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:
-
$P_N : \textbf{Ell}\rightarrow\textbf{Sets}$, where $\textbf{Ell}$ is the category of elliptic curves $E\rightarrow S$ over $S$ and morphisms are cartesian squares, and
$P_N(E/S) = \text{set of isomorphisms } \alpha : E[N]\rightarrow (\mathbb{Z}/N\mathbb{Z})^2 \text{ of determinant 1}$
-
$F_N : \textbf{Sch}\rightarrow\textbf{Sets}$, 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
Best Answer
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.