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The following questions seem to be related.

Firstly, let $E_1$ and $E_2$ denote elliptic curves. Silverman defines that $E_1$ and $E_2$ are isogeneous if and only if there exists a basepoint preserving regular map $E_1 \rightarrow E_2$ that is non-constant. Galbraith et. al. further require that the map be separable.

Question 1. Are these conditions equivalent?

Secondly, Galbraith et. al. (on the same page) state that a separable isogeny can be identified with its kernel. However, it seems to me that any non-constant isogeny can be identified with its kernel (up to isomorphism, if we view that isogeny as an object in the relevant coslice category.) This is because non-constant isogenies are surjective.

Question 2. Is separability really necessary for Galbraith et. al.'s statement to be correct?

I'm not sure if these are secretly the same question or not. If they're not related, feel free to comment and I'll happily break this into two different questions.

goblin GONE
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1 Answers1

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  1. I don't think they're equivalent, since you can use google to see that authors use the phrase "separably isogenous." I don't know how to construct a counterexample but Frobenius twists might work.

  2. I think Galbraith is taking the kernel on $\overline{\mathbb{F}_p}$-points, not the kernel as a group scheme. The kernel in this sense doesn't identify the isogeny unless the kernel group scheme is étale.

Qiaochu Yuan
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