Atkin--Lehner Quotients of Modular Curves at Primes of Bad Reduction

Abstract

Let $N$ be a square-free, positive integer and let $p\ge 5$ be a prime dividing $N$. In this work, we construct semistable models of Atkin--Lehner quotients of $X_0(N)$ over the Witt ring $W(\overline{\mathbb{F}}_p)$ by taking the corresponding quotients of the Deligne--Rapoport model of $X_0(N)$. We give an explicit description of the special fiber of the models and obtain genus formulae for the Atkin--Lehner quotients of $X_0(N)$.