By the Poincare reducibility theorem, every abelian variety is isogenous to a product of simple abelian subvarieties. If A is a modular abelian variety over Q, then A is isogenous to a product of simple abelian varieties Af attached to newforms. The Decomposition and Factorization commands compute such decompositions.
Given an abelian variety A, return a sequence [Bi] of simple modular abelian varieties, whose product is isogenous to A. Each Bi is equipped with an embedding into A such that the sum of the images of the Bi is equal to A. This embedding is the first element of the output of Embeddings, given a Bi.
The nth factor in Decomposition(A), denoted A(n) where A is an abelian variety.
> A := JZero(37) * JZero(22);
> D := Decomposition(A); D;
[
Modular abelian variety 37A of dimension 1, level 2*11*37 and
conductor 37 over Q,
Modular abelian variety 37B of dimension 1, level 2*11*37 and
conductor 37 over Q,
Modular abelian variety N(11,814,1)(11A) of dimension 1,
level 2*11*37 and conductor 11 over Q,
Modular abelian variety N(11,814,2)(11A) of dimension 1,
level 2*11*37 and conductor 11 over Q
]
> B := D[3];
> Embeddings(B);
[*
Homomorphism from N(11,814,1)(11A) to JZero(37) x JZero(22) given on
integral homology by:
[ 0 0 0 0 1 0 -1 3]
[ 0 0 0 0 0 1 -2 3]
*]
Given an abelian variety A, compute pairwise non-isogenous simple newform abelian varieties Af whose product, with multiplicities, is isomorphic to A. A list of pairs < B, [φ, ... ] > is returned, where B is an isogeny simple abelian variety and [φ, ... ] is a sequence of maps from B into A (whose length is the "multiplicity"), such that the product of all images of all B is isogenous to A, and the sum of the dimensions of the images of B is the dimension of A. Moreover, the B are pairwise non-isogenous. To obtain a list of the images of the B canonically embedded into A, use Decomposition(A).
> A := JZero(37) * JZero(22);
> Factorization(A);
[*
<Modular abelian variety 37A of dimension 1, level 37 and
conductor 37 over Q, [
Homomorphism N(37,814,1) from 37A to JZero(37) x JZero(22) given on
integral homology by:
[ 1 -1 1 0 0 0 0 0]
[ 1 -1 -1 1 0 0 0 0]
]>,
<Modular abelian variety 37B of dimension 1, level 37 and
conductor 37 over Q, [
Homomorphism N(37,814,1) from 37B to JZero(37) x JZero(22) given on
integral homology by:
[1 1 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
]>,
<Modular abelian variety 11A of dimension 1, level 11 and
conductor 11 over Q, [
Homomorphism N(11,814,1) from 11A to JZero(37) x JZero(22) given on
integral homology by:
[ 0 0 0 0 0 1 -2 3]
[ 0 0 0 0 1 -1 1 0],
Homomorphism N(11,814,2) from 11A to JZero(37) x JZero(22) given on
integral homology by:
[ 0 0 0 0 -1 0 2 -2]
[ 0 0 0 0 -1 2 -1 0]
]>
*]
The following commands use the elements of a commutative subring of endomorphisms to decompose a modular abelian variety A into a direct sum of abelian subvarieties by taking kernels (which are analogous to generalized eigenspaces).
Decompose an abelian variety A using the commutative ring of endomorphisms generated by the space of homomorphisms R of A.
Decompose an abelian variety A using the endomorphism φ of A.
> T2 := HeckeOperator(JZero(100),2);
> DecomposeUsing(T2);
[
Modular abelian variety of dimension 1 and level 2^2*5^2 over Q,
Modular abelian variety of dimension 5 and level 2^2*5^2 over Q,
Modular abelian variety of dimension 1 and level 2^2*5^2 over Q
]
> W := AtkinLehnerOperator(JZero(100),100);
> DecomposeUsing(W);
[
Modular abelian variety of dimension 3 and level 2^2*5^2 over Q,
Modular abelian variety of dimension 4 and level 2^2*5^2 over Q
]
> J := JZero(46); J;
Modular abelian variety JZero(46) of dimension 5 and level 2*23 over Q
> Decomposition(J);
[
Modular abelian variety 46A of dimension 1, level 2*23 and
conductor 2*23 over Q,
Modular abelian variety N(23,46,1)(23A) of dimension 2, level
2*23 and conductor 23^2 over Q,
Modular abelian variety N(23,46,2)(23A) of dimension 2, level
2*23 and conductor 23^2 over Q
]
Thus J decomposes as a product E x A x B, where
E is an elliptic curve of conductor 46, and A and B
are two isogenous images of J0(23).
> J(1); Modular abelian variety 46A of dimension 1, level 2*23 and conductor 2*23 over Q > Conductor(J(1)); 46 > Factorization(Conductor(J(2))); [ <23, 2> ]The Factorization command gives an explicit decomposition with embeddings of each factor into J0(46).
> Factorization(Conductor(J(2)));
[ <23, 2> ]
> Factorization(J);
[*
<Modular abelian variety 46A of dimension 1, level 2*23 and
conductor 2*23 over Q, [
Homomorphism from 46A to JZero(46) given on integral homology
by:
[ 1 0 -2 -1 -1 1 1 1 -2 1]
[ 0 1 -1 -1 0 0 0 1 -1 0]
]>,
<Modular abelian variety 23A of dimension 2, level 23 and
conductor 23^2 over Q, [
Homomorphism N(23,46,1) from 23A to JZero(46) given on integral
homology by:
[-1 1 -1 1 0 -1 -1 1 -1 2]
[ 0 0 -1 2 -2 -1 0 0 1 0]
[ 0 0 0 1 -2 0 0 1 0 0]
[ 0 1 0 -1 0 0 1 0 0 0],
Homomorphism N(23,46,2) from 23A to JZero(46) given on integral
homology by:
[ 0 -1 0 0 1 -1 0 1 0 0]
[-1 0 0 0 0 -1 2 -1 1 -1]
[-1 1 -1 0 0 0 2 -2 2 -2]
[ 0 0 -1 2 -1 0 0 0 0 -1]
]>
*]