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graded_ring.py
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graded_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 <jjermann2@gmail.com>
#
# 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 ZZ, QQ, infinity
from sage.rings.ring import CommutativeAlgebra
from sage.categories.all import CommutativeAlgebras
from sage.structure.unique_representation import UniqueRepresentation
from sage.misc.cachefunc import cached_method
from hecke_triangle_groups import HeckeTriangleGroup
from abstract_ring import FormsRing_abstract
def canonical_parameters(group, base_ring, red_hom):
r"""
Return a canonical version of the parameters.
EXAMPLES::
sage: canonical_parameters(4, ZZ, 1)
(Hecke triangle group for n = 4, Integer Ring, True)
"""
if (group == infinity):
group = HeckeTriangleGroup(infinity)
else:
try:
group = HeckeTriangleGroup(ZZ(group))
except TypeError:
group = HeckeTriangleGroup(group.n())
red_hom = bool(red_hom)
return (group, base_ring, red_hom)
class QMModularFormsRing(FormsRing_abstract, CommutativeAlgebra, UniqueRepresentation):
r"""
Graded ring of (Hecke) quasi meromorphic modular forms
for the given group and base ring
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, red_hom = False):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: (group, base_ring, red_hom) = canonical_parameters(4, ZZ, 1)
sage: QMModularFormsRing(4, ZZ, 1) == QMModularFormsRing(group, base_ring, red_hom)
True
"""
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom)
def __init__(self, group, base_ring, red_hom):
r"""
Return the graded ring of (Hecke) quasi meromorphic modular forms
for the given ``group`` and ``base_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 homogeneous spaces.
OUTPUT:
The corresponding graded ring of (Hecke) quasi meromorphic modular forms
for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: MR = QMModularFormsRing(4, ZZ, 1)
sage: MR
QuasiMeromorphicModularFormsRing(n=4) over Integer Ring
sage: MR.analytic_type()
quasi meromorphic modular
sage: MR.category()
Category of commutative algebras over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom)
CommutativeAlgebra.__init__(self, base_ring=self.coeff_ring(), category=CommutativeAlgebras(self.coeff_ring()))
self._analytic_type = self.AT(["quasi", "mero"])
class QWeakModularFormsRing(FormsRing_abstract, CommutativeAlgebra, UniqueRepresentation):
r"""
Graded ring of (Hecke) quasi weakly holomorphic modular forms
for the given group and base ring
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, red_hom = False):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: (group, base_ring, red_hom) = canonical_parameters(5, CC, 0)
sage: QWeakModularFormsRing(5, CC, 0) == QWeakModularFormsRing(group, base_ring, red_hom)
True
"""
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom)
def __init__(self, group, base_ring, red_hom):
r"""
Return the graded ring of (Hecke) quasi weakly holomorphic modular forms
for the given ``group`` and ``base_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 homogeneous spaces.
OUTPUT:
The corresponding graded ring of (Hecke) quasi weakly holomorphic modular forms
for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: MR = QWeakModularFormsRing(5, CC, 0)
sage: MR
QuasiWeakModularFormsRing(n=5) over Complex Field with 53 bits of precision
sage: MR.analytic_type()
quasi weakly holomorphic modular
sage: MR.category()
Category of commutative algebras over Fraction Field of Univariate Polynomial Ring in d over Complex Field with 53 bits of precision
"""
FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom)
CommutativeAlgebra.__init__(self, base_ring=self.coeff_ring(), category=CommutativeAlgebras(self.coeff_ring()))
self._analytic_type = self.AT(["quasi", "weak"])
class QModularFormsRing(FormsRing_abstract, CommutativeAlgebra, UniqueRepresentation):
r"""
Graded ring of (Hecke) quasi modular forms
for the given group and base ring
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, red_hom = False):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: (group, base_ring, red_hom) = canonical_parameters(6, ZZ, True)
sage: QModularFormsRing(6, ZZ, True) == QModularFormsRing(group, base_ring, red_hom)
True
"""
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom)
def __init__(self, group, base_ring, red_hom):
r"""
Return the graded ring of (Hecke) quasi modular forms
for the given ``group`` and ``base_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 homogeneous spaces.
OUTPUT:
The corresponding graded ring of (Hecke) quasi modular forms
for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: MR = QModularFormsRing(6, ZZ, True)
sage: MR
QuasiModularFormsRing(n=6) over Integer Ring
sage: MR.analytic_type()
quasi modular
sage: MR.category()
Category of commutative algebras over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom)
CommutativeAlgebra.__init__(self, base_ring=self.coeff_ring(), category=CommutativeAlgebras(self.coeff_ring()))
self._analytic_type = self.AT(["quasi", "holo"])
class QCuspFormsRing(FormsRing_abstract, CommutativeAlgebra, UniqueRepresentation):
r"""
Graded ring of (Hecke) quasi cusp forms
for the given group and base ring
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, red_hom = False):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: (group, base_ring, red_hom) = canonical_parameters(7, ZZ, 1)
sage: QCuspFormsRing(7, ZZ, 1) == QCuspFormsRing(group, base_ring, red_hom)
True
"""
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom)
def __init__(self, group, base_ring, red_hom):
r"""
Return the graded ring of (Hecke) quasi cusp forms
for the given ``group`` and ``base_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 homogeneous spaces.
OUTPUT:
The corresponding graded ring of (Hecke) quasi cusp forms
for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: MR = QCuspFormsRing(7, ZZ, 1)
sage: MR
QuasiCuspFormsRing(n=7) over Integer Ring
sage: MR.analytic_type()
quasi cuspidal
sage: MR.category()
Category of commutative algebras over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom)
CommutativeAlgebra.__init__(self, base_ring=self.coeff_ring(), category=CommutativeAlgebras(self.coeff_ring()))
self._analytic_type = self.AT(["quasi", "cusp"])
class MModularFormsRing(FormsRing_abstract, CommutativeAlgebra, UniqueRepresentation):
r"""
Graded ring of (Hecke) meromorphic modular forms
for the given group and base ring
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, red_hom = False):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: (group, base_ring, red_hom) = canonical_parameters(4, ZZ, 1)
sage: MModularFormsRing(4, ZZ, 1) == MModularFormsRing(group, base_ring, red_hom)
True
"""
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom)
def __init__(self, group, base_ring, red_hom):
r"""
Return the graded ring of (Hecke) meromorphic modular forms
for the given ``group`` and ``base_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 homogeneous spaces.
OUTPUT:
The corresponding graded ring of (Hecke) meromorphic modular forms
for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: MR = MModularFormsRing(4, ZZ, 1)
sage: MR
MeromorphicModularFormsRing(n=4) over Integer Ring
sage: MR.analytic_type()
meromorphic modular
sage: MR.category()
Category of commutative algebras over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom)
CommutativeAlgebra.__init__(self, base_ring=self.coeff_ring(), category=CommutativeAlgebras(self.coeff_ring()))
self._analytic_type = self.AT(["mero"])
class WeakModularFormsRing(FormsRing_abstract, CommutativeAlgebra, UniqueRepresentation):
r"""
Graded ring of (Hecke) weakly holomorphic modular forms
for the given group and base ring
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, red_hom = False):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: (group, base_ring, red_hom) = canonical_parameters(5, ZZ, 0)
sage: WeakModularFormsRing(5, ZZ, 0) == WeakModularFormsRing(group, base_ring, red_hom)
True
"""
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom)
def __init__(self, group, base_ring, red_hom):
r"""
Return the graded ring of (Hecke) weakly holomorphic modular forms
for the given ``group`` and ``base_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 homogeneous spaces.
OUTPUT:
The corresponding graded ring of (Hecke) weakly holomorphic modular forms
for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: MR = WeakModularFormsRing(5, ZZ, 0)
sage: MR
WeakModularFormsRing(n=5) over Integer Ring
sage: MR.analytic_type()
weakly holomorphic modular
sage: MR.category()
Category of commutative algebras over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom)
CommutativeAlgebra.__init__(self, base_ring=self.coeff_ring(), category=CommutativeAlgebras(self.coeff_ring()))
self._analytic_type = self.AT(["weak"])
class ModularFormsRing(FormsRing_abstract, CommutativeAlgebra, UniqueRepresentation):
r"""
Graded ring of (Hecke) modular forms
for the given group and base ring
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, red_hom = False):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: ModularFormsRing(3, ZZ, 0) == ModularFormsRing()
True
"""
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom)
def __init__(self, group, base_ring, red_hom):
r"""
Return the graded ring of (Hecke) modular forms
for the given ``group`` and ``base_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 homogeneous spaces.
OUTPUT:
The corresponding graded ring of (Hecke) modular forms
for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: MR = ModularFormsRing()
sage: MR
ModularFormsRing(n=3) over Integer Ring
sage: MR.analytic_type()
modular
sage: MR.category()
Category of commutative algebras over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom)
CommutativeAlgebra.__init__(self, base_ring=self.coeff_ring(), category=CommutativeAlgebras(self.coeff_ring()))
self._analytic_type = self.AT(["holo"])
class CuspFormsRing(FormsRing_abstract, CommutativeAlgebra, UniqueRepresentation):
r"""
Graded ring of (Hecke) cusp forms
for the given group and base ring
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, red_hom = False):
r"""
Return a (cached) instance with canonical parameters.
EXAMPLES::
sage: (group, base_ring, red_hom) = canonical_parameters(5, CC, True)
sage: CuspFormsRing(5, CC, True) == CuspFormsRing(group, base_ring, red_hom)
True
"""
(group, base_ring, red_hom) = canonical_parameters(group, base_ring, red_hom)
return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom)
def __init__(self, group, base_ring, red_hom):
r"""
Return the graded ring of (Hecke) cusp forms
for the given ``group`` and ``base_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 homogeneous spaces.
OUTPUT:
The corresponding graded ring of (Hecke) cusp forms
for the given ``group`` and ``base_ring``.
EXAMPLES::
sage: MR = CuspFormsRing(5, CC, True)
sage: MR
CuspFormsRing(n=5) over Complex Field with 53 bits of precision
sage: MR.analytic_type()
cuspidal
sage: MR.category()
Category of commutative algebras over Fraction Field of Univariate Polynomial Ring in d over Complex Field with 53 bits of precision
"""
FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom)
CommutativeAlgebra.__init__(self, base_ring=self.coeff_ring(), category=CommutativeAlgebras(self.coeff_ring()))
self._analytic_type = self.AT(["cusp"])