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, $p\ge 5$). We then compute the group of connected components of the fiber at $p$ of the N\'eron model of their Jacobians.