Modular units and cuspidal divisor classes on $X_0(n^2M)$ with $n|24$ and $M$ squarefree

TL;DR

This paper investigates the structure of cuspidal divisor classes on modular curves $X_0(N)$ for specific $N$, proving the equality of certain rational cuspidal subgroup types and characterizing modular units in terms of eta functions.

Contribution

It establishes the equality of two cuspidal divisor subgroups for $N=n^2M$ with $n|24$ and $M$ squarefree, and characterizes modular units as products of eta functions with specific parameters.

Findings

Proves $ ext{C}(N)( ext{Q}) = ext{C}_ ext{Q}(N)$ for specified $N$.

Characterizes modular units on $X_0(N)$ as products of eta functions.

Determines conditions for such products to be modular units.

Abstract

For a positive integer $N$, let $\mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $\mathscr C(N)(\mathbb Q):=\mathscr C(N)\cap J_0(N)(\mathbb Q)$ be its $\mathbb Q$-rational subgroup. Let also $\mathscr C_{\mathbb Q}(N)$ be the subgroup of $\mathscr C(N)(\mathbb Q)$ generated by $\mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $\mathscr C(N)(\mathbb Q)$ and $\mathscr C_{\mathbb Q}(N)$ are equal. To achieve this, we show that all modular units on $X_0(N)$ on such $N$ are products of functions of the form $\eta(m\tau+k/h)$, $mh^2|N$ and $k\in\mathbb Z$ and determine the necessary and sufficient conditions for products of such functions to be modular units on $X_0(N)$.