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Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with those two maps as columns.

A moduli problem for elliptic curves is then just a contravariant functor $\textbf{Ell}\rightarrow\textbf{Sets}$.

For example, we usual level $N$ moduli problem is the functor sending $E/S$ to the set of isomorphisms $E[N]\stackrel{\sim}{\longrightarrow}(\mathbb{Z}/N\mathbb{Z})_S^2$.

There are a ton of these functors, mostly coming from various cohomology theories, but the only such functors I can think of that land in the category of finite sets all have to do with torsion points on the elliptic curve.

Does anyone have any examples of a contravariant functor $F:\textbf{Ell}\rightarrow\textbf{Sets}$ such that for $E/S$ with $S$ connected, $F(E/S)$ is finite, and doesn't have to do with torsion data?

Ideally, the functor will actually land in the category of groups, be generically of some order $M\ge 3$, and always be of order $\le M$.

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    $\begingroup$ Aside from the fact that the phrase "doesn't have to do with torsion data" is vague, if you consider a $\mathbf{Z}[1/N]$-schemes $S$ that is sufficiently disconnected then the set of level-$N$ structures on a fixed elliptic curve $E$ over $S$ can be arbitrarily large and in particular not finite (akin to global sections of a constant sheaf on a disconnected space). So do you mean just that for elliptic curves over algebraically closed fields the associated set should be finite? If so, then how about assigning to any $E \rightarrow S$ the automorphism group? $\endgroup$
    – user28172
    Commented May 7, 2013 at 19:52

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Sure -- try the set of homomorphisms from $\pi_1^{\mathrm{et}}(E - O)$ to a fixed finite group G. This is a "non-abelian level structure" of the sort considered by de Jong and Pikaart

http://arxiv.org/abs/alg-geom/9501003

Rachel Davis, a 2013 Wisconsin Ph.D. working with Nigel Boston, wrote her thesis about this kind of stuff in the case of elliptic curves.

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    $\begingroup$ Hmm, so in that arxiv paper, they're using some symbols I'm not familiar with. In particular, in the first paragraph, what does $(\mathbb{Z}/\ltimes\mathbb{Z})^{\nvDash\partial}$, and $\text{Spec}(\mathbb{Z}[\nVdash/\#\mathbb{G}])$ mean? $\endgroup$
    – Will Chen
    Commented May 8, 2013 at 0:25
  • $\begingroup$ Also $\text{Spec}(\mathbb{Z}[\nVdash/\ltimes])$ in the statement of theorem 3.1.1 on the first page. $\endgroup$
    – Will Chen
    Commented May 8, 2013 at 0:27
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    $\begingroup$ Either that's a bizarre TeX error, or else I have a new entry for the question mathoverflow.net/questions/18593/… $\endgroup$
    – Marty
    Commented May 8, 2013 at 0:50
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    $\begingroup$ If you look at the source, you'll see that these are $(\mathbb{Z}/n\mathbb{Z})^{2g}$, Spec $\mathbb{Z}[1/\# G]$, and Spec $\mathbb{Z}[1/n]$, respectively. $\endgroup$
    – S. Carnahan ♦
    Commented May 8, 2013 at 1:27
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    $\begingroup$ Yes, in some sense it explains them all! This is a result of myself and McReynolds arxiv.org/abs/0909.1851 very much inspired by the old paper of Diaz, Donagi, and Harbater, "Every curve is a Hurwitz curve." $\endgroup$
    – JSE
    Commented May 9, 2013 at 1:55

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