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# 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()
    ...
    11 5
    12 1
    13 1
    14 6
    15 8
    16 1
    17 4
    18 1
    19 3
    20 6
    21 8
    22 25
    23 11
    24 8
    25 1
    26 21
    27 9
    28 36
    29 7
    30 192
    31 5
    32 8
    33 100
    34 48
    35 48
    36 12
    37 3
    38 135
    39 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_old`

  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

  `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)
        Traceback (most recent call last):
        ...
        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
        Traceback (most recent call last):
        ...
        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`

  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

  `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`

  An element of a finite subgroup of a modular abelian variety.

  INPUT:

  - `parent` - a finite subgroup of a modular abelian variety
  - `element` - a QQ vector space element that represents this element in terms of the ambient rational homology
  - `check` - bool (default: True) whether to check that element is in the appropriate vector space

  EXAMPLES: The following calls the TorsionPoint constructor implicitly:

      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: type(G.0)
      <class 'sage.modular.abvar.finite_subgroup.TorsionPoint'>

  `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'>

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