Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus

TL;DR

This paper investigates the rational torsion subgroup of the generalized Jacobian of a modular curve with cuspidal modulus, revealing it to be significantly smaller than that of the classical Jacobian when N is a prime power ≥ 5.

Contribution

It provides new insights into the structure of rational torsion points on the generalized Jacobian of modular curves with cuspidal divisors, especially for prime power levels.

Findings

Rational torsion points are smaller in the generalized Jacobian than in the classical Jacobian for N as a prime power ≥ 5.

The group of rational torsion points tends to be trivial or very limited in these cases.

The results highlight differences between generalized and classical Jacobians in the context of modular curves.

Abstract

We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much smaller than the classical Jacobian.