def __init__(self, domain, codomain, cat):
"""
Create a homspace.
INPUT:
- ``domain, codomain`` - modular abelian varieties
- ``cat`` - category
EXAMPLES::
sage: H = Hom(J0(11), J0(22)); H
Space of homomorphisms from Abelian variety J0(11) of dimension 1 to Abelian variety J0(22) of dimension 2
sage: Hom(J0(11), J0(11))
Endomorphism ring of Abelian variety J0(11) of dimension 1
sage: type(H)
<class 'sage.modular.abvar.homspace.Homspace_with_category'>
sage: H.homset_category()
Category of modular abelian varieties over Rational Field
"""
if not abelian_variety.is_ModularAbelianVariety(domain):
raise TypeError, "domain must be a modular abelian variety"
if not abelian_variety.is_ModularAbelianVariety(codomain):
raise TypeError, "codomain must be a modular abelian variety"
self._matrix_space = MatrixSpace(ZZ, 2 * domain.dimension(),
2 * codomain.dimension())
self._gens = None
HomsetWithBase.__init__(self, domain, codomain, cat)
def __init__(self, domain, codomain, cat):
"""
Create a homspace.
INPUT:
- ``domain, codomain`` - modular abelian varieties
- ``cat`` - category
EXAMPLES::
sage: H = Hom(J0(11), J0(22)); H
Space of homomorphisms from Abelian variety J0(11) of dimension 1 to Abelian variety J0(22) of dimension 2
sage: Hom(J0(11), J0(11))
Endomorphism ring of Abelian variety J0(11) of dimension 1
sage: type(H)
<class 'sage.modular.abvar.homspace.Homspace_with_category'>
sage: H.homset_category()
Category of modular abelian varieties over Rational Field
"""
if not abelian_variety.is_ModularAbelianVariety(domain):
raise TypeError, "domain must be a modular abelian variety"
if not abelian_variety.is_ModularAbelianVariety(codomain):
raise TypeError, "codomain must be a modular abelian variety"
self._matrix_space = MatrixSpace(ZZ,2*domain.dimension(), 2*codomain.dimension())
self._gens = None
HomsetWithBase.__init__(self, domain, codomain, cat)
def __init__(self, abvar, lattice, field_of_definition=QQbar, check=True):
"""
A finite subgroup of a modular abelian variety that is defined by a
given lattice.
INPUT:
- ``abvar`` - a modular abelian variety
- ``lattice`` - a lattice that contains the lattice of
abvar
- ``field_of_definition`` - the field of definition
of this finite group scheme
- ``check`` - bool (default: True) whether or not to
check that lattice contains the abvar lattice.
EXAMPLES::
sage: J = J0(11)
sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G
Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1
"""
if check:
if not is_FreeModule(lattice) or lattice.base_ring() != ZZ:
raise TypeError, "lattice must be a free module over ZZ"
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError, "abvar must be a modular abelian variety"
if not abvar.lattice().is_submodule(lattice):
lattice += abvar.lattice()
if lattice.rank() != abvar.lattice().rank():
raise ValueError, "lattice must contain the lattice of abvar with finite index"
FiniteSubgroup.__init__(self, abvar, field_of_definition)
self.__lattice = lattice
def __init__(self, abvar, field_of_definition=QQ):
"""
A finite subgroup of a modular abelian variety.
INPUT:
- ``abvar`` - a modular abelian variety
- ``field_of_definition`` - a field over which this
group is defined.
EXAMPLES: This is an abstract base class, so there are no instances
of this class itself.
::
sage: A = J0(37)
sage: G = A.torsion_subgroup(3); G
Finite subgroup with invariants [3, 3, 3, 3] over QQ of Abelian variety J0(37) of dimension 2
sage: type(G)
<class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice'>
sage: from sage.modular.abvar.finite_subgroup import FiniteSubgroup
sage: isinstance(G, FiniteSubgroup)
True
"""
if field_of_definition not in _Fields:
raise TypeError, "field_of_definition must be a field"
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError, "abvar must be a modular abelian variety"
Module_old.__init__(self, ZZ)
self.__abvar = abvar
self.__field_of_definition = field_of_definition
def __init__(self, abvar, field_of_definition=QQ):
"""
A finite subgroup of a modular abelian variety.
INPUT:
- ``abvar`` - a modular abelian variety
- ``field_of_definition`` - a field over which this
group is defined.
EXAMPLES: This is an abstract base class, so there are no instances
of this class itself.
::
sage: A = J0(37)
sage: G = A.torsion_subgroup(3); G
Finite subgroup with invariants [3, 3, 3, 3] over QQ of Abelian variety J0(37) of dimension 2
sage: type(G)
<class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice'>
sage: from sage.modular.abvar.finite_subgroup import FiniteSubgroup
sage: isinstance(G, FiniteSubgroup)
True
"""
if field_of_definition not in _Fields:
raise TypeError, "field_of_definition must be a field"
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError, "abvar must be a modular abelian variety"
Module_old.__init__(self, ZZ)
self.__abvar = abvar
self.__field_of_definition = field_of_definition
def __init__(self, abvar, n):
"""
Create the Hecke operator of index `n` acting on the
abelian variety abvar.
INPUT:
- ``abvar`` - a modular abelian variety
- ``n`` - a positive integer
EXAMPLES::
sage: J = J0(37)
sage: T2 = J.hecke_operator(2); T2
Hecke operator T_2 on Abelian variety J0(37) of dimension 2
sage: T2.parent()
Endomorphism ring of Abelian variety J0(37) of dimension 2
"""
n = ZZ(n)
if n <= 0:
raise ValueError("n must be positive")
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError("abvar must be a modular abelian variety")
self.__abvar = abvar
self.__n = n
sage.modules.matrix_morphism.MatrixMorphism_abstract.__init__(
self, abvar.Hom(abvar))
def __init__(self, abvar, n):
"""
Create the Hecke operator of index `n` acting on the
abelian variety abvar.
INPUT:
- ``abvar`` - a modular abelian variety
- ``n`` - a positive integer
EXAMPLES::
sage: J = J0(37)
sage: T2 = J.hecke_operator(2); T2
Hecke operator T_2 on Abelian variety J0(37) of dimension 2
sage: T2.parent()
Endomorphism ring of Abelian variety J0(37) of dimension 2
"""
n = ZZ(n)
if n <= 0:
raise ValueError, "n must be positive"
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError, "abvar must be a modular abelian variety"
self.__abvar = abvar
self.__n = n
sage.modules.matrix_morphism.MatrixMorphism_abstract.__init__(self, abvar.Hom(abvar))
def __init__(self, abvar, lattice, field_of_definition=QQbar, check=True):
"""
A finite subgroup of a modular abelian variety that is defined by a
given lattice.
INPUT:
- ``abvar`` - a modular abelian variety
- ``lattice`` - a lattice that contains the lattice of
abvar
- ``field_of_definition`` - the field of definition
of this finite group scheme
- ``check`` - bool (default: True) whether or not to
check that lattice contains the abvar lattice.
EXAMPLES::
sage: J = J0(11)
sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G
Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1
"""
if check:
if not is_FreeModule(lattice) or lattice.base_ring() != ZZ:
raise TypeError, "lattice must be a free module over ZZ"
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError, "abvar must be a modular abelian variety"
if not abvar.lattice().is_submodule(lattice):
lattice += abvar.lattice()
if lattice.rank() != abvar.lattice().rank():
raise ValueError, "lattice must contain the lattice of abvar with finite index"
FiniteSubgroup.__init__(self, abvar, field_of_definition)
self.__lattice = lattice
def __init__(self, abvar, field_of_definition=QQ):
"""
Initialize ``self``.
TESTS::
sage: A = J0(11)
sage: G = A.torsion_subgroup(2)
sage: TestSuite(G).run() # long time
"""
from sage.categories.category import Category
from sage.categories.fields import Fields
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.modules import Modules
if field_of_definition not in Fields():
raise TypeError("field_of_definition must be a field")
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError("abvar must be a modular abelian variety")
category = Category.join((Modules(ZZ), FiniteEnumeratedSets()))
Module.__init__(self, ZZ, category=category)
self.__abvar = abvar
self.__field_of_definition = field_of_definition
def __init__(self, abvar, field_of_definition=QQ):
"""
Initialize ``self``.
TESTS::
sage: from sage_modabvar import J0
sage: A = J0(11)
sage: G = A.torsion_subgroup(2)
sage: TestSuite(G).run() # long time
"""
from sage.categories.category import Category
from sage.categories.fields import Fields
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.modules import Modules
if field_of_definition not in Fields():
raise TypeError("field_of_definition must be a field")
if not abelian_variety.is_ModularAbelianVariety(abvar):
raise TypeError("abvar must be a modular abelian variety")
category = Category.join((Modules(ZZ), FiniteEnumeratedSets()))
Module.__init__(self, ZZ, category=category)
self.__abvar = abvar
self.__field_of_definition = field_of_definition
def intersection(self, other):
"""
Return the intersection of the finite subgroups self and other.
INPUT:
- ``other`` - a finite group
OUTPUT: a finite group
EXAMPLES::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9)
sage: G.intersection(H)
Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: W = E11a1.torsion_subgroup(15)
sage: G.intersection(W)
Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect subgroups of different abelian varieties.
::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5)
sage: G.intersection(H)
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect abelian varieties with subgroups::
sage: t = J0(33).hecke_operator(7)
sage: G = t.kernel()[0]; G
Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3
sage: A = J0(33).old_subvariety()
sage: A.intersection(G)
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: A.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: B = J0(33).new_subvariety()
sage: B.intersection(G)
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: B.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: A.intersection(B)[0]
Finite subgroup with invariants [3, 3] over QQ of Abelian subvariety of dimension 2 of J0(33)
"""
A = self.abelian_variety()
if abelian_variety.is_ModularAbelianVariety(other):
amb = other
B = other
M = B.lattice().scale(Integer(1) / self.exponent())
K = composite_field(self.field_of_definition(), other.base_field())
else:
amb = A
if not isinstance(other, FiniteSubgroup):
raise TypeError, "only addition of two finite subgroups is defined"
B = other.abelian_variety()
if A.ambient_variety() != B.ambient_variety():
raise TypeError, "finite subgroups must be in the same ambient product Jacobian"
M = other.lattice()
K = composite_field(self.field_of_definition(),
other.field_of_definition())
L = self.lattice()
if A != B:
# TODO: This might be way slower than what we could do if
# we think more carefully.
C = A + B
L = L + C.lattice()
M = M + C.lattice()
W = L.intersection(M).intersection(amb.vector_space())
return FiniteSubgroup_lattice(amb, W, field_of_definition=K)
def intersection(self, other):
"""
Return the intersection of the finite subgroups self and other.
INPUT:
- ``other`` - a finite group
OUTPUT: a finite group
EXAMPLES::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9)
sage: G.intersection(H)
Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: W = E11a1.torsion_subgroup(15)
sage: G.intersection(W)
Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect subgroups of different abelian varieties.
::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5)
sage: G.intersection(H)
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect abelian varieties with subgroups::
sage: t = J0(33).hecke_operator(7)
sage: G = t.kernel()[0]; G
Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3
sage: A = J0(33).old_subvariety()
sage: A.intersection(G)
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: A.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: B = J0(33).new_subvariety()
sage: B.intersection(G)
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: B.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: A.intersection(B)[0]
Finite subgroup with invariants [3, 3] over QQ of Abelian subvariety of dimension 2 of J0(33)
"""
A = self.abelian_variety()
if abelian_variety.is_ModularAbelianVariety(other):
amb = other
B = other
M = B.lattice().scale(Integer(1)/self.exponent())
K = composite_field(self.field_of_definition(), other.base_field())
else:
amb = A
if not isinstance(other, FiniteSubgroup):
raise TypeError, "only addition of two finite subgroups is defined"
B = other.abelian_variety()
if A.ambient_variety() != B.ambient_variety():
raise TypeError, "finite subgroups must be in the same ambient product Jacobian"
M = other.lattice()
K = composite_field(self.field_of_definition(), other.field_of_definition())
L = self.lattice()
if A != B:
# TODO: This might be way slower than what we could do if
# we think more carefully.
C = A + B
L = L + C.lattice()
M = M + C.lattice()
W = L.intersection(M).intersection(amb.vector_space())
return FiniteSubgroup_lattice(amb, W, field_of_definition=K)
def __call__(self, X):
"""
INPUT:
- ``X`` - abelian variety, finite group, or torsion
element
OUTPUT: abelian variety, finite group, torsion element
EXAMPLES: We apply morphisms to elements::
sage: t2 = J0(33).hecke_operator(2)
sage: G = J0(33).torsion_subgroup(2); G
Finite subgroup with invariants [2, 2, 2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3
sage: t2(G.0)
[(-1/2, 0, 1/2, -1/2, 1/2, -1/2)]
sage: t2(G.0) in G
True
sage: t2(G.1)
[(0, -1, 1/2, 0, 1/2, -1/2)]
sage: t2(G.2)
[(0, 0, 0, 0, 0, 0)]
sage: K = t2.kernel()[0]; K
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3
sage: t2(K.0)
[(0, 0, 0, 0, 0, 0)]
We apply morphisms to subgroups::
sage: t2 = J0(33).hecke_operator(2)
sage: G = J0(33).torsion_subgroup(2); G
Finite subgroup with invariants [2, 2, 2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3
sage: t2(G)
Finite subgroup with invariants [2, 2] over QQ of Abelian variety J0(33) of dimension 3
sage: t2.fcp()
(x - 1) * (x + 2)^2
We apply morphisms to abelian subvarieties::
sage: E11a0, E11a1, B = J0(33)
sage: t2 = J0(33).hecke_operator(2)
sage: t3 = J0(33).hecke_operator(3)
sage: E11a0
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: t3(E11a0)
Abelian subvariety of dimension 1 of J0(33)
sage: t3(E11a0).decomposition()
[
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)
]
sage: t3(E11a0) == E11a1
True
sage: t2(E11a0) == E11a0
True
sage: t3(E11a0) == E11a0
False
sage: t3(E11a0 + E11a1) == E11a0 + E11a1
True
We apply some Hecke operators to the cuspidal subgroup and split it
up::
sage: C = J0(33).cuspidal_subgroup(); C
Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3
sage: t2 = J0(33).hecke_operator(2); t2.fcp()
(x - 1) * (x + 2)^2
sage: (t2 - 1)(C)
Finite subgroup with invariants [5, 5] over QQ of Abelian variety J0(33) of dimension 3
sage: (t2 + 2)(C)
Finite subgroup with invariants [2, 2] over QQ of Abelian variety J0(33) of dimension 3
Same but on a simple new factor::
sage: C = J0(33)[2].cuspidal_subgroup(); C
Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
sage: t2 = J0(33)[2].hecke_operator(2); t2.fcp()
x - 1
sage: t2(C)
Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
"""
from abvar import is_ModularAbelianVariety
from finite_subgroup import FiniteSubgroup
if isinstance(X, TorsionPoint):
return self._image_of_element(X)
elif is_ModularAbelianVariety(X):
return self._image_of_abvar(X)
elif isinstance(X, FiniteSubgroup):
return self._image_of_finite_subgroup(X)
else:
raise TypeError("X must be an abelian variety or finite subgroup")
def __call__(self, X):
"""
INPUT:
- ``X`` - abelian variety, finite group, or torsion
element
OUTPUT: abelian variety, finite group, torsion element
EXAMPLES: We apply morphisms to elements::
sage: t2 = J0(33).hecke_operator(2)
sage: G = J0(33).torsion_subgroup(2); G
Finite subgroup with invariants [2, 2, 2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3
sage: t2(G.0)
[(-1/2, 0, 1/2, -1/2, 1/2, -1/2)]
sage: t2(G.0) in G
True
sage: t2(G.1)
[(0, -1, 1/2, 0, 1/2, -1/2)]
sage: t2(G.2)
[(0, 0, 0, 0, 0, 0)]
sage: K = t2.kernel()[0]; K
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3
sage: t2(K.0)
[(0, 0, 0, 0, 0, 0)]
We apply morphisms to subgroups::
sage: t2 = J0(33).hecke_operator(2)
sage: G = J0(33).torsion_subgroup(2); G
Finite subgroup with invariants [2, 2, 2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3
sage: t2(G)
Finite subgroup with invariants [2, 2] over QQ of Abelian variety J0(33) of dimension 3
sage: t2.fcp()
(x - 1) * (x + 2)^2
We apply morphisms to abelian subvarieties::
sage: E11a0, E11a1, B = J0(33)
sage: t2 = J0(33).hecke_operator(2)
sage: t3 = J0(33).hecke_operator(3)
sage: E11a0
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: t3(E11a0)
Abelian subvariety of dimension 1 of J0(33)
sage: t3(E11a0).decomposition()
[
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)
]
sage: t3(E11a0) == E11a1
True
sage: t2(E11a0) == E11a0
True
sage: t3(E11a0) == E11a0
False
sage: t3(E11a0 + E11a1) == E11a0 + E11a1
True
We apply some Hecke operators to the cuspidal subgroup and split it
up::
sage: C = J0(33).cuspidal_subgroup(); C
Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3
sage: t2 = J0(33).hecke_operator(2); t2.fcp()
(x - 1) * (x + 2)^2
sage: (t2 - 1)(C)
Finite subgroup with invariants [5, 5] over QQ of Abelian variety J0(33) of dimension 3
sage: (t2 + 2)(C)
Finite subgroup with invariants [2, 2] over QQ of Abelian variety J0(33) of dimension 3
Same but on a simple new factor::
sage: C = J0(33)[2].cuspidal_subgroup(); C
Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
sage: t2 = J0(33)[2].hecke_operator(2); t2.fcp()
x - 1
sage: t2(C)
Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
"""
from abvar import is_ModularAbelianVariety
from finite_subgroup import FiniteSubgroup
if isinstance(X, TorsionPoint):
return self._image_of_element(X)
elif is_ModularAbelianVariety(X):
return self._image_of_abvar(X)
elif isinstance(X, FiniteSubgroup):
return self._image_of_finite_subgroup(X)
else:
raise TypeError, "X must be an abelian variety or finite subgroup"