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