# Equivalent definitions of $\Gamma _1(N)$-structures on elliptic curves

In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), two equivalent definitions of a $\Gamma _1(N)$-structure (point of exact order $N$) are given on page 99 (page 55 in the pdf). While I see why one implication holds, I fail to understand the other one. Let me sum up the statement.

Let $S$ be an arbitrary scheme and $E$ be an elliptic curve over $S$. Let $N\geq 1$ be an integer. Assume that we are given a group homomorphism $$\phi: \mathbb{Z}/N\mathbb{Z}\rightarrow E[N](S)$$ such that the effective Cartier divisor in $E$ $$G:=\sum_{a \operatorname{mod} N}[\phi(a)]$$ is a subgroup-scheme of $E$ (ie. $\phi(1)$ is a point of exact order $N$).
Then, there exists an elliptic curve $E'$ over $S$ together with an $N$-isogeny of elliptic curves over $S$ $$E\xrightarrow {\pi}E'$$ such that we have an equality of effective Cartier divisors in $E$ $$\operatorname{Ker}\pi = G$$

I fail to understand how one can define such a morphism $\pi$. Actually, I am not even sure about how to define a suitable elliptic curve $E'$. Because I want the kernel of my morphism to be the given subgroup-scheme induced by $\phi$, I would like $E'$ to be, in some sense, "$E/G$". However, I am not even sure about the good definition of such an object, and whether it is an elliptic curve (over $S$).

Would someone see what is the correct argument here? Could someone please explain to me how $E'$ and $\pi$ are defined with respect to $\phi$ and $G$?

I thank you very much for your help.

• I am also interested in this answer, have you found one @Suzet? – user1728 Jul 31 '18 at 9:50
• @DistractedKerl Well, I didn't find out any precised/detailed arguments, however I had confirmation from my supervizor since that indeed, such a quotient object as $E/G$ can be defined in this context, together with a projection morphism $\pi: E\rightarrow E/G$ which turns out to be an $N$-isogeny between elliptic curves. Justifications for this may surely be found in SGA III, however as it is a rather wide source, I didn't try to find the actual precise statements involved. Basically, I went on reading the book, admitting that one may consider quotient group-schemes if needed. – Suzet Jul 31 '18 at 9:57
• Thank you for your comments! I'm reading the same book right now :) – user1728 Jul 31 '18 at 10:21
• Dear @Suzet, there is something I still don't understand perfectly, how are we to view $\ker \pi$ or even more explicitly $E[N]$ as a divisor on $E$? Could you provide me some insights on this definition please? – user1728 Aug 1 '18 at 12:52
• Hmm, as I understand it, the situation where a given closed subgroup-scheme happens to be an effective Cartier divisor is an exceptional situation. When Katz and Mazur write "an equality of effective Cartier divisors", I rather understand "the right hand side is an effective Cartier divisor, we have an equality between these two closed subschemes, and hence the left hand side happens to be an effective Cartier divisor too (as well as the right hand side happens to be a group-scheme)". I do not think there is a priori a Cartier divisor structure on $E[N]$ nor on $\operatorname{Ker}(\pi)$. – Suzet Aug 2 '18 at 13:42