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

## TL;DR

This paper completes the description of semistable models for modular curves linked to maximal subgroups of GL_2(F_p), detailing components and singularities for non-split Cartan and exceptional cases, aiding in Néron model computations.

## Contribution

It provides a comprehensive analysis of semistable models for modular curves associated with all maximal subgroups of GL_2(F_p), including new non-split Cartan and exceptional cases.

## Key findings

- Identified irreducible components and singularities of reductions mod p.
- Described complete local rings at singularities.
- Facilitated computation of Néron model component groups.

## 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 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 N\'eron model of the Jacobian.