Regular models of modular curves in prime level over ${\mathbb{Z}}_p^{\mathrm{ur}}$

Abstract

We give regular models for modular curves associated with (normalizer of) split and non-split Cartan subgroups of ${\mathrm{GL}}_2({\mathbb{F}}_p)$ (for $p$ any prime). We then compute the group of connected components of the fiber at $p$ of the Néron model of their Jacobians.