Decomposing and Factoring Abelian Varieties#
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 \({\mathbb{Q}}\), then \(A\) is isogenous to a product of simple abelian varieties \(A_f\) attached to newforms. The Decomposition and Factorization commands compute such decompositions.
Decomposition#
- Decomposition(A): ModAbVar -> SeqEnum#
Given an abelian variety \(A\), return a sequence \([B_i]\) of simple modular abelian varieties, whose product is isogenous to \(A\). Each \(B_i\) is equipped with an embedding into \(A\) such that the sum of the images of the \(B_i\) is equal to \(A\). This embedding is the first element of the output of
Embeddings, given a \(B_i\).
- A(n): RngIntElt, ModAbVar -> ModAbVar#
The \(n\)th factor in
Decomposition(A), denoted \(A(n)\) where \(A\) is an abelian variety.
- Example: Decomp Decomposition (ex-e61813)#
We decompose \(A=J_0(37)\times J_0(22)\), then find the embedding into \(A\) of a factor which is isogenous to \(J_0(11)\).
> 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] *]
Factorization#
- Factorisation(A): ModAbVar -> List#
- Factorization(A): ModAbVar -> List#
Given an abelian variety \(A\), compute pairwise non-isogenous simple newform abelian varieties \(A_f\) whose product, with multiplicities, is isomorphic to \(A\). A list of pairs \(\langle B,[\phi, \ldots]\rangle\) is returned, where \(B\) is an isogeny simple abelian variety and \([\phi, \ldots]\) 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).
- Example: Decomp Factorization (ex-189402)#
> 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] ]> *]
Decomposition with respect to an Endomorphism or a Commutative Ring#
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).
- DecomposeUsing(R): HomModAbVar -> SeqEnum#
Decompose an abelian variety \(A\) using the commutative ring of endomorphisms generated by the space of homomorphisms \(R\) of \(A\).
- DecomposeUsing(phi): MapModAbVar -> SeqEnum#
Decompose an abelian variety \(A\) using the endomorphism \(\phi\) of \(A\).
- Example: Decomp Decomposition With Respect To An Endomorphism Or A Commutative Ring (ex-27db11)#
> 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 ]
Additional Examples#
- Example: homology1 (ex-f1a775)#
We compute a decomposition of \(J_0(46)\) as a product of simple abelian subvarieties.
> 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 \(E\times A \times B\), where \(E\) is an elliptic curve of conductor \(46\), and \(A\) and \(B\) are two isogenous images of \(J_0(23)\).
> 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
Factorizationcommand gives an explicit decomposition with embeddings of each factor into \(J_0(46)\).> 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] ]> *]