Module: sage.modular.abvar.homology
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).
Author: William Stein (2007-03)
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: Homology
Functions: hecke_polynomial
self, n, [var=x]) |
Return the n-th Hecke polynomial in the given variable.
Input:
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: Homology_abvar
self, abvar, base) |
This is an abstract base class, so it is called implicitly in the following examples.
sage: H = J0(43).integral_homology() sage: type(H) <class 'sage.modular.abvar.homology.IntegralHomology'>
TESTS:
sage: H = J0(43).integral_homology() sage: loads(dumps(H)) == H True
Functions: abelian_variety,
ambient_hecke_module,
free_module,
gen,
gens,
hecke_bound,
hecke_matrix,
rank,
submodule
self) |
Return the abelian variety that this is the homology of.
sage: H = J0(48).homology() sage: H.abelian_variety() Abelian variety J0(48) of dimension 3
self) |
Return the ambient Hecke module that this homology is contained in.
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
self) |
Return the underlying free module of this homology group.
sage: H = J0(48).homology() sage: H.free_module() Ambient free module of rank 6 over the principal ideal domain Integer Ring
self, n) |
Return
th generator of self.
This is not yet implemented!
sage: H = J0(37).homology() sage: H.gen(0) # this will change Traceback (most recent call last): ... NotImplementedError: homology classes not yet implemented
self) |
Return generators of self.
This is not yet implemented!
sage: H = J0(37).homology() sage: H.gens() # this will change Traceback (most recent call last): ... NotImplementedError: homology classes not yet implemented
self) |
Return bound on the number of Hecke operators needed to generate
the Hecke algebra as a
-module acting on this space.
sage: J0(48).homology().hecke_bound() 16 sage: J1(15).homology().hecke_bound() 4
self, n) |
Return the matrix of the n-th Hecke operator acting on this homology group.
Input:
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
self) |
Return the rank as a module or vector space of this homology group.
sage: H = J0(5077).homology(); H Integral Homology of Abelian variety J0(5077) of dimension 422 sage: H.rank() 844
self, U, [check=True]) |
Return the submodule of this homology group given by
,
which should be a submodule of the free module associated to
this homology group.
Input:
NOTE: We do not check that U is invariant under all Hecke operators.
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
that
we just defined is not invariant under the Hecke operators, so
it is not a Hecke submodule - it is only a
-submodule.
sage: M.hecke_matrix(3) Traceback (most recent call last): ... ArithmeticError: subspace is not invariant under matrix
Special Functions: __init__,
_repr_
self) |
Return string representation of self. This must be defined in the derived class.
sage: H = J0(43).integral_homology() sage: from sage.modular.abvar.homology import Homology_abvar sage: Homology_abvar._repr_(H) Traceback (most recent call last): ... NotImplementedError: please override this in the derived class
Class: Homology_over_base
self, abvar, base_ring) |
Called when creating homology with coefficients not
or
.
Input:
sage: H = J0(23).homology(GF(5)); H Homology with coefficients in Finite Field of size 5 of Abelian variety J0(23) of dimension 2 sage: type(H) <class 'sage.modular.abvar.homology.Homology_over_base'>
TESTS:
sage: loads(dumps(H)) == H True
Functions: hecke_matrix
self, n) |
Return the matrix of the n-th Hecke operator acting on this homology group.
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
Special Functions: __init__,
_repr_
self) |
Return string representation of self.
sage: H = J0(23).homology(GF(5)) sage: H._repr_() 'Homology with coefficients in Finite Field of size 5 of Abelian variety J0(23) of dimension 2'
Class: Homology_submodule
self, ambient, submodule) |
Create a submodule of the homology of a modular abelian variety.
Input:
sage: H = J0(37).homology() sage: H.submodule([[1,0,0,0]]) Submodule of rank 1 of Integral Homology of Abelian variety J0(37) of dimension 2
TESTS:
sage: loads(dumps(H)) == H True
Functions: ambient_hecke_module,
free_module,
hecke_bound,
hecke_matrix,
rank
self) |
Return the ambient Hecke module that this homology is contained in.
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
self) |
Return the underlying free module of the homology group.
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]
self) |
Return a bound on the number of Hecke operators needed to generate the Hecke algebra acting on this homology group.
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
self, n) |
Return the matrix of the n-th Hecke operator acting on this homology group.
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
self) |
Return the rank of this homology group.
sage: d = J0(43).homology().decomposition(2) sage: [H.rank() for H in d] [2, 4]
Special Functions: __init__,
_repr_
self) |
String representation of this submodule of homology.
sage: H = J0(37).homology() sage: G = H.submodule([[1, 2, 3, 4]]) sage: G._repr_() 'Submodule of rank 1 of Integral Homology of Abelian variety J0(37) of dimension 2'
Class: IntegralHomology
self, abvar) |
Create the integral homology of a modular abelian variety.
Input:
sage: H = J0(23).integral_homology(); H Integral Homology of Abelian variety J0(23) of dimension 2 sage: type(H) <class 'sage.modular.abvar.homology.IntegralHomology'>
TESTS:
sage: loads(dumps(H)) == H True
Functions: hecke_matrix,
hecke_polynomial
self, n) |
Return the matrix of the n-th Hecke operator acting on this homology group.
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
self, n, [var=x]) |
Return the n-th Hecke polynomial on this integral homology group.
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
Special Functions: __init__,
_repr_
self) |
String representation of the integral homology.
sage: J0(23).integral_homology()._repr_() 'Integral Homology of Abelian variety J0(23) of dimension 2'
Class: RationalHomology
self, abvar) |
Create the rational homology of a modular abelian variety.
Input:
sage: H = J0(23).rational_homology(); H Rational Homology of Abelian variety J0(23) of dimension 2
TESTS:
sage: loads(dumps(H)) == H True
Functions: hecke_matrix,
hecke_polynomial
self, n) |
Return the matrix of the n-th Hecke operator acting on this homology group.
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
self, n, [var=x]) |
Return the n-th Hecke polynomial on this rational homology group.
sage: f = J0(43).rational_homology().hecke_polynomial(2) sage: f.base_ring() Rational Field sage: factor(f) (x + 2) * (x^2 - 2)
Special Functions: __init__,
_repr_
self) |
Return string representation of the rational homology.
sage: J0(23).rational_homology()._repr_() 'Rational Homology of Abelian variety J0(23) of dimension 2'