# examples of “exotic” moduli problems for elliptic curves?

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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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? – user28172 May 7 '13 at 19:52

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
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? – oxeimon May 8 '13 at 0:25
Also $\text{Spec}(\mathbb{Z}[\nVdash/\ltimes])$ in the statement of theorem 3.1.1 on the first page. – oxeimon May 8 '13 at 0:27
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. – S. Carnahan May 8 '13 at 1:27