Modular curves and Néron models of generalized Jacobians
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911869790322688 |
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| author | Jordan, Bruce W. Ribet, Kenneth A. Scholl, Anthony J. |
| author_facet | Jordan, Bruce W. Ribet, Kenneth A. Scholl, Anthony J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_13203 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Modular curves and Néron models of generalized Jacobians Jordan, Bruce W. Ribet, Kenneth A. Scholl, Anthony J. Algebraic Geometry Number Theory 14K30 (Primary), 11G18, 14G35 (Secondary) 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. |
| title | Modular curves and Néron models of generalized Jacobians |
| topic | Algebraic Geometry Number Theory 14K30 (Primary), 11G18, 14G35 (Secondary) |
| url | https://arxiv.org/abs/2207.13203 |