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 $\le M$.



examples of "exotic" moduli problems for elliptic curves?
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.


