The rational cuspidal subgroup of $J_0(p^{2}M)$ with $M$ squarefree

Abstract

For a positive integer $N$, let ${\mathcal{C}}_N(\mathbb{Q})$ be the rational cuspidal subgroup of $J_0(N)$ and $\mathcal{C}(N)$ be the rational cuspidal divisor class group of $X_0(N)$, which are both subgroups of the rational torsion subgroup of $J_0(N)$. We prove that two groups ${\mathcal{C}}_N(\mathbb{Q})$ and $\mathcal{C}(N)$ are equal when $N=p^{2}M$ for any prime $p$ and any squarefree integer $M$. To achieve this we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m,h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_0(N)$ under a mild assumption.

to appear in Mathematische Nachrichten