On component groups of $J_0(N)$ and degeneracy maps

For an integer $M>1$ and a prime $p \geq 5$ not dividing $M$, we study the kernel of the degeneracy map $\Phi _{Mp,p}^r \rightarrow \Phi _{Mp^r, p}$, where $\Phi _{Mp,p}$ and $\Phi _{Mp^r, p}$ are the component groups of $J_0(Mp)$ and $J_0(Mp^r)$, respectively. This is then used to determine the kernel of the degeneracy map $J_0(Mp)^2 \rightarrow J_0(Mp^2)$ when $J_0(M) =0$. We also compute the group structure of $\Phi _{Mp^2, p}$ in some cases.