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r"""

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

doctest:warning

...

DeprecationWarning: The default order on free modules has changed. The old ordering is in sage.modules.free_module.EchelonMatrixKey

See http://trac.sagemath.org/23878 for details.

[

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

"""

from __future__ import absolute_import

#*****************************************************************************

# This program is free software: you can redistribute it and/or modify

# it under the terms of the GNU General Public License as published by

# the Free Software Foundation, either version 2 of the License, or

# (at your option) any later version.

# http://www.gnu.org/licenses/

#*****************************************************************************

from sage.structure.richcmp import richcmp_method, richcmp, richcmp_not_equal

from sage.modular.hecke.all import HeckeModule_free_module

from sage.rings.all import Integer, ZZ, QQ, CommutativeRing

from .abvar import sqrt_poly

# TODO: we will probably also need homology that is *not* a Hecke module.

@richcmp_method

class Homology(HeckeModule_free_module):

"""

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

action.

"""

def hecke_polynomial(self, 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

"""

return self.hecke_matrix(n).charpoly(var)

class Homology_abvar(Homology):

"""

The homology of a modular abelian variety.

"""

def __init__(self, abvar, base):

"""

This is an abstract base class, so it is called implicitly in the

following examples.

EXAMPLES::

sage: H = J0(43).integral_homology()

sage: type(H)

TESTS::

sage: H = J0(43).integral_homology()

sage: loads(dumps(H)) == H

True

"""

if not isinstance(base, CommutativeRing):

raise TypeError("base ring must be a commutative ring")

HeckeModule_free_module.__init__(

self, base, abvar.level(), weight=2)

self.__abvar = abvar

def __richcmp__(self, other, op):

r"""

Compare ``self`` to ``other``.

EXAMPLES::

sage: J0(37).integral_homology() == J0(41).integral_homology()

False

sage: J0(37).integral_homology() == J0(37).rational_homology()

False

sage: J0(37).integral_homology() == loads(dumps(J0(37).integral_homology()))

True

"""

if not isinstance(other, Homology_abvar):

return NotImplemented

else:

return richcmp((self.abelian_variety(), self.base_ring()),

(other.abelian_variety(), other.base_ring()), op)

def _repr_(self):

"""

Return string representation of self. This must be defined in the

derived class.

EXAMPLES::

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

"""

raise NotImplementedError("please override this in the derived class")

def gens(self):

"""

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

"""

raise NotImplementedError("homology classes not yet implemented")

def gen(self, 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

"""

raise NotImplementedError("homology classes not yet implemented")

def abelian_variety(self):

"""

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

"""

return self.__abvar

def ambient_hecke_module(self):

"""

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

"""

return self

def free_module(self):

"""

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

"""

try:

return self.__free_module

except AttributeError:

M = self.base_ring()**self.rank()

self.__free_module = M

return M

def hecke_bound(self):

r"""

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()

sage: J1(15).homology().hecke_bound()

"""

return self.__abvar.modular_symbols(sign=1).hecke_bound()

def hecke_matrix(self, 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

"""

raise NotImplementedError

def rank(self):

"""

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

"""

return self.__abvar.dimension() * 2

def submodule(self, U, check=True):

r"""

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

"""

return Homology_submodule(self, U)

class IntegralHomology(Homology_abvar):

r"""

The integral homology `H_1(A,\ZZ)` of a modular

abelian variety.

"""

def __init__(self, abvar):

"""

Create the integral homology of a modular abelian variety.

INPUT:

- ``abvar`` - a modular abelian variety

EXAMPLES::

sage: H = J0(23).integral_homology(); H

Integral Homology of Abelian variety J0(23) of dimension 2

sage: type(H)

TESTS::

sage: loads(dumps(H)) == H

True

"""

Homology_abvar.__init__(self, abvar, ZZ)

def _repr_(self):

"""

String representation of the integral homology.

EXAMPLES::

sage: J0(23).integral_homology()._repr_()

'Integral Homology of Abelian variety J0(23) of dimension 2'

"""

return "Integral Homology of %s"%self.abelian_variety()

def hecke_matrix(self, n):

"""

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

homology group.

EXAMPLES::

sage: J0(48).integral_homology().hecke_bound()

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

"""

n = Integer(n)

return self.abelian_variety()._integral_hecke_matrix(n)

def hecke_polynomial(self, 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

"""

n = Integer(n)

M = self.abelian_variety().modular_symbols(sign=1)

f = (M.hecke_polynomial(n, var)**2).change_ring(ZZ)

return f

class RationalHomology(Homology_abvar):

r"""

The rational homology `H_1(A,\QQ)` of a modular

abelian variety.

"""

def __init__(self, abvar):

"""

Create the rational homology of a modular abelian variety.

INPUT:

- ``abvar`` - a modular abelian variety

EXAMPLES::

sage: H = J0(23).rational_homology(); H

Rational Homology of Abelian variety J0(23) of dimension 2

TESTS::

sage: loads(dumps(H)) == H

True

"""

Homology_abvar.__init__(self, abvar, QQ)

def _repr_(self):

"""

Return string representation of the rational homology.

EXAMPLES::

sage: J0(23).rational_homology()._repr_()

'Rational Homology of Abelian variety J0(23) of dimension 2'

"""

return "Rational Homology of %s"%self.abelian_variety()

def hecke_matrix(self, 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

"""

n = Integer(n)

return self.abelian_variety()._rational_hecke_matrix(n)

def hecke_polynomial(self, 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)

"""

f = self.hecke_operator(n).matrix().characteristic_polynomial(var)

return sqrt_poly(f)

#n = Integer(n)

#M = self.abelian_variety().modular_symbols(sign=1)

#f = M.hecke_polynomial(n, var)**2

#return f

class Homology_over_base(Homology_abvar):

r"""

The homology over a modular abelian variety over an arbitrary base

commutative ring (not `\ZZ` or

`\QQ`).

"""

def __init__(self, abvar, base_ring):

r"""

Called when creating homology with coefficients not

`\ZZ` or `\QQ`.

INPUT:

- ``abvar`` - a modular abelian variety

- ``base_ring`` - a commutative ring

EXAMPLES::

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)

TESTS::

sage: loads(dumps(H)) == H

True

"""

Homology_abvar.__init__(self, abvar, base_ring)

def _repr_(self):

"""

Return string representation of self.

EXAMPLES::

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'

"""

return "Homology with coefficients in %s of %s"%(self.base_ring(), self.abelian_variety())

def hecke_matrix(self, 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

"""

n = Integer(n)

return self.abelian_variety()._integral_hecke_matrix(n).change_ring(self.base_ring())

class Homology_submodule(Homology):

"""

A submodule of the homology of a modular abelian variety.

"""

def __init__(self, ambient, submodule):

"""

Create a submodule of the homology of a modular abelian variety.

INPUT:

- ``ambient`` - the homology of some modular abelian

variety with ring coefficients

- ``submodule`` - a submodule of the free module

underlying ambient

EXAMPLES::

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

"""

if not isinstance(ambient, Homology_abvar):

raise TypeError("ambient must be the homology of a modular abelian variety")

self.__ambient = ambient

#try:

# if not submodule.is_submodule(ambient):

# raise ValueError, "submodule must be a submodule of the ambient homology group"

#except AttributeError:

submodule = ambient.free_module().submodule(submodule)

self.__submodule = submodule

HeckeModule_free_module.__init__(

self, ambient.base_ring(), ambient.level(), weight=2)

def _repr_(self):

"""

String representation of this submodule of homology.

EXAMPLES::

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'

"""

return "Submodule of rank %s of %s" % (self.rank(), self.__ambient)

def __richcmp__(self, other, op):

r"""

Compare ``self`` to ``other``.

EXAMPLES::

sage: J0(37).homology().decomposition() # indirect doctest

[

Submodule of rank 2 of Integral Homology of Abelian variety J0(37) of dimension 2,

Submodule of rank 2 of Integral Homology of Abelian variety J0(37) of dimension 2

]

"""

if not isinstance(other, Homology_submodule):

return NotImplemented

lx = self.__ambient

rx = other.__ambient

if lx != rx:

return richcmp_not_equal(lx, rx, op)

return richcmp(self.__submodule, other.__submodule, op)

def ambient_hecke_module(self):

"""

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

"""

return self.__ambient

def free_module(self):

"""

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]

"""

return self.__submodule

def hecke_bound(self):

"""

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()

sage: d[1].hecke_bound()

"""

if self.rank() <= 2:

return ZZ(1)

else:

return self.__ambient.hecke_bound()

def hecke_matrix(self, 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 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

"""

n = Integer(n)

try:

return self.__hecke_matrix[n]

except AttributeError:

self.__hecke_matrix = {}

except KeyError:

pass

t = self.__ambient.hecke_matrix(n)

s = t.restrict(self.__submodule)

self.__hecke_matrix[n] = s

return s

def rank(self):

"""

Return the rank of this homology group.

EXAMPLES::

sage: d = J0(43).homology().decomposition(2)

sage: [H.rank() for H in d]

[2, 4]

"""

return self.__submodule.rank()

Coverage for local/lib/python2.7/site-packages/sage/modular/abvar/homology.py : 93%

116 statements 108 run 8 missing 0 excluded