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?


1 Answer 1


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 $\psi = \pi' \circ \phi$.

  • 1
    $\begingroup$ 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. $\endgroup$
    – A. GM
    Jun 24, 2018 at 14:47
  • $\begingroup$ Separability is needed for the uniqueness claim. There's actually a better (less general) proof of this fact I should have referenced, which is Theorem 4 on page 72/73/74 of Mumford's book. There he explicitly uses separability and it's needed to show that any two isogenies with the same kernel have isomorphic images. Without this you could compose with some Frobenius-like map and not change the kernel, I think. $\endgroup$ Jun 25, 2018 at 9:28
  • $\begingroup$ 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. $\endgroup$ Jun 25, 2018 at 9:31

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