# what exactly is the moduli functor for classifying elliptic curves with (full) level N structure?

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
• There's to say here, but let's start with a couple questions. Something's clearly wrong with your $P_N$ - is $\mathbb{Z}/N\mathbb{Z}$ supposed to be $(\mathbb{Z}/N\mathbb{Z})^2$ or something? Are you ultimately working over $\mathbb{C}$-schemes here? This is the only way $\Gamma(N)\backslash\scr{H}$ will be a "moduli scheme" after all. Aug 14, 2012 at 18:47
• Wow, that $H$ came out fancier than I intended. Let's try $\Gamma(N)\backslash \mathcal{H}$. Aug 14, 2012 at 18:48
• On second thought, I think I misspoke. I didn't quite understand $P_N$ correctly (in particular I missed the bit about the morphisms in $\mathbf{Ell}$ being Cartesian squares). It seems to me that $P_N$ should be representable iff $F_N$ is. You've stated one direction. For the other: suppose that $F_N$ is representable by the $\mathbb{C}$-scheme $X$. The identity map $X\to X$ gives rise to a universal elliptic curve $E\to X$ which ought to represent $P_N$. Aug 14, 2012 at 20:13
• cont'd: The K-M proof that representability implies rigidity uses constructions specific to elliptic curves. For example, in char. 0 choose $(E,\alpha) \in F_2(K)$ and let $E'$ be the twist of $E$ by a quadratic extension $L/K$. Quadratic twisting is invisible on 2-torsion (!), so $E'$ has an associated full level-2 structure $\alpha'$. Now $(E,\alpha), (E',\alpha')$ are distinct in $F_2(K)$ (check!) but the same in $F_2(L)$, so $F_2(K) \rightarrow F_2(L)$ is not injective. But $X(k) \rightarrow X(k')$ is injective for any scheme $X$ and field extension $k'/k$, so $F_2$ isn't representable. Aug 15, 2012 at 8:05
• @oxeimon: One should not think about elliptic curves in terms of those affine coordinate rings; best to always think in terms of the proper curve. Also, when you say that one "usually" views elliptic curves in terms of $\overline{K}$-points...once you learn about schemes, you can and should think entirely in terms of $K$-schemes. It is much clearer. Mumford's book on abelian varieties, Chapters 2 and 3, gives an excellent presentation of how to use scheme-theoretic methods in both a classical "variety" setting as well as a scheme-theoretic setting (to deal with inseparable isogenies, etc.). Aug 16, 2012 at 1:04

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.)