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

We give regular models for modular curves associated with (normalizer of) split and non-split Cartan subgroups of ${\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_p)$ (for $p$ any prime, $p\ge 5$). We then compute the group of connected components of the fiber at $p$ of the Néron model of their Jacobians.