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I would like to study the "modular curve" $Y(M,N)$, parametrizing an elliptic curve $E$ together with $p \in E[M]$ and $q \in E[N]$ (here and in the following $M$ divides $N$).

Let $\Gamma(M,N)$ be the subgroup of $SL_2(\mathbb{Z})$ of matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, where $a-1=b=0 \mod M$, $c=d-1=0 \mod N$. Observe that $\Gamma(M,N)$ contains $\Gamma(N)$, the principal congruence subgroup of level $N$.

Now define $Y(M,N)$ as the quotient of the upper half plane $\mathbb{H}$ by the action of $\Gamma(M,N)$.

Questions:

1) Does $Y(M,N)$ parametrize what I wrote in the first paragraph?

2) Is $Y(M,N)$ well-known and studied as $Y(N)$ is? Are there references?

3) For example, what are the irreducible components of $Y(M,N)$? Are they identified, via the restriction of the Weil pairing of $E[N] \times E[N]$, to the elements of $\mu_M$? EDIT: (See answer by David Loeffler) As observed below, this is clearly irreducible. The question is if this truly parametrizes triples $(E,p,q)$ as in the first paragraph, with fixed value of the Weil pairing.

4) Is there any hope of saying what is its genus? Is everything easily deducible from the known properties of the usual congruence subgroups and modular curves?

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1 Answer 1

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There seems to be a typo in your definition of the subgroup $\Gamma(M, N)$: you have $b = 0 \bmod N$ and also $b = 1 \bmod M$ which is clearly impossible if $M \mid N$. I suspect you want $c = 0 \bmod N$, $a = d = 1 \bmod N$ and $b = 0 \bmod M$. In that case, this curve is perhaps not as familiar as $\Gamma(N)$ but it is still very well known: see e.g. sections 1 and 2 of Kato's article "p-adic Hodge theory and values of zeta functions of modular forms" in Asterisque 295.

For the moduli space interpretation, you need to distinguish between the quotient of the upper half-plane (which is obviously irreducible and connected, but is naturally defined over $Q(\mu_M)$ rather than $Q$); and the curve you get by taking $\phi(M)$ copies of the former permuted by the Galois action, which is defined over $Q$. The latter (which is what Kato calls $Y(M, N)$) parametrizes elliptic curves with a point of $P$ of order M and a point $Q$ of order N which are linearly independent in the natural way (i.e. they generate a subgroup of order MN). The former parametrizes pairs where the Weil pairing of $P$ and $(N/M)Q$ is the specific root of unity $e^{2\pi i / M}$. This is not very different from the well-known case $M = N$.

Giving a general formula for the genus of the curve might be a bit messy, but note that conjugating $\Gamma(M, N)$ by $\begin{pmatrix} 1 & 0 \\ 0 & M \end{pmatrix}$ gives you the subgroup $\Gamma_1(N) \cap \Gamma_0(MN)$, which is intermediate between $\Gamma_0(MN)$ and $\Gamma_1(MN)$ and is thus one of the "$\Gamma_H$" groups, which there is a fair bit of literature on (e.g. there is a paper on dimension formulae for these by Jordi Quer; Quer's formulae are implemented in Sage).

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Thank you very much for your answer! I have troubles in finding Kato's paper for the moment. I am a bit puzzled by the notation. From what you say, what Kato calls $Y(M,N)$ is made of $\phi(M)$ irreducible components, whereas $Y(N)$ is normally irreducible, and the moduli space parametrizing $N$-torsion points of the elliptic curve is the one made of $\phi(N)$ components. –  OldMacdonaldHadaForm Jul 6 '12 at 8:03
    
I checked in the paper of Kato. It seems to me that $Y(M,N)$ is the connected one. Moreover there is still a typo in your conditions on $a,b,c,d$. I am fixing the conditions in my question. Thanks for your answer! –  OldMacdonaldHadaForm Jul 6 '12 at 8:34
    
@OldMacdonaldHadaForm: In your corrected version, you have switched the roles of $M$ and $N$. But what I wrote was correct in the case $M \mid N$, since the congruence on the bottom row and the fact that we're dealing with matrices of determinant 1 force $a - 1$ to be divisible by $N$. –  David Loeffler Jul 6 '12 at 9:03
    
Also, Kato's $Y(M, N)$ is definitely not connected unless $D = 1$, where $D := GCD(M, N)$. Look at the functor it represents: it's clear that the Weil pairing gives a surjection of $\mathbb{Q}$-schemes $Y(M, N) \to \mu_D$. –  David Loeffler Jul 6 '12 at 9:07
    
One last comment: Kato defines $Y(N)$ in such a way that $Y(N)(\mathbb{C})$ is isomorphic to $(\mathbb{Z} /N)^\times \times (\Gamma(N) \backslash \mathcal{H})$; see section 1.8 of his paper. So his $Y(N)$ and $Y(M, N)$ are definitely not geometrically connected. There is, as far as I know, no "nice" way to make $\Gamma(N) \backslash \mathcal{H}$ into an algebraic variety over $\mathbb{Q}$ that is naturally a moduli space for anything interesting. –  David Loeffler Jul 6 '12 at 9:15

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