# Ambient Jacobian Abelian Variety¶

TESTS:

sage: loads(dumps(J0(37))) == J0(37)
True
sage: loads(dumps(J1(13))) == J1(13)
True

sage.modular.abvar.abvar_ambient_jacobian.ModAbVar_ambient_jacobian(group)

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: 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


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

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

class sage.modular.abvar.abvar_ambient_jacobian.ModAbVar_ambient_jacobian_class(group)

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

ambient_variety()

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

decomposition(simple=True, bound=None)

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

EXAMPLES:

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)
]

degeneracy_map(level, t=1, check=True)

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 - bool (default: True); if True do some checks on the input

OUTPUT: a morphism

EXAMPLES:

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]

dimension()

Return the dimension of this modular abelian variety.

EXAMPLES:

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

group()

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

EXAMPLES:

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]

groups()

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

OUTPUT: tuple

EXAMPLES:

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


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Base class for modular abelian varieties

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Finite subgroups of modular abelian varieties