ``self``.
EXAMPLES::
sage: from graded_ring import QWeakModularFormsRing, ModularFormsRing, CuspFormsRing
sage: MR1 = QWeakModularFormsRing(base_ring=CC)
sage: MR2 = ModularFormsRing()
sage: MR3 = CuspFormsRing()
sage: MR3.has_coerce_map_from(MR2)
False
sage: MR1.has_coerce_map_from(MR2)
True
sage: MR2.has_coerce_map_from(MR3)
True
sage: MR3.has_coerce_map_from(ZZ)
False
sage: MR1.has_coerce_map_from(ZZ)
True
sage: from space import ModularForms, CuspForms
sage: MF2 = ModularForms(k=6, ep=-1)
sage: MF3 = CuspForms(k=12, ep=1)
sage: MR1.has_coerce_map_from(MF2)
True
sage: MR2.has_coerce_map_from(MF3)
True
"""
from space import FormsSpace_abstract
if ( isinstance(S, FormsRing_abstract)\
and self._group == S._group\
and self._analytic_type >= S._analytic_type\
and self.base_ring().has_coerce_map_from(S.base_ring()) ):
return True
# TODO: This case never occurs: remove it?
elif isinstance(S, FormsSpace_abstract):
return self._coerce_map_from_(S.graded_ring())
elif (self.AT("holo") <= self._analytic_type) and (self.coeff_ring().has_coerce_map_from(S)):
return True
else:
return False
def _an_element_(self):
r"""
Return an element of ``self``.
EXAMPLES::
sage: from graded_ring import CuspFormsRing
sage: from space import WeakModularForms
sage: CuspFormsRing().an_element()
f_rho^3*d - f_i^2*d
sage: CuspFormsRing().an_element() == CuspFormsRing().Delta()
True
sage: WeakModularForms().an_element()
O(q^5)
sage: WeakModularForms().an_element() == WeakModularForms().zero()
True
"""
return self(self.Delta())
def default_prec(self, prec = None):
r"""
Set the default precision ``prec`` for the Fourier expansion.
If ``prec=None`` (default) then the current default precision is returned instead.
Note: This is also used as the default precision for
the Fourier expansion when evaluating forms.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: from space import ModularForms
sage: MR = ModularFormsRing()
sage: MR.default_prec(3)
sage: MR.default_prec()
3
sage: MR.Delta().q_expansion_fixed_d()
q - 24*q^2 + O(q^3)
sage: MF = ModularForms(k=4)
sage: MF.default_prec(2)
sage: MF.E4()
1 + 240*q + O(q^2)
sage: MF.default_prec()
2
"""
if (prec is not None):
self._prec = ZZ(prec)
else:
return self._prec
def disp_prec(self, prec = None):
r"""
Set the maximal display precision to ``prec``.
If ``prec="max"`` the precision is set to the default precision.
If ``prec=None`` (default) then the current display precision is returned instead.
Note: This is used for displaying/representing (elements of)
``self`` as Fourier expansions.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(k=4)
sage: MF.default_prec(5)
sage: MF.disp_prec(3)
sage: MF.disp_prec()
3
sage: MF.E4()
1 + 240*q + 2160*q^2 + O(q^3)
sage: MF.disp_prec("max")
sage: MF.E4()
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + O(q^5)
"""
if (prec == "max"):
self._disp_prec = self._prec;
elif (prec is not None):
self._disp_prec = ZZ(prec)
else:
return self._disp_prec
def default_num_prec(self, prec = None):
r"""
Set the default numerical precision to ``prec`` (default: ``53``).
If ``prec=None`` (default) the current default numerical
precision is returned instead.
EXAMPLES::
sage: from space import ModularForms
sage: MF = ModularForms(k=6)
sage: MF.default_prec(20)
sage: MF.default_num_prec(10)
sage: MF.default_num_prec()
10
sage: E6 = MF.E6()
sage: E6(i) # rel tol 1e-4
-0.0020
sage: MF.default_num_prec(100)
sage: E6(i) # rel tol 1e-25
0.00000000000000000000000000000
sage: MF = ModularForms(group=5, k=4/3)
sage: F_rho = MF.F_rho()
sage: F_rho.q_expansion(prec=2)[1]
7/(100*d)
sage: MF.default_num_prec(10)
sage: F_rho.q_expansion_fixed_d(prec=2)[1] # rel tol 1e-1
9.9
sage: MF.default_num_prec(100)
sage: F_rho.q_expansion_fixed_d(prec=2)[1] # rel tol 1e-25
9.9259324351079591527601778294
"""
if (prec is not None):
self._num_prec = ZZ(prec)
else:
return self._num_prec
def change_ring(self, new_base_ring):
r"""
Return the same space as ``self`` but over a new base ring ``new_base_ring``.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().change_ring(CC)
ModularFormsRing(n=3) over Complex Field with 53 bits of precision
"""
return self.__class__.__base__(self._group, new_base_ring, self._red_hom)
def graded_ring(self):
r"""
Return the graded ring containing ``self``.
EXAMPLES::
sage: from graded_ring import ModularFormsRing, CuspFormsRing
sage: from space import CuspForms
sage: MR = ModularFormsRing(group=5)
sage: MR.graded_ring() == MR
True
sage: CF=CuspForms(k=12)
sage: CF.graded_ring() == CuspFormsRing()
False
sage: CF.graded_ring() == CuspFormsRing(red_hom=True)
True
sage: CF.subspace([CF.Delta()]).graded_ring() == CuspFormsRing(red_hom=True)
True
"""
return self.extend_type(ring=True)
def extend_type(self, analytic_type=None, ring=False):
r"""
Return a new space which contains (elements of) ``self`` with the analytic type
of ``self`` extended by ``analytic_type``, possibly extended to a graded ring
in case ``ring`` is ``True``.
INPUT:
- ``analytic_type`` - An ``AnalyticType`` or something which coerces into it (default: ``None``).
- ``ring`` - Whether to extend to a graded ring (default: ``False``).
OUTPUT:
The new extended space.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: from space import CuspForms
sage: MR = ModularFormsRing(group=5)
sage: MR.extend_type(["quasi", "weak"])
QuasiWeakModularFormsRing(n=5) over Integer Ring
sage: CF=CuspForms(k=12)
sage: CF.extend_type("holo")
ModularForms(n=3, k=12, ep=1) over Integer Ring
sage: CF.extend_type("quasi", ring=True)
QuasiCuspFormsRing(n=3) over Integer Ring
sage: CF.subspace([CF.Delta()]).extend_type()
CuspForms(n=3, k=12, ep=1) over Integer Ring
"""
if analytic_type == None:
analytic_type = self._analytic_type
else:
analytic_type = self._analytic_type.extend_by(analytic_type)
if (ring or not self.is_homogeneous()):
return FormsRing(analytic_type, group=self.group(), base_ring=self.base_ring(), red_hom=self.has_reduce_hom())
else:
return FormsSpace(analytic_type, group=self.group(), base_ring=self.base_ring(), k=self.weight(), ep=self.ep())
def reduce_type(self, analytic_type=None, degree=None):
r"""
Return a new space with analytic properties shared by both ``self`` and ``analytic_type``,
possibly reduced to its homogeneous space of the given ``degree`` (if ``degree`` is set).
Elements of the new space are contained in ``self``.
INPUT:
- ``analytic_type`` - An ``AnalyticType`` or something which coerces into it (default: ``None``).
- ``degree`` - ``None`` (default) or the degree of the homogeneous component to which
``self`` should be reduced.
OUTPUT:
The new reduced space.
EXAMPLES::
sage: from graded_ring import QModularFormsRing
sage: from space import QModularForms
sage: MR = QModularFormsRing()
sage: MR.reduce_type(["quasi", "cusp"])
QuasiCuspFormsRing(n=3) over Integer Ring
sage: MR.reduce_type("cusp", degree=(12,1))
CuspForms(n=3, k=12, ep=1) over Integer Ring
sage: MF=QModularForms(k=6)
sage: MF.reduce_type("holo")
ModularForms(n=3, k=6, ep=-1) over Integer Ring
sage: MF.reduce_type([])
ZeroForms(n=3, k=6, ep=-1) over Integer Ring
"""
if analytic_type == None:
analytic_type = self._analytic_type
else:
analytic_type = self._analytic_type.reduce_to(analytic_type)
if (degree == None and not self.is_homogeneous()):
return FormsRing(analytic_type, group=self.group(), base_ring=self.base_ring(), red_hom=self.has_reduce_hom())
elif (degree == None):
return FormsSpace(analytic_type, group=self.group(), base_ring=self.base_ring(), k=self.weight(), ep=self.ep())
else:
(weight, ep) = degree
if (self.is_homogeneous() and (weight != self.weight() or ep!=self.ep())):
analytic_type = self._analytic_type.reduce_to([])
return FormsSpace(analytic_type, group=self.group(), base_ring=self.base_ring(), k=weight, ep=ep)
def construction(self):
r"""
Return a functor that constructs ``self`` (used by the coercion machinery).
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().construction()
(ModularFormsRingFunctor(n=3), BaseFacade(Integer Ring))
"""
from functors import FormsRingFunctor, BaseFacade
return FormsRingFunctor(self._analytic_type, self._group, self._red_hom), BaseFacade(self._base_ring)
@cached_method
def group(self):
r"""
Return the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: MR = ModularFormsRing(group=7)
sage: MR.group()
Hecke triangle group for n = 7
sage: from space import CuspForms
sage: CF = CuspForms(group=7, k=4/5)
sage: CF.group()
Hecke triangle group for n = 7
"""
return self._group
@cached_method
def hecke_n(self):
r"""
Return the parameter ``n`` of the
(Hecke triangle) group of ``self``.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: MR = ModularFormsRing(group=7)
sage: MR.hecke_n()
7
sage: from space import CuspForms
sage: CF = CuspForms(group=7, k=4/5)
sage: CF.hecke_n()
7
"""
return self._group.n()
@cached_method
def base_ring(self):
r"""
Return base ring of ``self``.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().base_ring()
Integer Ring
sage: from space import CuspForms
sage: CuspForms(k=12, base_ring=AA).base_ring()
Algebraic Real Field
"""
return self._base_ring
@cached_method
def coeff_ring(self):
r"""
Return coefficient ring of ``self``.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().coeff_ring()
Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: from space import CuspForms
sage: CuspForms(k=12, base_ring=AA).coeff_ring()
Fraction Field of Univariate Polynomial Ring in d over Algebraic Real Field
"""
return self._coeff_ring
@cached_method
def pol_ring(self):
r"""
Return the underlying polynomial ring used
by ``self``.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().pol_ring()
Multivariate Polynomial Ring in x, y, z, d over Integer Ring
sage: from space import CuspForms
sage: CuspForms(k=12, base_ring=AA).pol_ring()
Multivariate Polynomial Ring in x, y, z, d over Algebraic Real Field
"""
return self._pol_ring
@cached_method
def rat_field(self):
r"""
Return the underlying rational field used by
``self`` to construct/represent elements.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().rat_field()
Fraction Field of Multivariate Polynomial Ring in x, y, z, d over Integer Ring
sage: from space import CuspForms
sage: CuspForms(k=12, base_ring=AA).rat_field()
Fraction Field of Multivariate Polynomial Ring in x, y, z, d over Algebraic Real Field
"""
return self._rat_field
@cached_method
def diff_alg(self):
r"""
Return the algebra of differential operators
(over QQ) which is used on rational functions
representing elements of ``self``.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().diff_alg()
Noncommutative Multivariate Polynomial Ring in X, Y, Z, dX, dY, dZ over Rational Field, nc-relations: {dY*Y: Y*dY + 1, dZ*Z: Z*dZ + 1, dX*X: X*dX + 1}
sage: from space import CuspForms
sage: CuspForms(k=12, base_ring=AA).diff_alg()
Noncommutative Multivariate Polynomial Ring in X, Y, Z, dX, dY, dZ over Rational Field, nc-relations: {dY*Y: Y*dY + 1, dZ*Z: Z*dZ + 1, dX*X: X*dX + 1}
"""
# We only use two operators for now which do not involve 'd', so for performance
# reason we choose FractionField(base_ring) instead of self.coeff_ring().
free_alg = FreeAlgebra(FractionField(ZZ),6,'X,Y,Z,dX,dY,dZ')
(X,Y,Z,dX,dY,dZ) = free_alg.gens()
diff_alg = free_alg.g_algebra({dX*X:1+X*dX,dY*Y:1+Y*dY,dZ*Z:1+Z*dZ})
return diff_alg
@cached_method
def _derivative_op(self):
r"""
Return the differential operator in ``self.diff_alg()``
corresponding to the derivative of forms.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing(group=7)._derivative_op()
-1/2*X^6*dY - 5/28*X^5*dZ + 1/7*X*Z*dX + 1/2*Y*Z*dY + 5/28*Z^2*dZ - 1/7*Y*dX
"""
(X,Y,Z,dX,dY,dZ) = self.diff_alg().gens()
return 1/self._group.n() * (X*Z-Y)*dX\
+ ZZ(1)/ZZ(2) * (Y*Z-X**(self._group.n()-1))*dY\
+ (self._group.n()-2) / (4*self._group.n()) * (Z**2-X**(self._group.n()-2))*dZ
@cached_method
def _serre_derivative_op(self):
r"""
Return the differential operator in ``self.diff_alg()``
corresponding to the serre derivative of forms.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing(group=8)._serre_derivative_op()
-1/2*X^7*dY - 3/16*X^6*dZ - 3/16*Z^2*dZ - 1/8*Y*dX
"""
(X,Y,Z,dX,dY,dZ) = self.diff_alg().gens()
return - 1/self._group.n() * Y*dX\
- ZZ(1)/ZZ(2) * X**(self._group.n()-1)*dY\
- (self._group.n()-2) / (4*self._group.n()) * (Z**2+X**(self._group.n()-2))*dZ
@cached_method
def has_reduce_hom(self):
r"""
Return whether the method ``reduce`` should reduce
homogeneous elements to the corresponding homogeneous space.
This is mainly used by binary operations on homogeneous
spaces which temporarily produce an element of ``self``
but want to consider it as a homogeneous element
(also see ``reduce``).
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().has_reduce_hom()
False
sage: ModularFormsRing(red_hom=True).has_reduce_hom()
True
sage: from space import ModularForms
sage: ModularForms(k=6).has_reduce_hom()
True
sage: ModularForms(k=6).graded_ring().has_reduce_hom()
True
"""
return self._red_hom
def is_homogeneous(self):
r"""
Return whether ``self`` is homogeneous component.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().is_homogeneous()
False
sage: from space import ModularForms
sage: ModularForms(k=6).is_homogeneous()
True
"""
return self._weight != None
def is_modular(self):
r"""
Return whether ``self`` only contains modular elements.
EXAMPLES::
sage: from graded_ring import QWeakModularFormsRing, CuspFormsRing
sage: QWeakModularFormsRing().is_modular()
False
sage: CuspFormsRing(group=7).is_modular()
True
sage: from space import QWeakModularForms, CuspForms
sage: QWeakModularForms(k=10).is_modular()
False
sage: CuspForms(group=7, k=12, base_ring=AA).is_modular()
True
"""
return not (self.AT("quasi") <= self._analytic_type)
def is_weakly_holomorphic(self):
r"""
Return whether ``self`` only contains weakly
holomorphic modular elements.
EXAMPLES::
sage: from graded_ring import QMModularFormsRing, QWeakModularFormsRing, CuspFormsRing
sage: QMModularFormsRing().is_weakly_holomorphic()
False
sage: QWeakModularFormsRing().is_weakly_holomorphic()
True
sage: from space import MModularForms, CuspForms
sage: MModularForms(k=10).is_weakly_holomorphic()
False
sage: CuspForms(group=7, k=12, base_ring=AA).is_weakly_holomorphic()
True
"""
return (self.AT("weak", "quasi") >= self._analytic_type)
def is_holomorphic(self):
r"""
Return whether ``self`` only contains holomorphic
modular elements.
EXAMPLES::
sage: from graded_ring import QWeakModularFormsRing, QModularFormsRing
sage: QWeakModularFormsRing().is_holomorphic()
False
sage: QModularFormsRing().is_holomorphic()
True
sage: from space import WeakModularForms, CuspForms
sage: WeakModularForms(k=10).is_holomorphic()
False
sage: CuspForms(group=7, k=12, base_ring=AA).is_holomorphic()
True
"""
return (self.AT("holo", "quasi") >= self._analytic_type)
def is_cuspidal(self):
r"""
Return whether ``self`` only contains cuspidal elements.
EXAMPLES::
sage: from graded_ring import QModularFormsRing, QCuspFormsRing
sage: QModularFormsRing().is_cuspidal()
False
sage: QCuspFormsRing().is_cuspidal()
True
sage: from space import ModularForms, QCuspForms
sage: ModularForms(k=12).is_cuspidal()
False
sage: QCuspForms(k=12).is_cuspidal()
True
"""
return (self.AT("cusp", "quasi") >= self._analytic_type)
def is_zerospace(self):
r"""
Return whether ``self`` is the (0-dimensional) zero space.
EXAMPLES::
sage: from graded_ring import ModularFormsRing
sage: ModularFormsRing().is_zerospace()
False
sage: from space import ModularForms, CuspForms
sage: ModularForms(k=12).is_zerospace()
False
sage: CuspForms(k=12).reduce_type([]).is_zerospace()
True
"""
return (self.AT(["quasi"]) >= self._analytic_type)
def analytic_type(self):
r"""
Return the analytic type of ``self``.
EXAMPLES::
sage: from graded_ring import QMModularFormsRing, QWeakModularFormsRing
sage: QMModularFormsRing().analytic_type()
quasi meromorphic modular
sage: QWeakModularFormsRing().analytic_type()
quasi weakly holomorphic modular
sage: from space import MModularForms, CuspForms
sage: MModularForms(k=10).analytic_type()
meromorphic modular
sage: CuspForms(group=7, k=12, base_ring=AA).analytic_type()
cuspidal
"""
return self._analytic_type
def homogeneous_space(self, k, ep):
r"""
Return the homogeneous component of degree (``k``, ``e``) of ``self``.
EXAMPLES::
sage: from graded_ring import QMModularFormsRing, QWeakModularFormsRing
sage: QMModularFormsRing(group=7).homogeneous_space(k=2, ep=-1)
QuasiMeromorphicModularForms(n=7, k=2, ep=-1) over Integer Ring
"""
return self.reduce_type(degree = (k,ep))
@cached_method
def J_inv(self):
r"""
Return the J-invariant (Hauptmodul) of the group of ``self``.
It is normalized such that ``J_inv(infinity) = infinity``,
it has real Fourier coefficients starting with ``d > 0`` and ``J_inv(i) = 1``
It lies in a (weak) extension of the graded ring of ``self``.
In case ``has_reduce_hom`` is ``True`` it is given as an element of
the corresponding homogeneous space.
EXAMPLES::
sage: from graded_ring import QMModularFormsRing, WeakModularFormsRing, CuspFormsRing
sage: MR = WeakModularFormsRing(group=7)
sage: J_inv = MR.J_inv()
sage: J_inv in MR
True
sage: CuspFormsRing(group=7).J_inv() == J_inv
True
sage: J_inv
f_rho^7/(f_rho^7 - f_i^2)
sage: QMModularFormsRing(group=7).J_inv() == QMModularFormsRing(group=7)(J_inv)
True
sage: from space import WeakModularForms, CuspForms
sage: MF = WeakModularForms(group=5, k=0)
sage: J_inv = MF.J_inv()
sage: J_inv in MF
True
sage: WeakModularFormsRing(group=5, red_hom=True).J_inv() == J_inv
True
sage: CuspForms(group=5, k=12).J_inv() == J_inv
True
sage: MF.disp_prec(3)
sage: J_inv
d*q^-1 + 79/200 + 42877/(640000*d)*q + 12957/(2000000*d^2)*q^2 + O(q^3)
sage: WeakModularForms().J_inv()
1/1728*q^-1 + 31/72 + 1823/16*q + 335840/27*q^2 + 16005555/32*q^3 + 11716352*q^4 + O(q^5)
"""
(x,y,z,d) = self._pol_ring.gens()
return self.extend_type("weak", ring=True)(x**self._group.n()/(x**self._group.n()-y**2)).reduce()
@cached_method
def j_inv(self):
r"""
Return the j-invariant (Hauptmodul) of the group of ``self``.
It is normalized such that ``j_inv(infinity) = infinity``,
and such that it has real Fourier coefficients starting with ``1``.
It lies in a (weak) extension of the graded ring of ``self``.
In case ``has_reduce_hom`` is ``True`` it is given as an element of
the corresponding homogeneous space.
EXAMPLES::
sage: from graded_ring import QMModularFormsRing, WeakModularFormsRing, CuspFormsRing
sage: MR = WeakModularFormsRing(group=7)
sage: j_inv = MR.j_inv()
sage: j_inv in MR
True
sage: CuspFormsRing(group=7).j_inv() == j_inv
True
sage: j_inv
f_rho^7/(f_rho^7*d - f_i^2*d)
sage: QMModularFormsRing(group=7).j_inv() == QMModularFormsRing(group=7)(j_inv)
True
sage: from space import WeakModularForms, CuspForms
sage: MF = WeakModularForms(group=5, k=0)
sage: j_inv = MF.j_inv()
sage: j_inv in MF
True
sage: WeakModularFormsRing(group=5, red_hom=True).j_inv() == j_inv
True
sage: CuspForms(group=5, k=12).j_inv() == j_inv
True
sage: MF.disp_prec(3)
sage: j_inv
q^-1 + 79/(200*d) + 42877/(640000*d^2)*q + 12957/(2000000*d^3)*q^2 + O(q^3)
sage: WeakModularForms().j_inv()
q^-1 + 744 + 196884*q + 21493760*q^2 + 864299970*q^3 + 20245856256*q^4 + O(q^5)
"""
(x,y,z,d) = self._pol_ring.gens()
return self.extend_type("weak", ring=True)(1/d*x**self._group.n()/(x**self._group.n()-y**2)).reduce()
@cached_method
def F_rho(self):
r"""
Return the generator ``F_rho`` of the graded ring of ``self``.
Up to the group action ``F_rho`` has exactly one simple zero at ``rho``. ``F_rho`` is
normalized such that its first nontrivial Fourier coefficient is ``1``.
The polynomial variable ``x`` exactly corresponds to ``F_rho``.
It lies in a (cuspidal) extension of the graded ring of ``self``.
In case ``has_reduce_hom`` is ``True`` it is given as an element of
the corresponding homogeneous space.
EXAMPLES::
sage: from graded_ring import QMModularFormsRing, ModularFormsRing, CuspFormsRing
sage: MR = ModularFormsRing(group=7)
sage: F_rho = MR.F_rho()
sage: F_rho in MR
True
sage: CuspFormsRing(group=7).F_rho() == F_rho
True
sage: F_rho
f_rho
sage: QMModularFormsRing(group=7).F_rho() == QMModularFormsRing(group=7)(F_rho)
True
sage: from space import ModularForms, CuspForms
sage: MF = ModularForms(group=5, k=4/3)
sage: F_rho = MF.F_rho()
sage: F_rho in MF
True
sage: ModularFormsRing(group=5, red_hom=True).F_rho() == F_rho
True
sage: CuspForms(group=5, k=12).F_rho() == F_rho
True
sage: MF.disp_prec(3)
sage: F_rho
1 + 7/(100*d)*q + 21/(160000*d^2)*q^2 + O(q^3)
sage: ModularForms(k=4).F_rho() == ModularForms(k=4).E4()
True
sage: ModularForms(k=4).F_rho()
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + O(q^5)
"""
(x,y,z,d) = self._pol_ring.gens()
return self.extend_type("holo", ring=True)(x).reduce()
@cached_method
def F_i(self):
r"""
Return the generator ``F_i`` of the graded ring of ``self``.
Up to the group action ``F_i`` has exactly one simple zero at ``i``. ``F_i`` is
normalized such that its first nontrivial Fourier coefficient is ``1``.
The polynomial variable ``y`` exactly corresponds to ``F_i``.
It lies in a (holomorphic) extension of the graded ring of ``self``.
In case ``has_reduce_hom`` is ``True`` it is given as an element of
the corresponding homogeneous space.
EXAMPLES::
sage: from graded_ring import QMModularFormsRing, ModularFormsRing, CuspFormsRing
sage: MR = ModularFormsRing(group=7)
sage: F_i = MR.F_i()
sage: F_i in MR
True
sage: CuspFormsRing(group=7).F_i() == F_i
True
sage: F_i
f_i
sage: QMModularFormsRing(group=7).F_i() == QMModularFormsRing(group=7)(F_i)
True
sage: from space import ModularForms, CuspForms