"""
Spaces of homomorphisms between modular abelian varieties
EXAMPLES:
First, we consider J0(37). This Jacobian has two simple factors,
corresponding to distinct newforms. These two intersect
nontrivially in J0(37).
::
sage: J = J0(37)
sage: D = J.decomposition() ; D
[
Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37),
Simple abelian subvariety 37b(1,37) of dimension 1 of J0(37)
]
sage: D[0].intersection(D[1])
(Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37),
Simple abelian subvariety of dimension 0 of J0(37))
As an abstract product, since these newforms are distinct, the
corresponding simple abelian varieties are not isogenous, and so
there are no maps between them. The endomorphism ring of the
corresponding product is thus isomorphic to the direct sum of the
endomorphism rings for each factor. Since the factors correspond to
abelian varieties of dimension 1, these endomorphism rings are each
isomorphic to ZZ.
::
sage: Hom(D[0],D[1]).gens()
()
sage: A = D[0] * D[1] ; A
Abelian subvariety of dimension 2 of J0(37) x J0(37)
sage: A.endomorphism_ring().gens()
(Abelian variety endomorphism of Abelian subvariety of dimension 2 of J0(37) x J0(37),
Abelian variety endomorphism of Abelian subvariety of dimension 2 of J0(37) x J0(37))
sage: [ x.matrix() for x in A.endomorphism_ring().gens() ]
[
[1 0 0 0] [0 0 0 0]
[0 1 0 0] [0 0 0 0]
[0 0 0 0] [0 0 1 0]
[0 0 0 0], [0 0 0 1]
]
However, these two newforms have a congruence between them modulo
2, which gives rise to interesting endomorphisms of J0(37).
::
sage: E = J.endomorphism_ring()
sage: E.gens()
(Abelian variety endomorphism of Abelian variety J0(37) of dimension 2,
Abelian variety endomorphism of Abelian variety J0(37) of dimension 2)
sage: [ x.matrix() for x in E.gens() ]
[
[1 0 0 0] [ 0 1 1 -1]
[0 1 0 0] [ 1 0 1 0]
[0 0 1 0] [ 0 0 -1 1]
[0 0 0 1], [ 0 0 0 1]
]
sage: (-1*E.gens()[0] + E.gens()[1]).matrix()
[-1 1 1 -1]
[ 1 -1 1 0]
[ 0 0 -2 1]
[ 0 0 0 0]
Of course, these endomorphisms will be reflected in the Hecke
algebra, which is in fact the full endomorphism ring of J0(37) in
this case::
sage: J.hecke_operator(2).matrix()
[-1 1 1 -1]
[ 1 -1 1 0]
[ 0 0 -2 1]
[ 0 0 0 0]
sage: T = E.image_of_hecke_algebra()
sage: T.gens()
(Abelian variety endomorphism of Abelian variety J0(37) of dimension 2,
Abelian variety endomorphism of Abelian variety J0(37) of dimension 2)
sage: [ x.matrix() for x in T.gens() ]
[
[1 0 0 0] [ 0 1 1 -1]
[0 1 0 0] [ 1 0 1 0]
[0 0 1 0] [ 0 0 -1 1]
[0 0 0 1], [ 0 0 0 1]
]
sage: T.index_in(E)
1
Next, we consider J0(33). In this case, we have both oldforms and
newforms. There are two copies of J0(11), one for each degeneracy
map from J0(11) to J0(33). There is also one newform at level 33.
The images of the two degeneracy maps are, of course, isogenous.
::
sage: J = J0(33)
sage: D = J.decomposition()
sage: D
[
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33),
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33),
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
]
sage: Hom(D[0],D[1]).gens()
(Abelian variety morphism:
From: Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
To: Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33),)
sage: Hom(D[0],D[1]).gens()[0].matrix()
[ 0 1]
[-1 0]
Then this gives that the component corresponding to the sum of the
oldforms will have a rank 4 endomorphism ring. We also have a rank
one endomorphism ring for the newform 33a (since it is again
1-dimensional), which gives a rank 5 endomorphism ring for J0(33).
::
sage: DD = J.decomposition(simple=False) ; DD
[
Abelian subvariety of dimension 2 of J0(33),
Abelian subvariety of dimension 1 of J0(33)
]
sage: A, B = DD
sage: A == D[0] + D[1]
True
sage: A.endomorphism_ring().gens()
(Abelian variety endomorphism of Abelian subvariety of dimension 2 of J0(33),
Abelian variety endomorphism of Abelian subvariety of dimension 2 of J0(33),
Abelian variety endomorphism of Abelian subvariety of dimension 2 of J0(33),
Abelian variety endomorphism of Abelian subvariety of dimension 2 of J0(33))
sage: B.endomorphism_ring().gens()
(Abelian variety endomorphism of Abelian subvariety of dimension 1 of J0(33),)
sage: E = J.endomorphism_ring() ; E.gens() # long time (3s on sage.math, 2011)
(Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3)
In this case, the image of the Hecke algebra will only have rank 3,
so that it is of infinite index in the full endomorphism ring.
However, if we call this image T, we can still ask about the index
of T in its saturation, which is 1 in this case.
::
sage: T = E.image_of_hecke_algebra() # long time
sage: T.gens() # long time
(Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3)
sage: T.index_in(E) # long time
+Infinity
sage: T.index_in_saturation() # long time
1
AUTHORS:
- William Stein (2007-03)
- Craig Citro, Robert Bradshaw (2008-03): Rewrote with modabvar overhaul
"""
from copy import copy
from sage.categories.homset import HomsetWithBase
from sage.misc.functional import parent
import abvar as abelian_variety
import morphism
import sage.rings.integer_ring
import sage.rings.all
from sage.rings.ring import Ring
from sage.categories.category import Category
from sage.categories.rings import Rings
from sage.matrix.matrix_space import MatrixSpace
from sage.matrix.constructor import Matrix, identity_matrix
from sage.structure.element import is_Matrix
ZZ = sage.rings.integer_ring.ZZ
class Homspace(HomsetWithBase):
"""
A space of homomorphisms between two modular abelian varieties.
"""
def __init__(self, domain, codomain, cat):
"""
Create a homspace.
INPUT:
- ``domain, codomain`` - modular abelian varieties
- ``cat`` - category
EXAMPLES::
sage: H = Hom(J0(11), J0(22)); H
Space of homomorphisms from Abelian variety J0(11) of dimension 1 to Abelian variety J0(22) of dimension 2
sage: Hom(J0(11), J0(11))
Endomorphism ring of Abelian variety J0(11) of dimension 1
sage: type(H)
<class 'sage.modular.abvar.homspace.Homspace_with_category'>
sage: H.homset_category()
Category of modular abelian varieties over Rational Field
"""
if not abelian_variety.is_ModularAbelianVariety(domain):
raise TypeError, "domain must be a modular abelian variety"
if not abelian_variety.is_ModularAbelianVariety(codomain):
raise TypeError, "codomain must be a modular abelian variety"
self._matrix_space = MatrixSpace(ZZ,2*domain.dimension(), 2*codomain.dimension())
self._gens = None
HomsetWithBase.__init__(self, domain, codomain, cat)
def __call__(self, M):
r"""
Create a homomorphism in this space from M. M can be any of the
following:
- a Morphism of abelian varieties
- a matrix of the appropriate size
(i.e. 2\*self.domain().dimension() x
2\*self.codomain().dimension()) whose entries are coercible
into self.base_ring()
- anything that can be coerced into self.matrix_space()
EXAMPLES::
sage: H = Hom(J0(11), J0(22))
sage: phi = H(matrix(ZZ,2,4,[5..12])) ; phi
Abelian variety morphism:
From: Abelian variety J0(11) of dimension 1
To: Abelian variety J0(22) of dimension 2
sage: phi.matrix()
[ 5 6 7 8]
[ 9 10 11 12]
sage: phi.matrix().parent()
Full MatrixSpace of 2 by 4 dense matrices over Integer Ring
::
sage: H = J0(22).Hom(J0(11)*J0(11))
sage: m1 = J0(22).degeneracy_map(11,1).matrix() ; m1
[ 0 1]
[-1 1]
[-1 0]
[ 0 -1]
sage: m2 = J0(22).degeneracy_map(11,2).matrix() ; m2
[ 1 -2]
[ 0 -2]
[ 1 -1]
[ 0 -1]
sage: m = m1.transpose().stack(m2.transpose()).transpose() ; m
[ 0 1 1 -2]
[-1 1 0 -2]
[-1 0 1 -1]
[ 0 -1 0 -1]
sage: phi = H(m) ; phi
Abelian variety morphism:
From: Abelian variety J0(22) of dimension 2
To: Abelian variety J0(11) x J0(11) of dimension 2
sage: phi.matrix()
[ 0 1 1 -2]
[-1 1 0 -2]
[-1 0 1 -1]
[ 0 -1 0 -1]
"""
if isinstance(M, morphism.Morphism):
if M.parent() is self:
return M
elif M.domain() == self.domain() and M.codomain() == self.codomain():
M = M.matrix()
else:
raise ValueError, "cannot convert %s into %s" % (M, self)
elif is_Matrix(M):
if M.base_ring() != ZZ:
M = M.change_ring(ZZ)
if M.nrows() != 2*self.domain().dimension() or M.ncols() != 2*self.codomain().dimension():
raise TypeError, "matrix has wrong dimension"
elif self.matrix_space().has_coerce_map_from(parent(M)):
M = self.matrix_space()(M)
else:
raise TypeError, "can only coerce in matrices or morphisms"
return morphism.Morphism(self, M)
def _coerce_impl(self, x):
"""
Coerce x into self, if possible.
EXAMPLES::
sage: J = J0(37) ; J.Hom(J)._coerce_impl(matrix(ZZ,4,[5..20]))
Abelian variety endomorphism of Abelian variety J0(37) of dimension 2
sage: K = J0(11) * J0(11) ; J.Hom(K)._coerce_impl(matrix(ZZ,4,[5..20]))
Abelian variety morphism:
From: Abelian variety J0(37) of dimension 2
To: Abelian variety J0(11) x J0(11) of dimension 2
"""
if self.matrix_space().has_coerce_map_from(parent(x)):
return self(x)
else:
return HomsetWithBase._coerce_impl(self, x)
def _repr_(self):
"""
String representation of a modular abelian variety homspace.
EXAMPLES::
sage: J = J0(11)
sage: End(J)._repr_()
'Endomorphism ring of Abelian variety J0(11) of dimension 1'
"""
return "Space of homomorphisms from %s to %s"%\
(self.domain(), self.codomain())
def _get_matrix(self, g):
"""
Given an object g, try to return a matrix corresponding to g with
dimensions the same as those of self.matrix_space().
INPUT:
- ``g`` - a matrix or morphism or object with a list
method
OUTPUT: a matrix
EXAMPLES::
sage: E = End(J0(11))
sage: E._get_matrix(matrix(QQ,2,[1,2,3,4]))
[1 2]
[3 4]
sage: E._get_matrix(J0(11).hecke_operator(2))
[-2 0]
[ 0 -2]
::
sage: H = Hom(J0(11) * J0(17), J0(22))
sage: H._get_matrix(tuple([8..23]))
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
[20 21 22 23]
sage: H._get_matrix(tuple([8..23]))
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
[20 21 22 23]
sage: H._get_matrix([8..23])
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
[20 21 22 23]
"""
try:
if g.parent() is self.matrix_space():
return g
except AttributeError:
pass
if isinstance(g, morphism.Morphism):
return g.matrix()
elif hasattr(g, 'list'):
return self.matrix_space()(g.list())
else:
return self.matrix_space()(g)
def free_module(self):
r"""
Return this endomorphism ring as a free submodule of a big
`\ZZ^{4nm}`, where `n` is the dimension of
the domain abelian variety and `m` the dimension of the
codomain.
OUTPUT: free module
EXAMPLES::
sage: E = Hom(J0(11), J0(22))
sage: E.free_module()
Free module of degree 8 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1 0 -3 1 1 1 -1 -1]
[ 0 1 -3 1 1 1 -1 0]
"""
self.calculate_generators()
V = ZZ**(4*self.domain().dimension() * self.codomain().dimension())
return V.submodule([ V(m.matrix().list()) for m in self.gens() ])
def gen(self, i=0):
"""
Return i-th generator of self.
INPUT:
- ``i`` - an integer
OUTPUT: a morphism
EXAMPLES::
sage: E = End(J0(22))
sage: E.gen(0).matrix()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
"""
self.calculate_generators()
if i > self.ngens():
raise ValueError, "self only has %s generators"%self.ngens()
return morphism.Morphism(self, self._gens[i])
def ngens(self):
"""
Return number of generators of self.
OUTPUT: integer
EXAMPLES::
sage: E = End(J0(22))
sage: E.ngens()
4
"""
self.calculate_generators()
return len(self._gens)
def gens(self):
"""
Return tuple of generators for this endomorphism ring.
EXAMPLES::
sage: E = End(J0(22))
sage: E.gens()
(Abelian variety endomorphism of Abelian variety J0(22) of dimension 2,
Abelian variety endomorphism of Abelian variety J0(22) of dimension 2,
Abelian variety endomorphism of Abelian variety J0(22) of dimension 2,
Abelian variety endomorphism of Abelian variety J0(22) of dimension 2)
"""
try:
return self._gen_morphisms
except AttributeError:
self.calculate_generators()
self._gen_morphisms = tuple([self.gen(i) for i in range(self.ngens())])
return self._gen_morphisms
def matrix_space(self):
"""
Return the underlying matrix space that we view this endomorphism
ring as being embedded into.
EXAMPLES::
sage: E = End(J0(22))
sage: E.matrix_space()
Full MatrixSpace of 4 by 4 dense matrices over Integer Ring
"""
return self._matrix_space
def calculate_generators(self):
"""
If generators haven't already been computed, calculate generators
for this homspace. If they have been computed, do nothing.
EXAMPLES::
sage: E = End(J0(11))
sage: E.calculate_generators()
"""
if self._gens is not None:
return
if (self.domain() == self.codomain()) and (self.domain().dimension() == 1):
self._gens = tuple([ identity_matrix(ZZ,2) ])
return
phi = self.domain()._isogeny_to_product_of_powers()
psi = self.codomain()._isogeny_to_product_of_powers()
H_simple = phi.codomain().Hom(psi.codomain())
im_gens = H_simple._calculate_product_gens()
M = phi.matrix()
Mt = psi.complementary_isogeny().matrix()
R = ZZ**(4*self.domain().dimension()*self.codomain().dimension())
gens = R.submodule([ (M*self._get_matrix(g)*Mt).list()
for g in im_gens ]).saturation().basis()
self._gens = tuple([ self._get_matrix(g) for g in gens ])
def _calculate_product_gens(self):
"""
For internal use.
Calculate generators for self, assuming that self is a product of
simple factors.
EXAMPLES::
sage: E = End(J0(37))
sage: E.gens()
(Abelian variety endomorphism of Abelian variety J0(37) of dimension 2,
Abelian variety endomorphism of Abelian variety J0(37) of dimension 2)
sage: [ x.matrix() for x in E.gens() ]
[
[1 0 0 0] [ 0 1 1 -1]
[0 1 0 0] [ 1 0 1 0]
[0 0 1 0] [ 0 0 -1 1]
[0 0 0 1], [ 0 0 0 1]
]
sage: E._calculate_product_gens()
[
[1 0 0 0] [0 0 0 0]
[0 1 0 0] [0 0 0 0]
[0 0 0 0] [0 0 1 0]
[0 0 0 0], [0 0 0 1]
]
"""
Afactors = self.domain().decomposition(simple=False)
Bfactors = self.codomain().decomposition(simple=False)
matrix_space = self.matrix_space()
if len(Afactors) == 1 and len(Bfactors) == 1:
Asimples = Afactors[0].decomposition()
Bsimples = Bfactors[0].decomposition()
if len(Asimples) == 1 and len(Bsimples) == 1:
gens = self._calculate_simple_gens()
else:
gens = []
phi_matrix = Afactors[0]._isogeny_to_product_of_simples().matrix()
psi_t_matrix = Bfactors[0]._isogeny_to_product_of_simples().complementary_isogeny().matrix()
for i in range(len(Asimples)):
for j in range(len(Bsimples)):
hom_gens = Asimples[i].Hom(Bsimples[j]).gens()
for sub_gen in hom_gens:
sub_mat = sub_gen.matrix()
M = copy(self.matrix_space().zero_matrix())
M.set_block(sub_mat.nrows()*i, sub_mat.ncols()*j, sub_mat)
gens.append(phi_matrix * M * psi_t_matrix)
else:
gens = []
cur_row = 0
for Afactor in Afactors:
cur_row += Afactor.dimension() * 2
cur_col = 0
for Bfactor in Bfactors:
cur_col += Bfactor.dimension() * 2
Asimple = Afactor[0]
Bsimple = Bfactor[0]
if Asimple.newform_label() == Bsimple.newform_label():
for sub_gen in Afactor.Hom(Bfactor).gens():
sub_mat = sub_gen.matrix()
M = copy(self.matrix_space().zero_matrix())
M.set_block(cur_row - sub_mat.nrows(),
cur_col - sub_mat.ncols(),
sub_mat)
gens.append(M)
return gens
def _calculate_simple_gens(self):
"""
Calculate generators for self, where both the domain and codomain
for self are assumed to be simple abelian varieties. The saturation
of the span of these generators in self will be the full space of
homomorphisms from the domain of self to its codomain.
EXAMPLES::
sage: H = Hom(J0(11), J0(22)[0])
sage: H._calculate_simple_gens()
[
[1 0]
[1 1]
]
sage: J = J0(11) * J0(33) ; J.decomposition()
[
Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(33),
Simple abelian subvariety 11a(1,33) of dimension 1 of J0(11) x J0(33),
Simple abelian subvariety 11a(3,33) of dimension 1 of J0(11) x J0(33),
Simple abelian subvariety 33a(1,33) of dimension 1 of J0(11) x J0(33)
]
sage: J[0].Hom(J[1])._calculate_simple_gens()
[
[ 0 -1]
[ 1 -1]
]
sage: J[0].Hom(J[2])._calculate_simple_gens()
[
[-1 0]
[-1 -1]
]
sage: J[0].Hom(J[0])._calculate_simple_gens()
[
[1 0]
[0 1]
]
sage: J[1].Hom(J[2])._calculate_simple_gens()
[
[ 0 -4]
[ 4 0]
]
::
sage: J = J0(23) ; J.decomposition()
[
Simple abelian variety J0(23) of dimension 2
]
sage: J[0].Hom(J[0])._calculate_simple_gens()
[
[1 0 0 0] [ 0 1 -1 0]
[0 1 0 0] [ 0 1 -1 1]
[0 0 1 0] [-1 2 -2 1]
[0 0 0 1], [-1 1 0 -1]
]
sage: J.hecke_operator(2).matrix()
[ 0 1 -1 0]
[ 0 1 -1 1]
[-1 2 -2 1]
[-1 1 0 -1]
::
sage: H = Hom(J0(11), J0(22)[0])
sage: H._calculate_simple_gens()
[
[1 0]
[1 1]
]
"""
A = self.domain()
B = self.codomain()
if A.newform_label() != B.newform_label():
return []
f = A._isogeny_to_newform_abelian_variety()
g = B._isogeny_to_newform_abelian_variety().complementary_isogeny()
Af = f.codomain()
ls = Af._calculate_endomorphism_generators()
Mf = f.matrix()
Mg = g.matrix()
return [ Mf * self._get_matrix(e) * Mg for e in ls ]
class EndomorphismSubring(Homspace, Ring):
def __init__(self, A, gens=None):
"""
A subring of the endomorphism ring.
INPUT:
- ``A`` - an abelian variety
- ``gens`` - (default: None); optional; if given
should be a tuple of the generators as matrices
EXAMPLES::
sage: J0(23).endomorphism_ring()
Endomorphism ring of Abelian variety J0(23) of dimension 2
sage: sage.modular.abvar.homspace.EndomorphismSubring(J0(25))
Endomorphism ring of Abelian variety J0(25) of dimension 0
sage: E = J0(11).endomorphism_ring()
sage: type(E)
<class 'sage.modular.abvar.homspace.EndomorphismSubring_with_category'>
sage: E.category()
Join of Category of hom sets in Category of sets and Category of rings
sage: E.homset_category()
Category of modular abelian varieties over Rational Field
sage: TestSuite(E).run(skip=["_test_elements"])
TESTS:
The following tests against a problem on 32 bit machines that
occured while working on trac ticket #9944::
sage: sage.modular.abvar.homspace.EndomorphismSubring(J1(12345))
Endomorphism ring of Abelian variety J1(12345) of dimension 5405473
"""
self._J = A.ambient_variety()
self._A = A
homset_cat = A.category()
cat = Category.join([homset_cat.hom_category(),Rings()])
Ring.__init__(self, A.base_ring())
Homspace.__init__(self, A, A, cat=homset_cat)
self._refine_category_(Rings())
if gens is None:
self._gens = None
else:
self._gens = tuple([ self._get_matrix(g) for g in gens ])
self._is_full_ring = gens is None
def _repr_(self):
"""
Return the string representation of self.
EXAMPLES::
sage: J0(31).endomorphism_ring()._repr_()
'Endomorphism ring of Abelian variety J0(31) of dimension 2'
sage: J0(31).endomorphism_ring().image_of_hecke_algebra()._repr_()
'Subring of endomorphism ring of Abelian variety J0(31) of dimension 2'
"""
if self._is_full_ring:
return "Endomorphism ring of %s" % self._A
else:
return "Subring of endomorphism ring of %s" % self._A
def abelian_variety(self):
"""
Return the abelian variety that this endomorphism ring is attached
to.
EXAMPLES::
sage: J0(11).endomorphism_ring().abelian_variety()
Abelian variety J0(11) of dimension 1
"""
return self._A
def index_in(self, other, check=True):
"""
Return the index of self in other.
INPUT:
- ``other`` - another endomorphism subring of the
same abelian variety
- ``check`` - bool (default: True); whether to do some
type and other consistency checks
EXAMPLES::
sage: R = J0(33).endomorphism_ring()
sage: R.index_in(R)
1
sage: J = J0(37) ; E = J.endomorphism_ring() ; T = E.image_of_hecke_algebra()
sage: T.index_in(E)
1
sage: J = J0(22) ; E = J.endomorphism_ring() ; T = E.image_of_hecke_algebra()
sage: T.index_in(E)
+Infinity
"""
if check:
if not isinstance(other, EndomorphismSubring):
raise ValueError, "other must be a subring of an endomorphism ring of an abelian variety."
if not (self.abelian_variety() == other.abelian_variety()):
raise ValueError, "self and other must be endomorphisms of the same abelian variety"
M = self.free_module()
N = other.free_module()
if M.rank() < N.rank():
return sage.rings.all.Infinity
return M.index_in(N)
def index_in_saturation(self):
"""
Given a Hecke algebra T, compute its index in its saturation.
EXAMPLES::
sage: End(J0(23)).image_of_hecke_algebra().index_in_saturation()
1
sage: End(J0(44)).image_of_hecke_algebra().index_in_saturation()
2
"""
A = self.abelian_variety()
d = A.dimension()
M = ZZ**(4*d**2)
gens = [ x.matrix().list() for x in self.gens() ]
R = M.submodule(gens)
return R.index_in_saturation()
def discriminant(self):
"""
Return the discriminant of this ring, which is the discriminant of
the trace pairing.
.. note::
One knows that for modular abelian varieties, the
endomorphism ring should be isomorphic to an order in a
number field. However, the discriminant returned by this
function will be `2^n` ( `n =`
self.dimension()) times the discriminant of that order,
since the elements are represented as 2d x 2d
matrices. Notice, for example, that the case of a one
dimensional abelian variety, whose endomorphism ring must
be ZZ, has discriminant 2, as in the example below.
EXAMPLES::
sage: J0(33).endomorphism_ring().discriminant()
-64800
sage: J0(46).endomorphism_ring().discriminant() # long time (6s on sage.math, 2011)
24200000000
sage: J0(11).endomorphism_ring().discriminant()
2
"""
g = self.gens()
M = Matrix(ZZ,len(g), [ (g[i]*g[j]).trace()
for i in range(len(g)) for j in range(len(g)) ])
return M.determinant()
def image_of_hecke_algebra(self, check_every=1):
"""
Compute the image of the Hecke algebra inside this endomorphism
subring.
We simply calculate Hecke operators up to the Sturm bound, and look
at the submodule spanned by them. While computing, we can check to
see if the submodule spanned so far is saturated and of maximal
dimension, in which case we may be done. The optional argument
check_every determines how many Hecke operators we add in before
checking to see if this condition is met.
EXAMPLES::
sage: E = J0(33).endomorphism_ring()
sage: E.image_of_hecke_algebra()
Subring of endomorphism ring of Abelian variety J0(33) of dimension 3
sage: E.image_of_hecke_algebra().gens()
(Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3,
Abelian variety endomorphism of Abelian variety J0(33) of dimension 3)
sage: [ x.matrix() for x in E.image_of_hecke_algebra().gens() ]
[
[1 0 0 0 0 0] [ 0 2 0 -1 1 -1] [ 0 0 1 -1 1 -1]
[0 1 0 0 0 0] [-1 -2 2 -1 2 -1] [ 0 -1 1 0 1 -1]
[0 0 1 0 0 0] [ 0 0 1 -1 3 -1] [ 0 0 1 0 2 -2]
[0 0 0 1 0 0] [-2 2 0 1 1 -1] [-2 0 1 1 1 -1]
[0 0 0 0 1 0] [-1 1 0 2 0 -3] [-1 0 1 1 0 -1]
[0 0 0 0 0 1], [-1 1 -1 1 1 -2], [-1 0 0 1 0 -1]
]
sage: J0(33).hecke_operator(2).matrix()
[-1 0 1 -1 1 -1]
[ 0 -2 1 0 1 -1]
[ 0 0 0 0 2 -2]
[-2 0 1 0 1 -1]
[-1 0 1 1 -1 -1]
[-1 0 0 1 0 -2]
"""
try:
return self.__hecke_algebra_image
except AttributeError:
pass
A = self.abelian_variety()
if not A.is_hecke_stable():
raise ValueError, "ambient variety is not Hecke stable"
M = A.modular_symbols()
d = A.dimension()
EndVecZ = ZZ**(4*d**2)
if d == 1:
self.__hecke_algebra_image = EndomorphismSubring(A, [[1,0,0,1]])
return self.__hecke_algebra_image
V = EndVecZ.submodule([A.hecke_operator(1).matrix().list()])
for n in range(2,M.sturm_bound()+1):
V += EndVecZ.submodule([ A.hecke_operator(n).matrix().list() ])
self.__hecke_algebra_image = EndomorphismSubring(A, V.basis())
return self.__hecke_algebra_image
