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 QThe 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 φand ψof 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 φand ψto 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, by 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 φof abelian varieties and a morphism from the codomain of φto 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