Cuspidal Class Number Formulae for Intermediate Modular Curves of Prime Level

Abstract

Let $X$ be a modular curve lying between $X_{0}(p)$ and $X_{1}(p)$, where $p \ge 13$ is prime. The cuspidal group of $X$ is the subgroup of its Jacobian generated by the equivalence classes of differences of cusps. In this paper, we give explicit formulae for the order of the cuspidal group, and two subgroups of it defined over the rationals.