Ambient Jacobian abelian variety¶

sage.modular.abvar.abvar_ambient_jacobian.ModAbVar_ambient_jacobian(group)[source]¶

Return the ambient Jacobian attached to a given congruence subgroup.

The result is cached using a weakref. This function is called internally by modular abelian variety constructors.

INPUT:

group – a congruence subgroup

OUTPUT: a modular abelian variety attached

EXAMPLES:

Sage

sage: import sage.modular.abvar.abvar_ambient_jacobian as abvar_ambient_jacobian
sage: A = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
sage: A
Abelian variety J0(11) of dimension 1
sage: B = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
sage: A is B
True

Python

>>> from sage.all import *
>>> import sage.modular.abvar.abvar_ambient_jacobian as abvar_ambient_jacobian
>>> A = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(Integer(11)))
>>> A
Abelian variety J0(11) of dimension 1
>>> B = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(Integer(11)))
>>> A is B
True

Sage Live

import sage.modular.abvar.abvar_ambient_jacobian as abvar_ambient_jacobian
A = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
A
B = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
A is B

You can get access to and/or clear the cache as follows:

Sage

sage: abvar_ambient_jacobian._cache = {}
sage: B = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
sage: A is B
False

Python

>>> from sage.all import *
>>> abvar_ambient_jacobian._cache = {}
>>> B = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(Integer(11)))
>>> A is B
False

Sage Live

abvar_ambient_jacobian._cache = {}
B = abvar_ambient_jacobian.ModAbVar_ambient_jacobian(Gamma0(11))
A is B

class sage.modular.abvar.abvar_ambient_jacobian.ModAbVar_ambient_jacobian_class(group)[source]¶

Bases: ModularAbelianVariety_modsym_abstract

An ambient Jacobian modular abelian variety attached to a congruence subgroup.

ambient_variety()[source]¶

Return the ambient modular abelian variety that contains self. Since self is a Jacobian modular abelian variety, this is just self.

OUTPUT: abelian variety

EXAMPLES:

Sage

sage: A = J0(17)
sage: A.ambient_variety()
Abelian variety J0(17) of dimension 1
sage: A is A.ambient_variety()
True

Python

>>> from sage.all import *
>>> A = J0(Integer(17))
>>> A.ambient_variety()
Abelian variety J0(17) of dimension 1
>>> A is A.ambient_variety()
True

Sage Live

A = J0(17)
A.ambient_variety()
A is A.ambient_variety()

decomposition(simple=True, bound=None)[source]¶

Decompose this ambient Jacobian as a product of abelian subvarieties, up to isogeny.

EXAMPLES:

Sage

sage: J0(33).decomposition(simple=False)
[
Abelian subvariety of dimension 2 of J0(33),
Abelian subvariety of dimension 1 of J0(33)
]
sage: J0(33).decomposition(simple=False)[1].is_simple()
True
sage: J0(33).decomposition(simple=False)[0].is_simple()
False
sage: J0(33).decomposition(simple=False)
[
Abelian subvariety of dimension 2 of J0(33),
Simple abelian subvariety 33a(None,33) of dimension 1 of J0(33)
]
sage: J0(33).decomposition(simple=True)
[
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33),
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33),
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
]

Python

>>> from sage.all import *
>>> J0(Integer(33)).decomposition(simple=False)
[
Abelian subvariety of dimension 2 of J0(33),
Abelian subvariety of dimension 1 of J0(33)
]
>>> J0(Integer(33)).decomposition(simple=False)[Integer(1)].is_simple()
True
>>> J0(Integer(33)).decomposition(simple=False)[Integer(0)].is_simple()
False
>>> J0(Integer(33)).decomposition(simple=False)
[
Abelian subvariety of dimension 2 of J0(33),
Simple abelian subvariety 33a(None,33) of dimension 1 of J0(33)
]
>>> J0(Integer(33)).decomposition(simple=True)
[
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33),
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33),
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
]

Sage Live

J0(33).decomposition(simple=False)
J0(33).decomposition(simple=False)[1].is_simple()
J0(33).decomposition(simple=False)[0].is_simple()
J0(33).decomposition(simple=False)
J0(33).decomposition(simple=True)

degeneracy_map(level, t=1, check=True)[source]¶

Return the \(t\)-th degeneracy map from self to J(level). Here \(t\) must be a divisor of either level/self.level() or self.level()/level.

INPUT:

level – integer (multiple or divisor of level of self)
t – divisor of quotient of level of self and level
check – boolean (default: True); if True do some checks on the input

OUTPUT: a morphism

EXAMPLES:

Sage

sage: J0(11).degeneracy_map(33)
Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [1]
sage: J0(11).degeneracy_map(33).matrix()
[ 0 -3  2  1 -2  0]
[ 1 -2  0  1  0 -1]
sage: J0(11).degeneracy_map(33,3).matrix()
[-1  0  0  0  1 -2]
[-1 -1  1 -1  1  0]
sage: J0(33).degeneracy_map(11,1).matrix()
[ 0  1]
[ 0 -1]
[ 1 -1]
[ 0  1]
[-1  1]
[ 0  0]
sage: J0(11).degeneracy_map(33,1).matrix() * J0(33).degeneracy_map(11,1).matrix()
[4 0]
[0 4]

Python

>>> from sage.all import *
>>> J0(Integer(11)).degeneracy_map(Integer(33))
Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [1]
>>> J0(Integer(11)).degeneracy_map(Integer(33)).matrix()
[ 0 -3  2  1 -2  0]
[ 1 -2  0  1  0 -1]
>>> J0(Integer(11)).degeneracy_map(Integer(33),Integer(3)).matrix()
[-1  0  0  0  1 -2]
[-1 -1  1 -1  1  0]
>>> J0(Integer(33)).degeneracy_map(Integer(11),Integer(1)).matrix()
[ 0  1]
[ 0 -1]
[ 1 -1]
[ 0  1]
[-1  1]
[ 0  0]
>>> J0(Integer(11)).degeneracy_map(Integer(33),Integer(1)).matrix() * J0(Integer(33)).degeneracy_map(Integer(11),Integer(1)).matrix()
[4 0]
[0 4]

Sage Live

J0(11).degeneracy_map(33)
J0(11).degeneracy_map(33).matrix()
J0(11).degeneracy_map(33,3).matrix()
J0(33).degeneracy_map(11,1).matrix()
J0(11).degeneracy_map(33,1).matrix() * J0(33).degeneracy_map(11,1).matrix()

dimension()[source]¶

Return the dimension of this modular abelian variety.

EXAMPLES:

Sage

sage: J0(2007).dimension()
221
sage: J1(13).dimension()
2
sage: J1(997).dimension()
40920
sage: J0(389).dimension()
32
sage: JH(389,[4]).dimension()
64
sage: J1(389).dimension()
6112

Python

>>> from sage.all import *
>>> J0(Integer(2007)).dimension()
221
>>> J1(Integer(13)).dimension()
2
>>> J1(Integer(997)).dimension()
40920
>>> J0(Integer(389)).dimension()
32
>>> JH(Integer(389),[Integer(4)]).dimension()
64
>>> J1(Integer(389)).dimension()
6112

Sage Live

J0(2007).dimension()
J1(13).dimension()
J1(997).dimension()
J0(389).dimension()
JH(389,[4]).dimension()
J1(389).dimension()

group()[source]¶

Return the group that this Jacobian modular abelian variety is attached to.

EXAMPLES:

Sage

sage: J1(37).group()
Congruence Subgroup Gamma1(37)
sage: J0(5077).group()
Congruence Subgroup Gamma0(5077)
sage: J = GammaH(11,[3]).modular_abelian_variety(); J
Abelian variety JH(11,[3]) of dimension 1
sage: J.group()
Congruence Subgroup Gamma_H(11) with H generated by [3]

Python

>>> from sage.all import *
>>> J1(Integer(37)).group()
Congruence Subgroup Gamma1(37)
>>> J0(Integer(5077)).group()
Congruence Subgroup Gamma0(5077)
>>> J = GammaH(Integer(11),[Integer(3)]).modular_abelian_variety(); J
Abelian variety JH(11,[3]) of dimension 1
>>> J.group()
Congruence Subgroup Gamma_H(11) with H generated by [3]

Sage Live

J1(37).group()
J0(5077).group()
J = GammaH(11,[3]).modular_abelian_variety(); J
J.group()

groups()[source]¶

Return the tuple of congruence subgroups attached to this ambient Jacobian. This is always a tuple of length 1.

OUTPUT: tuple

EXAMPLES:

Sage

sage: J0(37).groups()
(Congruence Subgroup Gamma0(37),)

Python

>>> from sage.all import *
>>> J0(Integer(37)).groups()
(Congruence Subgroup Gamma0(37),)

Sage Live

J0(37).groups()

newform_decomposition(names=None)[source]¶

Return the newforms of the simple subvarieties in the decomposition of self as a product of simple subvarieties, up to isogeny.

OUTPUT: an array of newforms

EXAMPLES:

Sage

sage: J0(81).newform_decomposition('a')
[q - 2*q^4 + O(q^6), q - 2*q^4 + O(q^6), q + a0*q^2 + q^4 - a0*q^5 + O(q^6)]

sage: J1(19).newform_decomposition('a')
[q - 2*q^3 - 2*q^4 + 3*q^5 + O(q^6),
 q + a1*q^2 + (-1/9*a1^5 - 1/3*a1^4 - 1/3*a1^3 + 1/3*a1^2 - a1 - 1)*q^3 + (4/9*a1^5 + 2*a1^4 + 14/3*a1^3 + 17/3*a1^2 + 6*a1 + 2)*q^4 + (-2/3*a1^5 - 11/3*a1^4 - 10*a1^3 - 14*a1^2 - 15*a1 - 9)*q^5 + O(q^6)]

Python

>>> from sage.all import *
>>> J0(Integer(81)).newform_decomposition('a')
[q - 2*q^4 + O(q^6), q - 2*q^4 + O(q^6), q + a0*q^2 + q^4 - a0*q^5 + O(q^6)]

>>> J1(Integer(19)).newform_decomposition('a')
[q - 2*q^3 - 2*q^4 + 3*q^5 + O(q^6),
 q + a1*q^2 + (-1/9*a1^5 - 1/3*a1^4 - 1/3*a1^3 + 1/3*a1^2 - a1 - 1)*q^3 + (4/9*a1^5 + 2*a1^4 + 14/3*a1^3 + 17/3*a1^2 + 6*a1 + 2)*q^4 + (-2/3*a1^5 - 11/3*a1^4 - 10*a1^3 - 14*a1^2 - 15*a1 - 9)*q^5 + O(q^6)]

Sage Live

J0(81).newform_decomposition('a')
J1(19).newform_decomposition('a')