Semistable reduction of modular curves associated with maximal subgroups in prime level

Abstract

We complete the description of semistable models for modular curves associated with maximal subgroups of ${\mathrm{GL}}_2({\mathbb{F}}_p)$ (for $p$ any prime, $p>5)$. That is, in the new cases of non-split Cartan modular curves and exceptional subgroups, we identify the irreducible components and singularities of the reduction $\mathrm{mod}p$, and the complete local rings at the singularities. We review the case of split Cartan modular curves. This description suffices for computing the group of connected components of the fibre at $p$ of the Néron model of the Jacobian.