The $\mathbb{Q}$-rational cuspidal group of $J_{1}(2p)$

Abstract

Let $p$ be a prime not equal to 2 or 3. In this paper we study the $\mathbb{Q}$-rational cuspidal group $\mathcal{C}_{\mathbb{Q}}$ of the jacobian $J_{1}(2p)$ of the modular curve $X_{1}(2p)$. We prove that the group $\mathcal{C}_{\mathbb{Q}}$ is generated by the $\mathbb{Q}$-rational cusps. We determine the order of $\mathcal{C}_{\mathbb{Q}}$, and give numerical tables for all $p\leq127$. These tables give also other cuspidal class numbers for the modular curves $X_{1}(2p)$ and $X_{1}(p)$. We give a basis of the group of the principal divisors supported on the $\mathbb{Q}$-rational cusps, and using this we determine the explicit structure of $\mathcal{C}_{\mathbb{Q}}$ for all $p\leq127$. We determine the structure of the Sylow $p$-subgroup of $\mathcal{C}_{\mathbb{Q}}$, and the explicit structure for all $p\leq4001$.