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

TL;DR

This paper proves the equality of two rational cuspidal subgroups of modular Jacobians for specific levels and characterizes modular units on $X_0(N)$ using generalized Dedekind eta functions.

Contribution

It establishes the equality of the rational cuspidal subgroup and the divisor class group for levels of the form $p^2M$, and characterizes modular units as products of specific eta-based functions.

Findings

$ ext{C}_N( ext{Q}) = ext{C}(N)$ for $N=p^2M$ with prime $p$ and squarefree $M$

All modular units on $X_0(N)$ can be expressed as products of functions $F_{m,h}$

Provides necessary and sufficient conditions for such products to be modular units

Abstract

For a positive integer $N$, let $\mathscr{C}_N(\mathbb{Q})$ be the rational cuspidal subgroup of $J_0(N)$ and $\mathscr{C}(N)$ 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 $\mathscr{C}_N(\mathbb{Q})$ and $\mathscr{C}(N)$ are equal when $N=p^2M$ 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.