The rational cuspidal subgroup of J_0(N)

Hwajong Yoo; Myungjun Yu

paper

The rational cuspidal subgroup of J_0(N)

arXiv:2504.12564

Abstract

For a positive integer $N$ , let $J_{0} (N)$ be the Jacobian of the modular curve $X_{0} (N)$ . In this paper we completely determine the structure of the rational cuspidal subgroup of $J_{0} (N)$ when the largest perfect square dividing $N$ is either an odd prime power or a product of two odd prime powers. Indeed, we prove that the rational cuspidal divisor class group of $X_{0} (N)$ is the whole rational cuspidal subgroup of $J_{0} (N)$ for such an $N$ , and the structure of the former group is already determined by the first author in [14].