Finite subgroups of modular abelian varieties¶

Sage can compute with fairly general finite subgroups of modular abelian varieties. Elements of finite order are represented by equivalence classes of elements in $H_1(A,\QQ)$ modulo $H_1(A,\ZZ)$ . A finite subgroup can be defined by giving generators and via various other constructions. Given a finite subgroup, one can compute generators, as well as the structure as an abstract group. Arithmetic on subgroups is also supported, including adding two subgroups together, checking inclusion, etc.

TODO: Intersection, action of Hecke operators.

AUTHORS:

William Stein (2007-03)

EXAMPLES:

sage: J = J0(33)
sage: C = J.cuspidal_subgroup()
sage: C
Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3
sage: C.order()
100
sage: C.gens()
[[(1/10, 0, 1/10, 1/10, 1/10, 3/10)], [(0, 1/5, 1/10, 0, 1/10, 9/10)], [(0, 0, 1/2, 0, 1/2, 1/2)]]
sage: C.0 + C.1
[(1/10, 1/5, 1/5, 1/10, 1/5, 6/5)]
sage: 10*(C.0 + C.1)
[(0, 0, 0, 0, 0, 0)]
sage: G = C.subgroup([C.0 + C.1]); G
Finite subgroup with invariants [10] over QQbar of Abelian variety J0(33) of dimension 3
sage: G.gens()
[[(1/10, 1/5, 1/5, 1/10, 1/5, 1/5)]]
sage: G.order()
10
sage: G <= C
True
sage: G >= C
False

We make a table of the order of the cuspidal subgroup for the first few levels:

sage: for N in range(11,40): print N, J0(N).cuspidal_subgroup().order()
...
5
1
1
6
8
1
4
1
3
6
8
25
11
8
1
21
9
36
7
192
5
8
100
48
48
12
3
135
56

TESTS:

sage: G = J0(11).finite_subgroup([[1/3,0], [0,1/5]]); G
Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1
sage: loads(dumps(G)) == G
True
sage: loads(dumps(G.0)) == G.0
True

class sage.modular.abvar.finite_subgroup.FiniteSubgroup(abvar, field_of_definition=Rational Field)¶

Bases: sage.modules.module.Module

abelian_variety()¶

Return the abelian variety that this is a finite subgroup of.

EXAMPLES:

sage: J = J0(42)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(42) of dimension 5
sage: G.abelian_variety()
Abelian variety J0(42) of dimension 5

exponent()¶

Return the exponent of this finite abelian group.

OUTPUT: Integer

EXAMPLES:

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: G.exponent()
4

field_of_definition()¶

Return the field over which this finite modular abelian variety subgroup is defined. This is a field over which this subgroup is defined.

EXAMPLES:

sage: J = J0(42)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(42) of dimension 5
sage: G.field_of_definition()
Rational Field

gen(n)¶

Return $n^{th}$ generator of self.

EXAMPLES:

sage: J = J0(23)
sage: C = J.torsion_subgroup(3)
sage: C.gens()
[[(1/3, 0, 0, 0)], [(0, 1/3, 0, 0)], [(0, 0, 1/3, 0)], [(0, 0, 0, 1/3)]]
sage: C.gen(0)
[(1/3, 0, 0, 0)]
sage: C.gen(3)
[(0, 0, 0, 1/3)]
sage: C.gen(4)
...
IndexError: list index out of range

Negative indices wrap around:

sage: C.gen(-1)
[(0, 0, 0, 1/3)]

gens()¶

Return generators for this finite subgroup.

EXAMPLES: We list generators for several cuspidal subgroups:

sage: J0(11).cuspidal_subgroup().gens()
[[(0, 1/5)]]
sage: J0(37).cuspidal_subgroup().gens()
[[(0, 0, 0, 1/3)]]
sage: J0(43).cuspidal_subgroup().gens()
[[(0, 1/7, 0, 6/7, 0, 5/7)]]
sage: J1(13).cuspidal_subgroup().gens()
[[(1/19, 0, 0, 9/19)], [(0, 1/19, 1/19, 18/19)]]
sage: J0(22).torsion_subgroup(6).gens()
[[(1/6, 0, 0, 0)], [(0, 1/6, 0, 0)], [(0, 0, 1/6, 0)], [(0, 0, 0, 1/6)]]

intersection(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)

invariants()¶

Return elementary invariants of this abelian group, by which we mean a nondecreasing (immutable) sequence of integers $n_i$ , $1 \leq i \leq k$ , with $n_i$ dividing $n_{i+1}$ , and such that this group is abstractly isomorphic to $\ZZ/n_1\ZZ \times\cdots\times \ZZ/n_k\ZZ.$

EXAMPLES:

sage: J = J0(38)
sage: C = J.cuspidal_subgroup(); C
Finite subgroup with invariants [3, 45] over QQ of Abelian variety J0(38) of dimension 4
sage: v = C.invariants(); v
[3, 45]
sage: v[0] = 5
...
ValueError: object is immutable; please change a copy instead.
sage: type(v[0])
<type 'sage.rings.integer.Integer'>

sage: C * 3
Finite subgroup with invariants [15] over QQ of Abelian variety J0(38) of dimension 4

An example involving another cuspidal subgroup:

sage: C = J0(22).cuspidal_subgroup(); C
Finite subgroup with invariants [5, 5] over QQ of Abelian variety J0(22) of dimension 2
sage: C.lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1/5 1/5 4/5   0]
[  0   1   0   0]
[  0   0   1   0]
[  0   0   0 1/5]
sage: C.invariants()
[5, 5]

is_subgroup(other)¶

Return True exactly if self is a subgroup of other, and both are defined as subgroups of the same ambient abelian variety.

EXAMPLES:

sage: C = J0(22).cuspidal_subgroup()
sage: H = C.subgroup([C.0])
sage: K = C.subgroup([C.1])
sage: H.is_subgroup(K)
False
sage: K.is_subgroup(H)
False
sage: K.is_subgroup(C)
True
sage: H.is_subgroup(C)
True

lattice()¶

Return the lattice corresponding to this subgroup in the rational homology of the modular Jacobian product. The elements of the subgroup are represented by vectors in the ambient vector space (the rational homology), and this returns the lattice they span. EXAMPLES:

sage: J = J0(33); C = J[0].cuspidal_subgroup(); C
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: C.lattice()
Free module of degree 6 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1/5 13/5   -2 -4/5    2 -1/5]
[   0    3   -2   -1    2    0]

order()¶

Return the order (number of elements) of this finite subgroup.

EXAMPLES:

sage: J = J0(42)
sage: C = J.cuspidal_subgroup()
sage: C.order()
2304

subgroup(gens)¶

Return the subgroup of self spanned by the given generators, which all must be elements of self.

EXAMPLES:

sage: J = J0(23)
sage: G = J.torsion_subgroup(11); G
Finite subgroup with invariants [11, 11, 11, 11] over QQ of Abelian variety J0(23) of dimension 2

We create the subgroup of the 11-torsion subgroup of $J_0(23)$ generated by the first $11$ -torsion point:

sage: H = G.subgroup([G.0]); H
Finite subgroup with invariants [11] over QQbar of Abelian variety J0(23) of dimension 2
sage: H.invariants()
[11]

We can also create a subgroup from a list of objects that coerce into the ambient rational homology.

sage: H == G.subgroup([[1/11,0,0,0]])
True

class sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice(abvar, lattice, field_of_definition=Algebraic Field, check=True)¶

Bases: sage.modular.abvar.finite_subgroup.FiniteSubgroup

lattice()¶

Return lattice that defines this finite subgroup.

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
sage: G.lattice()
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/3   0]
[  0 1/5]

class sage.modular.abvar.finite_subgroup.TorsionPoint(parent, element, check=True)¶

Bases: sage.structure.element.ModuleElement

additive_order()¶

Return the additive order of this element.

EXAMPLES:

sage: J = J0(11); G = J.finite_subgroup([[1/3,0], [0,1/5]])
sage: G.0.additive_order()
3
sage: G.1.additive_order()
5
sage: (G.0 + G.1).additive_order()
15
sage: (3*G.0).additive_order()
1

element()¶

Return an underlying QQ-vector space element that defines this element of a modular abelian variety. This is a vector in the ambient Jacobian variety’s rational homology.

EXAMPLES: We create some elements of $J_0(11)$ :

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
sage: G.0.element()
(1/3, 0)

The underlying element is a vector over the rational numbers:

sage: v = (G.0-G.1).element(); v
(1/3, -1/5)
sage: type(v)
<type 'sage.modules.vector_rational_dense.Vector_rational_dense'>

Finite subgroups of modular abelian varieties¶

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