This is question from Silverman's Advanced topics in the arithmetic of elliptic curves, p. 106. Let $E_1$ and $E_2$ be elliptic curves. What does

isogeny $φ:E_1→E_2$ is determined by its kernel, at least up to an automorphism of $E_1$ and $E_2$

mean? Once $\text{ker}\, φ$ is given, what can we say about $φ$ and what does it have to do with $\text{Aut}(E_1)$ and $\text{Aut}(E_2)$ ? (To me, $\text{Aut}$ is nothing to do with $φ$).


1 Answer 1


Here's an analogue in group theory that is pretty arrow-theoretic (meaning it's about general nonsense with mappings that you see in many categories having quotients) and thus applies to modules, vector spaces, and so on.

Let $f \colon G_1 \to G_2$ and $f' \colon G_1 \to G_2$ be surjective group homomorphisms between the same groups with the same kernel $K$. Are $f$ and $f'$ related? They induce isomorphisms $\overline{f}, \overline{f'} \colon G_1/K \to G_2$. Then $F = \overline{f} \circ \overline{f'}^{-1} \colon G_2 \to G_2$ is an automorphism of $G_2$ and $F \circ \overline{f'} = \overline{f}$ as isomorphisms $G_1/K \to G_2$, and pulling back to $G_1$ we get $F \circ f' = f$ as homomorphisms $G_1 \to G_2$. Thus the homomorphisms $f$ and $f'$ with the same kernel are the "same" map up to an automorphism of $G_2$.

  • $\begingroup$ Hello, I really appreciated your online notes on (infinite) Galois theory. What does 'arrow-theoretic' mean? I have a good knowledge of many of the basics of category theory but have never seen this (presumably categorical) term $\endgroup$
    – FShrike
    Commented Oct 17, 2022 at 19:03
  • 1
    $\begingroup$ I updated my answer with a parenthetical description. I could have said "categorical" instead. $\endgroup$
    – KCd
    Commented Oct 17, 2022 at 19:07
  • $\begingroup$ Ah ok, thanks. I asked in case it referred to a specific discipline that I might want to learn more about $\endgroup$
    – FShrike
    Commented Oct 17, 2022 at 19:09

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