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Modular curves and Néron models of generalized Jacobians
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 5 / May 2024
- Published online by Cambridge University Press:
- 26 March 2024, pp. 945-981
- Print publication:
- May 2024
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- Article
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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.
