# factorisation of morphisms of abelian varieties

Corollary 4.11 of Silverman's book The Arithmetic of elliptic curves (p. 73) says

Let $$\phi:E_1 \rightarrow E_2 \text{ and } \psi:E_1 \rightarrow E_3$$ be nonconstant isogenies, and assume that $\phi$ is separable, If $$\ker \phi \subset \ker \psi$$ then there is a unique isogeny $$\lambda: E_2 \rightarrow E_3$$ satisfying $\psi = \lambda\circ \phi$.

Does this result extend to abelian varieties of higher dimensions?

Yes it does. If $\phi : A_{1} \to A_{2}$ and $\psi: A_{1} \to A_{3}$ are isogenies of abelian varieties as in your question, then you may form a quotient $A_{2}/\phi(\ker \psi)$ with quotient map $\pi : A_{2} \to A_{2}/\phi(\ker \psi)$. The composite map $\pi \circ \phi$ then has kernel exactly $\ker \psi$, and so its image is $A_{2}/\phi(\ker \psi)$ is isomorphic to $A_{3}$ by Corollary 1, pg. 118 of Mumford's book on Abelian Varieties. Post-composing $\pi$ with this isomorphism then gets you an isogeny $\pi' : A_{2} \to A_{3}$ such that $\psi = \pi' \circ \phi$.
• Thanks for your answer. I have two questions: (1) I don't see where you use the fact that $\phi$ is separable. (2) It is possible to extend this result for all algebraic varieties? (If yes, I guess the proof is totally different). Thanks again. Jun 24, 2018 at 14:47
• As for the generalization, the general idea is of dividing a variety by a collection of automorphisms acting on the variety (so in the case of abelian varieties these are the automorphisms $P \mapsto Q + P$ for each fixed $Q$ in the kernel), and there is a version of this on pg. 66 of Mumford's book. The quotient map and codomain variety are again determined up to isomorphism. Jun 25, 2018 at 9:31