# Generality of the Manin-Drinfel'd theorem

The Manin-Drinfeld theorem asserts that for a modular curve $X_0(N)$ and Jacobian $J_0(N)$ with the former being embdedded in the latter under the map that takes $i\infty$ to $0$, the cusps are torsion.

The proof Manin-Drinfeld theorem seems to use Hecke operators and thus seems to be valid only for congruence subgroups of $PSL_2(\mathbb Z)$. Is there possibly a finite-index subgroup of $PSL_2(\mathbb Z)$ such that the statement of Manin-Drinfeld theorem is still true for the quotient?

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You might have better luck with this question at mathoverflow.net – Noah Snyder Nov 24 '10 at 20:38

I like your question! I wish I could give a full answer, but I'm not familiar enough with the Manin-Drinfeld theorem. After I finish a couple weeks of work, I hope I can return and think about it in detail.

I would guess the answer is yes, and I have some candidates you could try. The Fermat curves can be identified with compactified quotients $\Gamma\backslash\mathcal{H}$ for finite index subgroups $\Gamma$ in the following way:

The modular function $\lambda(z) = \frac{\theta_2^4(q)}{\theta_3^4(q)}=16 q^{1/2} -128q +704q^{3/2} + \ldots$ is a Hauptmodul of the genus 0 congruence subgroup $\Gamma(2)$. (I.e., it is invariant under the action of $\Gamma(2)$ by fractional linear transformation, and parametrizes the genus 0 modular surface $X_{\Gamma(2)} = \Gamma(2) \backslash \mathcal{H} \cup \{0,1,i_\infty\}$.)

We define Hauptmoduln $x:= \sqrt[n]{\lambda}$ and $y:= \sqrt[n]{1-\lambda}$, which determine finite index genus 0 subgroups $\Gamma_x$ and $\Gamma_y$. Then $\Gamma:= \Gamma_x \cap \Gamma_y$ is subgroup with compactified quotient $X_{\Gamma}: x^n + y^n = 1$, and $\Gamma$ is a noncongruence subgroup for $n \neq 1,2,4,8$.

This construction is from a survey paper by Ling Long. We can use essentially the same idea to obtain quotient curves of the Fermat curves: If we set $x:= -\sqrt[5]{\lambda}$ and $y:= \sqrt{1-\lambda}$, we obtain a noncongruence subgroup $\Gamma$ with model $y^2=x^5+1$, which is a nice genus 2 curve, and its Jacobian has CM.

If you can verify what happens in this case, I would be interested to hear about it.

They give the same construction of the Fermat curves as I gave above, attributing it to Fricke and Klein. They also cite a result of Rohrlich which establishes that the cusps of $\Gamma$ map to torsion points of the Jacobian.
So the answer is yes; the noncongruence groups $\Gamma$ corresponding to the Fermat curves also satisfy the statement of the Manin-Drinfeld theorem.