Computing the Cuspidal Subgroup of the Modular Jacobian $J_H\left(p\right)$

Abstract

For a fixed prime $p$ congruent to $1$ modulo $4$ we may define the modular curve $X_H\left(p\right)$ associated to the subgroup of non-zero squares modulo $p$. This curve has four cusps and we consider the subgroup of the Jacobian $J_H\left(p\right)$ of $X_H\left(p\right)$ generated by these points, which we will call the cuspidal subgroup of $J_H\left(p\right)$. This is a finite subgroup by the results of Manin and Drinfeld, and lies inside the $\mathbb{Q}\left(\sqrt{p}\right)$-rational torsion subgroup. In this paper we compute the cuspidal subgroup for all such curves of genus $g$, $2\le g\le 10$, namely those with $p\in \{29,37,41,53,61,73\}$, and compare this with $J_H{\left(\mathbb{Q}\left(\sqrt{p}\right)\right)}_{\text{tors}}$.

References in corpus (1)

• Another look at rational torsion of modular Jacobians