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r""" 

Graded rings of modular forms for Hecke triangle groups 

 

AUTHORS: 

 

- Jonas Jermann (2013): initial version 

 

""" 

from __future__ import absolute_import 

 

#***************************************************************************** 

# 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, n=None): 

r""" 

Return a canonical version of the parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import canonical_parameters 

sage: canonical_parameters(4, ZZ, 1) 

(Hecke triangle group for n = 4, Integer Ring, True, 4) 

sage: canonical_parameters(infinity, RR, 0) 

(Hecke triangle group for n = +Infinity, Real Field with 53 bits of precision, False, +Infinity) 

""" 

 

if not (n is None): 

group = n 

 

if (group == infinity): 

group = HeckeTriangleGroup(infinity) 

else: 

try: 

group = HeckeTriangleGroup(ZZ(group)) 

except TypeError: 

group = HeckeTriangleGroup(group.n()) 

 

red_hom = bool(red_hom) 

n = group.n() 

 

return (group, base_ring, red_hom, n) 

 

 

class QuasiMeromorphicModularFormsRing(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, n=None): 

r""" 

Return a (cached) instance with canonical parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, QuasiMeromorphicModularFormsRing) 

sage: (group, base_ring, red_hom, n) = canonical_parameters(4, ZZ, 1) 

sage: QuasiMeromorphicModularFormsRing(4, ZZ, 1) == QuasiMeromorphicModularFormsRing(group, base_ring, red_hom, n) 

True 

""" 

 

(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

 

def __init__(self, group, base_ring, red_hom, n): 

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 spaces of homogeneous elements. 

 

OUTPUT: 

 

The corresponding graded ring of (Hecke) quasi meromorphic modular forms 

for the given ``group`` and ``base_ring``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing 

sage: MR = QuasiMeromorphicModularFormsRing(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 Integer Ring 

sage: MR in MR.category() 

True 

 

sage: QuasiMeromorphicModularFormsRing(n=infinity) 

QuasiMeromorphicModularFormsRing(n=+Infinity) over Integer Ring 

""" 

 

FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

CommutativeAlgebra.__init__(self, base_ring=base_ring, category=CommutativeAlgebras(base_ring)) 

self._analytic_type = self.AT(["quasi", "mero"]) 

 

class QuasiWeakModularFormsRing(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, n=None): 

r""" 

Return a (cached) instance with canonical parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, QuasiWeakModularFormsRing) 

sage: (group, base_ring, red_hom, n) = canonical_parameters(5, CC, 0) 

sage: QuasiWeakModularFormsRing(5, CC, 0) == QuasiWeakModularFormsRing(group, base_ring, red_hom, n) 

True 

""" 

 

(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

 

def __init__(self, group, base_ring, red_hom, n): 

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 spaces of homogeneous elements. 

 

OUTPUT: 

 

The corresponding graded ring of (Hecke) quasi weakly holomorphic modular forms 

for the given ``group`` and ``base_ring``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiWeakModularFormsRing 

sage: MR = QuasiWeakModularFormsRing(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 Complex Field with 53 bits of precision 

sage: MR in MR.category() 

True 

""" 

 

FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

CommutativeAlgebra.__init__(self, base_ring=base_ring, category=CommutativeAlgebras(base_ring)) 

self._analytic_type = self.AT(["quasi", "weak"]) 

 

class QuasiModularFormsRing(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, n=None): 

r""" 

Return a (cached) instance with canonical parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, QuasiModularFormsRing) 

sage: (group, base_ring, red_hom, n) = canonical_parameters(6, ZZ, True) 

sage: QuasiModularFormsRing(6, ZZ, True) == QuasiModularFormsRing(group, base_ring, red_hom, n) 

True 

""" 

 

(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

 

def __init__(self, group, base_ring, red_hom, n): 

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 spaces of homogeneous elements. 

 

OUTPUT: 

 

The corresponding graded ring of (Hecke) quasi modular forms 

for the given ``group`` and ``base_ring``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing 

sage: MR = QuasiModularFormsRing(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 Integer Ring 

sage: MR in MR.category() 

True 

""" 

 

FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

CommutativeAlgebra.__init__(self, base_ring=base_ring, category=CommutativeAlgebras(base_ring)) 

self._analytic_type = self.AT(["quasi", "holo"]) 

 

class QuasiCuspFormsRing(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, n=None): 

r""" 

Return a (cached) instance with canonical parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, QuasiCuspFormsRing) 

sage: (group, base_ring, red_hom, n) = canonical_parameters(7, ZZ, 1) 

sage: QuasiCuspFormsRing(7, ZZ, 1) == QuasiCuspFormsRing(group, base_ring, red_hom, n) 

True 

""" 

 

(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

 

def __init__(self, group, base_ring, red_hom, n): 

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 spaces of homogeneous elements. 

 

OUTPUT: 

 

The corresponding graded ring of (Hecke) quasi cusp forms 

for the given ``group`` and ``base_ring``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiCuspFormsRing 

sage: MR = QuasiCuspFormsRing(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 Integer Ring 

sage: MR in MR.category() 

True 

""" 

 

FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

CommutativeAlgebra.__init__(self, base_ring=base_ring, category=CommutativeAlgebras(base_ring)) 

self._analytic_type = self.AT(["quasi", "cusp"]) 

 

class MeromorphicModularFormsRing(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, n=None): 

r""" 

Return a (cached) instance with canonical parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, MeromorphicModularFormsRing) 

sage: (group, base_ring, red_hom, n) = canonical_parameters(4, ZZ, 1) 

sage: MeromorphicModularFormsRing(4, ZZ, 1) == MeromorphicModularFormsRing(group, base_ring, red_hom, n) 

True 

""" 

 

(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

 

def __init__(self, group, base_ring, red_hom, n): 

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 spaces of homogeneous elements. 

 

OUTPUT: 

 

The corresponding graded ring of (Hecke) meromorphic modular forms 

for the given ``group`` and ``base_ring``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import MeromorphicModularFormsRing 

sage: MR = MeromorphicModularFormsRing(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 Integer Ring 

sage: MR in MR.category() 

True 

""" 

 

FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

CommutativeAlgebra.__init__(self, base_ring=base_ring, category=CommutativeAlgebras(base_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, n=None): 

r""" 

Return a (cached) instance with canonical parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, WeakModularFormsRing) 

sage: (group, base_ring, red_hom, n) = canonical_parameters(5, ZZ, 0) 

sage: WeakModularFormsRing(5, ZZ, 0) == WeakModularFormsRing(group, base_ring, red_hom, n) 

True 

""" 

 

(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

 

def __init__(self, group, base_ring, red_hom, n): 

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 spaces of homogeneous elements. 

 

OUTPUT: 

 

The corresponding graded ring of (Hecke) weakly holomorphic modular forms 

for the given ``group`` and ``base_ring``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing 

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 Integer Ring 

sage: MR in MR.category() 

True 

""" 

 

FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

CommutativeAlgebra.__init__(self, base_ring=base_ring, category=CommutativeAlgebras(base_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, n=None): 

r""" 

Return a (cached) instance with canonical parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: ModularFormsRing(3, ZZ, 0) == ModularFormsRing() 

True 

""" 

 

(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

 

def __init__(self, group, base_ring, red_hom, n): 

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 spaces of homogeneous elements. 

 

OUTPUT: 

 

The corresponding graded ring of (Hecke) modular forms 

for the given ``group`` and ``base_ring``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing 

sage: MR = ModularFormsRing() 

sage: MR 

ModularFormsRing(n=3) over Integer Ring 

sage: MR.analytic_type() 

modular 

sage: MR.category() 

Category of commutative algebras over Integer Ring 

sage: MR in MR.category() 

True 

""" 

 

FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

CommutativeAlgebra.__init__(self, base_ring=base_ring, category=CommutativeAlgebras(base_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, n=None): 

r""" 

Return a (cached) instance with canonical parameters. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import (canonical_parameters, CuspFormsRing) 

sage: (group, base_ring, red_hom, n) = canonical_parameters(5, CC, True) 

sage: CuspFormsRing(5, CC, True) == CuspFormsRing(group, base_ring, red_hom, n) 

True 

""" 

 

(group, base_ring, red_hom, n) = canonical_parameters(group, base_ring, red_hom, n) 

return super(FormsRing_abstract,cls).__classcall__(cls, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

 

def __init__(self, group, base_ring, red_hom, n): 

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 spaces of homogeneous elements. 

 

OUTPUT: 

 

The corresponding graded ring of (Hecke) cusp forms 

for the given ``group`` and ``base_ring``. 

 

EXAMPLES:: 

 

sage: from sage.modular.modform_hecketriangle.graded_ring import CuspFormsRing 

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 Complex Field with 53 bits of precision 

sage: MR in MR.category() 

True 

 

sage: CuspFormsRing(n=infinity, base_ring=CC, red_hom=True) 

CuspFormsRing(n=+Infinity) over Complex Field with 53 bits of precision 

""" 

 

FormsRing_abstract.__init__(self, group=group, base_ring=base_ring, red_hom=red_hom, n=n) 

CommutativeAlgebra.__init__(self, base_ring=base_ring, category=CommutativeAlgebras(base_ring)) 

self._analytic_type = self.AT(["cusp"])