So, when people say, "the moduli problem of classifying elliptic curves over $\mathbb{C}$ with level $N$ structure", there are usually two associated functors I've seen:


1. $P\_N : \textbf{Ell}\rightarrow\textbf{Sets}$, where $\textbf{Ell}$ is the category of elliptic curves $E\rightarrow S$ over $S$ and morphisms are cartesian squares, and


$P\_N(E/S) = \text{set of isomorphisms } \alpha : E[N]\rightarrow (\mathbb{Z}/N\mathbb{Z})^2 \text{ of determinant 1}$
2. $F\_N : \textbf{Sch}\rightarrow\textbf{Sets}$, where


$F\_N(S) = \text{set of isomorphism classes of pairs } (E/S,\alpha) \text{ with } \alpha\in P\_N(E/S) $


I apologize for the length of this post, but this has been terribly confusing for me.


Alright, so I know that for $N\ge 3$, *both* functors are representable by the modular scheme $Y(N) := \Gamma(N)\backslash\mathcal{H}$ which are fine moduli schemes, and that's got something to do with the fact that there are no automorphisms of elliptic curves $E/S$ fixing any $\alpha\in P\_N(E/S)$.


However, in the case $N = 1,2$, the modular curve $Y(2) := \Gamma(2)\backslash\mathcal{H}$ only gives you a coarse moduli scheme.


How should I think of the relation between the two above functors? In a way, a representing object $E/S$ for $P\_N$ gives you both the universal elliptic curve $E$ and the base moduli scheme $S$ in one fell swoop. However, the functor $P\_N$ doesn't seem like a naturally phrased moduli problem, since being able to represent $P\_N$ just says:


"there is an elliptic curve $E/S$ such that for any other elliptic curve $E'/S'$, a level structure on $E'/S'$ is equivalent to a morphism $S'\rightarrow S$ such that $E'\cong E\times\_S S'$."


(after thinking about it for a bit, it seems you can show that $P\_N$ representable $\Longrightarrow$ $F\_N$ representable)


On the other hand, the functor $F\_N$ is much more natural, in that a representing object for $F\_N$ much more clearly parametrizes elliptic curves with level structure. However, Peter Bruin's article (<http://user.math.uzh.ch/bruin/moduli.pdf>) and Katz/Mazur's book (specifically thm's 3.6 and 4.7) both seem to imply that if $P\_N$ is not rigid (eg, $N = 1,2$), then even if $F\_N$ is representable by an object $M$, the object $M$ might not carry a universal family. I am further confused by the fact that wikipedia (in the section on Fine Moduli Spaces) says that the universal family exists and must correspond to $\text{id}\_M\in\text{Hom}(M,M)$.


I'm assuming wikipedia is wrong.


If wikipedia is wrong, then in the case $N = 2$, is the functor $F\_2$ representable? If it is, is $\Gamma(2)\backslash\mathcal{H}$ the representing object?


When people talk of the moduli problem of classifying elliptic curves with full level $N$ structure, which functor are they referring to?


...onto stacks...


For $N = 1,2$, there is no fine moduli scheme, and hence at least $P\_N$ is not representable. However, is there a fine moduli *stack*? (does that mean anything?)


In general, would I be correct in saying that a stack for a moduli problem is basically just the moduli functor itself? Can you make this more precise? (though I guess you'd have to replace "isomorphism classes of..." with the objects themselves)


Are there *meaningful* moduli problems that aren't stacks?


Thanks for bearing with me


* will



what exactly is the moduli functor for classifying elliptic curves with (full) level N structure?
Your $F\_N$ is the functor people would usually mean when they talk about the functor classifying elliptic curves with full level N structure (though it's a bit nicer if you replace $(Z/nZ )^2$ with $\mu\_n \times Z/nZ$, so that the determinant takes values in $\mu\_n$ on both sides.)


Wikipedia is not wrong. (Wikipedia is surprisingly seldom wrong!) Your F\_2 is indeed not representable by a scheme, which is to say that the scheme known as Y(2) is not a fine moduli space. To say it was a fine moduli space would be precisely to say it represents the functor in question. That's why Y(2) doesn't have to have a universal family over it.


Yes -- the stack is the same thing as the functor; but we're keeping track of certain facts about that functor when we say it's a stack. For that matter, in the case where the functor is representable by a scheme, the scheme is *also* the same thing as a functor; but in that case we don't think of it as or refer to it as a functor, because that would make us seem very pretentious.


