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

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 $W({\bar{\mathbb{F}}}_p)$ 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 $X_0(N)$.