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

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

Python

>>> from sage.all import *
>>> J = J0(Integer(43))
>>> H = J.integral_homology()
>>> H
Integral Homology of Abelian variety J0(43) of dimension 3
>>> H.hecke_matrix(Integer(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]
>>> H.base_ring()
Integer Ring
>>> 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
]
>>> a = d[Integer(0)]
>>> a.hecke_matrix(Integer(5))
[-4  0]
[ 0 -4]
>>> a.T(Integer(7))
Hecke operator T_7 on
 Submodule of rank 2 of Integral Homology of Abelian variety J0(43) of dimension 3

Sage Live

J = J0(43)
H = J.integral_homology()
H
H.hecke_matrix(19)
H.base_ring()
d = H.decomposition(); d
a = d[0]
a.hecke_matrix(5)
a.T(7)

class sage.modular.abvar.homology.Homology(base_ring, level, weight, category=None)[source]¶

Bases: HeckeModule_free_module

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

hecke_polynomial(n, var='x')[source]¶

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

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

Python

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

Sage Live

H = J0(43).integral_homology(); H
f = H.hecke_polynomial(3); f
parent(f)
H.hecke_polynomial(3,'w')

class sage.modular.abvar.homology.Homology_abvar(abvar, base)[source]¶

Bases: Homology

The homology of a modular abelian variety.

abelian_variety()[source]¶

Return the abelian variety that this is the homology of.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> H = J0(Integer(48)).homology()
>>> H.abelian_variety()
Abelian variety J0(48) of dimension 3

Sage Live

H = J0(48).homology()
H.abelian_variety()

ambient_hecke_module()[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> H = J0(Integer(48)).homology(); H
Integral Homology of Abelian variety J0(48) of dimension 3
>>> H.ambient_hecke_module()
Integral Homology of Abelian variety J0(48) of dimension 3

Sage Live

H = J0(48).homology(); H
H.ambient_hecke_module()

free_module()[source]¶

Return the underlying free module of this homology group.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> H = J0(Integer(48)).homology()
>>> H.free_module()
Ambient free module of rank 6 over the principal ideal domain Integer Ring

Sage Live

H = J0(48).homology()
H.free_module()

gen(n)[source]¶

Return \(n\)-th generator of self.

This is not yet implemented!

EXAMPLES:

Sage

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

Python

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

Sage Live

H = J0(37).homology()
H.gen(0)    # this will change

gens()[source]¶

Return generators of self.

This is not yet implemented!

EXAMPLES:

Sage

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

Python

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

Sage Live

H = J0(37).homology()
H.gens()    # this will change

hecke_bound()[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> J0(Integer(48)).homology().hecke_bound()
16
>>> J1(Integer(15)).homology().hecke_bound()
32

Sage Live

J0(48).homology().hecke_bound()
J1(15).homology().hecke_bound()

hecke_matrix(n)[source]¶

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

INPUT:

n – positive integer

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

EXAMPLES:

Sage

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]

Python

>>> from sage.all import *
>>> H = J0(Integer(23)).integral_homology()
>>> H.hecke_matrix(Integer(3))
[-1 -2  2  0]
[ 0 -3  2 -2]
[ 2 -4  3 -2]
[ 2 -2  0  1]

Sage Live

H = J0(23).integral_homology()
H.hecke_matrix(3)

The matrix is over the coefficient ring:

Sage

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 with I = 1*I

Python

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

Sage Live

J = J0(23)
J.homology(QQ[I]).hecke_matrix(3).parent()

rank()[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> H = J0(Integer(5077)).homology(); H
Integral Homology of Abelian variety J0(5077) of dimension 422
>>> H.rank()
844

Sage Live

H = J0(5077).homology(); H
H.rank()

submodule(U, check=True)[source]¶

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

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

Python

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

Sage Live

H = J0(23).homology(); H
F = H.free_module()
U = F.span([[1,2,3,4]])
M = H.submodule(U); M

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

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

Python

>>> from sage.all import *
>>> M.hecke_matrix(Integer(3))
Traceback (most recent call last):
...
ArithmeticError: subspace is not invariant under matrix

Sage Live

M.hecke_matrix(3)

class sage.modular.abvar.homology.Homology_over_base(abvar, base_ring)[source]¶

Bases: Homology_abvar

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

hecke_matrix(n)[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> t = J1(Integer(13)).homology(GF(Integer(3))).hecke_matrix(Integer(3)); t
[1 2 2 1]
[1 0 2 0]
[0 0 0 1]
[0 0 2 1]
>>> t.base_ring()
Finite Field of size 3

Sage Live

t = J1(13).homology(GF(3)).hecke_matrix(3); t
t.base_ring()

class sage.modular.abvar.homology.Homology_submodule(ambient, submodule)[source]¶

Bases: Homology

A submodule of the homology of a modular abelian variety.

ambient_hecke_module()[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> H = J0(Integer(48)).homology(); H
Integral Homology of Abelian variety J0(48) of dimension 3
>>> 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
]
>>> d[Integer(0)].ambient_hecke_module()
Integral Homology of Abelian variety J0(48) of dimension 3

Sage Live

H = J0(48).homology(); H
d = H.decomposition(); d
d[0].ambient_hecke_module()

free_module()[source]¶

Return the underlying free module of the homology group.

EXAMPLES:

Sage

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]

Python

>>> from sage.all import *
>>> H = J0(Integer(48)).homology()
>>> K = H.decomposition()[Integer(1)]; K
Submodule of rank 4 of Integral Homology of Abelian variety J0(48) of dimension 3
>>> 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]

Sage Live

H = J0(48).homology()
K = H.decomposition()[1]; K
K.free_module()

hecke_bound()[source]¶

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

EXAMPLES:

Sage

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
]

Python

>>> from sage.all import *
>>> d = J0(Integer(43)).homology().decomposition(Integer(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
]

Sage Live

d = J0(43).homology().decomposition(2); d

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

Sage

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

Python

>>> from sage.all import *
>>> d[Integer(0)].hecke_bound()
1
>>> d[Integer(1)].hecke_bound()
8

Sage Live

d[0].hecke_bound()
d[1].hecke_bound()

hecke_matrix(n)[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> d = J0(Integer(125)).homology(GF(Integer(17))).decomposition(Integer(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
]
>>> t = d[Integer(0)].hecke_matrix(Integer(17)); t
[16 15 15  0]
[ 0  5  0  2]
[ 2  0  5 15]
[ 0 15  0 16]
>>> t.base_ring()
Finite Field of size 17
>>> t.fcp()
(x^2 + 13*x + 16)^2

Sage Live

d = J0(125).homology(GF(17)).decomposition(2); d
t = d[0].hecke_matrix(17); t
t.base_ring()
t.fcp()

rank()[source]¶

Return the rank of this homology group.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> d = J0(Integer(43)).homology().decomposition(Integer(2))
>>> [H.rank() for H in d]
[2, 4]

Sage Live

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

class sage.modular.abvar.homology.IntegralHomology(abvar)[source]¶

Bases: Homology_abvar

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

hecke_matrix(n)[source]¶

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

EXAMPLES:

Sage

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

Python

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

Sage Live

J0(48).integral_homology().hecke_bound()
t = J1(13).integral_homology().hecke_matrix(3); t
t.base_ring()

hecke_polynomial(n, var='x')[source]¶

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

EXAMPLES:

Sage

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

Python

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

Sage Live

f = J0(43).integral_homology().hecke_polynomial(2)
f.base_ring()
factor(f)

class sage.modular.abvar.homology.RationalHomology(abvar)[source]¶

Bases: Homology_abvar

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

hecke_matrix(n)[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> t = J1(Integer(13)).homology(QQ).hecke_matrix(Integer(3)); t
[-2  2  2 -2]
[-2  0  2  0]
[ 0  0  0 -2]
[ 0  0  2 -2]
>>> t.base_ring()
Rational Field
>>> t = J1(Integer(13)).homology(GF(Integer(3))).hecke_matrix(Integer(3)); t
[1 2 2 1]
[1 0 2 0]
[0 0 0 1]
[0 0 2 1]
>>> t.base_ring()
Finite Field of size 3

Sage Live

t = J1(13).homology(QQ).hecke_matrix(3); t
t.base_ring()
t = J1(13).homology(GF(3)).hecke_matrix(3); t
t.base_ring()

hecke_polynomial(n, var='x')[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> f = J0(Integer(43)).rational_homology().hecke_polynomial(Integer(2))
>>> f.base_ring()
Rational Field
>>> factor(f)
(x + 2) * (x^2 - 2)

Sage Live

f = J0(43).rational_homology().hecke_polynomial(2)
f.base_ring()
factor(f)