Question

Let $G(N)$ be the congruence subgroup

$\big\{ \begin{pmatrix} a&b \\ c&d \end{pmatrix} \in SL_2(\mathbb{Z}) \ \ | \ \ a \equiv d \mod N \textrm{ and } b \equiv c\equiv 0 \mod N \big\}$.

$G(N) \backslash \mathbb{H}$ seems to parametrize elliptic curves with a cyclic subgroup and some extra data (I think its automorphism group is $PSL_2(\mathbb{Z} / N \mathbb{Z})$, intuitively it is moving cyclic subgroups but fixing their Weil pairings).

Has this quotient been considered already somewhere in the literature? Ideally I would like to know exactly what it is the moduli space for, and how it relates to $\Gamma_0(N) \backslash \mathbb{H}$.

Answer 1

Here's a very clunky answer:

For a particular elliptic curve $E$, the collection of bases $\{P,Q\}$ of the $N$-torsion $E[N]$ is acted upon (diagonally) by the group $(\mathbb{Z}/N\mathbb{Z})^\times$. This group has a subgroup $G$ consisting of elements $x$ with a lifting to an element $y\in (\mathbb{Z}/N^2\mathbb{Z})^\times$ such that $y^2\equiv 1\pmod{N^2}$. (Of course, the group $G$ has a better description since this lifting is automatic at odd primes, etc. - but I'll stick with this one because its relation to the problem is more transparent).

I think that the moduli problem classifies isomorphism classes of pairs $(E,O)$ where $O$ is a $G$-orbit of bases of the full $N$-torsion.

Answer 2

A disjoint union of copies of this space parameterizes elliptic curves with a choice of two disjoint (except for the identity) $N$-element cyclic subgroups and an isomorphism between them.

Proof: First we show that any two disjoint, except for the identity, subgroups must form a basis for the level $N$ structure. This is clear because, if $\alpha$ is a generator of the first subgroup, and $\beta$ a generator of the second, then any relation between them must take the form $k\alpha+l \beta=0$, so $k \alpha = -l \beta$, so the subgroups must have nontrivial intersection.

Clearly, an element in $SL_2(\mathbb Z/N)$ fixes such a choice if and only if it is scalar modulo $n$. Thus the moduli space of structures of this type is the moduli space of elliptic curves with full level $N$ modulo the scalars in $SL_2(\mathbb Z/N)$. The moduli space of elliptic curves with full level $N$ is a disjoint union of some copies of the modular curve $X(N)$, so the moduli space of elliptic curves with that structure is $X(N)$ modulo the scalars in $SL_2(\mathbb Z/N)$.