Modular interpretation of modular curves

Let $\Gamma$ be a congruence subgroup of level $N$. What is the modular interpretation of $\Gamma\backslash \mathcal{H}^*$, by that I mean what are the elliptic curves + additional structure it parametrizes.

I know the answer when $\Gamma$ is one of the classical groups $\Gamma(N),\Gamma_0(N)$ or $\Gamma_1(N)$ and I'm looking for a more general picture.

Edit : So I guess after thinking about it I have a little a bit of an idea of how this would work but i'm really not sure. Here it goes :

We know that $\Gamma \backslash \mathcal{H} = \overline{\Gamma} \backslash X(N)$ where $N$ is the level of $\Gamma$ and $\overline{\Gamma}$ is the subgroup of $SL_2(\mathbb{Z}/N\mathbb{Z})$.

Furthermore the modular interpretation for $X(N)$ is that it parametrises pairs $(E,(P,Q))$ where

• $E$ is an elliptic curve over $\mathbb{C}$ and $(P,Q)$ is a base of the $N$-torsion of $E$ such that the weil pairing $e(P,Q)$ is equal to $\zeta_n$
• $(E,(P,Q))$ and $(E',(P',Q'))$ are equivalent if $E$ and $E'$ are isomorphic and that the isomorphism between the two sends $(P,Q)$ to $(P',Q')$

So the question can be reformulated as (modulo everything I said before is correct) how does $SL_2(\mathbb{Z}/N\mathbb{Z})$ act on $(E,(P,Q))$. The obvious thing would be that for $\gamma = \left (\begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right ) \in SL_2(\mathbb{Z}/N\mathbb{Z}) $ we have $\gamma (E,(P,Q)) = (E,(aP + bQ, cP + dQ))$

Then the modular interpretation of $\Gamma \backslash \mathcal{H}$ would be pairs $(E,(P,Q)$ as before which are equivalent if the isomorphism $E \to E'$ sends $(P,Q)$ to $(P'',Q'')$ where $(P'',Q'')$ can be obtained from $(P',Q')$ by a transformation in $\overline{\Gamma}$

As I said I'm not really sure if that is the correct way to think about things especially since I haven't really seen things explained this way before which probably mean it's false (or it was obvious). So I'm looking for someone to tell me if this interpretation is correct and how is it usually done ?

Edit 2 Obviously a good sanity check would be to see if I'm able this way to get the moduli interpretation of $X_0(N)$ and $X_1(N)$ back but so far I haven't be able to do so (but I haven't tried very hard).

Solution 1:

To a complex number $\tau \in \Bbb H$, associate the lattice $\Lambda_\tau = \langle \tau, 1 \rangle$, the elliptic curve $E_\tau = \Bbb C/\Lambda_\tau$, and the two $N$-torsion points $P_N(\tau) = \frac 1N \tau \pmod{\Lambda_\tau}$ and $Q_N(\tau) = \frac 1N \pmod {\Lambda_\tau}$.

For $\gamma = \begin{pmatrix}a&b\\c&d\end{pmatrix} \in SL_2(\Bbb Z)$ and $\tau \in \Bbb H$, we have $\gamma.\tau = \frac {a\tau+b}{c\tau+d}$, and an isomorphism $ E_\tau \to E_{\gamma.\tau}$ induced by $z \mapsto z/(c\tau+d)$, since $\Lambda_\tau = \langle \tau ; 1 \rangle = \langle a\tau+b ; c\tau+d \rangle$, which is homothetic to $\langle \gamma.\tau,1\rangle$.

Now, let $F(\tau) \in X(N)$ be the isomorphism class of the triplet $(E_\tau,P_N(\tau),Q_N(\tau))$. $F(\gamma.\tau) = (E_{\gamma.\tau},P_N(\gamma.\tau),Q_N(\gamma.\tau)) = (E_\tau,\phi_{\gamma,\tau}^{-1}(P_N(\gamma.\tau)),\phi_{\gamma,\tau}^{-1}(Q_N(\gamma.\tau))) = (E_\tau,\frac aN\tau+ \frac bN,\frac cN\tau+ \frac dN) = (E_\tau, aP_N+bQ_N,cP_N+dQ_N)$.

Now it is pretty clear that for most $\tau$, $F(\gamma.\tau) = F(\tau)$ if and only if $\gamma = \pm I_2 \pmod N$, and that the $(P_N(\gamma.\tau),Q_N(\gamma.\tau))$ are the $\Bbb Z/N \Bbb Z$-bases of $E_\tau[N]$ with some specific Weil pairing (the specific value is unimportant).

At this point we have exactly your action of $SL_2(\Bbb Z/N \Bbb Z)$ on $E(P,Q)$ that you described. However, $-I_2$ acts trivially on $\Bbb H$ and not on the $\Bbb Z/N \Bbb Z$-bases of $E_\tau$. Instead of looking at $(P,Q)$ we may look instead at the orbit $\{(P,Q),(-P,-Q)\}$, on which we have a natural action by $SL_2(\Bbb Z/N \Bbb Z)/\{\pm I_2\}$. Then you can start making pictures describing $X(N)$ using those objects.

The above is functorial in $N$ : we have a multiplication-by-$M$ maps : $\mu_M : E_\tau[MN] \to E_\tau[N]$, which induces maps $\mu_M : (E_\tau,P,Q) \in X(MN) \mapsto (E_\tau,\mu_M(P),\mu_M(Q)) \in X(N)$. As for the groups, we have the reduction modulo $N$ maps $SL_2(\Bbb Z/MN \Bbb Z) \to SL_2(\Bbb Z/N \Bbb Z)$, and $(\gamma \pmod N).(\mu_M(E,P,Q)) = \mu_M(\gamma.(E,P,Q))$.

In order to recover other moduli problems, you have to take a suitable quotient :

If you want to classify the isomorphism classes of $(E,P)$ where $P$ is of order exactly $N$, then we simply have to forget about $Q$. Let $G_1$ be the subgroup of $SL_2(\Bbb Z/N \Bbb Z)$ fixing $P$ ($a=1,b=0$), then there is a natural correspondance between those isomorphism classes and the isomorphism classes of the $G_1$-orbits of the $(E,P)$. From there you obtain that with $\Gamma_1(N) = \{\begin{pmatrix}a&b\\c&d\end{pmatrix}, a=1 \pmod N,b=0 \pmod N\}$, $\Gamma_1(N) \backslash \Bbb H$ classifies those isomorphism classes.

As for the isomorphism classes of cyclic subgroups of order $N$, you need to blur the difference between $P$ and $aP$, so you want to pick $G_0$ to be the subgroup of $SL_2(\Bbb Z/N \Bbb Z)$ such that $\gamma P$ is a multiple of $P$, and you obtain the condition $b=0$, from which you recover $\Gamma_0$(N).