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

Abstract

We consider the generalized Jacobian ${\tilde{J}}_0(N)$ 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 $\ge 5$, we exhibit that the group of rational torsion points ${\tilde{J}}_0(N)(\mathbb{Q}{)}_{\text{Tor}}$ tends to be much smaller than the classical Jacobian.