Modular curves and Néron models of generalized Jacobians

Let X be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring R, and m a modulus on X, given by a closed subscheme of X which is geometrically reduced. The generalized Jacobian J_ m of X with respect to m is then an extension of the Jacobian of X by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of X over R. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves X₀ (N) with moduli supported on the cusps.