Let N be a prime integer. We know that the element $c=(0)-(\infty)$ generates the torsion subgroup of $J\_0(N)$ and it has order Num( (N-1)/12). Now, there is a natural map $\pi^\*:J\_0(N) \rightarrow J\_1(N)$, coming from the covering map $\pi:X\_1(N) \rightarrow X\_0(N)$. My question is what is the image of c under this map? Specifically, is it possible for $\pi^\*(c)=0$?



Image of the cuspidal subgroup of J_0(N) in J_1(N)
The fact you mention about $(0) - (\infty)$ generating the torsion subgroup of $J\_0(N)$ is Theorem 1 (Ogg's conjecture) on the first page of Mazur's paper "The Eisenstein Ideal". I recommend you actually read this paper. If you get as far as page 2, you will find a "Theorem 2 (twisted Ogg's conjecture)" which concerns the Shimura subgroup $\Sigma$. The construction of this subgroup *essentially* identifies it with the kernel of $J\_0(N) \rightarrow J\_1(N)$, and Proposition 11.6 of *ibid*. shows that $\Sigma$ is of multiplicative type, so BCnrd's remarks apply to the general case.


