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

Abstract

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

to appear on the Journal of Algebra

References in corpus (1)

• Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus