Modular curves and N\'eron models of generalized Jacobians

TL;DR

This paper describes the Néron models of generalized Jacobians of algebraic curves over discrete valuation rings, extending Raynaud's work, and provides explicit computations for modular curves with cusp moduli.

Contribution

It generalizes Raynaud's description of Néron models from usual Jacobians to generalized Jacobians with respect to a modulus, including modular curves.

Findings

Explicit description of Néron models for generalized Jacobians.

Character and component groups of the special fiber are characterized.

Computations for modular curves $X_0(N)$ with cusp moduli are provided.

Abstract

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 N\'eron 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.