Module: sage.modular.abvar.homspace
Spaces of homomorphisms between modular abelian varieties.
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 1 0 0] [0 0 0 0] [0 0 0 0], [0 0 0 0] [0 0 0 0] [0 0 1 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 0 0] [0 0 1 0] [0 0 0 1], [ 0 1 1 -1] [ 1 0 1 0] [ 0 0 -1 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 0 0] [0 0 1 0] [0 0 0 1], [ 0 1 1 -1] [ 1 0 1 0] [ 0 0 -1 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() (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() sage: T.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: T.index_in(E) +Infinity sage: T.index_in_saturation() 1
Author Log:
Class: EndomorphismSubring
self, A, [gens=None]) |
A subring of the endomorphism ring.
Input:
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: type(J0(11).endomorphism_ring()) <class 'sage.modular.abvar.homspace.EndomorphismSubring'>
Functions: abelian_variety,
discriminant,
image_of_hecke_algebra,
index_in,
index_in_saturation
self) |
Return the abelian variety that this endomorphism ring is attached to.
sage: J0(11).endomorphism_ring().abelian_variety() Abelian variety J0(11) of dimension 1
self) |
Return the discriminant of this ring, which is the discriminant of the trace pairing.
sage: J0(33).endomorphism_ring().discriminant() -64800 sage: J0(46).endomorphism_ring().discriminant() 24200000000
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.
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 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1], [ 0 2 0 -1 1 -1] [-1 -2 2 -1 2 -1] [ 0 0 1 -1 3 -1] [-2 2 0 1 1 -1] [-1 1 0 2 0 -3] [-1 1 -1 1 1 -2], [ 0 0 1 -1 1 -1] [ 0 -1 1 0 1 -1] [ 0 0 1 0 2 -2] [-2 0 1 1 1 -1] [-1 0 1 1 0 -1] [-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]
self, other, [check=True]) |
Return the index of self in other.
Input:
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
self) |
Given a Hecke algebra T, compute its index in its saturation.
sage: End(J0(23)).image_of_hecke_algebra().index_in_saturation() 1 sage: End(J0(44)).image_of_hecke_algebra().index_in_saturation() 2
Special Functions: __init__,
_repr_
self) |
Return the string representation of self.
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'
Class: Homspace
self, domain, codomain, cat) |
Create a homspace.
Input:
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'> sage: H.homset_category() Category of modular abelian varieties over Rational Field
Functions: calculate_generators,
free_module,
gen,
gens,
matrix_space,
ngens
self) |
If generators haven't already been computed, calculate generators for this homspace. If they have been computed, do nothing.
sage: E = End(J0(11)) sage: E.calculate_generators()
self) |
Return this endomorphism ring as a free submodule of a big
,
where
is the dimension of the domain abelian variety and
the
dimension of the codomain.
Output: free module
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, [i=0]) |
Return i-th generator of self.
Input:
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) |
Return tuple of generators for this endomorphism ring.
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)
self) |
Return the underlying matrix space that we view this endomorphism ring as being embedded into.
sage: E = End(J0(22)) sage: E.matrix_space() Full MatrixSpace of 4 by 4 dense matrices over Integer Ring
self) |
Return number of generators of self.
Output: integer
sage: E = End(J0(22)) sage: E.ngens() 4
Special Functions: __call__,
__init__,
_calculate_product_gens,
_calculate_simple_gens,
_coerce_impl,
_get_matrix,
_repr_
self, M) |
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()
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]
self) |
For internal use.
Calculate generators for self, assuming that self is a product of simple factors.
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 0 0] [0 0 1 0] [0 0 0 1], [ 0 1 1 -1] [ 1 0 1 0] [ 0 0 -1 1] [ 0 0 0 1]] sage: E._calculate_product_gens() [[1 0 0 0] [0 1 0 0] [0 0 0 0] [0 0 0 0], [0 0 0 0] [0 0 0 0] [0 0 1 0] [0 0 0 1]]
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.
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 0 0] [0 0 1 0] [0 0 0 1], [ 0 1 -1 0] [ 0 1 -1 1] [-1 2 -2 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]]
self, x) |
Coerce x into self, if possible.
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
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:
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]
self) |
String representation of a modular abelian variety homspace.
sage: J = J0(11) sage: End(J)._repr_() 'Endomorphism ring of Abelian variety J0(11) of dimension 1'