Cuspidal $\mathbb{Q}$-Rational Torsion Subgroup of $J(\Gamma)$ of Level $P$

Abstract

For $p\!\gt \!3$ an odd prime, let $\Gamma$ be a congruence subgroup between $\Gamma_1(p)$ and $\Gamma_0(p)$. In this article, we give an explicit basis for the group of modular units on $X(\Gamma)$ that have divisors defined over $\mathbb{Q}$. As an application, we determine the order of the cuspidal $\mathbb{Q}$-rational torsion subgroup of $J(\Gamma)$ generated by the divisor classes of cuspidal divisors of degree $0$ defined over $\mathbb{Q}$.