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



reference request for the finiteness of cuspidal subgroup of $X_0(N)$?
This is the Manin-Drinfeld theorem.See [related MO question](https://mathoverflow.net/questions/124502/does-the-manin-drinfeld-theorem-hold-over-number-fields) 


