Rational Torsion on the Generalized Jacobian of a Modular Curve With Cuspidal Modulus

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 $≥ 5$, we exhibit that the
group of rational torsion points $\widetilde{J}_{0}(N)(\{Q})_{Tor}$
tends to be much smaller than the classical Jacobian.

2010 Mathematics Subject Classification: Primary 14H40; Secondary 11G16, 11F03, 14G35.

Keywords and Phrases: Generalized Jacobian, torsion points, modular units, cuspidal divisor class.

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