One can take arbitrary finite direct sums of modular abelian varieties. We do not write A+B for the direct sum, since it is already used for the sum of A and B inside a common ambient abelian variety, and this sum need not be direct, unless A∩B = 0.
The direct sum D of abelian varieties A and B, together with the embedding maps from A into D and B into D, respectively, and the projection maps from D onto A and B, respectively. It is not possible to take the direct sum of abelian varieties with different signs.
The direct sum D of the sequence X of modular abelian varieties, together with a list containing the embedding maps from each modular abelian variety of X into D and a list containing the projection maps from D onto each modular abelian variety in X. It is not possible to take the direct sum of abelian varieties with different signs.
The direct sum of n copies of the abelian variety A. If n=0, the zero subvariety of A. If n is negative, the ( - n)-th power of the dual of A.
> J := JZero(65);
> D := Decomposition(J); D;
[
Modular abelian variety 65A of dimension 1, level 5*13 and
conductor 5*13 over Q,
Modular abelian variety 65B of dimension 2, level 5*13 and
conductor 5^2*13^2 over Q,
Modular abelian variety 65C of dimension 2, level 5*13 and
conductor 5^2*13^2 over Q
]
> A := D[1];
> B := D[2];
> A*B;
Modular abelian variety 65A x 65B of dimension 3 and level 5*13
over Q
Homomorphism from 65A to 65A x 65B given on integral homology by:
[1 0 0 0 0 0]
[0 1 0 0 0 0]
Homomorphism from 65B to 65A x 65B given on integral homology by:
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
Homomorphism from 65A x 65B to 65A (not printing 6x2 matrix)
Homomorphism from 65A x 65B to 65B (not printing 6x4 matrix)
> M := JZero(11,4);M;
Modular motive JZero(11,4) of dimension 2 and level 11 over Q
> P := A*M; P;
Modular motive 65A x JZero(11,4) of dimension 3 and level 5*11*13 over Q
The product also returns inclusions of each factor into the
product and projection from the product onto each factor.
> C,f,g := A*B; > f; [* Homomorphism from 65A to 65A x 65B given on integral homology by: [1 0 0 0 0 0] [0 1 0 0 0 0], Homomorphism from 65B to 65A x 65B given on integral homology by: [0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1] *]
Here we compare direct sums of abelian varieties to the sum of of abelian varieties in a common ambient abelian variety. Thus if A is as above, then
> Dimension(A); 1 > Dimension(A*A); 2 > Dimension(A+A); 1
If you take a direct sum of abelian varieties that are defined over different base rings, then Magma will first attempt to express them over a common over-ring.
> A := JZero(11); > B := BaseExtend(JZero(14),CyclotomicField(3)); > C := A*B; C; Modular abelian variety JZero(11) x JZero(14) of dimension 2 and level 2*7*11 over Q(zeta_3)The above would not work if CyclotomicField(3) were replaced by GF(3), since the base ring Q of A is not contained in GF(3).
The sum A+B is the sum of A and B inside a common ambient abelian variety. This sum need not be direct, unless the intersection of A and B is 0.
The sum of the images of the abelian varieties A and B in a common ambient abelian variety.
The sum of the modular abelian varieties in the sequence X.
The sum D of the images of the morphisms φ and ψ of abelian varieties in their common codomain, a morphism from D into their common codomain, and a list containing a morphism from the domain of each of φ and ψ to D. If the codomains are not the same, then the homomorphisms are replaced by homomorphisms into an appropriate direct sum of codomains.
The sum D of the images of the morphisms of abelian varieties in the list X in their common codomain, a morphism from D into their common codomain, and a list containing a morphism from the domain of each morphism in X to D. If not all codomains of the elements of X are the same, then the homomorphisms are replaced by homomorphisms into an appropriate direct sum of codomains.
Return true and a list of embeddings into a common abelian variety, if one can be found using Embeddings(A) for all abelian varieties A in the sequence X.
Two abelian varieties cannot, by themselves, be intersected without choosing an embedding of both varieties in a common ambient abelian variety. The algorithm for computing an intersection is to compute the kernel of a certain homomorphism.
Intersections are computed in Magma by finding a homomorphism whose kernel is isomorphic to the intersection. For example, if f:A to C and g:B to C are injective homomorphisms, then the intersection of their images is isomorphic to the kernel of f - g.
As mentioned above, kernels of morphisms of abelian varieties are frequently not themselves abelian varieties. Instead a kernel is an extension of an abelian variety by a finite group of components. Likewise, intersections of abelian varieties are often not abelian varieties.
The intersection commands also take a sequence of abelian varieties or list of morphisms in order to facilitate computation of n-fold intersections, for any positive integer n.
Given abelian varieties A and B or a sequence X of abelian varieties, compute a finite lift G of the component group of the intersection, the connected component of the intersection C, and a map from the abelian variety that contains C to the abelian variety that contains G. The relevant intersection is C + G. The elements of X are replaced by their images via their modular embedding map. All the elements of X must be embedded in the same abelian variety.
Given a sequence X of morphisms from abelian varieties into a common abelian variety, compute a finite lift G of the component group of the intersection, the connected component C of the intersection, and a map from the abelian variety that contains C to the abelian variety that contains G. The morphisms in X do not have to be injective.
Given abelian varieties A and B or a sequence X of abelian varieties compute the group of components of the intersection of A and B or the varieties in X. (For more details, see the discussion of kernels in Section Kernels).
> D := Decomposition(JZero(65)); > G := ComponentGroupOfIntersection(D); G; Finitely generated subgroup of abelian variety with invariants [ 2 ] > FieldOfDefinition(G); Rational FieldThe quotient of D[1] by this subgroup of order 2 is an elliptic curve over Q isogenous to D[1], but not isomorphic to D[1].
> B := D[1]/G; B; Modular abelian variety of dimension 1 and level 5*13 over Q > IsIsomorphic(D[1],B); false
Next we compute some non-finite intersections.
> A := D[1] + D[2]; > B := D[1] + D[3]; > A meet B; Finitely generated subgroup of abelian variety with invariants [ 2, 2, 2 ] Modular abelian variety of dimension 1 and level 5*13 over Q Homomorphism from modular abelian variety of dimension 1 to modular abelian variety of dimension 6 given on integral homology by: [ 1 -1 0 0 0 0 1 -1 0 0 0 -1] [ 0 0 1 -1 1 -1 0 0 1 -1 1 0] Homomorphism from modular abelian variety of dimension 6 to JZero(65) (not printing 12x10 matrix)We can also intersect images of morphisms.
> f := ModularEmbedding(A); > g := ModularEmbedding(B); > _, C := IntersectionOfImages([* f, g *]); > C eq D[1]; true
> J := JZero(431);
> IsPrime(431);
true
> A := Decomposition(J)[1];
> B := Decomposition(J)[2];
> G, C := A meet B;
> G;
{ 0 }: finitely generated subgroup of abelian variety with
invariants []
> C;
Modular abelian variety ZERO of dimension 0 and level 431 over Q
> Newform(A) - Newform(B);
-2*q^3 + 4*q^5 + 2*q^6 - 4*q^7 + O(q^8)
If B is an abelian subvariety of A (or some natural image of B lies in A), then the quotient A/B is an abelian variety. Also, the cokernel of a homomorphism of abelian varieties is an abelian variety.
The quotient of the abelian variety A by a natural image B' of the abelian variety B. Here B' is the image of B under the modular embedding composed with the modular parameterization to A.
The cokernel of the morphism φ of abelian varieties and a morphism from the codomain of φ to the cokernel.
We compute a 2-dimensional quotient of the 3-dimensional abelian variety J0(33) using the Hecke operator T2.
> J := JZero(33);
> T := HeckeOperator(J,2);
> Factorization(CharacteristicPolynomial(T));
[
<x - 1, 2>,
<x + 2, 4>
]
> C := ConnectedKernel(T-1);
> B,psi := J/C;
> B;
Modular abelian variety of dimension 2 and level 3*11 over Q