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 $\Gamma\_0(N) \backslash \mathbb{H}$.



What does this quotient of the upper half plane parametrize?
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.


