Let $E$ be a supersingular elliptic curve, $\mathsf{O}$ be the endomorphism ring of $E$, and $I$ be a left $\mathsf{O}$-ideal. In Voight's Quaternion Algebras, he defines the map $\phi_I$ as the isogeny from $E$ to $E / E[I]$, where $E[I]$ is the scheme-theoretic intersection $$E[I] := \bigcap_{\alpha \in I} E[\alpha],$$ and $E[\alpha]$ is $\ker \alpha$ as a group scheme.
The author then goes on to prove Lemma 42.2.7:
The pullback map $$\phi_I^* : \operatorname{Hom}(E_I, E) \to I$$ defined by $\psi \mapsto \psi \phi_I$ is an isomorphism of left $\mathsf{O}$-modules.
In the proof, the first sentence is "The image of $\operatorname{Hom}(E_I, E)$ under precomposition by $\phi_I$ lands in $\operatorname{End}(E) = \mathsf{O}$ and factors through $\phi_I$ so lands in $I$ by definition."
I'm not sure what definition the author is referring to at the end of the sentence. It seems that the statement we want is that $I$ is exactly the set of endomorphisms whose kernel contains $E[I]$, but it isn't clear to me why there can't be an endomorphism outside of $I$ with this property. I'm moderately (but not very) familiar with the language of schemes, but not familiar at all with group schemes, so I suspect this is coming from some universal property that $\phi_I$ enjoys as the map from $E \to E[I]$ where $E[I]$ is defined as a scheme-theoretic intersection, but I'm not sure. Can anyone explain which definition is giving the result that the image of the pullback lands in $I$?
Best Answer
Re-found my own question, and I'll add an answer just to get this off of the unanswered queue:
As far as I can tell, the quoted statement is equivalent to showing that the set of $\alpha \in O$ such that $E[I] \subset \ker \alpha$ is exactly $I$. Voight defines $I(H) := \{\alpha \in O : \alpha(P) = 0 \text{ for all } P \in H\}$, and then proves in a later Proposition (42.2.16) that $I(E[I]) = I$, which is the statement we want here.
That proof is more complicated than saying that this is true by definition, but doesn't seem to rely on Lemma 42.2.7, so I think Lemma 42.2.7 should really come after Proposition 42.2.16 logically. That would fix the problem presented here.
If anyone has another explanation though, I would still love to hear it!