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I've seen stated offhand in many sources that the cuspidal subgroup of the Jacobian of $X_0(N)$ is finite.

Do they mean that the subgroup of the jacobian generated by $\mathbb{Q}$-rational cusps is finite, or do they mean the subgroup generated by all cusps is finite?

I know that Ogg (and also Mazur) does the case $N = p$ prime in a rather explicit manner (in this case all cusps are rational), but says that the case for $N$ composite is more complicated and doesn't state any results to that end.

Ideally, it would be nice to see a moduli-theoretic explanation of this, if it exists.

thanks,

  • will
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  • $\begingroup$ I have put the number theory tag as this question is of interest to number theorists as well. $\endgroup$
    – Arijit
    Commented Apr 23, 2013 at 10:01
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    $\begingroup$ The statement is one about all cusps (not necessarily $\mathbb{Q}$-rational ones). The original paper by Drinfeld, "Two theorems on modular curves", is very readable. $\endgroup$
    – David Loeffler
    Commented Apr 24, 2013 at 7:54
  • $\begingroup$ @David Loeffler - I've managed to find that paper, though I'm having trouble understanding his "proof of assertion 1". Given any $g\in M$ and $\gamma\in\text{PSL}_2(\mathbb{Z})$, how does he conclude that $g\gamma(i\infty) = \gamma h(i\infty)$ for some $h\in\Gamma(N)$? $\endgroup$
    – Will Chen
    Commented Apr 24, 2013 at 11:16
  • $\begingroup$ @David Loeffler - Also, I don't understand how he deduces that $\omega|_g = \omega$ in his proof of assertion 2. Any chance you could help me with that? $\endgroup$
    – Will Chen
    Commented Apr 24, 2013 at 11:38
  • $\begingroup$ See also R. Elkik, "Le théorème de Manin-Drinfeld." Astérisque, pages 59-67, 1990. Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). The proof given there follows an idea of Deligne and uses mixed Hodge structures. $\endgroup$
    – Damian Rössler
    Commented May 2, 2013 at 10:30

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This is the Manin-Drinfeld theorem.See related MO question

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