# 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`](modabvar.md#function-modabvar-embed), 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 $J_0(11)$. ```magma > 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)` ```magma > A := JZero(37) * JZero(22); > Factorization(A); [* , , *] ``` ## 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)` ```magma > 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. ```magma > 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 $J_0(23)$. ```magma > 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 $J_0(46)$. ```magma > Factorization(Conductor(J(2))); [ <23, 2> ] > Factorization(J); [* , *] ```