Why can I take the quotient of a relative elliptic curve by a finite locally free subgroup?

Question

I am currently reading Katz and Mazur’s Arithmetic moduli of elliptic curves and I am puzzled by a statement in the discussion of the $[\Gamma_0(N)]$ moduli problem in Chapter 3.

The authors define a $\Gamma_0(N)$-structure on a relative elliptic curve $E/S$ as an isogeny $E \rightarrow E’$ of degree $N$, whose kernel is cyclic in the sense that it has a generator fppf-locally on $S$.

Equivalently, the authors add, it is the datum of a finite flat $S$-subgroup scheme $C \subset E[N]$, locally free of degree $N$ which is cyclic in the above sense.

The equivalence is obviously given by the fact that given a relative elliptic curve $E/S$ and a finite locally free cyclic subgroup $C$ of $E/S$, we can form the quotient elliptic curve $E/C$. But why is this true?

This fact seems to be taken for granted in Katz-Mazur, and I didn’t see any reference given in the book (and the same seems to hold for Brian Conrad’s article on generalized elliptic curves).

I understand this question as two-fold:

1. why is the abelian fppf-sheaf $E/C$ on $Sch/S$ representable at all?

2. even if this sheaf is representable, why is it representable by an elliptic curve?

I would tentatively guess that the answer to 1) lies in the tools developed to do algebraic geometry beyond schemes, which I know very little about: stacks, algebraic spaces… But I haven’t been able so far to pinpoint what was actually needed.

To answer 2), one could first show that $E/C$ is flat over $S$ (but how?), and then all that remains to do is the case where $S$ is an algebraically closed field, but even this step is not straightforward (although it seems more tractable than the rest) when $C$ is not étale (and thus cannot directly be seen as a group of automorphisms).

Answer 1

A reference for 1. is https://stacks.math.columbia.edu/tag/07S7

We can then answer question 2. like this:

The quotient morphism $E\to E/C$ is faithfully flat (this is a part of the conclusion of the lemma above), which implies that $E/C$ is flat over $S$ because $E$ is flat over $S$. The quotient scheme $E/C$ gets an induced group scheme structure over $S$, because $C\subset E$ was a normal subgroup scheme. Again using that $E\to E/C$ is faithfully flat, we learn that the morphism $E/C\to S$ is proper, and its fibers are geometrically reduced and (geometrically) connected of dimension 1. Since $E/C$ is a flat group scheme, the fact that its fibers are geometrically reduced implies that it is smooth, which means that $E/C\to S$ is a family of elliptic curves.

Another reference that discusses quotients of abelian schemes by finite group subschemes (though only over a field) is Mumford's 'Abelian varieties', chapters 12-13.