Hyperelliptic and trigonal modular curves in characteristic $p$

Let $X_{\mathrm{\Delta }}(N)$ be an intermediate modular curve of level $N$, meaning that there exist (possibly trivial) morphisms $X_1(N)\to X_{\mathrm{\Delta }}(N)\to X_0(N)$. For all such intermediate modular curves, we give an explicit description of all primes $p$ such that $X_{\mathrm{\Delta }}(N{)}_{{\bar{\mathbb{F}}}_p}$ is either hyperelliptic or trigonal. Furthermore we also determine all primes $p$ such that $X_{\mathrm{\Delta }}(N{)}_{{\mathbb{F}}_p}$ is trigonal. This is done by first using the Castelnuovo-Severi inequality to establish a bound $N_0$ such that if $X_0(N{)}_{{\bar{\mathbb{F}}}_p}$ is hyperelliptic or trigonal, then $N\le N_0$. To deal with the remaining small values of $N$, we develop a method based on the careful study of the canonical ideal to determine, for a fixed curve $X_{\mathrm{\Delta }}(N)$, all the primes $p$ such that the $X_{\mathrm{\Delta }}(N{)}_{{\bar{\mathbb{F}}}_p}$ is trigonal or hyperelliptic. Furthermore, using similar methods, we show that $X_{\mathrm{\Delta }}(N{)}_{{\bar{\mathbb{F}}}_p}$ is not a smooth plane quintic, for any $N$ and any $p$.