The rational torsion subgroup of J0(N)

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

“…We also expect the following, which is [23, Conjecture 1.3]. Conjecture For any positive integer N , we have $$$$\begin{equation*} {\mathcal {C}}_N({\mathbf {Q}})={\mathcal {C}}(N).$$…”

“…In this paper, we are primarily concerned with Conjecture 1.3. (For Conjecture1.2, see [23]. ) To the best knowledge of the authors, Conjecture1.3 is only known when N is one of the following cases: (1) N is small enough. (2)$$N=2^r M$$ with $$0\le r \le 3$$ and M odd squarefree. (3)$$N=n^2 M$$ with $$n \hspace*{1.42262pt}|\hspace*{1.42262pt}24$$ and M squarefree. The second case is obvious as all the cusps of $$X_0(N)$$ are defined over Q , and so $${\mathcal {C}}(N)={\mathcal {C}}_N({\mathbf {Q}})={\mathcal {C}}_N$$.…”