Computing the Cuspidal Subgroup of the Modular Jacobian $J_{H}\left( p \right)$

TL;DR

This paper computes the cuspidal subgroup of the Jacobian for certain modular curves associated with primes congruent to 1 mod 4, for genus 2 to 10, and compares it with the rational torsion subgroup over quadratic fields.

Contribution

It provides explicit computations of the cuspidal subgroup for all such curves with genus between 2 and 10, extending previous theoretical results.

Findings

Cuspidal subgroup computed for p in {29, 37, 41, 53, 61, 73}

Comparison with the torsion subgroup over quadratic fields

Results support conjectures on torsion structures in modular Jacobians

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 $g$, $2 \leq g \leq 10$, namely those with $p \in \{ 29, 37, 41, 53, 61, 73 \}$, and compare this with $J_{H}\left( \mathbb{Q} \left( \sqrt{p} \right) \right)_{\text{tors}}$.