Modular curve parametrizing two cyclic subgroups of an elliptic curve

Question

The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing an elliptic curve $E$, together with two cyclic subgroups $G$ and $H$, of order respectively $M$ and $N$, of the group of degree $0$ line bundles modulo linear equivalence on $E$.

(note that the subgroup generated by $G$ and $H$ need not be of rank $2$, although it will be so generically)

When $N=1$, $Y_0(M,N)$ is just the well known modular curve $Y_0(M)$.

Questions:

1) Are there references for this moduli space/modular curve? Is this well studied/well known?

2) I think of this as the fiber product $Y_0(M) \times_{\mathcal{M}_{1,1}} Y_0(N)$ (the fiber product over the moduli space of elliptic curves). Is this correct?

3) What can we say about the curve $Y_0(M,N)$? Is it irreducible? Are there some kind of formulas for its genus as there are for $Y_0(M)$?

Answer 1

$Y_0(M,N)$ can be reinterpreted as the moduli space of diagrams $E_1 \to E \leftarrow E_2$ of elliptic curves, where the arrows are cyclic isogenies of degree $M$ and $N$. From this viewpoint, it is naturally a fiber product of $Y_0(N)$ with $Y_0(M)$, as you suggested. It is in general a disjoint union of modular curves, because the kernels of dual isogenies may have isomorphism types and restrictions of the Weil pairing that are not unique.

In good characteristic, you may enumerate components by finding all abelian groups generated by a cyclic group of order $M$ and a cyclic group of order $N$, and finding all equivalence classes of $\mu_{(M,N)}$-valued symplectic forms on these groups. In characteristic $p$, you have to work out cases, depending on how many powers of $p$ divide $M$ and $N$. For example, if $M=N=p$, then $Y_0(p)$ is two copies of the affine line, glued at supersingular points, and the fiber product is four copies of the affine line, also glued at those points.

Most of the information you want can be extracted from Katz and Mazur's Arithmetic Moduli of Elliptic Curves, although I don't know where a table of genera of components can be found. You may have to explicitly enumerate elliptic points and cusp orders, then use Hurwitz.