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# Homology of modular abelian varieties

Sage can compute with homology groups associated to modular abelian varieties with coefficients in any commutative ring. Supported operations include computing matrices and characteristic polynomials of Hecke operators, rank, and rational decomposition as a direct sum of factors (obtained by cutting out kernels of Hecke operators).

AUTHORS:

- William Stein (2007-03)

EXAMPLES:

    sage: J = J0(43)
    sage: H = J.integral_homology()
    sage: H
    Integral Homology of Abelian variety J0(43) of dimension 3
    sage: H.hecke_matrix(19)
    [ 0  0 -2  0  2  0]
    [ 2 -4 -2  0  2  0]
    [ 0  0 -2 -2  0  0]
    [ 2  0 -2 -4  2 -2]
    [ 0  2  0 -2 -2  0]
    [ 0  2  0 -2  0  0]
    sage: H.base_ring()
    Integer Ring
    sage: d = H.decomposition(); d
    [
    Submodule of rank 2 of Integral Homology of Abelian variety J0(43) of dimension 3,
    Submodule of rank 4 of Integral Homology of Abelian variety J0(43) of dimension 3
    ]
    sage: a = d[0]
    sage: a.hecke_matrix(5)
    [-4  0]
    [ 0 -4]
    sage: a.T(7)
    Hecke operator T_7 on Submodule of rank 2 of Integral Homology of Abelian variety J0(43) of dimension 3

class `sage.modular.abvar.homology.``Homology`(base_ring, level, weight)

  Bases: `sage.modular.hecke.module.HeckeModule_free_module`

  A homology group of an abelian variety, equipped with a Hecke action.

  `hecke_polynomial`(n, var='x')

    Return the n-th Hecke polynomial in the given variable.

    INPUT:

    - `n` - positive integer
    - `var` - string (default: ‘x’) the variable name

    OUTPUT: a polynomial over ZZ in the given variable

    EXAMPLES:

        sage: H = J0(43).integral_homology(); H
        Integral Homology of Abelian variety J0(43) of dimension 3
        sage: f = H.hecke_polynomial(3); f
        x^6 + 4*x^5 - 16*x^3 - 12*x^2 + 16*x + 16
        sage: parent(f)
        Univariate Polynomial Ring in x over Integer Ring
        sage: H.hecke_polynomial(3,'w')
        w^6 + 4*w^5 - 16*w^3 - 12*w^2 + 16*w + 16

class `sage.modular.abvar.homology.``Homology_abvar`(abvar, base)

  Bases: `sage.modular.abvar.homology.Homology`

  The homology of a modular abelian variety.

  `abelian_variety`()

    Return the abelian variety that this is the homology of.

    EXAMPLES:

        sage: H = J0(48).homology()
        sage: H.abelian_variety()
        Abelian variety J0(48) of dimension 3

  `ambient_hecke_module`()

    Return the ambient Hecke module that this homology is contained in.

    EXAMPLES:

        sage: H = J0(48).homology(); H
        Integral Homology of Abelian variety J0(48) of dimension 3
        sage: H.ambient_hecke_module()
        Integral Homology of Abelian variety J0(48) of dimension 3

  `free_module`()

    Return the underlying free module of this homology group.

    EXAMPLES:

        sage: H = J0(48).homology()
        sage: H.free_module()
        Ambient free module of rank 6 over the principal ideal domain Integer Ring

  `gen`(n)

    Return \(n^{th}\) generator of self.

    This is not yet implemented!

    EXAMPLES:

        sage: H = J0(37).homology()
        sage: H.gen(0)    # this will change
        Traceback (most recent call last):
        ...
        NotImplementedError: homology classes not yet implemented

  `gens`()

    Return generators of self.

    This is not yet implemented!

    EXAMPLES:

        sage: H = J0(37).homology()
        sage: H.gens()    # this will change
        Traceback (most recent call last):
        ...
        NotImplementedError: homology classes not yet implemented

  `hecke_bound`()

    Return bound on the number of Hecke operators needed to generate the Hecke algebra as a \(\ZZ\)-module acting on this space.

    EXAMPLES:

        sage: J0(48).homology().hecke_bound()
        16
        sage: J1(15).homology().hecke_bound()
        32

  `hecke_matrix`(n)

    Return the matrix of the n-th Hecke operator acting on this homology group.

    INPUT:

    - `n` - a positive integer

    OUTPUT: a matrix over the coefficient ring of this homology group

    EXAMPLES:

        sage: H = J0(23).integral_homology()
        sage: H.hecke_matrix(3)
        [-1 -2  2  0]
        [ 0 -3  2 -2]
        [ 2 -4  3 -2]
        [ 2 -2  0  1]

    The matrix is over the coefficient ring:

        sage: J = J0(23)
        sage: J.homology(QQ[I]).hecke_matrix(3).parent()
        Full MatrixSpace of 4 by 4 dense matrices over Number Field in I with defining polynomial x^2 + 1

  `rank`()

    Return the rank as a module or vector space of this homology group.

    EXAMPLES:

        sage: H = J0(5077).homology(); H
        Integral Homology of Abelian variety J0(5077) of dimension 422
        sage: H.rank()
        844

  `submodule`(U, check=True)

    Return the submodule of this homology group given by \(U\), which should be a submodule of the free module associated to this homology group.

    INPUT:

    - `U` - submodule of ambient free module (or something that defines one)
    - `check` - currently ignored.

    Note

    We do not check that U is invariant under all Hecke operators.

    EXAMPLES:

        sage: H = J0(23).homology(); H
        Integral Homology of Abelian variety J0(23) of dimension 2
        sage: F = H.free_module()
        sage: U = F.span([[1,2,3,4]])
        sage: M = H.submodule(U); M
        Submodule of rank 1 of Integral Homology of Abelian variety J0(23) of dimension 2

    Note that the submodule command doesn’t actually check that the object defined is a homology group or is invariant under the Hecke operators. For example, the fairly random \(M\) that we just defined is not invariant under the Hecke operators, so it is not a Hecke submodule - it is only a \(\ZZ\)-submodule.

        sage: M.hecke_matrix(3)
        Traceback (most recent call last):
        ...
        ArithmeticError: subspace is not invariant under matrix

class `sage.modular.abvar.homology.``Homology_over_base`(abvar, base_ring)

  Bases: `sage.modular.abvar.homology.Homology_abvar`

  The homology over a modular abelian variety over an arbitrary base commutative ring (not \(\ZZ\) or \(\QQ\)).

  `hecke_matrix`(n)

    Return the matrix of the n-th Hecke operator acting on this homology group.

    EXAMPLES:

        sage: t = J1(13).homology(GF(3)).hecke_matrix(3); t
        [0 0 2 1]
        [1 1 0 2]
        [1 1 0 0]
        [0 1 2 1]
        sage: t.base_ring()
        Finite Field of size 3

class `sage.modular.abvar.homology.``Homology_submodule`(ambient, submodule)

  Bases: `sage.modular.abvar.homology.Homology`

  A submodule of the homology of a modular abelian variety.

  `ambient_hecke_module`()

    Return the ambient Hecke module that this homology is contained in.

    EXAMPLES:

        sage: H = J0(48).homology(); H
        Integral Homology of Abelian variety J0(48) of dimension 3
        sage: d = H.decomposition(); d
        [
        Submodule of rank 2 of Integral Homology of Abelian variety J0(48) of dimension 3,
        Submodule of rank 4 of Integral Homology of Abelian variety J0(48) of dimension 3
        ]
        sage: d[0].ambient_hecke_module()
        Integral Homology of Abelian variety J0(48) of dimension 3

  `free_module`()

    Return the underlying free module of the homology group.

    EXAMPLES:

        sage: H = J0(48).homology()
        sage: K = H.decomposition()[1]; K
        Submodule of rank 4 of Integral Homology of Abelian variety J0(48) of dimension 3
        sage: K.free_module()
        Free module of degree 6 and rank 4 over Integer Ring
        Echelon basis matrix:
        [ 1  0  0  0  0  0]
        [ 0  1  0  0  1 -1]
        [ 0  0  1  0 -1  1]
        [ 0  0  0  1  0 -1]

  `hecke_bound`()

    Return a bound on the number of Hecke operators needed to generate the Hecke algebra acting on this homology group.

    EXAMPLES:

        sage: d = J0(43).homology().decomposition(2); d
        [
        Submodule of rank 2 of Integral Homology of Abelian variety J0(43) of dimension 3,
        Submodule of rank 4 of Integral Homology of Abelian variety J0(43) of dimension 3
        ]

    Because the first factor has dimension 2 it corresponds to an elliptic curve, so we have a Hecke bound of 1.

        sage: d[0].hecke_bound()
        1
        sage: d[1].hecke_bound()
        8

  `hecke_matrix`(n)

    Return the matrix of the n-th Hecke operator acting on this homology group.

    EXAMPLES:

        sage: d = J0(125).homology(GF(17)).decomposition(2); d
        [
        Submodule of rank 4 of Homology with coefficients in Finite Field of size 17 of Abelian variety J0(125) of dimension 8,
        Submodule of rank 4 of Homology with coefficients in Finite Field of size 17 of Abelian variety J0(125) of dimension 8,
        Submodule of rank 8 of Homology with coefficients in Finite Field of size 17 of Abelian variety J0(125) of dimension 8
        ]
        sage: t = d[0].hecke_matrix(17); t
        [16 15 15  0]
        [ 0  5  0  2]
        [ 2  0  5 15]
        [ 0 15  0 16]
        sage: t.base_ring()
        Finite Field of size 17
        sage: t.fcp()
        (x^2 + 13*x + 16)^2

  `rank`()

    Return the rank of this homology group.

    EXAMPLES:

        sage: d = J0(43).homology().decomposition(2)
        sage: [H.rank() for H in d]
        [2, 4]

class `sage.modular.abvar.homology.``IntegralHomology`(abvar)

  Bases: `sage.modular.abvar.homology.Homology_abvar`

  The integral homology \(H_1(A,\ZZ)\) of a modular abelian variety.

  `hecke_matrix`(n)

    Return the matrix of the n-th Hecke operator acting on this homology group.

    EXAMPLES:

        sage: J0(48).integral_homology().hecke_bound()
        16
        sage: t = J1(13).integral_homology().hecke_matrix(3); t
        [ 0  0  2 -2]
        [-2 -2  0  2]
        [-2 -2  0  0]
        [ 0 -2  2 -2]
        sage: t.base_ring()
        Integer Ring

  `hecke_polynomial`(n, var='x')

    Return the n-th Hecke polynomial on this integral homology group.

    EXAMPLES:

        sage: f = J0(43).integral_homology().hecke_polynomial(2)
        sage: f.base_ring()
        Integer Ring
        sage: factor(f)
        (x + 2)^2 * (x^2 - 2)^2

class `sage.modular.abvar.homology.``RationalHomology`(abvar)

  Bases: `sage.modular.abvar.homology.Homology_abvar`

  The rational homology \(H_1(A,\QQ)\) of a modular abelian variety.

  `hecke_matrix`(n)

    Return the matrix of the n-th Hecke operator acting on this homology group.

    EXAMPLES:

        sage: t = J1(13).homology(QQ).hecke_matrix(3); t
        [ 0  0  2 -2]
        [-2 -2  0  2]
        [-2 -2  0  0]
        [ 0 -2  2 -2]
        sage: t.base_ring()
        Rational Field
        sage: t = J1(13).homology(GF(3)).hecke_matrix(3); t
        [0 0 2 1]
        [1 1 0 2]
        [1 1 0 0]
        [0 1 2 1]
        sage: t.base_ring()
        Finite Field of size 3

  `hecke_polynomial`(n, var='x')

    Return the n-th Hecke polynomial on this rational homology group.

    EXAMPLES:

        sage: f = J0(43).rational_homology().hecke_polynomial(2)
        sage: f.base_ring()
        Rational Field
        sage: factor(f)
        (x + 2) * (x^2 - 2)

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