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
[
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 sage.modular.hecke.all import HeckeModule_free_module
from sage.rings.all import Integer, ZZ, QQ, is_CommutativeRing
import abvar
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)
<class 'sage.modular.abvar.homology.IntegralHomology_with_category'>
TESTS::
sage: H = J0(43).integral_homology()
sage: loads(dumps(H)) == H
True
"""
if not is_CommutativeRing(base):
raise TypeError, "base ring must be a commutative ring"
HeckeModule_free_module.__init__(
self, base, abvar.level(), weight=2)
self.__abvar = abvar
def __cmp__(self, other):
r"""
Compare self to other.
EXAMPLE::
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 cmp(type(self), type(other))
else:
return cmp((self.abelian_variety(), self.base_ring()), (other.abelian_variety(), other.base_ring()))
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()
16
sage: J1(15).homology().hecke_bound()
32
"""
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)
<class 'sage.modular.abvar.homology.IntegralHomology_with_category'>
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()
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
"""
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 abvar.sqrt_poly(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)
<class 'sage.modular.abvar.homology.Homology_over_base_with_category'>
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
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 __cmp__(self, other):
r"""
Compare self to other.
EXAMPLE::
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 cmp(type(self), type(other))
c = cmp(self.__ambient, other.__ambient)
if c: return c
return cmp(self.__submodule, other.__submodule)
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()
1
sage: d[1].hecke_bound()
8
"""
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 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()
