Modular curves and Néron models of generalized Jacobians

• Published in: arXiv.org
• Main Authors: Jordan, Bruce W, Ribet, Kenneth A, Scholl, Anthony J
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 28.09.2023
• Subjects: Jacobians; Toruses
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2207.13203

Let \(X\) be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring \(R\), and \(\mathfrak{m}\) a modulus on \(X\), given by a closed subscheme of \(X\) which is geometrically reduced. The generalized Jacobian \(J_\mathfrak{m}\) of \(X\) with respect to \(\mathfrak{m}\) 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 \(X_0(N)\) with moduli supported on the cusps.