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We consider the generalized Jacobian $\widetilde{J}$ of the modular curve$X_0(N)$ of level $N$ with respect to a reduced divisor consisting of allcusps. Supposing $N$ is square free, we explicitly determine the structure ofthe $\mathbb{Q}$-rational torsion points on $\widetilde{J}$ up to $6$-primarytorsion. The result turns out to be very different from the case of prime powerlevel previously studied by Yang and the second author. We also obtain ananalogous result for Drinfeld modular curves. Our proof relies on similarresults for classical Jacobians due to Ohta, Papikian and the first author. Wealso discuss the Hecke action on $\widetilde{J}$ and its Eisenstein property.

Generalized Jacobians of modular and Drinfeld modular curves